# Robust approach for comparing two dependent normal populations through Wald-
type tests based on Rényi’s pseudodistance estimators

María Jaenadaa,Elena Castillab, Nirian Martínc, and Leandro Pardoa $a$, Dep.
of Statistics and Operations Research, Complutense University of Madrid;$b$,
Dep. of Applied Mathematics, Materials Science and Engineering, and Electronic
Technology, Rey Juan Carlos University; $c$, Dep. of Financial and Actuarial
Economics & Statistics, Complutense University of Madrid; $\ast$, Correponding
author: Nirian Martín<EMAIL_ADDRESS>
###### Abstract

Since the two seminal papers by Fisher (1915, 1921) were published, the test
under a fixed value correlation coefficient null hypothesis for the bivariate
normal distribution constitutes an important statistical problem. In the
framework of asymptotic robust statistics, it remains being a topic of great
interest to be investigated. For this and other tests, focused on paired
correlated normal random samples, Rényi’s pseudodistance estimators are
proposed, their asymptotic distribution is established and an iterative
algorithm is provided for their computation. From them the Wald-type test
statistics are constructed for different problems of interest and their
influence function is theoretically studied. For testing null correlation in
different contexts, an extensive simulation study and two real data based
examples support the robust properties of our proposal.

##  1 Introduction

In parametric estimation the role of divergence measures is very intuitive:
minimizing a suitable divergence measure between the data and the assumed
model in order to estimate the unknown parameters. These estimators are called
“minimum divergence estimators” (MDEs). There is a growing body of literature
that recognizes the importance of MDEs on the basis of their robustness,
without a significant loss of efficiency, in comparison with the maximum
likelihood estimator (MLE). [5] showed that the minimum Hellinger distance
estimator that minimizes Hellinger distance between the modelled parametric
density and its non-parametric estimator is robust against small perturbation
in the underlying model. Other interesting results in relation to the MDEs can
be seen in [36], [34, 35], [23], [27], [4], [7] and references therein.

In the case of continuous models, it is convenient to consider families of
divergence measures for which non-parametric estimators of the unknown density
function are needed. For instance, the theory developed by the cited paper of
Beran needs a non-parametric estimator of the unknown density function. From
this perspective, the density power divergence (DPD) family, leading to the
minimum density power divergence estimators (MDPDEs), is a good example. For
more details see [4]. However, there is another important family of divergence
measures which neither needs non-parametric estimators, the Rényi’s
pseudodistances (RPDs). This family of pseudodistances will be considered in
this paper.

Let $X_{1},\ldots,X_{n}$ be a random sample of size $n$ from a population $X$,
having true and unknown density function $g,$ modelled by a parametric family
of densities $f_{\boldsymbol{\theta}}$ with
$\boldsymbol{\theta}\in\Theta\subset\mathbb{R}^{p}$. The RPD between the
densities $f_{\boldsymbol{\theta}}$ and $g$ is given, for a tuning parameter
$\alpha>0$, by

$\displaystyle R_{\alpha}\left(f_{\boldsymbol{\theta}},g\right)$
$\displaystyle=\frac{1}{\alpha+1}\log\int_{-\infty}^{+\infty}f_{\boldsymbol{\theta}}^{\alpha+1}(x)dx$
$\displaystyle+\frac{1}{\alpha\left(\alpha+1\right)}\log\int_{-\infty}^{+\infty}g^{\alpha+1}(x)dx-\frac{1}{\alpha}\log\int_{-\infty}^{+\infty}f_{\boldsymbol{\theta}}^{\alpha}(x)g(x)dx.$
(1)

The RPD was considered for the first time in [22]. [15] used the RPD under the
name of $\gamma$-cross entropy. Due to the resemblance with the Rényi
divergence ([32]), [7] named it RPD.

The RPD can be extended for $\alpha=0$ taking continuous limits on the left
yielding the expression

$R_{\alpha=0}\left(f_{\boldsymbol{\theta}},g\right)=\lim_{\alpha\downarrow
0}R_{\alpha}\left(f_{\boldsymbol{\theta}},g\right)=\int_{-\infty}^{+\infty}g(x)\log\frac{g(x)}{f_{\theta}(x)}dx,$

i.e., the RPD coincides with the Kullback-Leibler divergence (KLD) between $g$
and $f_{\boldsymbol{\theta}}$, at $\alpha=0$ (see [27]).

[7] established that the RPD is positive for any two densities and for all
values of the tuning parameter $\alpha>0$,
$R_{\alpha}\left(f_{\boldsymbol{\theta}},g\right)\geq 0$ and further
$R_{\alpha}\left(f_{\boldsymbol{\theta}},g\right)=0$ if and only if
$f_{\boldsymbol{\theta}}=g$. This property suggests the definition of the
minimum RPD estimators (MRPDEs) as the minimizer of the RPD between the
assumed distribution and the empirical distribution of the data. Therefore,
the MRPDE for the unknown parameter $\boldsymbol{\theta}$, based on the random
sample $X_{1},\ldots,X_{n}$,
$\widehat{\boldsymbol{\theta}}_{R,\alpha}=\widehat{\boldsymbol{\theta}}_{R,\alpha}(X_{1},\ldots,X_{n})$,
is given, for a tuning parameter $\alpha>0$, by

$\widehat{\boldsymbol{\theta}}_{R,\alpha}=\arg\sup_{\boldsymbol{\theta}\in\Theta}\sum\limits_{i=1}^{n}w_{\alpha}(\boldsymbol{\theta})f_{\boldsymbol{\theta}}^{\alpha}(X_{i}),$
(2)

where the weight is defined as
$w_{\alpha}(\boldsymbol{\theta})=\kappa_{\alpha}^{-\frac{\alpha}{\alpha+1}}(\boldsymbol{\theta})$
with

$\kappa_{\alpha}(\boldsymbol{\theta})=\mathrm{E}[f_{\boldsymbol{\theta}}^{\alpha}(X)]=\int_{-\infty}^{+\infty}f_{\boldsymbol{\theta}}^{\alpha+1}(x)dx.$
(3)

Note that the value $\alpha=0$ was defined as the KLD and hence, the MRPDE
coincides with the MLE at $\alpha=0$.

The estimating equations, based on (2), are given by

$\sum_{i=1}^{n}\boldsymbol{\Psi}_{\alpha}(x_{i};\boldsymbol{\theta})=\boldsymbol{0}_{p},$
(4)

where $\boldsymbol{0}_{p}$ is the null column vector of dimension $p$ and

$\displaystyle\boldsymbol{\Psi}_{\alpha}(x_{i};\boldsymbol{\theta})$
$\displaystyle=f_{\boldsymbol{\theta}}^{\alpha}(x_{i})\left(\boldsymbol{u}_{\boldsymbol{\theta}}(x_{i})-\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)\right),$
$\displaystyle\boldsymbol{u}_{\boldsymbol{\theta}}(x_{i})$
$\displaystyle=\tfrac{\partial}{\partial\boldsymbol{\theta}}\log
f_{\boldsymbol{\theta}}(x_{i})=\frac{\tfrac{\partial}{\partial\boldsymbol{\theta}}f_{\boldsymbol{\theta}}(x_{i})}{f_{\boldsymbol{\theta}}(x_{i})},$
$\displaystyle\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\frac{\tfrac{\partial}{\partial\boldsymbol{\theta}}\log\kappa_{\alpha}(\boldsymbol{\theta})}{\alpha+1}=\frac{\boldsymbol{\xi}_{\alpha}(\boldsymbol{\theta})}{\kappa_{\alpha}(\boldsymbol{\theta})}=\left(c_{\alpha,1}\left(\boldsymbol{\theta}\right),\ldots,c_{\alpha,p}\left(\boldsymbol{\theta}\right)\right)^{T},$
(5)

where $\kappa_{\alpha}(\boldsymbol{\theta})$ is given by (3) and

$\boldsymbol{\xi}_{\alpha}(\boldsymbol{\theta})=\frac{1}{\alpha+1}\tfrac{\partial}{\partial\boldsymbol{\theta}}\kappa_{\alpha}(\boldsymbol{\theta})=\mathrm{E}[f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)]=\int_{-\infty}^{+\infty}f_{\boldsymbol{\theta}}^{\alpha+1}(x)\boldsymbol{u}_{\boldsymbol{\theta}}(x)dx.$
(6)

The MRPDE is an $M$-estimator and thus it asymptotic distribution and
influence function (IF) can be obtained based on the asymptotic theory of the
$M$-estimators. [7] studied the asymptotic properties and robustness of the
MRPDEs. In relation with the asymptotic distribution they got

$\sqrt{n}(\widehat{\boldsymbol{\theta}}_{R,\alpha}-\boldsymbol{\theta}_{0})\underset{n\rightarrow\infty}{\overset{\mathcal{L}}{\rightarrow}}\mathcal{N}\left(\boldsymbol{0}_{p},\boldsymbol{V}_{\alpha}\left(\boldsymbol{\theta}_{0}\right)\right),$
(7)

where $\boldsymbol{\theta}_{0}$ is the true unknown value of
$\boldsymbol{\theta}$ and

$\boldsymbol{V}_{\alpha}\left(\boldsymbol{\theta}\right)=\boldsymbol{S}_{\alpha}^{-1}\left(\boldsymbol{\theta}\right)\boldsymbol{K}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{S}_{\alpha}^{-1}\left(\boldsymbol{\theta}\right),$
(8)

with

$\displaystyle\boldsymbol{S}_{\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=-\mathrm{E}\left[\frac{\partial\boldsymbol{\Psi}_{\alpha}^{T}\left(X;\boldsymbol{\theta}\right)}{\partial\boldsymbol{\theta}}\right],$
(9) $\displaystyle\boldsymbol{K}_{\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\mathrm{E}\left[\boldsymbol{\Psi}_{\alpha}\left(X;\boldsymbol{\theta}\right)\boldsymbol{\Psi}_{\alpha}^{T}\left(X;\boldsymbol{\theta}\right)\right].$
(10)

The new result given in Section 2 provides a simplified version which is very
useful in practice.

At the same time [7] established that the IF of the functional of the MRPDE of
$\boldsymbol{\theta}$, $\boldsymbol{T}_{\alpha}$, is given by
$\mathcal{IF}\left(x,\boldsymbol{T}_{\alpha},F_{\boldsymbol{\theta}}\right)=\boldsymbol{S}_{\alpha}^{-1}\left(\boldsymbol{\theta}\right)\boldsymbol{\Psi}_{\alpha}\left(x,\boldsymbol{\theta}\right)$.
In aforementioned paper an application was presented to the multiple
regression model (MRM) with random covariates. [37] used RP in order to define
new robustness and efficiency measures. In the same vein, [8] introduced Wald-
type tests based on the minimum RPD estimators for the MRM and its extension
for Generalized Linear models was presented in [21]. Further, [9] studied the
MRPDE for the linear regression model in the ultra-high dimensional set-up.

##  2 Simplified version of the asymptotic variance-covariance matrix of
Rényi’s pseudodistance estimators

This is a short but very important section as it establishes for the first
time new and short expressions of
$\boldsymbol{S}_{\alpha}\left(\boldsymbol{\theta}\right)$ and
$\boldsymbol{K}_{\alpha}\left(\boldsymbol{\theta}\right)$, given in (9) and
(10), in terms of a scalar $\kappa_{\alpha}(\boldsymbol{\theta})$, a vector
$\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)$, and a matrix
$\boldsymbol{J}_{\alpha}\left(\boldsymbol{\theta}\right)$, whose calculation
of any distribution is exactly the same as the one developed for MDPDEs, so
the complexity of the construction of the theory based on MRPDEs is not higher
than the MDPDEs.

###### Theorem 1

The expression of the variance-covariance matrix in the asymptotic
distribution, (7), is given by (8) where

$\displaystyle\boldsymbol{S}_{\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\boldsymbol{J}_{\alpha}\left(\boldsymbol{\theta}\right)-\kappa_{\alpha}(\boldsymbol{\theta})\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})\boldsymbol{c}_{\alpha}^{T}(\boldsymbol{\theta}),$
(11) $\displaystyle\boldsymbol{K}_{\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\boldsymbol{S}_{2\alpha}\left(\boldsymbol{\theta}\right)+\kappa_{2\alpha}(\boldsymbol{\theta})\left(\boldsymbol{c}_{2\alpha}(\boldsymbol{\theta})-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})\right)\left(\boldsymbol{c}_{2\alpha}(\boldsymbol{\theta})-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})\right)^{T},$
(12)

with

$\boldsymbol{J}_{\alpha}\left(\boldsymbol{\theta}\right)=\mathrm{E}[f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)]=\int_{-\infty}^{+\infty}f_{\boldsymbol{\theta}}^{\alpha+1}(x)\boldsymbol{u}_{\boldsymbol{\theta}}(x)\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(x)dx,$
(13)

and the expressions of $\kappa_{\alpha}(\boldsymbol{\theta})$ and
$\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)$ were given by (3)
and (5) respectively.

Proof. See Appendix A.

##  3 Minimum Rényi pseudodistance estimators for two dependent populations
with normal distribution

In the previous results univariate case was considered, but it is
straightforward to extend it for the multivariate set-up. In this paper we are
considering the bidimensional normal distribution model, and so in the
following the role of $x$ is replaced by $(x,y)$ and all the integrals are in
$\mathbb{R}^{2}$. In addition, we are going to get Wald-type test statistics
for testing different composite null hypothesis regarding the model
parameters.

Let $(X,Y)$ be a bidimensional normal model with density function

$f_{\boldsymbol{\theta}}(x,y)=\tfrac{1}{2\pi\sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}}\exp\left\\{-\tfrac{1}{2(1-\rho^{2})}\left[\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}+\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}-2\rho\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)\right]\right\\},$
(14)

$\sigma_{1},\sigma_{2}>0$, $\mu_{1},\mu_{2}\in\mathbb{R}$ and $-1<\rho<1$, and
we shall denote by

$\boldsymbol{\theta}=\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)^{T}$
(15)

the model parameters belonging to the parameter space
$\Theta=\mathbb{R}^{2}\times\mathbb{R}_{+}^{2}\times(-1,1)$.

We are interested, on the basis of a random sample of size $n$,
$(X_{1},Y_{1}),\ldots,\allowbreak(X_{n},Y_{n})$, in obtaining the MRPDE for
$\boldsymbol{\theta}$, as well as the asymptotic distribution. Further, we aim
to develop Wald-type tests, in the bidimensional normal model, based on MRPDE.
Some preliminary results from which proofs the reader could find many clues
were presented in [25].

###### Proposition 2

For the bidimensional normal model, (14), the vector of score functions is
given by

$\boldsymbol{u}_{\boldsymbol{\theta}}(x,y)=(u_{\mu_{1}}(x,y),u_{\mu_{2}}(x,y),u_{\sigma_{1}}(x,y),u_{{}_{\sigma_{2}}}(x,y),u_{\rho}(x,y))^{T},$
(16)

where

$\displaystyle u_{\mu_{1}}(x,y)$
$\displaystyle=\frac{1}{\sigma_{1}\left(1-\rho^{2}\right)}\left[\frac{x-\mu_{1}}{\sigma_{1}}-\rho\frac{y-\mu_{2}}{\sigma_{2}}\right],$
$\displaystyle u_{\mu_{2}}(x,y)$
$\displaystyle=\frac{1}{\sigma_{2}\left(1-\rho^{2}\right)}\left[\frac{y-\mu_{2}}{\sigma_{2}}-\rho\frac{x-\mu_{1}}{\sigma_{1}}\right],$
$\displaystyle u_{\sigma_{1}}(x,y)$
$\displaystyle=-\frac{1}{\sigma_{1}}-\frac{1}{\sigma_{1}(1-\rho^{2})}\left[\rho\frac{x-\mu_{1}}{\sigma_{1}}\frac{y-\mu_{2}}{\sigma_{2}}-\left(\frac{x-\mu_{1}}{\sigma_{1}}\right)^{2}\right],$
$\displaystyle u_{\sigma_{2}}(x,y)$
$\displaystyle=-\frac{1}{\sigma_{2}}-\frac{1}{\sigma_{2}(1-\rho^{2})}\left[\rho\frac{x-\mu_{1}}{\sigma_{1}}\frac{y-\mu_{2}}{\sigma_{2}}-\left(\frac{y-\mu_{2}}{\sigma_{2}}\right)^{2}\right],$
$\displaystyle u_{\rho}(x,y)$
$\displaystyle=\frac{1}{(1-\rho^{2})}\left[\rho+\frac{x-\mu_{1}}{\sigma_{1}}\frac{y-\mu_{2}}{\sigma_{2}}\right]$
$\displaystyle\hskip
5.69046pt-\frac{\rho}{(1-\rho^{2})^{2}}\left[\left(\frac{x-\mu_{1}}{\sigma_{1}}\right)^{2}+\left(\frac{y-\mu_{2}}{\sigma_{2}}\right)^{2}-2\rho\frac{x-\mu_{1}}{\sigma_{1}}\frac{y-\mu_{2}}{\sigma_{2}}\right].$

###### Proposition 3

For the bidimensional normal model, (14), the expressions of (6) and (3) are
given by

$\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)=\left(c_{\alpha}(\mu_{1}),c_{\alpha}(\mu_{2}),c_{\alpha}(\sigma_{1}),c_{\alpha}(\sigma_{2}),c_{\alpha}(\rho)\right)^{T}=\begin{pmatrix}\boldsymbol{c}_{1,\alpha}\left(\boldsymbol{\theta}\right)\\\
\boldsymbol{c}_{2,\alpha}\left(\boldsymbol{\theta}\right)\end{pmatrix},$ (17)

where

$\boldsymbol{c}_{1,\alpha}\left(\boldsymbol{\theta}\right)=\boldsymbol{0}_{2},\hskip
8.5359pt\boldsymbol{c}_{2,\alpha}\left(\boldsymbol{\theta}\right)=\frac{\alpha}{\alpha+1}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}\begin{pmatrix}-1\\\
-1\\\
\frac{\rho}{1-\rho^{2}}\end{pmatrix}=\frac{\alpha}{\alpha+1}\begin{pmatrix}-\frac{1}{\sigma_{1}}\\\
-\frac{1}{\sigma_{2}}\\\ \frac{\rho}{1-\rho^{2}}\end{pmatrix},$

with

$\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}=\mathrm{diag}\\{\sigma_{1},\sigma_{2},1\\},$
(18)

and

$\kappa_{\alpha}(\boldsymbol{\theta})=\frac{1}{k^{\alpha}(\boldsymbol{\theta})\left(\alpha+1\right)},$
(19)

with

$k(\boldsymbol{\theta})=2\pi\sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}.$ (20)

In the following theorem we shall present the expressions of the matrices
$\boldsymbol{K}_{\alpha}\left(\boldsymbol{\theta}\right)$ and
$\boldsymbol{S}_{\alpha}\left(\boldsymbol{\theta}\right)$, defined in (12) and
(11). But first, it is necessary to provide the following result.

###### Proposition 4

For the bidimensional normal model, (14), we have the following results
concerning with the integrals of the cross product for the score functions

$\boldsymbol{J}_{\alpha}\left(\boldsymbol{\theta}\right)=\left(\begin{array}[]{cc}\boldsymbol{J}_{1,\alpha}\left(\boldsymbol{\theta}\right)&\boldsymbol{0}_{2\times
3}\\\ \boldsymbol{0}_{3\times
2}&\boldsymbol{J}_{2,\alpha}\left(\boldsymbol{\theta}\right)\end{array}\right),$

where

$\displaystyle\boldsymbol{J}_{1,\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\frac{1}{k^{\alpha}(\boldsymbol{\theta})(\alpha+1)^{2}}\boldsymbol{D}_{1,\sigma_{1},\sigma_{2}}^{-1}\boldsymbol{J}_{1}(\rho)\boldsymbol{D}_{1,\sigma_{1},\sigma_{2}}^{-1},$
(21) $\displaystyle\boldsymbol{J}_{1}(\rho)$
$\displaystyle=\frac{1}{1-\rho^{2}}\left(\begin{array}[]{cc}1&-\rho\\\
-\rho&1\end{array}\right),$ (24)
$\displaystyle\boldsymbol{J}_{2,\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\frac{1}{k^{\alpha}(\boldsymbol{\theta})(\alpha+1)^{3}}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}\boldsymbol{J}_{2,\alpha}(\rho)\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1},$
$\displaystyle\boldsymbol{J}_{2,\alpha}(\rho)$
$\displaystyle=\frac{1}{1-\rho^{2}}\begin{pmatrix}\alpha^{2}-\rho^{2}(\alpha^{2}+1)+2&\alpha^{2}-\rho^{2}(\alpha^{2}+1)&-\rho(\alpha^{2}+1)\\\
\alpha^{2}-\rho^{2}(\alpha^{2}+1)&\alpha^{2}-\rho^{2}(\alpha^{2}+1)+2&-\rho(\alpha^{2}+1)\\\
-\rho(\alpha^{2}+1)&-\rho(\alpha^{2}+1)&\frac{\rho^{2}(\alpha^{2}+1)+1}{1-\rho^{2}}\end{pmatrix},$

with

$\boldsymbol{D}_{1,\sigma_{1},\sigma_{2}}=\mathrm{diag}\\{\sigma_{1},\sigma_{2}\\},$
(25)

$\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}$ is given by (18) and
$k(\boldsymbol{\theta})$ by (20).

###### Theorem 5

For the bidimensional normal model, (14), we have the following results
concerning with the expectations of the estimating equations

$\boldsymbol{S}_{\alpha}(\boldsymbol{\boldsymbol{\theta}})=\left(\begin{array}[]{cc}\boldsymbol{S}_{1,\alpha}(\boldsymbol{\boldsymbol{\theta}})&\boldsymbol{0}_{2\times
3}\\\ \boldsymbol{0}_{3\times
2}&\boldsymbol{S}_{2,\alpha}(\boldsymbol{\boldsymbol{\theta}})\end{array}\right),$

with

$\boldsymbol{S}_{1,\alpha}(\boldsymbol{\boldsymbol{\theta}})=\boldsymbol{J}_{1,\alpha}\left(\boldsymbol{\theta}\right),\hskip
8.5359pt\boldsymbol{S}_{2,\alpha}(\boldsymbol{\boldsymbol{\theta}})=\frac{1}{k^{\alpha}(\boldsymbol{\theta})(\alpha+1)^{3}}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}\boldsymbol{S}_{2,1}(\rho)\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1},$

and

$\boldsymbol{S}_{2,1}(\rho)=\frac{1}{1-\rho^{2}}\begin{pmatrix}2-\rho^{2}&-\rho^{2}&-\rho\\\
-\rho^{2}&2-\rho^{2}&-\rho\\\
-\rho&-\rho&\frac{1+\rho^{2}}{1-\rho^{2}}\end{pmatrix}.$ (26)

On the other hand

$\boldsymbol{K}_{\alpha}\left(\boldsymbol{\theta}\right)=\left(\begin{array}[]{cc}\boldsymbol{K}_{1,\alpha}\left(\boldsymbol{\theta}\right)&\boldsymbol{0}_{2\times
3}\\\ \boldsymbol{0}_{3\times
2}&\boldsymbol{K}_{2,\alpha}\left(\boldsymbol{\theta}\right)\end{array}\right)$

where

$\displaystyle\boldsymbol{K}_{1,\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\boldsymbol{J}_{1,2\alpha}\left(\boldsymbol{\theta}\right),$
$\displaystyle\boldsymbol{K}_{2,\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\tfrac{1}{k^{2\alpha}(\boldsymbol{\theta})(2\alpha+1)^{3}\left(\alpha+1\right)^{2}}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}\boldsymbol{K}_{2,\alpha}(\rho)\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1},$

with

$\boldsymbol{K}_{2,\alpha}(\rho)=(\alpha+1)^{2}\boldsymbol{S}_{2,1}(\rho)+\alpha^{2}\boldsymbol{S}_{2,2}(\rho),$
(27)

and

$\boldsymbol{S}_{2,2}(\rho)=\tfrac{1}{1-\rho^{2}}\begin{pmatrix}1-\rho^{2}&1-\rho^{2}&-\rho\\\
1-\rho^{2}&1-\rho^{2}&-\rho\\\
-\rho&-\rho&\frac{\rho^{2}}{1-\rho^{2}}\end{pmatrix}.$ (28)

For both, $\boldsymbol{J}_{1,\alpha}\left(\boldsymbol{\theta}\right)$ is given
by (21), $\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}$ by (18) and
$k(\boldsymbol{\theta})$ by (20).

Proof. See Appendix B.

Inverting the diagonal blocks of
$\boldsymbol{S}_{\alpha}(\boldsymbol{\boldsymbol{\theta}})$, we obtain

$\boldsymbol{S}_{1,\alpha}^{-1}(\boldsymbol{\boldsymbol{\theta}})=k^{\alpha}(\boldsymbol{\theta})\left(\alpha+1\right)^{2}\boldsymbol{D}_{1,\sigma_{1},\sigma_{2}}\boldsymbol{J}_{1}^{-1}(\rho)\boldsymbol{D}_{1,\sigma_{1},\sigma_{2}},$
(29)

where

$\boldsymbol{J}_{1}^{-1}(\rho)=\left(\begin{array}[]{cc}1&\rho\\\
\rho&1\end{array}\right)$

and

$\boldsymbol{S}_{2,\alpha}^{-1}(\boldsymbol{\boldsymbol{\theta}})=k^{\alpha}(\boldsymbol{\theta})\left(\alpha+1\right)^{3}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}\boldsymbol{S}_{2,1}^{-1}(\rho)\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}},$
(30)

where

$\boldsymbol{S}_{2,1}^{-1}(\rho)=\frac{1}{2}\begin{pmatrix}1&\rho^{2}&\rho(1-\rho^{2})\\\
\rho^{2}&1&\rho(1-\rho^{2})\\\
\rho(1-\rho^{2})&\rho(1-\rho^{2})&2(1-\rho^{2})^{2}\end{pmatrix}.$ (31)

Therefore, (8) is given by

$\boldsymbol{\boldsymbol{V}}_{\alpha}\boldsymbol{\left({\boldsymbol{\theta}}\right)=}\left(\begin{array}[]{cc}\boldsymbol{V}_{1,\alpha}\boldsymbol{\left({\boldsymbol{\theta}}\right)}&\boldsymbol{0}_{2\times
3}\\\ \boldsymbol{0}_{3\times
2}&\boldsymbol{V}_{2,\alpha}\boldsymbol{\left({\boldsymbol{\theta}}\right)}\end{array}\right),$
(32)

where

$\displaystyle\boldsymbol{V}_{1,\alpha}({\boldsymbol{\theta}})$
$\displaystyle=\boldsymbol{S}_{1,\alpha}^{-1}\left(\boldsymbol{\theta}\right)\boldsymbol{K}_{1,\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{S}_{1,\alpha}^{-1}\left(\boldsymbol{\theta}\right)=\frac{\left(\alpha+1\right)^{4}}{\left(2\alpha+1\right)^{2}}\boldsymbol{D}_{1,\sigma_{1},\sigma_{2}}\boldsymbol{J}_{1}^{-1}(\rho)\boldsymbol{D}_{1,\sigma_{1},\sigma_{2}},$
(33) $\displaystyle\boldsymbol{V}_{2,\alpha}({\boldsymbol{\theta}})$
$\displaystyle=\boldsymbol{S}_{2,\alpha}^{-1}\left(\boldsymbol{\theta}\right)\boldsymbol{K}_{2,\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{S}_{2,\alpha}^{-1}\left(\boldsymbol{\theta}\right)=\frac{\left(\alpha+1\right)^{4}}{\left(2\alpha+1\right)^{3}}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}\boldsymbol{V}_{2,\alpha}\left(\rho\right)\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}},$
(34)

with

$\boldsymbol{V}_{2,\alpha}\left(\rho\right)=\boldsymbol{S}_{2,1}^{-1}(\rho)\boldsymbol{K}_{2,\alpha}(\rho)\boldsymbol{S}_{2,1}^{-1}(\rho)=(\alpha+1)^{2}\boldsymbol{S}_{2,1}^{-1}(\rho)+\alpha^{2}\boldsymbol{S}_{2,1}^{-1}(\rho)\boldsymbol{S}_{2,2}(\rho)\boldsymbol{S}_{2,1}^{-1}(\rho).$
(35)

Based on the previous results we have the following Theorem.

###### Theorem 6

For the bidimensional normal model, (14), the MRPDE for $\boldsymbol{\theta}$,

$\widehat{\boldsymbol{\theta}}_{R,\alpha}=(\widehat{\mu}_{1,R,\alpha},\widehat{\mu}_{2,R,\alpha},\widehat{\sigma}_{1,R,\alpha},\widehat{\sigma}_{2,R,\alpha},\widehat{\rho}_{R,\alpha})^{T},$
(36)

is obtained as a solution of

$\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\left(\boldsymbol{u}_{\boldsymbol{\theta}}(X_{i},Y_{i})-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})\right)=\boldsymbol{0}_{5},$

with

$w_{i,\boldsymbol{\theta}}=\exp\left\\{\tfrac{1}{2(1-\rho^{2})}\left[(\tfrac{X_{i}-\mu_{1}}{\sigma_{1}})^{2}+(\tfrac{Y_{i}-\mu_{2}}{\sigma_{2}})^{2}-2\rho\tfrac{X_{i}-\mu_{1}}{\sigma_{1}}\tfrac{Y_{i}-\mu_{2}}{\sigma_{2}}\right]\right\\},$
(37)

$\boldsymbol{u}_{\boldsymbol{\theta}}(X_{i},Y_{i})$ is given in Proposition 2
and $\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})$ in Proposition 3. The
corresponding asymptotic distribution is

$\sqrt{n}(\widehat{\boldsymbol{\theta}}_{R,\alpha}-\boldsymbol{\theta}_{0})\underset{n\rightarrow\infty}{\overset{\mathcal{L}}{\rightarrow}}\mathcal{N}\left(\boldsymbol{0}_{p},\boldsymbol{V}_{\alpha}\left(\boldsymbol{\theta}_{0}\right)\right),$
(38)

where $\boldsymbol{\theta}_{0}$ is the true unknown value of (15) and
$\boldsymbol{V}_{\alpha}\left({\boldsymbol{\theta}}\right)$ was given in (32).

The following algorithm is useful for computing the MRPDE of
$\boldsymbol{\theta}$ given in Theorem 6. It works iteratively for a sequence
of increasing values of the tuning parameter,
$\alpha\in\\{\alpha_{k}\\}_{k=0}^{K}$ with $\alpha_{k}=\frac{k}{K}$, having a
very simple iterative scheme and converging rapidly to the final optimal
value. As the MLEs have an explicit expression, the tuning parameter
$\alpha_{0}=0$ initializes the iterations. Herein the following parameter
transformation is considered

$\boldsymbol{\vartheta}=\left(\mu_{1},\mu_{2},\zeta_{1}^{2},\zeta_{2}^{2},\rho\right)^{T}\hskip
8.5359pt\text{where}\hskip
8.5359pt\sigma_{j}^{2}=(\alpha+1)\zeta_{j}^{2},\quad j=1,2.$

The strength of the algorithm is its simplicity for estimating in a chained
way and with the semi-explicit expressions given in the inner iterations, with
expression which mimics the MLEs as weighted version (see the corresponding
proof given in Appendix D). The updating recursive elements comprise only of
the weights, as the name Iteratively Reweighted Moments Algorithm suggests.
The semi-explicit expressions given in the inner iterations of Algorithm 1 are
particular cases of (6) and (7) of [38] with $N$ equals $2$. However, our
proposed algorithm is different from the one proposed in the Monte-Carlo
simulations of Toma and Leoni-Aubin (2015) in two features. First, their
algorithm does not consider outer iterations as the estimation is initialized
always with the MLE and second, they do not consider the reparameterization of
the variance-covariance matrix. Both are crucial features, first one to save
iterations when a grid of the tuning parameter is considered, taking into
account the smoothness of the objective function. The second one justifies our
proposed Iteratively Reweighted Moments Algorithm to be an EM algorithm. It is
well-known that some M-estimators are obtained from estimating equations
associated to scale mixture of normal distributions, and their corresponding
iterative reweighted algorithm is an EM algorithm. Under weak conditions, each
step of IRLS increases the objective function and the solution of the
iterative reweighted algorithm converges to a local maximum of the likelihood
function. The minimum pseudodistance distance estimators for bivariate normal
populations falls within this class of M-estimators. From Algorithm 1 and
taking into account the Bayesian interpretation of the EM algorithm (see for
example Section 12.2 in [24], we obtain the following way of understanding the
weights update or the E step. Let us consider the sample
$(X_{i}^{(k)},Y_{i}^{(k)},Z_{i}^{(k)})$, $i=1,\ldots,n$, with
$(X_{i}^{(k)},Y_{i}^{(k)})$ being a bivariate normal distribution with
parameters
$\boldsymbol{\vartheta}_{k}=(\mu_{1},\mu_{2},\zeta_{1,k}^{2},\zeta_{2,k}^{2},\rho)^{T}$,
where $\zeta_{j,k}^{2}=\frac{1}{\alpha_{k}+1}\sigma_{j}^{2}$, $j=1,2$, and
$Z_{i}^{(k)}$ being an auxiliary random variable (a latent random variable for
the usual EM algorithm with missing data). Assuming that $Z_{i}^{(k)}$ is a
random variable degenerated at $\frac{\alpha_{k}}{\alpha_{k}+1}$ (prior
distribution), it holds
$\left((X_{i}^{(k)},Y_{i}^{(k)})\left|Z_{i}^{(k)}=z\right.\right)$ behaves as
a bivariate normal distribution with parameters
$(\mu_{1},\mu_{2},\frac{\zeta_{1,k}^{2}}{z},\frac{\zeta_{2,k}^{2}}{z},\rho)^{T}$,
while the weight is the expectation of the posterior distribution (line 13 of
Algorithm 1),

$\widehat{\varpi}_{i,\alpha_{k}}=E\left[Z_{i}^{(k)}\left|(X_{i}^{(k)},Y_{i}^{(k)})\right.\right],\quad
i=1,\ldots,n.$

The estimator of $\boldsymbol{\vartheta}_{k}$ is updated in the M step (line
14 of Algorithm 1). While parameters $\zeta_{j,k}^{2}$, $j=1,2$, produce a
shrinkage effect on the original variance parameters $\sigma_{j}^{2}$,
$j=1,2$, the estimates associated with small values the weights,
$\widehat{\varpi}_{i,\alpha_{k}}^{(k)}$, $i=1,\ldots,n$, produce a down-
weighting effect on estimates of $\boldsymbol{\vartheta}_{k}$ to prevent from
high value of the Mahalanobis distance (line 11) for the $i$-th observation
and $k$-th tuning parameter $\alpha_{k}$, $k=1,\ldots,n$.

Algorithm 1 Iteratively Reweighted Moments Algorithm for MRPDE of
$\boldsymbol{\vartheta}$

1:$K$, $\\{\alpha_{k}=\frac{k}{K}\\}_{k=0}^{K}$,
$\\{(X_{i},Y_{i})\\}_{i=1}^{n}$, $\xi$;

2:$k\leftarrow 0$, $r_{k=0}\leftarrow 0$ $\triangleright$ Inizialization

3:$\mathrm{vec}\\{\widehat{\varpi}_{i,\alpha_{k=0}=0,(r_{k=0}=0)}\\}_{i=1}^{n}=\boldsymbol{1}_{n}$

4:Compute $\widehat{\boldsymbol{\vartheta}}_{\alpha_{k=0}=0,(r_{k=0}=0)}$:
$\displaystyle\widehat{\mu}_{1,R,\alpha_{k=0}=0,(r_{k=0}=0)}$
$\displaystyle=\frac{\sum_{i=1}^{n}X_{i}}{n},\quad\widehat{\zeta}_{1,R,\alpha_{k=0}}^{2}=\frac{\sum_{i=1}^{n}(X_{i}-\widehat{\mu}_{1,R,\alpha_{0}=0,(r_{k=0}=0)})^{2}}{n},$
$\displaystyle\widehat{\mu}_{2,R,\alpha_{k=0}=0,(r_{k=0}=0)}$
$\displaystyle=\frac{\sum_{i=1}^{n}Y_{i}}{n},\quad\widehat{\zeta}_{2,R,\alpha_{k=0}}^{2}=\frac{\sum_{i=1}^{n}(Y_{i}-\widehat{\mu}_{2,R,\alpha_{0}=0,(r_{k=0}=0)})^{2}}{n},$
$\displaystyle\widehat{\rho}_{R,\alpha_{k=0}=0,(r_{k=0}=0)}$
$\displaystyle=\frac{\sum_{i=1}^{n}\frac{X_{i}-\widehat{\mu}_{1,R,\alpha_{k=0}=0,(r_{k=0}=0)}}{\widehat{\zeta}_{1,R,\alpha_{k=0}=0,(r_{k=0}=0)}}\frac{Y_{i}-\widehat{\mu}_{2,R,\alpha_{0}=0,(r_{k=0}=0)}}{\widehat{\zeta}_{2,R,\alpha_{k=0}=0,(r_{k=0}=0)}}}{n};$

5:while $k<K$ do $\triangleright$ Outer loop starts

6: $k\leftarrow k+1$, $r_{k}\leftarrow 0$

7:
$\mathrm{vec}\\{\widehat{\varpi}_{i,\alpha_{k},(r_{k}=0)}\\}_{i=1}^{n}\leftarrow\mathrm{vec}\\{\widehat{\varpi}_{i,\alpha_{k-1,r_{k-1}}}\\}_{i=1}^{n}$

8:
$\widehat{\boldsymbol{\vartheta}}_{\alpha_{k},(r_{k}=0)}=\widehat{\boldsymbol{\vartheta}}_{\alpha_{k-1,r_{k-1}}}$
$\triangleright$ Inner loop starts

9: repeat

10:
$\mathrm{vec}\\{\widehat{X}_{i,\alpha_{k},(r_{k})}\\}_{i=1}^{n}\leftarrow\mathrm{vec}\left\\{\tfrac{X_{i}-\widehat{\mu}_{1,R,\alpha_{k},(r_{k})}}{\widehat{\zeta}_{1,R,\alpha_{k},(r_{k})}}\right\\}_{i=1}^{n}$

11:
$\mathrm{vec}\\{\widehat{Y}_{i,\alpha_{k},(r_{k})}\\}_{i=1}^{n}\leftarrow\mathrm{vec}\left\\{\tfrac{Y_{i}-\widehat{\mu}_{2,R,\alpha_{k},(r_{k})}}{\widehat{\zeta}_{2,R,\alpha_{k},(r_{k})}}\right\\}_{i=1}^{n}$

12:
$\mathrm{vec}\\{d_{i,\alpha_{k},(r_{k})}^{2}\\}_{i=1}^{n}\leftarrow\mathrm{vec}\left\\{\tfrac{\widehat{X}_{i,\alpha_{k},(r_{k})}^{2}+\widehat{Y}_{i,\alpha_{k},(r_{k})}^{2}-2\widehat{\rho}_{1,R,\alpha_{k},(r_{k})}\widehat{X}_{i,\alpha_{k},(r_{k})}\widehat{Y}_{i,\alpha_{k},(r_{k})}}{1-\widehat{\rho}_{1,R,\alpha_{k},(r_{k})}^{2}}\right\\}_{i=1}^{n}$

13: $r_{k}\leftarrow r_{k}+1,$

14:
$\mathrm{vec}\\{\widehat{\varpi}_{i,\alpha_{k},(r_{k})}\\}_{i=1}^{n}\leftarrow\mathrm{vec}\\{\exp^{-\frac{\alpha_{h}}{\alpha_{h}+1}}(\frac{1}{2}d_{i,\alpha_{k},(r_{k}-1)}^{2})\\}_{i=1}^{n}$

15: Compute $\widehat{\boldsymbol{\vartheta}}_{\alpha_{k},(r_{k})}$:
$\displaystyle t_{\alpha_{k},(r_{k})}$
$\displaystyle=\sum_{i=1}^{n}\widehat{\varpi}_{i,\alpha_{k},(r_{k})},$
$\displaystyle\widehat{\mu}_{1,R,\alpha_{k},(r_{k})}$
$\displaystyle=\frac{\sum_{i=1}^{n}\widehat{\varpi}_{i,\alpha_{k},(r_{k})}X_{i}}{t_{\alpha_{k},(r_{k})}},\quad\widehat{\zeta}_{1,R,\alpha_{k},(r_{k})}^{2}=\frac{\sum_{i=1}^{n}\widehat{\varpi}_{i,\alpha_{k},(r_{k})}(X_{i}-\widehat{\mu}_{1,R,\alpha_{k},(r_{k})})^{2}}{t_{\alpha_{k},(r_{k})}},$
$\displaystyle\widehat{\mu}_{2,R,\alpha_{k},(r_{k})}$
$\displaystyle=\frac{\sum_{i=1}^{n}\widehat{\varpi}_{i,\alpha_{k},(r_{k})}Y_{i}}{t_{\alpha_{k},(r_{k})}},\quad\widehat{\zeta}_{2,R,\alpha_{k},(r_{k})}^{2}=\frac{\sum_{i=1}^{n}\widehat{\varpi}_{i,\alpha_{k},(r_{k})}(Y_{i}-\widehat{\mu}_{2,R,\alpha_{k},(r_{k})})^{2}}{t_{\alpha_{k},(r_{k})}},$
$\displaystyle\widehat{\rho}_{R,\alpha_{k},(r_{k})}$
$\displaystyle=\frac{\sum_{i=1}^{n}\widehat{\varpi}_{i,\alpha_{k},(r_{k})}\tfrac{X_{i}-\widehat{\mu}_{1,R,\alpha_{k},(r_{k})}}{\widehat{\zeta}_{1,R,\alpha_{k},(r_{k})}}\tfrac{Y_{i}-\widehat{\mu}_{2,R,\alpha_{k},(r_{k})}}{\widehat{\zeta}_{2,R,\alpha_{k},(r_{k})}}}{t_{\alpha_{k},(r_{k})}}$

16: until
$\left\|\widehat{\boldsymbol{\vartheta}}_{\alpha_{k},(r_{k})}-\widehat{\boldsymbol{\vartheta}}_{\alpha_{k},(r_{k}-1)}\right\|_{2}<\xi$
$\triangleright$ Inner loop ends

17:end while$\triangleright$ Outer loop ends

18:$\\{\widehat{\boldsymbol{\vartheta}}_{\alpha_{k}}=\widehat{\boldsymbol{\vartheta}}_{\alpha_{k},(r_{k})}\\}_{k=0}^{K}$.

It is clear that due to the invariance property of the MRPDEs, it holds

$\widehat{\sigma}_{j,R,\alpha}=\sqrt{\alpha+1}\widehat{\zeta}_{j,R,\alpha},\quad
j=1,2.$

##  4 Wald-type tests based on Rényi’s pseudodistance estimators

Based on the asymptotic distribution of the MRPDE for $\boldsymbol{\theta}$,
$\widehat{\boldsymbol{\theta}}_{R,\alpha}$, given in Theorem 6, we present
Wald-type tests for testing composite null hypothesis regarding bidimensional
normal model parameters.

The restricted parameter space
$\Theta_{0}\subset\Theta=\mathbb{R}^{2}\times\mathbb{R}_{+}^{2}\times(-1,1)$,
is often defined by a set of $r$ restrictions of the form

$\boldsymbol{m}(\boldsymbol{\theta})=\boldsymbol{0}_{r},$ (39)

where $\boldsymbol{\theta}$ is (15) and
$\boldsymbol{m}:\;\Theta\rightarrow\mathbb{R}^{r}$ (see Serfling 1980). Assume
that the $5\times r$ matrix

$\boldsymbol{M}({\boldsymbol{\theta}})=\frac{\partial\boldsymbol{m}^{T}(\boldsymbol{\theta})}{\partial{\boldsymbol{\theta}}}$
(40)

exists and is continuous in ${\boldsymbol{\theta}}$, and
rank$\left(\boldsymbol{M}({\boldsymbol{\theta}})\right)=r$, where $r\leq 5$.
Let $(X_{1},Y_{1}),\ldots,\allowbreak(X_{n},Y_{n})$ be a random sample of size
$n$ from a distribution modelled by the bidimensional normal model probability
density function $f_{\boldsymbol{\theta}}(x,y)$, where
${\boldsymbol{\theta}}\in\Theta$. Our interest is in testing the hypothesis

$H_{0}:{\boldsymbol{\theta}}\in\Theta_{0}\quad\text{against}\quad
H_{1}:\boldsymbol{\theta}\notin\Theta_{0},$ (41)

where
$\Theta_{0}=\\{\boldsymbol{\theta}\in\Theta:\boldsymbol{m}(\boldsymbol{\theta})=\boldsymbol{0}\\}$.

###### Definition 1

Let $\widehat{\boldsymbol{\theta}}_{R,\alpha}$ be the MDPDE of
${\boldsymbol{\theta}}$. The family of proposed Wald-type test statistics for
testing the null hypothesis in (41) is given by

$W_{n,\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})=n\boldsymbol{m}^{T}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\left(\boldsymbol{M}^{T}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\boldsymbol{V}_{\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\boldsymbol{M}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\right)^{-1}\boldsymbol{m}(\widehat{\boldsymbol{\theta}}_{R,\alpha}),$
(42)

where the matrix $\boldsymbol{V}_{\alpha}$ is as in (32) and the functions
$\boldsymbol{m}(\cdot)$ and $\boldsymbol{M}(\cdot)$ are defined in (39) and
(40).

###### Theorem 7

The asymptotic null distribution of the proposed Wald-type test statistics
given in (42) is chi-square with $r$ degrees of freedom, $\chi^{2}_{r}$.

Proof. See Appendix C.

We will reject the null hypothesis in (39) if
$W_{n,\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})>\chi_{r,\varsigma}^{2}$,
where $\chi_{r,\varsigma}^{2}$ is the upper percentage point of order
$\varsigma$ of the $\chi_{r}^{2}$ distribution. Based on Definition 1 and
Theorem 7, the following subsections are devoted to derive a variety
hypothesis tests for the bidimensional normal model. The proofs of the stated
formulas are given in Appendix F.

###### Case 1

(Comparing means of two dependent populations with normal distribution).
If we are interested in testing

$H_{0}:\mu_{1}=\mu_{2},$ (43)

the corresponding Wald-type test statistics based on MRPDEs is

$W_{n,\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})=n\frac{\left(2\alpha+1\right)^{2}}{\left(\alpha+1\right)^{4}}\frac{(\widehat{\mu}_{1,R,\alpha}-\widehat{\mu}_{2,R,\alpha})^{2}}{(\widehat{\sigma}_{1,R,\alpha}-\widehat{\sigma}_{2,R,\alpha})^{2}+2(1-\widehat{\rho}_{R,\alpha})\widehat{\sigma}_{1,R,\alpha}\widehat{\sigma}_{2,R,\alpha}},$
(44)

and its asymptotic distribution is a chi-squared distribution with one degree
of freedom under (43).

In Case 1 a non-standard Behrens–Fisher problem is covered, i.e., a comparison
of the means of two populations which may possess not only different
variances, but also a non-null correlation. It is of great interest to be
aware that formulating the same problem as a paired test constructed taking
the difference of both populations, $V=X-Y$, as a single population problem
for testing $H_{0}:\mu_{V}=0$, with an unknown variance $\sigma_{V}^{2}$, the
same value of the Wald-type test statistics

$W_{n,\alpha}(\widehat{\mu}_{V,R,\alpha},\widehat{\sigma}_{V,R,\alpha})=n\frac{\left(2\alpha+1\right)^{2}}{\left(\alpha+1\right)^{4}}\frac{\widehat{\mu}_{V,R,\alpha}^{2}}{\widehat{\sigma}_{V,R,\alpha}^{2}}$

is obtained
($W_{n,\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})=W_{n,\alpha}(\widehat{\mu}_{V,R,\alpha},\widehat{\sigma}_{V,R,\alpha})$)
from the invariance property of the Rényi’s pseudodistance estimators, since
$\mu_{V}=\mu_{1}-\mu_{2}$,
$\sigma_{V}^{2}=\sigma_{1}^{2}+\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}=(\sigma_{1}-\sigma_{2})^{2}+2(1-\rho)\sigma_{1}\sigma_{2}$.

The most efficient classic procedure to address this problem is the paired
$t$-test, i.e

$T_{V}=\sqrt{n}\frac{\overline{V}_{n}}{S_{V,n-1}},$

where

$\overline{V}_{n}=\overline{X}_{n}-\overline{Y}_{n},$ $\displaystyle
S_{V,n-1}^{2}$
$\displaystyle=\frac{1}{n-1}\sum_{i=1}^{n}(V_{i}-\overline{V}_{n})^{2}=S_{X,n-1}^{2}+S_{Y,n-1}^{2}-2S_{XY,n-1},$
$\displaystyle S_{X,n-1}^{2}$
$\displaystyle=\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X}_{n})^{2}=\frac{n}{n-1}\widehat{\sigma}_{1,R,\alpha=0}^{2},$
$\displaystyle S_{Y,n-1}^{2}$
$\displaystyle=\frac{1}{n-1}\sum_{i=1}^{n}(Y_{i}-\overline{Y}_{n})^{2}=\frac{n}{n-1}\widehat{\sigma}_{2,R,\alpha=0}^{2},$
$\displaystyle S_{XY,n-1}$
$\displaystyle=\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X}_{n})(Y_{i}-\overline{Y}_{n})=\frac{n}{n-1}\widehat{\rho}_{R,\alpha=0}\widehat{\sigma}_{2,R,\alpha=0}\widehat{\sigma}_{1,R,\alpha=0}.$

Its exact distribution is a Student-$t$ with $n-1$ degrees of freedom,
$t_{n-1}$.

###### Case 2

(Comparing variances of two dependent populations with normal distribution).
If we are interested in testing

$H_{0}:\sigma_{1}=\sigma_{2}.$ (45)

the corresponding Wald-type test statistics based on MRPDEs is

$W_{n,\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})=n\frac{\left(2\alpha+1\right)^{3}}{\left(\alpha+1\right)^{6}}\frac{(\widehat{\sigma}_{1,R,\alpha}-\widehat{\sigma}_{2,R,\alpha})^{2}}{\beta_{\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})},$
(46)

where

$\beta_{\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})=\left[\frac{1}{4}\left(\tfrac{\alpha}{\alpha+1}\right)^{2}+\frac{1}{2}\right](\widehat{\sigma}_{1,R,\alpha}-\widehat{\sigma}_{2,R,\alpha})^{2}+(1-\widehat{\rho}_{R,\alpha}^{2})\widehat{\sigma}_{1,R,\alpha}\widehat{\sigma}_{2,R,\alpha}.$
(47)

The asymptotic distribution of (46) is a chi-squared distribution with $1$
degree of freedom under (45).

###### Case 3

(Fixing a value of the for correlation coefficient of two normal populations).
If we are interested in testing

$H_{0}:\rho=\rho_{0},$ (48)

the corresponding Wald-type test statistics based on MRPDEs is

$W_{n,\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})=n\frac{\left(2\alpha+1\right)^{3}}{\left(\alpha+1\right)^{6}}\frac{(\widehat{\rho}_{R,\alpha}-\rho_{0})^{2}}{(1-\widehat{\rho}_{R,\alpha}^{2})^{2}},$
(49)

and its asymptotic distribution is a chi-squared distribution with $1$ degree
of freedom under (48).

The classic Wald and Rao test statistics are given by

$\displaystyle W_{n,\alpha=0}(\widehat{\boldsymbol{\theta}}_{R,\alpha=0})$
$\displaystyle=n\frac{(\widehat{\rho}_{R,\alpha=0}-\rho_{0})^{2}}{(1-\widehat{\rho}_{R,\alpha=0}^{2})^{2}},$
$\displaystyle R_{n,\alpha=0}(\widehat{\boldsymbol{\theta}}_{R,\alpha=0})$
$\displaystyle=n\frac{(\widehat{\rho}_{R,\alpha=0}-\rho_{0})^{2}}{(1-\rho_{0}\widehat{\rho}_{R,\alpha=0})^{2}},$

where

$\widehat{\rho}_{R,\alpha=0}=\frac{\sum_{i=1}^{n}(X_{i}-\bar{X}_{n})(Y_{i}-\bar{Y}_{n})}{\sqrt{\sum_{i=1}^{n}(X_{i}-\bar{X}_{n})^{2}}\sqrt{\sum_{i=1}^{n}(Y_{i}-\bar{Y}_{n})^{2}}}=R_{XY},$

but

$W_{n,\alpha=0}^{\prime}(\widehat{\boldsymbol{\theta}}_{R,\alpha=0})=n\frac{(\widehat{\rho}_{R,\alpha=0}-\rho_{0})^{2}}{(1-\rho_{0}^{2})^{2}}$

convergences more rapidly to the chi-square distribution with $1$ degree of
freedom (see [1]). The extension of
$W_{n,\alpha=0}^{\prime}(\widehat{\boldsymbol{\theta}}_{R,\alpha=0})$ to

$W_{n,\alpha}^{\prime}(\widehat{\boldsymbol{\theta}}_{R,\alpha})=n\frac{\left(2\alpha+1\right)^{3}}{\left(\alpha+1\right)^{6}}\frac{(\widehat{\rho}_{R,\alpha}-\rho_{0})^{2}}{(1-\rho_{0}^{2})^{2}},$
(50)

is directly obtained from the same proof of
$W_{n,\alpha=0}(\widehat{\boldsymbol{\theta}}_{R,\alpha=0})$, since in (87)
$\widehat{\rho}_{R,\alpha=0}$ can be replaced by $\rho_{0}$. In the particular
case of fixing $\rho_{0}=0$ under the null, (48), it holds
$W_{n,\alpha=0}^{\prime}(\widehat{\boldsymbol{\theta}}_{R,\alpha=0})=R_{n,\alpha=0}(\widehat{\boldsymbol{\theta}}_{R,\alpha=0})$

###### Case 4

(Comparing means and variances of two dependent populations with normal
distribution).
If we are interested in testing

$H_{0}:\mu_{1}=\mu_{2}\text{ and }\sigma_{1}=\sigma_{2},$ (51)

the corresponding Wald-type test statistics based on MRPDEs is

$\displaystyle
W_{n,\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})=n\frac{\left(2\alpha+1\right)^{2}}{\left(\alpha+1\right)^{4}}\bigg{(}$
$\displaystyle\frac{(\widehat{\mu}_{1,R,\alpha}-\widehat{\mu}_{2,R,\alpha})^{2}}{(\widehat{\sigma}_{1,R,\alpha}-\widehat{\sigma}_{1,R,\alpha})^{2}+2(1-\widehat{\rho}_{R,\alpha}^{2})\widehat{\sigma}_{1,R,\alpha},\widehat{\sigma}_{2,R,\alpha}}$
(52)
$\displaystyle+\frac{\left(2\alpha+1\right)(\widehat{\sigma}_{1,R}^{\alpha}-\widehat{\sigma}_{2,R}^{\alpha})^{2}}{\left(\alpha+1\right)^{2}\beta_{\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})}\bigg{)},$

and its asymptotic distribution is a chi-squared distribution with $2$ degrees
of freedom under (51).

###### Case 5

(Fixing a value for covariance of two normal populations).
If we are interested in testing

$H_{0}:\sigma_{1}\sigma_{2}\rho=\sigma_{12,0},$ (53)

where $\sigma_{12,0}\in\mathbb{R}$, the corresponding Wald-type test
statistics based on MRPDEs is

$W_{n,\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})=n\frac{\left(2\alpha+1\right)^{3}}{\left(\alpha+1\right)^{4}}\frac{\left(\widehat{\sigma}_{1,R,\alpha}\widehat{\sigma}_{2,R,\alpha}\widehat{\rho}_{R,\alpha}-\sigma_{12,0}\right)^{2}}{\widehat{\sigma}_{1,R,\alpha}^{2}\widehat{\sigma}_{2,R,\alpha}^{2}\left[\left(\alpha+1\right)^{2}(\widehat{\rho}_{R,\alpha}^{2}+1)+\frac{\alpha^{2}}{2}\widehat{\rho}_{R,\alpha}^{2}\right]}.$
(54)

The asymptotic distribution of (54) is a chi-squared distribution with $1$
degree of freedom under (53).

###### Case 6

(Fixing values for means of two dependent populations with normal
distribution).
If we are interested in testing

$H_{0}:\mu_{1}=\mu_{1,0}\text{ and }\mu_{2}=\mu_{2,0}\text{,}$ (55)

the corresponding Wald-type test statistics based on MRPDEs is

$\displaystyle
W_{n,\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})=n\dfrac{\left(2\alpha+1\right)^{2}}{\left(\alpha+1\right)^{4}}\bigg{[}$
$\displaystyle\frac{\left(\dfrac{\widehat{\mu}_{1,R,\alpha}-\mu_{1,0}}{\widehat{\sigma}_{1,R,\alpha}}-\dfrac{\widehat{\mu}_{2,R,\alpha}-\mu_{2,0}}{\widehat{\sigma}_{2,R,\alpha}}\right)^{2}}{1-\widehat{\rho}_{R,\alpha}^{2}}$
(56)
$\displaystyle\frac{+2(1-\widehat{\rho}_{R,\alpha})\left(\frac{\widehat{\mu}_{1,R,\alpha}-\mu_{1,0}}{\widehat{\sigma}_{1,R,\alpha}}\right)\left(\frac{\widehat{\mu}_{2,R,\alpha}-\mu_{2,0}}{\widehat{\sigma}_{2,R,\alpha}}\right)}{1-\widehat{\rho}_{R,\alpha}^{2}}\bigg{]}$

The asymptotic distribution of (56) is a chi squared distribution with $2$
degrees of freedom under (55).

###### Case 7

(Fixing values for variances and covariance of two dependent populations with
normal distribution).
If we are interested in testing

$H_{0}:\sigma_{1}=\sigma_{1,0}\text{, }\sigma_{2}=\sigma_{2,0},\text{
}\sigma_{1}\sigma_{2}\rho=\sigma_{12,0},$ (57)

the corresponding Wald-type test statistics based on MRPDEs is

$W_{n,\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})=n\frac{(2\alpha+1)^{3}}{(\alpha+1)^{4}}\boldsymbol{w}_{\alpha}^{T}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\boldsymbol{V}_{2,\alpha}^{-1}(\widehat{\rho}_{R,\alpha})\boldsymbol{w}_{\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha}),$
(58)

where

$\displaystyle\boldsymbol{w}_{\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})$
$\displaystyle=\boldsymbol{D}_{2,\widehat{\sigma}_{1,R,\alpha},\widehat{\sigma}_{2,R,\alpha}}^{-1}\left(\boldsymbol{M}_{22}^{T}(\boldsymbol{\theta})\right)^{-1}\boldsymbol{m}(\widehat{\boldsymbol{\theta}}_{R,\alpha})$
$\displaystyle=\begin{pmatrix}1-\frac{\sigma_{1,0}}{\widehat{\sigma}_{1,R,\alpha}}\\\
1-\frac{\sigma_{2,0}}{\widehat{\sigma}_{2,R,\alpha}}\\\
\widehat{\rho}_{R,\alpha}-\frac{\sigma_{12,0}}{\widehat{\sigma}_{1,R,\alpha}\widehat{\sigma}_{2,R,\alpha}}-\widehat{\rho}_{R,\alpha}\left(2-\frac{\sigma_{1,0}}{\widehat{\sigma}_{1,R,\alpha}}-\frac{\sigma_{2,0}}{\widehat{\sigma}_{2,R,\alpha}}\right)\end{pmatrix},$

with

$\boldsymbol{V}_{2,\alpha}^{-1}(\widehat{\rho}_{R,\alpha})=(\alpha+1)^{2}\boldsymbol{S}_{2,1}(\widehat{\rho}_{R,\alpha})\left[\boldsymbol{S}_{2,1}(\widehat{\rho}_{R,\alpha})+\left(\frac{\alpha}{\alpha+1}\right)^{2}\boldsymbol{S}_{2,2}(\widehat{\rho}_{R,\alpha})\right]^{-1}\boldsymbol{S}_{2,1}(\widehat{\rho}_{R,\alpha}),$

$\boldsymbol{S}_{2,1}(\widehat{\rho}_{R,\alpha})$ given by (26) and
$\boldsymbol{S}_{2,2}(\widehat{\rho}_{R,\alpha})$ by (28). The asymptotic
distribution of (58) is a chi squared with $3$ degrees of freedom under (57).

##  5 Study of the Influence Function

In the precedent sections we have developed the MRPDE for
$\boldsymbol{\theta}$ in the bidimensional normal model, as well as Wald-type
tests based on MRPDE, as a robust alternative to the MLE and classic Wald-type
tests. In this section, we will theoretically justify the robustness of the
proposed estimators through the study of its Influence Function (IF). The IF
([16]) for any estimator defined in terms of an statistical functional
$\boldsymbol{T}(F)$ from the true distribution $F$, is defined as

$\mathcal{IF}(t,\boldsymbol{T},F)=\lim_{\varepsilon\downarrow
0}\frac{\boldsymbol{T}((1-\varepsilon)F+\varepsilon\Delta_{t})-\boldsymbol{T}(F)}{\varepsilon}=\left.\frac{\partial\boldsymbol{T}(F_{\varepsilon})}{\partial\varepsilon}\right|_{\varepsilon=0^{+}},$
(59)

with $\varepsilon$ being the contamination proportion and $\Delta_{t}$ being
the degenerate distribution at the contamination point $t$. Thus, the IF, as a
function of $t$, measures the standardized asymptotic bias caused by the
infinitesimal contamination at the point $t$. The maximum of this IF over $t$
indicates the extent of bias due to contamination and so smaller its value,
the more robust the estimator is. Note that, in this context, the statistical
functional $\boldsymbol{T}_{\alpha}$ corresponding to the MRPDE is defined as
the minimizer of $R_{\alpha}(f_{\boldsymbol{\theta}},g)$ in (1).

IF for MLE in the bidimensional normal model has been widely studied in
literature. For example, [12] presented the IF for the Pearson’s correlation
coefficient $\rho$. A proof was given years later by [11]. IFs for the mean
and variance can be found, in [30] and [19], among others.

In [7] (Theorem 5), the IF of Renyi’s pseudodistances-based estimators was
provided in a general form and particularized to some particular models. [10]
generalized this result to the case of independent not identically distributed
observations. Based on these results, in Theorem 8 we present the IF
associated to the MRPDE of $\boldsymbol{\theta}$ the bidimensional normal
model. A detailed proof of the following result is provided in Appendix E.

###### Theorem 8

Let us consider the bidimensional normal model (14). The IF associated to the
MRPDE of $\boldsymbol{\theta}$ is given by

$\mathcal{IF}((x,y)^{T},\boldsymbol{T}_{\alpha},F_{\boldsymbol{\theta}})=(\mathcal{IF}_{\alpha}(\mu_{1}),\mathcal{IF}_{\alpha}(\mu_{2}),\mathcal{IF}_{\alpha}(\sigma_{1}),\mathcal{IF}_{\alpha}(\sigma_{2}),\mathcal{IF}_{\alpha}(\rho))^{T},$

where

$\displaystyle\mathcal{IF}_{\alpha}(\mu_{1})$
$\displaystyle=(\alpha+1)^{2}w_{\boldsymbol{\theta}}^{-\alpha/(1-\rho^{2})}(x,y)(x-\mu_{1}),$
(60) $\displaystyle\mathcal{IF}_{\alpha}(\mu_{2})$
$\displaystyle=(\alpha+1)^{2}w_{\boldsymbol{\theta}}^{-\alpha/(1-\rho^{2})}(x,y)(y-\mu_{2}),$
(61) $\displaystyle\mathcal{IF}_{\alpha}(\sigma_{1})$
$\displaystyle=\frac{\left(\alpha+1\right)^{3}}{2}w_{\boldsymbol{\theta}}^{-\alpha}(x,y)\tfrac{\sigma_{1}}{1-\rho^{2}}\left[\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}-\rho^{2}\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}-(1-\rho^{2})(1+2\rho^{2})\tfrac{1}{\alpha+1}\right],$
(62) $\displaystyle\mathcal{IF}_{\alpha}(\sigma_{2})$
$\displaystyle=\frac{\left(\alpha+1\right)^{3}}{2}w_{\boldsymbol{\theta}}^{-\alpha}(x,y)\tfrac{\sigma_{2}}{1-\rho^{2}}\left[\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}-\rho^{2}\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}-(1-\rho^{2})(1+2\rho^{2})\tfrac{1}{\alpha+1}\right],$
(63) $\displaystyle\mathcal{IF}_{\alpha}(\rho)$
$\displaystyle=\left(\alpha+1\right)^{3}w_{\boldsymbol{\theta}}^{-\alpha}(x,y)\left\\{-\tfrac{\rho}{2}\left[\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}+\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}\right]+\tfrac{x-\mu_{1}}{\sigma_{1}}\tfrac{y-\mu_{2}}{\sigma_{2}}\right\\},$
(64)

with

$w_{\boldsymbol{\theta}}(x,y)=\exp\left\\{\tfrac{1}{2(1-\rho^{2})}\left[(\tfrac{x-\mu_{1}}{\sigma_{1}})^{2}+(\tfrac{y-\mu_{2}}{\sigma_{2}})^{2}-2\rho\tfrac{x-\mu_{1}}{\sigma_{1}}\tfrac{y-\mu_{2}}{\sigma_{2}}\right]\right\\}.$

###### Remark 1

In particular, for $\alpha=0$ (MLE),

$\displaystyle\mathcal{IF}_{\alpha=0}(\mu_{1})$ $\displaystyle=x-\mu_{1},$
$\displaystyle\mathcal{IF}_{\alpha=0}(\mu_{2})$ $\displaystyle=y-\mu_{2},$
$\displaystyle\mathcal{IF}_{\alpha=0}(\sigma_{1})$
$\displaystyle=\tfrac{\sigma_{1}}{2(1-\rho^{2})}\left[\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}-\rho^{2}\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}\right]-\tfrac{\sigma_{1}}{2}(1+2\rho^{2}),$
$\displaystyle\mathcal{IF}_{\alpha=0}(\sigma_{2})$
$\displaystyle=\tfrac{\sigma_{2}}{2(1-\rho^{2})}\left[\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}-\rho^{2}\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}\right]-\tfrac{\sigma_{1}}{2}(1+2\rho^{2}),$
$\mathcal{IF}_{\alpha=0}(\rho)=-\tfrac{\rho}{2}\left[\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}+\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}\right]+\tfrac{x-\mu_{1}}{\sigma_{1}}\tfrac{y-\mu_{2}}{\sigma_{2}}.$

The IF (65) is bounded for positive values of the parameter $\alpha$,
$\alpha>0$, and is unbounded at the MLE, $\alpha=0$.

|
---|---
|
|
Figure 1: $\mathcal{IF}_{\alpha}(\mu_{1})$ (above),
$\mathcal{IF}_{\alpha}(\sigma_{1})$ (middle) and $\mathcal{IF}_{\alpha}(\rho)$
(below) for $\alpha=0$ (left) and $\alpha=0.3$ (right), with
$\boldsymbol{\theta}=(1,2,1,1.5,0.3)^{T}$.

Once we have computed the IF for the minimum RP estimators, we can define and
study the IF for the Wald-type test statistics defined in (42). As noted by
[10], when the corresponding IF is identically zero and is therefore necessary
to consider the second order IF of the proposed Wald-type tests functional
$W_{\alpha}$.

###### Theorem 9

Let us consider the bidimensional normal model (14). The second order IF of
the proposed Wald-type test functionals for testing simple null hypothesis in
(41) is given by

$\displaystyle\mathcal{IF}_{2}((x,y)^{T},W_{\alpha},F_{\boldsymbol{\theta}})$
$\displaystyle=2(\mathcal{IF}((x,y)^{T},\boldsymbol{T}_{\alpha},F_{\boldsymbol{\theta}}))^{T}\boldsymbol{M}(\boldsymbol{\theta})\left(\boldsymbol{M}^{T}(\boldsymbol{\theta})\boldsymbol{V}_{\alpha}(\boldsymbol{\theta})\boldsymbol{M}(\boldsymbol{\theta})\right)^{-1}\boldsymbol{M}^{T}(\boldsymbol{\theta})\mathcal{IF}((x,y)^{T},\boldsymbol{T}_{\alpha},F_{\boldsymbol{\theta}}),$
(65)

where
$\mathcal{IF}((x,y)^{T},\boldsymbol{T}_{\alpha},F_{\boldsymbol{\theta}})$ is
given in Theorem 8.

Note that the second-order IF of the proposed Wald-type tests is a quadratic
function of the corresponding IF of the MRPDE. Therefore, the boundedness of
the IF of MRPDE at $\alpha>0$ also indicates the boundedness of the IF of the
Wald-type test functionals, implying its robustness against contamination.

|
---|---
Figure 2: $\mathcal{IF}_{2}((x,y)^{T},W_{\alpha},F_{\boldsymbol{\theta}})$ for
testing $H_{0}$: $\sigma_{1}=\sigma_{2}$, with $\alpha=0$ (left) and
$\alpha=0.3$ (right), and $\boldsymbol{\theta}=(1,2,1,1.5,0.3)^{T}$.

##  6 Simulation study

It is well-known that the Morgan-Pitman test is best unbiased and best
invariant test statistic for testing equality of variances (see [26], [28] and
[17]). The idea behind the Morgan-Pitman test allows us to include for testing
equality of variances not only the family of test statistics given in Case 2
but also the family of test statistics given in Case 3 for the data
$(u_{1},v_{1}),\ldots,(u_{n},v_{n})$, where $U=X+Y$ and $V=X-Y$ are
transformed variables. As $\mathrm{Cov}[U,V]=\sigma_{1}^{2}-\sigma_{2}^{2}$,
testing $H_{0}$: $\sigma_{1}=\sigma_{2}$ given in (45) (Case 2), matches
$H_{0}$: $\rho_{UV}=0$, from (48) with $\rho_{0}=0$ (Case 3). We have compared
these two families of test statistics, first (46) in Tables 2, 10, 14, 18,
rewritten as

$W_{n,\alpha}(\widehat{\gamma}_{R,\alpha},\widehat{\rho}_{R,\alpha})=n\frac{\left(2\alpha+1\right)^{3}}{\left(\alpha+1\right)^{6}}\frac{(\widehat{\gamma}_{R,\alpha}-1)^{2}}{\beta_{\alpha}(\widehat{\gamma}_{R,\alpha},\widehat{\rho}_{R,\alpha})},$
(66)

where

$\displaystyle\widehat{\gamma}_{R,\alpha}$
$\displaystyle=\frac{\widehat{\sigma}_{1,R,\alpha}}{\widehat{\sigma}_{2,R,\alpha}},$
$\displaystyle\beta_{\alpha}(\widehat{\gamma}_{R,\alpha},\widehat{\rho}_{R,\alpha})$
$\displaystyle=\frac{1}{4}\left[\left(\tfrac{\alpha}{\alpha+1}\right)^{2}+2\right](\widehat{\gamma}_{R,\alpha}-1)^{2}+(1-\widehat{\rho}_{R,\alpha}^{2})\widehat{\gamma}_{R,\alpha}.$
(67)

and second (50) in Tables 4, 12, 16, 20, rewritten as

$W_{n,\alpha}^{\prime}(\widehat{\rho}_{UV,R,\alpha})=n\frac{\left(2\alpha+1\right)^{3}}{\left(\alpha+1\right)^{6}}\widehat{\rho}_{UV,R,\alpha}^{2}.$
(68)

A third one, (49), was also considered but do not present the results here as
the corresponding results were very bad in comparison with (50). In addition,
the exact Morgan-Pitman test,

$T_{MP}=\widehat{\rho}_{UV,R,\alpha=0}\sqrt{\frac{n-2}{1-\widehat{\rho}_{UV,R,\alpha=0}^{2}}},$
(69)

is taken into account, whose exact distribution is a Student $t$ with $n-2$
degrees of freedom ($t_{n-2}$) under $H_{0}$, with

$\widehat{\rho}_{UV,R,\alpha=0}=\frac{\sum_{i=1}^{n}(U_{i}-\bar{U}_{n})(V_{i}-\bar{V}_{n})}{\sqrt{\sum_{i=1}^{n}(U_{i}-\bar{U}_{n})^{2}}\sqrt{\sum_{i=1}^{n}(V_{i}-\bar{V}_{n})^{2}}}=R_{UV},$

being the Pearson correlation coefficient, i.e. the MLE of $\rho_{UV}$.
Furthermore, we included the simulated significance level of the Morgan Pitman
test described in (69) in the aforemetioned tables and calculated by
simulation
$\mathrm{MSE}(\widehat{\gamma}_{R,\alpha})=\left|\widehat{\gamma}_{R,\alpha}-1\right|$
in Tables 1, 9, 13, 17 as well as
$\mathrm{MSE}(\widehat{\gamma}_{R,\alpha})=\left|\widehat{\rho}_{R,\alpha}\right|$
in Tables 3, 11, 15, 19.

So as to evaluate the performance of the proposed Wald-type tests, we
considered the bidimensional normal model (14) for the true parameters values
$\mu_{1}=\mu_{2}=0$, $\sigma_{1}=\sigma_{2}=1$ and the different correlations
between the normal variables $\rho\in\\{0,0.\allowbreak 3,0.\allowbreak
6,0.\allowbreak 9\\}.$ Additionally, in order to evaluate the robustness of
the Wald-type tests, we analysed ten different scenarios of contamination:

  * •

Pure data

  * •

Slightly contaminated data : We replace a $5\%,10\%$ and $20\%$ of the samples
by a bidimensional normal distribution, substituting the true parameter values
$\sigma_{1}^{\prime}=\sigma_{2}^{\prime}=1$ by
$\sigma_{1}^{\prime}=\sigma_{2}^{\prime}=\sqrt{3}.$

  * •

Contaminated data : We replace a $5\%,10\%$ and $20\%$ of the samples by a
bidimensional Student $t$ distribution with $d=5$ degrees of freedom.

  * •

Heavily contaminated data : We replace a $5\%,10\%$ and $20\%$ of the samples
by a bidimensional normal distribution, substituting the true parameter value
$\sigma_{2}^{\prime}=1$ by $\sigma_{2}^{\prime}=5.$

|  |  | slightly | regular | heavily
---|---|---|---|---|---
$\rho$ | $\alpha$ | pure | $0.05$ | $0.10$ | $0.20$ | $0.05$ | $0.10$ | $0.20$ | $0.05$ | $0.10$ | $0.20$
$0$ | $0$ | 0.169 | 0.175 | 0.182 | 0.190 | 0.173 | 0.179 | 0.186 | 0.279 | 0.380 | 0.522
| $0.1$ | 0.170 | 0.173 | 0.178 | 0.184 | 0.172 | 0.175 | 0.178 | 0.209 | 0.281 | 0.432
| $0.2$ | 0.176 | 0.177 | 0.182 | 0.187 | 0.178 | 0.179 | 0.181 | 0.187 | 0.220 | 0.320
| $0.3$ | 0.187 | 0.187 | 0.192 | 0.198 | 0.188 | 0.189 | 0.192 | 0.191 | 0.209 | 0.267
| $0.5$ | 0.223 | 0.224 | 0.229 | 0.236 | 0.228 | 0.227 | 0.230 | 0.226 | 0.234 | 0.264
| $0.7$ | 0.307 | 0.327 | 0.313 | 0.320 | 0.304 | 0.384 | 0.313 | 0.330 | 0.317 | 0.351
$0.3$ | $0$ | 0.161 | 0.169 | 0.175 | 0.182 | 0.164 | 0.168 | 0.176 | 0.278 | 0.376 | 0.523
| $0.1$ | 0.162 | 0.167 | 0.170 | 0.176 | 0.162 | 0.164 | 0.169 | 0.204 | 0.273 | 0.430
| $0.2$ | 0.168 | 0.171 | 0.174 | 0.180 | 0.168 | 0.168 | 0.173 | 0.181 | 0.210 | 0.313
| $0.3$ | 0.178 | 0.181 | 0.184 | 0.189 | 0.177 | 0.177 | 0.183 | 0.184 | 0.199 | 0.256
| $0.5$ | 0.213 | 0.216 | 0.219 | 0.225 | 0.214 | 0.213 | 0.218 | 0.218 | 0.226 | 0.252
| $0.7$ | 0.290 | 0.290 | 0.304 | 0.329 | 0.298 | 0.289 | 0.295 | 0.290 | 0.312 | 0.353
$0.6$ | $0$ | 0.133 | 0.142 | 0.146 | 0.150 | 0.139 | 0.142 | 0.147 | 0.268 | 0.371 | 0.522
| $0.1$ | 0.134 | 0.140 | 0.142 | 0.146 | 0.137 | 0.139 | 0.140 | 0.178 | 0.252 | 0.417
| $0.2$ | 0.139 | 0.143 | 0.145 | 0.149 | 0.141 | 0.143 | 0.143 | 0.154 | 0.181 | 0.288
| $0.3$ | 0.147 | 0.151 | 0.154 | 0.157 | 0.149 | 0.152 | 0.152 | 0.156 | 0.169 | 0.227
| $0.5$ | 0.176 | 0.179 | 0.184 | 0.189 | 0.176 | 0.181 | 0.183 | 0.183 | 0.191 | 0.220
| $0.7$ | 0.241 | 0.248 | 0.253 | 0.254 | 0.237 | 0.245 | 0.244 | 0.250 | 0.268 | 0.289
$0.9$ | $0$ | 0.074 | 0.077 | 0.079 | 0.081 | 0.075 | 0.077 | 0.081 | 0.240 | 0.367 | 0.525
| $0.1$ | 0.074 | 0.076 | 0.077 | 0.078 | 0.075 | 0.075 | 0.077 | 0.099 | 0.162 | 0.348
| $0.2$ | 0.077 | 0.078 | 0.079 | 0.080 | 0.077 | 0.077 | 0.078 | 0.082 | 0.098 | 0.170
| $0.3$ | 0.081 | 0.082 | 0.084 | 0.084 | 0.081 | 0.081 | 0.083 | 0.085 | 0.092 | 0.120
| $0.5$ | 0.097 | 0.098 | 0.101 | 0.102 | 0.099 | 0.097 | 0.099 | 0.101 | 0.107 | 0.121
| $0.7$ | 0.131 | 0.131 | 0.136 | 0.143 | 0.137 | 0.130 | 0.135 | 0.136 | 0.148 | 0.204
Table 1: Simulated mean square error of the MRPDE for ratio of variances, $\widehat{\gamma}_{R,\alpha}$, when $n=25$ |  |  | slightly | regular | heavily
---|---|---|---|---|---
$\rho$ | $\alpha$ | pure | $0.05$ | $0.10$ | $0.20$ | $0.05$ | $0.10$ | $0.20$ | $0.05$ | $0.10$ | $0.20$
$0$ | $0$ | 0.059 | 0.070 | 0.081 | 0.093 | 0.067 | 0.074 | 0.089 | 0.352 | 0.585 | 0.852
| $0.1$ | 0.058 | 0.063 | 0.072 | 0.079 | 0.062 | 0.064 | 0.069 | 0.172 | 0.362 | 0.699
| $0.2$ | 0.059 | 0.064 | 0.069 | 0.075 | 0.063 | 0.064 | 0.066 | 0.094 | 0.180 | 0.438
| $0.3$ | 0.064 | 0.068 | 0.071 | 0.079 | 0.067 | 0.067 | 0.072 | 0.081 | 0.124 | 0.276
| $0.5$ | 0.085 | 0.091 | 0.094 | 0.103 | 0.093 | 0.089 | 0.097 | 0.100 | 0.115 | 0.182
| $0.7$ | 0.144 | 0.148 | 0.155 | 0.167 | 0.154 | 0.146 | 0.159 | 0.157 | 0.173 | 0.217
| MP | 0.051 | 0.062 | 0.071 | 0.082 | 0.059 | 0.065 | 0.080 | 0.341 | 0.572 | 0.844
$0.3$ | $0$ | 0.061 | 0.075 | 0.082 | 0.092 | 0.062 | 0.072 | 0.086 | 0.365 | 0.596 | 0.860
| $0.1$ | 0.060 | 0.066 | 0.071 | 0.078 | 0.056 | 0.060 | 0.069 | 0.179 | 0.361 | 0.708
| $0.2$ | 0.061 | 0.064 | 0.067 | 0.075 | 0.058 | 0.058 | 0.068 | 0.097 | 0.181 | 0.433
| $0.3$ | 0.065 | 0.066 | 0.070 | 0.079 | 0.062 | 0.063 | 0.073 | 0.081 | 0.123 | 0.271
| $0.5$ | 0.089 | 0.093 | 0.097 | 0.104 | 0.088 | 0.088 | 0.098 | 0.101 | 0.122 | 0.181
| $0.7$ | 0.146 | 0.152 | 0.156 | 0.172 | 0.148 | 0.143 | 0.154 | 0.158 | 0.181 | 0.223
| MP | 0.052 | 0.063 | 0.071 | 0.082 | 0.054 | 0.062 | 0.075 | 0.352 | 0.584 | 0.852
$0.6$ | $0$ | 0.057 | 0.072 | 0.081 | 0.090 | 0.067 | 0.075 | 0.090 | 0.394 | 0.631 | 0.883
| $0.1$ | 0.056 | 0.064 | 0.071 | 0.075 | 0.060 | 0.064 | 0.070 | 0.175 | 0.368 | 0.713
| $0.2$ | 0.058 | 0.064 | 0.068 | 0.072 | 0.060 | 0.062 | 0.070 | 0.095 | 0.167 | 0.430
| $0.3$ | 0.063 | 0.069 | 0.074 | 0.077 | 0.065 | 0.067 | 0.074 | 0.085 | 0.116 | 0.265
| $0.5$ | 0.088 | 0.093 | 0.102 | 0.109 | 0.088 | 0.091 | 0.099 | 0.101 | 0.120 | 0.183
| $0.7$ | 0.148 | 0.148 | 0.160 | 0.171 | 0.143 | 0.151 | 0.155 | 0.157 | 0.179 | 0.226
| MP | 0.047 | 0.062 | 0.069 | 0.076 | 0.057 | 0.064 | 0.079 | 0.380 | 0.616 | 0.875
$0.9$ | $0$ | 0.064 | 0.078 | 0.084 | 0.091 | 0.068 | 0.077 | 0.092 | 0.465 | 0.719 | 0.932
| $0.1$ | 0.061 | 0.068 | 0.073 | 0.075 | 0.062 | 0.065 | 0.073 | 0.125 | 0.282 | 0.647
| $0.2$ | 0.060 | 0.067 | 0.073 | 0.072 | 0.063 | 0.063 | 0.071 | 0.075 | 0.111 | 0.280
| $0.3$ | 0.066 | 0.069 | 0.079 | 0.077 | 0.068 | 0.068 | 0.075 | 0.073 | 0.094 | 0.162
| $0.5$ | 0.089 | 0.093 | 0.105 | 0.110 | 0.097 | 0.092 | 0.102 | 0.100 | 0.118 | 0.146
| $0.7$ | 0.147 | 0.151 | 0.165 | 0.171 | 0.157 | 0.153 | 0.163 | 0.158 | 0.185 | 0.217
| MP | 0.051 | 0.064 | 0.069 | 0.077 | 0.057 | 0.063 | 0.078 | 0.448 | 0.706 | 0.927
Table 2: Simulated significance level for testing equal variances through $W_{n,\alpha}(\widehat{\gamma}_{R,\alpha},\widehat{\rho}_{R,\alpha})$ given by (66) and the Morgan-Pitman test, when $n=25$ |  |  | slightly | regular | heavily
---|---|---|---|---|---
$\rho$ | $\alpha$ | pure | $0.05$ | $0.10$ | $0.20$ | $0.05$ | $0.10$ | $0.20$ | $0.05$ | $0.10$ | $0.20$
$0$ | $0$ | 0.165 | 0.171 | 0.177 | 0.185 | 0.169 | 0.174 | 0.181 | 0.317 | 0.448 | 0.627
| $0.1$ | 0.167 | 0.169 | 0.174 | 0.180 | 0.168 | 0.170 | 0.174 | 0.225 | 0.322 | 0.515
| $0.2$ | 0.172 | 0.173 | 0.177 | 0.183 | 0.173 | 0.174 | 0.177 | 0.193 | 0.238 | 0.371
| $0.3$ | 0.181 | 0.182 | 0.187 | 0.192 | 0.183 | 0.183 | 0.187 | 0.192 | 0.218 | 0.297
| $0.5$ | 0.213 | 0.214 | 0.218 | 0.224 | 0.217 | 0.215 | 0.219 | 0.220 | 0.234 | 0.272
| $0.7$ | 0.270 | 0.272 | 0.277 | 0.284 | 0.275 | 0.273 | 0.278 | 0.277 | 0.291 | 0.320
$0.3$ | $0$ | 0.165 | 0.173 | 0.178 | 0.185 | 0.168 | 0.173 | 0.180 | 0.325 | 0.453 | 0.636
| $0.1$ | 0.166 | 0.171 | 0.174 | 0.180 | 0.166 | 0.169 | 0.173 | 0.227 | 0.321 | 0.520
| $0.2$ | 0.171 | 0.175 | 0.178 | 0.184 | 0.171 | 0.173 | 0.177 | 0.194 | 0.236 | 0.370
| $0.3$ | 0.181 | 0.184 | 0.187 | 0.192 | 0.181 | 0.182 | 0.187 | 0.194 | 0.216 | 0.294
| $0.5$ | 0.213 | 0.217 | 0.220 | 0.225 | 0.213 | 0.213 | 0.220 | 0.223 | 0.234 | 0.271
| $0.7$ | 0.270 | 0.275 | 0.278 | 0.287 | 0.271 | 0.270 | 0.277 | 0.280 | 0.293 | 0.322
$0.6$ | $0$ | 0.163 | 0.172 | 0.177 | 0.183 | 0.169 | 0.172 | 0.179 | 0.342 | 0.477 | 0.664
| $0.1$ | 0.165 | 0.170 | 0.173 | 0.178 | 0.168 | 0.169 | 0.172 | 0.225 | 0.324 | 0.534
| $0.2$ | 0.171 | 0.175 | 0.177 | 0.181 | 0.172 | 0.174 | 0.175 | 0.192 | 0.230 | 0.370
| $0.3$ | 0.180 | 0.183 | 0.186 | 0.191 | 0.181 | 0.184 | 0.184 | 0.193 | 0.212 | 0.290
| $0.5$ | 0.212 | 0.215 | 0.218 | 0.225 | 0.212 | 0.215 | 0.218 | 0.221 | 0.232 | 0.271
| $0.7$ | 0.270 | 0.272 | 0.280 | 0.287 | 0.268 | 0.275 | 0.276 | 0.279 | 0.294 | 0.323
$0.9$ | $0$ | 0.166 | 0.173 | 0.178 | 0.183 | 0.168 | 0.172 | 0.180 | 0.382 | 0.542 | 0.729
| $0.1$ | 0.167 | 0.171 | 0.174 | 0.176 | 0.167 | 0.167 | 0.172 | 0.199 | 0.283 | 0.514
| $0.2$ | 0.172 | 0.175 | 0.178 | 0.179 | 0.172 | 0.171 | 0.175 | 0.179 | 0.201 | 0.294
| $0.3$ | 0.181 | 0.183 | 0.187 | 0.189 | 0.182 | 0.180 | 0.184 | 0.186 | 0.198 | 0.235
| $0.5$ | 0.215 | 0.216 | 0.223 | 0.228 | 0.217 | 0.214 | 0.219 | 0.220 | 0.232 | 0.254
| $0.7$ | 0.285 | 0.286 | 0.296 | 0.307 | 0.289 | 0.284 | 0.294 | 0.294 | 0.311 | 0.344
Table 3: Simulated mean square error of the MRPDE for correlation coefficient, $\widehat{\rho}_{R,\alpha}$, when $n=25$ |  |  | slightly | regular | heavily
---|---|---|---|---|---
$\rho$ | $\alpha$ | pure | $0.05$ | $0.10$ | $0.20$ | $0.05$ | $0.10$ | $0.20$ | $0.05$ | $0.10$ | $0.20$
$0$ | $0$ | 0.053 | 0.065 | 0.074 | 0.085 | 0.062 | 0.069 | 0.082 | 0.345 | 0.576 | 0.847
| $0.1$ | 0.053 | 0.058 | 0.065 | 0.072 | 0.057 | 0.059 | 0.063 | 0.162 | 0.351 | 0.689
| $0.2$ | 0.053 | 0.058 | 0.063 | 0.068 | 0.057 | 0.058 | 0.061 | 0.087 | 0.170 | 0.425
| $0.3$ | 0.058 | 0.060 | 0.064 | 0.071 | 0.060 | 0.061 | 0.065 | 0.073 | 0.115 | 0.264
| $0.5$ | 0.076 | 0.082 | 0.085 | 0.095 | 0.085 | 0.080 | 0.090 | 0.092 | 0.106 | 0.168
| $0.7$ | 0.132 | 0.137 | 0.143 | 0.154 | 0.141 | 0.135 | 0.146 | 0.144 | 0.160 | 0.203
| MP | 0.051 | 0.062 | 0.071 | 0.082 | 0.059 | 0.065 | 0.080 | 0.341 | 0.572 | 0.844
$0.3$ | $0$ | 0.055 | 0.067 | 0.074 | 0.084 | 0.056 | 0.065 | 0.079 | 0.357 | 0.588 | 0.855
| $0.1$ | 0.055 | 0.059 | 0.063 | 0.072 | 0.051 | 0.054 | 0.062 | 0.170 | 0.350 | 0.699
| $0.2$ | 0.055 | 0.058 | 0.060 | 0.067 | 0.051 | 0.051 | 0.062 | 0.088 | 0.170 | 0.422
| $0.3$ | 0.058 | 0.059 | 0.062 | 0.072 | 0.055 | 0.056 | 0.066 | 0.073 | 0.113 | 0.258
| $0.5$ | 0.080 | 0.084 | 0.088 | 0.094 | 0.079 | 0.080 | 0.088 | 0.091 | 0.111 | 0.168
| $0.7$ | 0.134 | 0.138 | 0.143 | 0.157 | 0.137 | 0.132 | 0.141 | 0.144 | 0.167 | 0.208
| MP | 0.052 | 0.063 | 0.071 | 0.082 | 0.054 | 0.062 | 0.075 | 0.352 | 0.584 | 0.852
$0.6$ | $0$ | 0.049 | 0.064 | 0.072 | 0.080 | 0.060 | 0.067 | 0.082 | 0.386 | 0.621 | 0.878
| $0.1$ | 0.048 | 0.057 | 0.063 | 0.066 | 0.053 | 0.056 | 0.064 | 0.164 | 0.353 | 0.705
| $0.2$ | 0.049 | 0.056 | 0.060 | 0.064 | 0.052 | 0.055 | 0.060 | 0.087 | 0.155 | 0.417
| $0.3$ | 0.055 | 0.060 | 0.065 | 0.068 | 0.057 | 0.058 | 0.065 | 0.075 | 0.104 | 0.251
| $0.5$ | 0.077 | 0.082 | 0.089 | 0.098 | 0.077 | 0.082 | 0.087 | 0.089 | 0.106 | 0.168
| $0.7$ | 0.135 | 0.134 | 0.148 | 0.159 | 0.130 | 0.138 | 0.142 | 0.146 | 0.166 | 0.211
| MP | 0.047 | 0.062 | 0.069 | 0.076 | 0.057 | 0.064 | 0.079 | 0.380 | 0.616 | 0.875
$0.9$ | $0$ | 0.055 | 0.067 | 0.072 | 0.081 | 0.060 | 0.067 | 0.081 | 0.453 | 0.710 | 0.929
| $0.1$ | 0.052 | 0.059 | 0.063 | 0.064 | 0.053 | 0.055 | 0.063 | 0.112 | 0.268 | 0.635
| $0.2$ | 0.051 | 0.057 | 0.061 | 0.063 | 0.052 | 0.053 | 0.060 | 0.064 | 0.098 | 0.265
| $0.3$ | 0.055 | 0.059 | 0.065 | 0.066 | 0.057 | 0.057 | 0.064 | 0.064 | 0.082 | 0.148
| $0.5$ | 0.080 | 0.084 | 0.094 | 0.100 | 0.088 | 0.082 | 0.091 | 0.089 | 0.106 | 0.140
| $0.7$ | 0.152 | 0.158 | 0.171 | 0.182 | 0.161 | 0.157 | 0.168 | 0.164 | 0.191 | 0.236
| MP | 0.051 | 0.064 | 0.069 | 0.077 | 0.057 | 0.063 | 0.078 | 0.448 | 0.706 | 0.927
Table 4: Simulated significance level for testing null correlation coefficient
through $W_{n,\alpha}^{\prime}(\widehat{\rho}_{UV,R,\alpha})$ given by (68)
and the Morgan-Pitman test, when $n=25$

We repeated the same schema for a nominal type I error,
$\varsigma=0.\allowbreak 05$, for different sample sizes
$n\in\\{15,25,50,100\\}$, but in the main document only the case of $n=25$ is
presented (the remaining sizes are included in the Appendix G). We report, for
the different values of the tuning parameter $\alpha\in\\{0,0.\allowbreak
1,0.\allowbreak 2,0.\allowbreak 3,0.\allowbreak 5,0.\allowbreak 7\\}$, the
simulated mean square error (MSE) committed in the estimation of
$\gamma=\sigma_{1}/\sigma_{2}$ and $\rho$ as well as the simulated
significance level of the tests, computed as the number of times the null
hypothesis is rejected out of the total simulated samples $R=15,000$.

With pure data, as expected, the MSEs and closeness of the simulated
significance level of both asymptotic tests, (66) and (68), to the nominal
significance level, $\varsigma=0.\allowbreak 05$, is improved as the sample
size, $n$, increases. For MSEs under contamination $\alpha\in\\{0.\allowbreak
1,0.\allowbreak 2\\}$ tuning parameters outperform the MSEs with $\alpha=0$
but the greatest improvement under contamination is for the simulated
significance levels of $W_{n,\alpha}^{\prime}(\widehat{\rho}_{UV,R,\alpha})$,
given in (68), when $\alpha=0.\allowbreak 2$, since it is always better than
any other, included the well-known Morgan-Pitman test, for all the considered
scenarios.

For MSEs under contamination, MRPDEs with $\alpha\geq 0.\allowbreak 1$
outperform the MSEs with $\alpha=0$ (the MLE) for all the considered
scenarios. Furthermore, the effect of contamination is accented for low sample
sizes. The greater contamination, the greater the optimal choice of the tuning
parameter is, although moderate values of $\alpha$ over $0.2$ generally offer
a suitable trade-off between efficiency and robustness. Conversely,
contamination is quite difficult to measure in real-life datasets, and
therefore an optimality criterion for the choice of the best tuning parameter
is of great practical interest. Several criteria have been proposed in the
robustness literature for choosing optimal values of the DPD tuning parameter,
that can be straightforward extended for the RP divergence. [39] proposed a
useful data-based procedure based on the minimization of an estimate of the
asymptotic MSE, given by

$\widehat{\text{MSE}}(\alpha)=(\widehat{\boldsymbol{\theta}}_{R,\alpha}-\boldsymbol{\theta}_{P})^{T}(\widehat{\boldsymbol{\theta}}_{R,\alpha}-\boldsymbol{\theta}_{P})+\frac{1}{n}\operatorname{tr}(\boldsymbol{V}_{\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})),$
(70)

where the $\operatorname{tr}$ denotes the trace of the matrix
$\boldsymbol{V}_{\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})$ given in
(8) and $\boldsymbol{\theta}_{P}$ is a pilot estimator used for assessing the
bias. Naturally, the previous criterion was considerably pilot-dependent, as
the pilot invariably draws the optimal estimator towards itself. [2] improved
the method by iteratively updating the pilot with the optimal estimate
obtained, and so the process was repeated until there was no further change in
the optimal estimate. The iterative algorithm empirically shown to alleviate
the pilot dependency of the optimal choice. Then, Basak et al. algorithm could
be adopted in real life problems for choosing the best MRPDE of the bivariate
normal distribution, iteratively minimizing formula (70). The MLE or MRPDE
with moderate values of the tuning parameter $\alpha=0.2$ can be used as
initial pilot estimators. Further, applying transformed Wald-type test
statistics, $W_{n,\alpha}^{\prime}(\widehat{\rho}_{UV,R,\alpha})$ given in
(68) entails a clear improvement in terms of significance levels under
contamination. In this case, we would empirically recommend the choice of
$\alpha=0.\allowbreak 2$ since it is better than any other, included the well-
known Morgan-Pitman test, in most of the considered scenarios.

##  7 Illustrative examples

###  7.1 Cork data set: comparing means or variances

Originally studied in [31], there is a well-known and publicity available real
data set, the cork data set. It is included in several R packages ([29]), in
particular in agridat as a box.cork data. The data report the weights of cork
boring of the trunk of $28$ trees in the north, east, west and south sides.
Rao pointed out that there exist positive correlation between the reported
pairs of $4$ variables, and sometimes it is assumed that they follow a normal
distribution. Four-dimensional normality is an arguable issue since using the
R package MVN, in four out of five tests could multivariate normality be
rejected with significance level $0.05$, as shown in Table 5 (left hand side).
We focussed on the two variables devoted to east and south sides respectively,
as in [40, 41], performed two-dimensional normal tests and this time all the
tests rejected according to Table 5 (central columns). We did not study the
robustness of an estimator and test statistic as in [40], in the sense of
being resistant to data coming from distribution different from the bivariate
normal as required for the original data. Our proposed estimator and test
statistic are robust in the sense of being resistant to outliers once
normality is being assumed. Having this in mind, the data were transformed
using the base $e$ logarithm and as shown in Table 5 (right hand side) and in
four out five tests could not be multivariate normality rejected. In addition,
outliers were studied for transformed data through the scatter-plot with
confidence ellipses shown in Figure 3, concluding that observations $18$ and
$16$ were suspicious to be outliers.

Taking the root of the Wald-type test statistics given in Cases 1, 2 and 3, as
well as the Wald-type test given in Section 6 based on transformed variables,
here we are going to provide alternative but equivalent expressions in
practice, having the same $p$-value. From the original data,
$(x_{1},y_{1}),\ldots,(x_{n},y_{n})$, the $z$-type test statistic for equal
variances and based on MRPDEs, is given by

$\displaystyle
Z_{n,\alpha}(\widehat{\gamma}_{R,\alpha},\widehat{\rho}_{R,\alpha})$
$\displaystyle=\mathrm{sign}(\widehat{\gamma}_{R,\alpha}-1)\sqrt{W_{n,\alpha}(\widehat{\gamma}_{R,\alpha},\widehat{\rho}_{R,\alpha})}$
(71)
$\displaystyle=\sqrt{n}\left(\frac{\sqrt{2\alpha+1}}{\alpha+1}\right)^{3}\frac{\widehat{\gamma}_{R,\alpha}-1}{\sqrt{\beta_{\alpha}(\widehat{\gamma}_{R,\alpha},\widehat{\rho}_{R,\alpha})}},$

with $\beta_{\alpha}(\widehat{\gamma}_{R,\alpha},\widehat{\rho}_{R,\alpha})$
given by (67), has a standard normal asymptotic distribution. From the
transformed data $(u_{1},v_{1}),\ldots,(u_{n},v_{n})$, where $U=X+Y$ and
$V=X-Y$, the $z$-type test statistic for equal variances and based on MRPDEs,
is given by

$Z_{n,\alpha}^{\prime}(\widehat{\rho}_{UV,R,\alpha})=\mathrm{sign}(\widehat{\rho}_{UV,R,\alpha})\sqrt{W_{n,\alpha}^{\prime}(\widehat{\rho}_{UV,R,\alpha})}=\sqrt{n}\left(\frac{\sqrt{2\alpha+1}}{\alpha+1}\right)^{3}\widehat{\rho}_{UV,R,\alpha}.$
(72)

Notice that the paired $t$-test for the same test (null correlation), with
exact distribution $t_{n-2}$, is the Morgan-Pitman test $T_{MP}$ given by
(69). The $z$-type test statistic for equal means and based on MRPDEs and the
transformed data $v_{1},\ldots,v_{n}$, where $V=X-Y$, is given by

$Z_{n,\alpha}(\widehat{\mu}_{V,R,\alpha},\widehat{\sigma}_{V,R,\alpha})=\mathrm{sign}(\widehat{\mu}_{V,R,\alpha})\sqrt{W_{n,\alpha}(\widehat{\mu}_{V,R,\alpha},\widehat{\sigma}_{V,R,\alpha})}=\sqrt{n}\frac{2\alpha+1}{\left(\alpha+1\right)^{2}}\frac{\widehat{\mu}_{V,R,\alpha}}{\widehat{\sigma}_{V,R,\alpha}}.$
(73)

Notice that the paired $t$-test for the same test (equal means), with exact
distribution $t_{n-1}$, is given by

$T_{V}=\sqrt{\frac{n-1}{n}}Z_{n,\alpha=0}(\widehat{\mu}_{V,R,\alpha},\widehat{\sigma}_{V,R,\alpha}).$

The results of the classic exact tests, $T_{V}$ for testing (43) or $T_{MP}$
for (45), with two-sided alternative, are summarized in Table 6. The decision,
with $0.05$ significance level, is opposite for both versions of the data,
since the null hypothesis cannot be rejected for the full data set, while it
is rejected for the outliers deleted data.

Table 5: Normality tests for three versions of the corn data set | 4-dimensional | 2-dimensional |  $\log$ 2-dimensional
---|---|---|---
normality test | value |  $p$-value | value |  $p$-value | value |  $p$-value
1.- Doornik-Hansen | 16.123 | 0.041 | 9.833 | 0.043 | 2.490 | 0.646
2.- Henze-Zirkler | 0.999 | 0.011 | 1.236 | 0.003 | 0.784 | 0.053
3.- Royston | 12.161 | 0.003 | 11.784 | 0.002 | 5.564 | 0.047
4.- E-statistic | 1.276 | 0.007 | 1.473 | 0.001 | 0.931 | 0.053
5a.- Mardia: Skewness | 20.890 | 0.404 | 10.231 | 0.037 | 2.622 | 0.623
5b.- Mardia: Kurtosis | -0.398 | 0.690 | 0.899 | 0.369 | -0.795 | 0.427
Figure 3: Scatter-plot of east and south variables, with confidence ellipses, for log-transformed cork data set. Table 6: Classic exact tests of equal means or equal variances for the log-transformed corn data set with respect to full data or ouliers deleted data. | full data | outliers deleted data
---|---|---
classic exact t-test | value |  $p$-value | value |  $p$-value
Paired t-test (equal means), $T_{V}$ | -1.454 | 0.157 | -2.233 | 0.035
Morgan-Pitman test (equal variances), $T_{MP}$ | -1.656 | 0.110 | -3.033 | 0.005

The advantage of these new expressions, (71)-(73), is that one-sided tests can
be considered, apart from the two sided ones (as in the example given in
Section 7.2). Based on (71)-(73), in Figures 4-5 the values of the test
statistics (left hand side) and the values of the estimates of parameters used
to construct the test statistics (right hand sides) are shown, in solid lines
the ones associated with the full log-transformed cork data set and in dashed
lines the ones associated with the outliers deleted log-transformed cork data
set. All the left hand side figures suggest rejecting the null, equal means or
variances, as an appropriate decision with 0.05 significance level.

|
---|---
|
Figure 4: Wald-type tests (left) and estimates (right) for log-transformed cork data set: Case 2 (above) and Case 3 (below) |
---|---
Figure 5: Wald-type tests in Case 1 (left) and mean diference estimates
(right) for log-transformed cork data set

###  7.2 Lactate levels data set: fixing a positive correlation coefficient

[18] studied the one sided test (48), $H_{0}:\rho=\rho_{0}$ vs.
$H_{0}:\rho>\rho_{0}$, where $\rho_{0}=0$, for lactate levels measured in the
blood and the cerebrospinal fluid on $13$ female subjects. The study was done
with a newly proposed robust test statistic, but in the sense of being
resistant to data coming from distribution different from the bivariate
normal. Using the R package MVN, in none of tests could multivariate normality
be rejected with significance level $0.05$, as shown in Table 7. Again, we
highlight that our proposed estimators and test statistics are robust in the
sense of being resistant to outliers once normality is being assumed. Deleting
the two most influential observations, i.e. taking observations $1$ and $7$ as
influential (rather than outliers), the sample Pearson correlation is modified
from $0.572$ for the full data to $0.471$ for the influential observations
deleted data (see Figure 6) and according to Table 8, using the Morgan-Pitman
exact test statistic, the decision of being accepted a positive correlation
with $0.05$ significance level is modified to not being possible to be
accepted. With Figure 7 we try to test whether with the $Z$-test statistic
based on MRPDE of $\rho$, (72), a positive correlation could be accepted for
the lactate levels data and actually it suggest as desirable decision nor
being possible to reject it.

Figure 6: Scatter-plot of CSF and blood variables, with confidence ellipses, for lactate levels data set. Table 7: Normality tests for lactate levels data set normality test | value |  $p$-value
---|---|---
1.- Doornik-Hansen | 1.436 | 0.838
2.- Henze-Zirkler | 0.352 | 0.461
3.- Royston | 0.923 | 0.642
4.- E-statistic | 0.620 | 0.509
5a.- Mardia: Skewness | 1.814 | 0.770
5b.- Mardia: Kurtosis | -0.774 | 0.439
|
---|---
Figure 7: Wald-type tests (left) and estimates (right) for the lactate levels data set. Table 8: Classic exact test of uncorrelation for the lactate levels data set with respect to full data or influential observations deleted data. | full data | infl. obs. deleted data
---|---|---
classic exact t-test | value |  $p$-value | value |  $p$-value
one-sided positive correlation with $T_{MP}$ | 2.313 | 0.020 |  1.601 | 0.072

##  8 Concluding Remarks

In practice, it is very important finding out a robust estimator and test
statistic which does not loose much efficiency. Under the null hypothesis
$\rho=0$, the Morgan-Pitman exact test is the most efficient one but among the
classic asymptotic tests there are several versions we should know. However,
we would like to highlight that in comparative studies of recent papers the
most competitive one, the Rao test given in Case 3 (Section 4),
$R_{n,\alpha=0}(\widehat{\boldsymbol{\theta}}_{R,\alpha=0})$, is not often
being recognized. Since the MRPDEs are regulated through a positive $\alpha$
tuning parameter, being the tuning parameter $\alpha=0$ the cornerstone as
being the most efficient one out of all possible values of $\alpha\geq 0$. In
case of having a poor efficiency for the asymptotic test statistic with the
null tuning parameter of the MRPDEs, the test statistics constructed with the
other values of the tuning parameters will increase such lack of efficiency
and the obtained robustness could not compensate such drawback. This is just
what happened with the Wald-type test statistic
$W_{n,\alpha=0}(\widehat{\boldsymbol{\theta}}_{R,\alpha=0})$, given in Case 3
(Section 4) for testing $\rho=0$. Further, we used a modified version of the
Wald-type test statistic
$W_{n,\alpha=0}^{\prime}(\widehat{\boldsymbol{\theta}}_{R,\alpha=0})$, given
in Case 3 (Section 4), which matches the Rao test statistic only when
$\rho_{0}=0$, and has provided for $\alpha=0$ magnificent results in
efficiency and also for $\alpha>0$ strong robust properties. The improvements
and properties are shown by simulation for the specific null hypothesis
$\rho=0$, but proven in the framework of the developed general theory.

## Acknowledgement

This research is supported by the Spanish Grants PGC2018-095194-B-100 and
FPU/018240 (M. Jaenada). E. Castilla, M.Jaenada, N. Martín and L. Pardo are
members of the Instituto de Matematica Interdisciplinar, Complutense
University of Madrid.

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##  Appendix A Proof of Theorem 1

$\displaystyle\tfrac{\partial}{\partial\boldsymbol{\theta}}\boldsymbol{\Psi_{\alpha}^{T}}(X;\boldsymbol{\theta})$
$\displaystyle=\tfrac{\partial}{\partial\boldsymbol{\theta}}f_{\boldsymbol{\theta}}^{\alpha}(X)\left(\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)-\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)\right)+f_{\boldsymbol{\theta}}^{\alpha}(X)\left(\tfrac{\partial}{\partial\boldsymbol{\theta}}\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)-\tfrac{\partial}{\partial\boldsymbol{\theta}}\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)\right)$
$\displaystyle=\alpha
f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\left(\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)-\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)\right)+f_{\boldsymbol{\theta}}^{\alpha}(X)\left(\tfrac{\partial}{\partial\boldsymbol{\theta}}\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)-\tfrac{\partial}{\partial\boldsymbol{\theta}}\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)\right)$
$\displaystyle=\alpha
f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)-\alpha
f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)+f_{\boldsymbol{\theta}}^{\alpha}(X)\tfrac{\partial}{\partial\boldsymbol{\theta}}\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)-f_{\boldsymbol{\theta}}^{\alpha}(X)\tfrac{\partial}{\partial\boldsymbol{\theta}}\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right),$
$\displaystyle
E\left[\tfrac{\partial}{\partial\boldsymbol{\theta}}\boldsymbol{\Psi_{\alpha}^{T}}(X;\boldsymbol{\theta})\right]$
$\displaystyle=\alpha
E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)\right]-\alpha
E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\right]\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)$
$\displaystyle+E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\tfrac{\partial}{\partial\boldsymbol{\theta}}\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)\right]-E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\right]\tfrac{\partial}{\partial\boldsymbol{\theta}}\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\alpha
E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)\right]-\alpha
E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\right]\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)$
$\displaystyle+E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\tfrac{\partial}{\partial\boldsymbol{\theta}}\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)\right]-E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\right]\tfrac{\partial}{\partial\boldsymbol{\theta}}\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\alpha
E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)\right]-\alpha
E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\right]\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)$
$\displaystyle+(\alpha+1)E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\right]\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)-(\alpha+1)E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)\right]$
$\displaystyle=E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\right]\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)-E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)\right]$
$\displaystyle=E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\right]\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)-E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)\right],$
$\displaystyle\boldsymbol{S}_{\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=-E\left[\tfrac{\partial}{\partial\boldsymbol{\theta}}\boldsymbol{\Psi_{\alpha}^{T}}(x;\boldsymbol{\theta})\right]=E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)\right]-E\left[f_{\boldsymbol{\theta}}^{\alpha}(X)\right]\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\boldsymbol{J}_{\alpha}\left(\boldsymbol{\theta}\right)-\kappa_{\alpha}(\boldsymbol{\theta})\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right),$

$\displaystyle\boldsymbol{K}_{\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\mathrm{E}\left[\left(f_{\boldsymbol{\theta}}^{\alpha}(X)\left(\boldsymbol{u}_{\boldsymbol{\theta}}(X)-\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)\right)\right)\left(f_{\boldsymbol{\theta}}^{\alpha}(X)\left(\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)-\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)\right)\right)\right]$
$\displaystyle=\mathrm{E}\left[f_{\boldsymbol{\theta}}^{2\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)\right]+\mathrm{E}\left[f_{\boldsymbol{\theta}}^{2\alpha}(X)\right]\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)$
$\displaystyle-\mathrm{E}\left[f_{\boldsymbol{\theta}}^{2\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}(X)\right]\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)-\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)\mathrm{E}\left[f_{\boldsymbol{\theta}}^{2\alpha}(X)\boldsymbol{u}_{\boldsymbol{\theta}}^{T}(X)\right]$
$\displaystyle=\boldsymbol{J}_{2\alpha}\left(\boldsymbol{\theta}\right)+\kappa_{2\alpha}(\boldsymbol{\theta})\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)$
$\displaystyle-\kappa_{2\alpha}(\boldsymbol{\theta})\boldsymbol{c}_{2\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)-\kappa_{2\alpha}(\boldsymbol{\theta})\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{2\alpha}^{T}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\boldsymbol{S}_{2\alpha}\left(\boldsymbol{\theta}\right)+\kappa_{2\alpha}(\boldsymbol{\theta})\left(\boldsymbol{c}_{2\alpha}(\boldsymbol{\theta})-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})\right)\left(\boldsymbol{c}_{2\alpha}(\boldsymbol{\theta})-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})\right)^{T},$

where

$\displaystyle\boldsymbol{S}_{2\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\boldsymbol{J}_{2\alpha}\left(\boldsymbol{\theta}\right)-\kappa_{2\alpha}(\boldsymbol{\theta})\boldsymbol{c}_{2\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{2\alpha}^{T}\left(\boldsymbol{\theta}\right)$
$\displaystyle\left(\boldsymbol{c}_{2\alpha}(\boldsymbol{\theta})-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})\right)\left(\boldsymbol{c}_{2\alpha}(\boldsymbol{\theta})-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})\right)^{T}$
$\displaystyle=\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)-\boldsymbol{c}_{2\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{\alpha}^{T}\left(\boldsymbol{\theta}\right)-\boldsymbol{c}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{2\alpha}^{T}\left(\boldsymbol{\theta}\right)+\boldsymbol{c}_{2\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{2\alpha}^{T}\left(\boldsymbol{\theta}\right).$

##  Appendix B Proof of Theorem 5

We shall follow 1 as well as Propositions 2, 3, 4. About the first partition,
since
$\boldsymbol{c}_{1,\alpha}\left(\boldsymbol{\theta}\right)=\boldsymbol{0}_{2}$
it is trivial that

$\displaystyle\boldsymbol{S}_{1,\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\boldsymbol{J}_{1,\alpha}\left(\boldsymbol{\theta}\right),$
$\displaystyle\boldsymbol{K}_{1,\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\boldsymbol{J}_{1,2\alpha}\left(\boldsymbol{\theta}\right).$

On the other hand,

$\displaystyle\boldsymbol{S}_{2,\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\boldsymbol{J}_{2,\alpha}\left(\boldsymbol{\theta}\right)-\kappa_{\alpha}(\boldsymbol{\theta})\boldsymbol{c}_{2,\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{c}_{2,\alpha}^{T}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\frac{1}{k^{\alpha}(\boldsymbol{\theta})(\alpha+1)^{3}}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}\boldsymbol{J}_{2,\alpha}(\rho)\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}$
$\displaystyle-\frac{\alpha^{2}}{k^{\alpha}(\boldsymbol{\theta})\left(\alpha+1\right)^{3}}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}\boldsymbol{S}_{2,2}\left(\rho\right)\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}$
$\displaystyle=\frac{1}{k^{\alpha}(\boldsymbol{\theta})(\alpha+1)^{3}}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}\left[\boldsymbol{J}_{2,\alpha}(\rho)-\alpha^{2}\boldsymbol{S}_{2,2}\left(\rho\right)\right]\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1},$

where

$\displaystyle\boldsymbol{J}_{2,\alpha}(\rho)-\alpha^{2}\boldsymbol{S}_{2,2}\left(\rho\right)$
$\displaystyle=\frac{1}{1-\rho^{2}}\left[\begin{pmatrix}\alpha^{2}-\rho^{2}(\alpha^{2}+1)+2&\alpha^{2}-\rho^{2}(\alpha^{2}+1)&-\rho(\alpha^{2}+1)\\\
\alpha^{2}-\rho^{2}(\alpha^{2}+1)&\alpha^{2}-\rho^{2}(\alpha^{2}+1)+2&-\rho(\alpha^{2}+1)\\\
-\rho(\alpha^{2}+1)&-\rho(\alpha^{2}+1)&\frac{\rho^{2}(\alpha^{2}+1)+1}{1-\rho^{2}}\end{pmatrix}\right.$
$\displaystyle\left.-\alpha^{2}\begin{pmatrix}1-\rho^{2}&1-\rho^{2}&-\rho\\\
1-\rho^{2}&1-\rho^{2}&-\rho\\\
-\rho&-\rho&\frac{\rho^{2}}{1-\rho^{2}}\end{pmatrix}\right]$
$\displaystyle=\frac{1}{1-\rho^{2}}\begin{pmatrix}2-\rho^{2}&-\rho^{2}&-\rho\\\
-\rho^{2}&2-\rho^{2}&-\rho\\\
-\rho&-\rho&\frac{\rho^{2}+1}{1-\rho^{2}}\end{pmatrix}=\boldsymbol{S}_{2,1}(\rho).$

In addition,

$\boldsymbol{c}_{2\alpha}(\boldsymbol{\theta})-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})=\frac{\alpha}{(2\alpha+1)(\alpha+1)}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}\begin{pmatrix}-1\\\
-1\\\ \frac{\rho}{1-\rho^{2}}\end{pmatrix}$
$\left(\boldsymbol{c}_{2\alpha}(\boldsymbol{\theta})-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})\right)\left(\boldsymbol{c}_{2\alpha}(\boldsymbol{\theta})-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})\right)^{T}=\frac{\alpha^{2}}{(2\alpha+1)^{2}(\alpha+1)^{2}}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}\boldsymbol{S}_{2,2}\left(\rho\right)\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}$
$\displaystyle\boldsymbol{K}_{\alpha}\left(\boldsymbol{\theta}\right)$
$\displaystyle=\boldsymbol{S}_{2\alpha}\left(\boldsymbol{\theta}\right)+\kappa_{2\alpha}(\boldsymbol{\theta})\left(\boldsymbol{c}_{2\alpha}(\boldsymbol{\theta})-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})\right)\left(\boldsymbol{c}_{2\alpha}(\boldsymbol{\theta})-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})\right)^{T}$
$\displaystyle=\frac{1}{k^{2\alpha}(\boldsymbol{\theta})(2\alpha+1)^{3}}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}\boldsymbol{S}_{2,1}(\rho)\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}+\kappa_{2\alpha}(\boldsymbol{\theta})\frac{\alpha^{2}}{(2\alpha+1)^{2}(\alpha+1)^{2}}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}\boldsymbol{S}_{2,2}\left(\rho\right)\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}$
$\displaystyle=\frac{1}{k^{2\alpha}(\boldsymbol{\theta})(2\alpha+1)^{3}}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}\boldsymbol{S}_{2,1}(\rho)\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}+\frac{\alpha^{2}}{k^{2\alpha}(\boldsymbol{\theta})(2\alpha+1)^{3}(\alpha+1)^{2}}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}\boldsymbol{S}_{2,2}\left(\rho\right)\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}$
$\displaystyle=\frac{1}{k^{2\alpha}(\boldsymbol{\theta})(2\alpha+1)^{3}}\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}\left((\alpha+1)^{2}\boldsymbol{S}_{2,1}(\rho)+\alpha^{2}\boldsymbol{S}_{2,2}\left(\rho\right)\right)\boldsymbol{D}_{2,\sigma_{1},\sigma_{2}}^{-1}.$

##  Appendix C Proof of Theorem 7

Let ${\boldsymbol{\theta}}_{0}\in\Theta_{0}$ be the true value of
${\boldsymbol{\theta}}$. Using a Taylor series expansion we get

$\displaystyle\boldsymbol{m}(\widehat{\boldsymbol{\theta}}_{R,\alpha})$
$\displaystyle=\boldsymbol{m}({\boldsymbol{\theta}}_{0})+\boldsymbol{M}^{T}({\boldsymbol{\theta}}_{0})(\widehat{\boldsymbol{\theta}}_{R,\alpha}-{\boldsymbol{\theta}}_{0})+o_{p}\left(\left\|\widehat{\boldsymbol{\theta}}_{R,\alpha}-{\boldsymbol{\theta}}_{0}\right\|\right)$
$\displaystyle=\boldsymbol{M}^{T}({\boldsymbol{\theta}}_{0})(\widehat{\boldsymbol{\theta}}_{R,\alpha}-{\boldsymbol{\theta}}_{0})+o_{p}\left(\left\|\widehat{\boldsymbol{\theta}}_{R,\alpha}-{\boldsymbol{\theta}}_{0}\right\|\right),$
(74)

because from equation (39) we have
$\boldsymbol{m}({\boldsymbol{\theta}}_{0})=\boldsymbol{0}_{r}$. Now, under
$H_{0}$,

$n^{1/2}(\widehat{\boldsymbol{\theta}}_{R,\alpha}-{\boldsymbol{\theta}}_{0})\underset{n\rightarrow\infty}{\overset{\mathcal{L}}{\longrightarrow}}\mathcal{N}(\boldsymbol{0}_{p},\boldsymbol{V}_{\alpha}\left({\boldsymbol{\theta}}_{0}\right)).$

Therefore, from equation (74) we get, under $H_{0}$,

$n^{1/2}\boldsymbol{m}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\underset{n\rightarrow\infty}{\overset{\mathcal{L}}{\longrightarrow}}\mathcal{N}(\boldsymbol{0}_{r},\boldsymbol{M}^{T}\left({\boldsymbol{\theta}}_{0}\right)\boldsymbol{V}_{\alpha}\left({\boldsymbol{\theta}}_{0}\right)\boldsymbol{M}\left(\boldsymbol{\theta}_{0}\right)).$

As rank$\left(\boldsymbol{M}({\boldsymbol{\theta}})\right)=r$, we get

$n\boldsymbol{m}^{T}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\left(\boldsymbol{M}^{T}({\boldsymbol{\theta}}_{0})\boldsymbol{V}_{\alpha}\left({\boldsymbol{\theta}}_{0}\right)\boldsymbol{M}({\boldsymbol{\theta}}_{0})\right)^{-1}\boldsymbol{m}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\underset{n\rightarrow\infty}{\overset{\mathcal{L}}{\longrightarrow}}\chi_{r}^{2}.$

Now
$\boldsymbol{M}^{T}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\boldsymbol{V}_{\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\boldsymbol{M}(\widehat{\boldsymbol{\theta}}_{R,\alpha})$
is a consistent estimator of

$\boldsymbol{M}^{T}({\boldsymbol{\theta}}_{0})\boldsymbol{V}_{\alpha}\left({\boldsymbol{\theta}}_{0}\right)\boldsymbol{M}({\boldsymbol{\theta}}_{0}).$

Hence, under $H_{0}$,

$W_{n,\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\underset{n\rightarrow\infty}{\overset{\mathcal{L}}{\longrightarrow}}\chi_{r}^{2}.$

##  Appendix D Proof of the formulas of the inner iterations of Algorithm 1

Taking into account Theorem 6 and the components of
$\boldsymbol{u}_{\boldsymbol{\theta}}(x,y)-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})$,
given by

$\displaystyle u_{\mu_{1}}(x,y)-c_{\alpha}(\mu_{1})=$
$\displaystyle\tfrac{1}{\sigma_{1}\left(1-\rho^{2}\right)}\left[\tfrac{x-\mu_{1}}{\sigma_{1}}-\rho\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)\right],$
$\displaystyle u_{\mu_{2}}(x,y)-c_{\alpha}(\mu_{2})=$
$\displaystyle\tfrac{1}{\sigma_{2}\left(1-\rho^{2}\right)}\left[\tfrac{y-\mu_{2}}{\sigma_{2}}-\rho\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)\right],$
$\displaystyle u_{\sigma_{1}}(x,y)-c_{\alpha}(\sigma_{1})=$
$\displaystyle-\tfrac{1}{\sigma_{1}}\left\\{\tfrac{1}{\alpha+1}+\tfrac{1}{1-\rho^{2}}\left[\rho\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)-\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}\right]\right\\},$
$\displaystyle u_{\sigma_{2}}(x,y)-c_{\alpha}(\sigma_{2})=$
$\displaystyle-\tfrac{1}{\sigma_{2}}\left\\{\tfrac{1}{\alpha+1}+\tfrac{1}{1-\rho^{2}}\left[\rho\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)-\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\\},$
$\displaystyle u_{\rho}(x,y)-c_{\alpha}(\rho)=$
$\displaystyle\tfrac{1}{(1-\rho^{2})^{2}}\left\\{(1+\rho^{2})\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)-\rho\left[\tfrac{1-\rho^{2}}{\alpha+1}+\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}+\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\\},$

the estimating equations are

$\displaystyle\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}-\rho\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{Y}_{i}$
$\displaystyle=0,$ (75)
$\displaystyle\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{Y}_{i}-\rho\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}$
$\displaystyle=0,$ (76)
$\displaystyle\frac{1-\rho^{2}}{\alpha+1}\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}+\rho\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}\widetilde{Y}_{i}-\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}^{2}$
$\displaystyle=0,$ (77)
$\displaystyle\frac{1-\rho^{2}}{\alpha+1}\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}+\rho\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}\widetilde{Y}_{i}-\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{Y}_{i}^{2}$
$\displaystyle=0,$ (78)
$\displaystyle(1+\rho^{2})\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}\widetilde{Y}_{i}+\rho\frac{1-\rho^{2}}{\alpha+1}\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}-\rho\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}^{2}-\rho\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{Y}_{i}^{2}$
$\displaystyle=0,$ (79) $\displaystyle w_{i,\boldsymbol{\theta}}$
$\displaystyle=\exp\left\\{\tfrac{1}{2(1-\rho^{2})}\left[\widetilde{X}_{i}^{2}+\widetilde{Y}_{i}^{2}-2\rho\widetilde{X}_{i}\widetilde{Y}_{i}\right]\right\\},$
$\displaystyle\widetilde{X}_{i}$
$\displaystyle=\tfrac{X_{i}-\mu_{1}}{\sigma_{1}},\qquad\widetilde{Y}_{i}=\tfrac{Y_{i}-\mu_{2}}{\sigma_{2}}.$

Since $\rho\in(-1,1)$, from (75)-(76) it holds

$\displaystyle\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}$
$\displaystyle=$ $\displaystyle 0$ (80)
$\displaystyle\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{Y}_{i}$
$\displaystyle=$ $\displaystyle 0,$ (81)

from (77)-(78)

$\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}^{2}=\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{Y}_{i}^{2},$
(82)

with

$\displaystyle\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}^{2}$
$\displaystyle=\frac{1-\rho^{2}}{\alpha+1}\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}+\rho\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}\widetilde{Y}_{i},$
$\displaystyle\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{Y}_{i}^{2}$
$\displaystyle=\frac{1-\rho^{2}}{\alpha+1}\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}+\rho\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}\widetilde{Y}_{i}.$

Replacing both in (79) we get

$(1+\rho^{2})\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}\widetilde{Y}_{i}+\rho\frac{1-\rho^{2}}{\alpha+1}\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}-2\rho\left(\frac{1-\rho^{2}}{\alpha+1}\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}+\rho\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}\widetilde{Y}_{i}\right)=0,$

i.e.

$\rho=\frac{(\alpha+1)\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}\widetilde{Y}_{i}}{\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}}.$
(83)

Hence

$\displaystyle\frac{1-\rho^{2}}{\alpha+1}\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}+\rho\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}\widetilde{Y}_{i}$
$\displaystyle=\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}^{2},$
$\displaystyle\frac{1-\rho^{2}}{\alpha+1}\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}+\frac{\rho^{2}}{\alpha+1}\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}$
$\displaystyle=\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}^{2},$
$\displaystyle\frac{1}{\alpha+1}\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}$
$\displaystyle=\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}^{2},$
(84)

and

$\displaystyle\frac{1-\rho^{2}}{\alpha+1}\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}+\rho\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{X}_{i}\widetilde{Y}_{i}$
$\displaystyle=\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{Y}_{i}^{2},$
$\displaystyle\frac{1-\rho^{2}}{\alpha+1}\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}+\frac{\rho^{2}}{\alpha+1}\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}$
$\displaystyle=\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{Y}_{i}^{2},$
$\displaystyle\frac{1}{\alpha+1}\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}$
$\displaystyle=\sum_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\widetilde{Y}_{i}^{2}.$
(85)

Finally,

$\displaystyle\mu_{1}$
$\displaystyle=\frac{\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}X_{i}}{\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}},\quad\mu_{2}=\frac{\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}Y_{i}}{\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}},$
$\displaystyle\frac{\sigma_{1}^{2}}{\alpha+1}$
$\displaystyle=\frac{\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\left(X_{i}-\mu_{1}\right)^{2}}{\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}},\quad\frac{\sigma_{2}^{2}}{\alpha+1}=\frac{\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\left(Y_{i}-\mu_{2}\right)^{2}}{\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}},$
$\displaystyle\rho$
$\displaystyle=\frac{(\alpha+1)\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}\frac{X_{i}-\mu_{1}}{\sigma_{1}}\frac{Y_{i}-\mu_{2}}{\sigma_{2}}}{\sum\limits_{i=1}^{n}w_{i,\boldsymbol{\theta}}^{-\alpha}},$

from which are derived the main formulas of the inner iterations of the
Iteratively Reweighted Moments Algorithm.

##  Appendix E Proof of Theorem 8

From Theorem 5 in [7], the IF associated to the MRPDE of $\boldsymbol{\theta}$
is given by

$\mathcal{IF}((x,y)^{T},\boldsymbol{T}_{\alpha},F_{\boldsymbol{\theta}})=\boldsymbol{S}_{\alpha}^{-1}\left(\boldsymbol{\theta}\right)f_{\boldsymbol{\theta}}^{\alpha}(x,y)\left[\boldsymbol{u}_{\boldsymbol{\theta}}(x,y)-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})\right],$

where

$\boldsymbol{S}_{\alpha}^{-1}\left(\boldsymbol{\theta}\right)=\begin{pmatrix}\boldsymbol{S}_{1,\alpha}^{-1}(\boldsymbol{\boldsymbol{\theta}})&\boldsymbol{0}_{2\times
3}\\\ \boldsymbol{0}_{3\times
2}&\boldsymbol{S}_{2,\alpha}^{-1}(\boldsymbol{\boldsymbol{\theta}})\end{pmatrix}$

is the inverse of the matrix,
$\boldsymbol{S}_{\alpha}\left(\boldsymbol{\theta}\right)$, defined in Theorem
5. The expressions of
$\boldsymbol{S}_{1,\alpha}^{-1}(\boldsymbol{\boldsymbol{\theta}})$ and
$\boldsymbol{S}_{2,\alpha}^{-1}(\boldsymbol{\boldsymbol{\theta}})$ are given
in (29) and (30) respectively and the ones of
$\boldsymbol{u}_{\boldsymbol{\theta}}(x,y)-\boldsymbol{c}_{\alpha}(\boldsymbol{\theta})$
in (75)-(78). On one hand

$\begin{pmatrix}\mathcal{IF}_{\alpha}(\mu_{1})\\\
\mathcal{IF}_{\alpha}(\mu_{2})\end{pmatrix}=\frac{k^{\alpha}(\boldsymbol{\theta})\left(\alpha+1\right)^{2}}{1-\rho^{2}}f_{\boldsymbol{\theta}}^{\alpha}(x,y)\left(\begin{array}[]{cc}\sigma_{1}&\sigma_{1}\rho\\\
\sigma_{2}\rho&\sigma_{2}\end{array}\right)\begin{pmatrix}\tfrac{x-\mu_{1}}{\sigma_{1}}-\rho\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)\\\
\tfrac{y-\mu_{2}}{\sigma_{2}}-\rho\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)\end{pmatrix},$

where

$\left(\begin{array}[]{cc}\sigma_{1}&\sigma_{1}\rho\\\
\sigma_{2}\rho&\sigma_{2}\end{array}\right)\begin{pmatrix}\tfrac{x-\mu_{1}}{\sigma_{1}}-\rho\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)\\\
\tfrac{y-\mu_{2}}{\sigma_{2}}-\rho\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)\end{pmatrix}=(1-\rho^{2})\begin{pmatrix}x-\mu_{1}\\\
y-\mu_{2}\end{pmatrix},$

and

$\displaystyle f_{\boldsymbol{\theta}}^{\alpha}(x,y)$
$\displaystyle=\frac{w_{\alpha,\boldsymbol{\theta}}^{-\alpha/(1-\rho^{2})}(x,y)}{k^{\alpha}(\boldsymbol{\theta})}$
$\displaystyle=\frac{1}{k^{\alpha}(\boldsymbol{\theta})}\exp\left\\{-\tfrac{\alpha}{2(1-\rho^{2})}\left[(\tfrac{x-\mu_{1}}{\sigma_{1}})^{2}+(\tfrac{y-\mu_{2}}{\sigma_{2}})^{2}-2\rho\tfrac{x-\mu_{1}}{\sigma_{1}}\tfrac{y-\mu_{2}}{\sigma_{2}}\right]\right\\},$

hence we obtain (60)-(61). On the other hand, we have

$\displaystyle\begin{pmatrix}\mathcal{IF}_{\alpha}(\sigma_{1})\\\
\mathcal{IF}_{\alpha}(\sigma_{2})\end{pmatrix}$
$\displaystyle=k^{\alpha}(\boldsymbol{\theta})\frac{\left(\alpha+1\right)^{3}}{2}f_{\boldsymbol{\theta}}^{\alpha}(x,y)\begin{pmatrix}\sigma_{1}&\sigma_{1}\rho^{2}&\sigma_{1}\rho(1-\rho^{2})\\\
\sigma_{2}\rho^{2}&\sigma_{2}&\sigma_{2}\rho(1-\rho^{2})\end{pmatrix}$
$\displaystyle\times\begin{pmatrix}-\left\\{\tfrac{1}{\alpha+1}+\tfrac{1}{1-\rho^{2}}\left[\rho\tfrac{(x-\mu_{1})(y-\mu_{2})}{\sigma_{1}\sigma_{2}}-\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}\right]\right\\}\\\
-\left\\{\tfrac{1}{\alpha+1}+\tfrac{1}{1-\rho^{2}}\left[\rho\tfrac{(x-\mu_{1})(y-\mu_{2})}{\sigma_{1}\sigma_{2}}-\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\\}\\\
\tfrac{1}{(1-\rho^{2})^{2}}\left\\{(1+\rho^{2})\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)-\rho\left[\tfrac{1-\rho^{2}}{\alpha+1}+\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}+\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\\}\end{pmatrix},$

where

$\displaystyle\begin{pmatrix}\sigma_{1}&\sigma_{1}\rho^{2}&\sigma_{1}\rho(1-\rho^{2})\\\
\sigma_{2}\rho^{2}&\sigma_{2}&\sigma_{2}\rho(1-\rho^{2})\end{pmatrix}$
$\displaystyle\cdot\begin{pmatrix}-\left\\{\tfrac{1}{\alpha+1}+\tfrac{1}{1-\rho^{2}}\left[\rho\tfrac{(x-\mu_{1})(y-\mu_{2})}{\sigma_{1}\sigma_{2}}-\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}\right]\right\\}\\\
-\left\\{\tfrac{1}{\alpha+1}+\tfrac{1}{1-\rho^{2}}\left[\rho\tfrac{(x-\mu_{1})(y-\mu_{2})}{\sigma_{1}\sigma_{2}}-\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\\}\\\
\tfrac{1}{(1-\rho^{2})^{2}}\left\\{(1+\rho^{2})\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)-\rho\left[\tfrac{1-\rho^{2}}{\alpha+1}+\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}+\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\\}\end{pmatrix}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}\tfrac{\sigma_{1}}{1-\rho^{2}}\left[\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}-\rho^{2}\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}-(1-\rho^{2})(1+2\rho^{2})\tfrac{1}{\alpha+1}\right]\\\
\tfrac{\sigma_{2}}{1-\rho^{2}}\left[\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}-\rho^{2}\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}-(1-\rho^{2})(1+2\rho^{2})\tfrac{1}{\alpha+1}\right]\end{pmatrix}.$

Hence, we obtain (62)-(63). Finally,

$\displaystyle\mathcal{IF}_{\alpha}(\rho)$
$\displaystyle=k^{\alpha}(\boldsymbol{\theta})\frac{\left(\alpha+1\right)^{3}}{2}f_{\boldsymbol{\theta}}^{\alpha}(x,y)\begin{pmatrix}\rho(1-\rho^{2})&\rho(1-\rho^{2})&2(1-\rho^{2})^{2}\end{pmatrix}$
$\displaystyle\times\begin{pmatrix}-\left\\{\tfrac{1}{\alpha+1}+\tfrac{1}{1-\rho^{2}}\left[\rho\tfrac{x-\mu_{1}}{\sigma_{1}}\tfrac{y-\mu_{2}}{\sigma_{2}}-\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}\right]\right\\}\\\
-\left\\{\tfrac{1}{\alpha+1}+\tfrac{1}{1-\rho^{2}}\left[\rho\tfrac{x-\mu_{1}}{\sigma_{1}}\tfrac{y-\mu_{2}}{\sigma_{2}}-\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\\}\\\
\tfrac{1}{(1-\rho^{2})^{2}}\left\\{(1+\rho^{2})\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)-\rho\left[\tfrac{1-\rho^{2}}{\alpha+1}+\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}+\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\\}\end{pmatrix}$

where

$\displaystyle\begin{pmatrix}\rho(1-\rho^{2})&\rho(1-\rho^{2})&2(1-\rho^{2})^{2}\end{pmatrix}$
$\displaystyle\cdot\begin{pmatrix}-\left\\{\tfrac{1}{\alpha+1}+\tfrac{1}{1-\rho^{2}}\left[\rho\tfrac{x-\mu_{1}}{\sigma_{1}}\tfrac{y-\mu_{2}}{\sigma_{2}}-\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}\right]\right\\}\\\
-\left\\{\tfrac{1}{\alpha+1}+\tfrac{1}{1-\rho^{2}}\left[\rho\tfrac{x-\mu_{1}}{\sigma_{1}}\tfrac{y-\mu_{2}}{\sigma_{2}}-\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\\}\\\
\tfrac{1}{(1-\rho^{2})^{2}}\left\\{(1+\rho^{2})\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)-\rho\left[\tfrac{1-\rho^{2}}{\alpha+1}+\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}+\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\\}\end{pmatrix}$
$\displaystyle=-\rho\left[\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}+\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)^{2}\right]+2\left(\tfrac{x-\mu_{1}}{\sigma_{1}}\right)\left(\tfrac{y-\mu_{2}}{\sigma_{2}}\right),$

from which it follows (64).

##  Appendix F Proofs of cases of testing problems (Section 4)

Case 1 (Comparing means of two dependent populations with normal
distributions).
If we consider the function

$m\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)=\mu_{1}-\mu_{2},$

the null hypothesis can be given by
$m\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)=0$. In this case

$\displaystyle\boldsymbol{M}^{T}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)$
$\displaystyle=\left(\begin{array}[]{ccccc}1&-1&0&0&0\end{array}\right),$
$\displaystyle\left(\boldsymbol{M}^{T}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\boldsymbol{V}_{\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\boldsymbol{M}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\right)^{-1}$
$\displaystyle=\left(\left(\begin{array}[]{cc}1&-1\end{array}\right)\boldsymbol{V}_{1,\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\left(\begin{array}[]{c}1\\\
-1\end{array}\right)\right)^{-1}$
$\displaystyle=\frac{\left(2\alpha+1\right)^{2}}{\left(\alpha+1\right)^{4}}(\widehat{\sigma}_{1,R,\alpha}^{2}-2\widehat{\rho}_{R,\alpha}\widehat{\sigma}_{1,R,\alpha}\widehat{\sigma}_{2,R,\alpha}+\widehat{\sigma}_{2,R,\alpha}^{2})^{-1}$
$\displaystyle=\frac{\left(2\alpha+1\right)^{2}}{\left(\alpha+1\right)^{4}}\big{[}(\widehat{\sigma}_{1,R,\alpha}-\widehat{\sigma}_{2,R,\alpha})^{2}$
$\displaystyle\hskip
51.21504pt+2(1-\widehat{\rho}_{R,\alpha})\widehat{\sigma}_{1,R,\alpha},\widehat{\sigma}_{2,R,\alpha}\big{]},$

where $\boldsymbol{V}_{1,\alpha}(\cdot)$ is given by (33). Therefore, (44) is
obtained.

Case 2 (Comparing variances of two dependent populations with normal
distributions).
If we consider the function

$m\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)=\sigma_{1}-\sigma_{2},$

the null hypothesis can be given by
$m\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)=0$. In this case

$\boldsymbol{M}^{T}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)=\left(\begin{array}[]{ccccc}0&0&1&-1&0\end{array}\right)$

and taking

$\displaystyle\boldsymbol{M}^{T}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)\boldsymbol{V}_{\alpha}(\boldsymbol{\theta})\boldsymbol{M}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)$
$\displaystyle=\left(\begin{array}[]{ccc}\sigma_{1}&-\sigma_{2}&0\end{array}\right)\boldsymbol{V}_{2,\alpha}(\rho)\begin{pmatrix}\sigma_{1}\\\
-\sigma_{2}\\\
0\end{pmatrix}=\dfrac{\left(\alpha+1\right)^{4}}{\left(2\alpha+1\right)^{3}}b_{\alpha}(\boldsymbol{\theta}),$

we denote

$\displaystyle b_{\alpha}(\boldsymbol{\theta})$
$\displaystyle=\left(\alpha+1\right)^{2}b_{1,\alpha}(\boldsymbol{\theta})+\alpha^{2}b_{2,\alpha}(\boldsymbol{\theta})$
(86)
$\displaystyle=\frac{2\left(\alpha+1\right)^{2}+\alpha^{2}}{4}(\sigma_{1}-\sigma_{2})^{2}+\left(\alpha+1\right)^{2}(1-\rho^{2})\sigma_{1}\sigma_{2}$
$\displaystyle=\frac{1}{4}\left(\alpha+1\right)^{2}\left\\{\left[2+\left(\tfrac{\alpha}{\alpha+1}\right)^{2}\right](\sigma_{1}-\sigma_{2})^{2}+4(1-\rho^{2})\sigma_{1}\sigma_{2}\right\\},$

where

$\displaystyle b_{1,\alpha}(\boldsymbol{\theta})$
$\displaystyle=\frac{1}{2}\left(\begin{array}[]{cc}\sigma_{1}&-\sigma_{2}\end{array}\right)\begin{pmatrix}1&\rho^{2}\\\
\rho^{2}&1\end{pmatrix}\left(\begin{array}[]{c}\sigma_{1}\\\
-\sigma_{2}\end{array}\right)$
$\displaystyle=\frac{1}{2}\left(\sigma_{1}^{2}+\sigma_{2}^{2})-\rho^{2}\sigma_{1}\sigma_{2}\right)$
$\displaystyle=\frac{1}{2}(\sigma_{1}-\sigma_{2})^{2}+(1-\rho^{2})\sigma_{1}\sigma_{2},$
$\displaystyle b_{2,\alpha}(\boldsymbol{\theta})$
$\displaystyle=\frac{1}{4}\tfrac{1}{1-\rho^{2}}\left(\begin{array}[]{ccc}\sigma_{1}&-\sigma_{2}&0\end{array}\right)\begin{pmatrix}1&\rho^{2}&\rho(1-\rho^{2})\\\
\rho^{2}&1&\rho(1-\rho^{2})\\\
\rho(1-\rho^{2})&\rho(1-\rho^{2})&2(1-\rho^{2})^{2}\end{pmatrix}$
$\displaystyle\times\begin{pmatrix}1-\rho^{2}&1-\rho^{2}&-\rho\\\
1-\rho^{2}&1-\rho^{2}&-\rho\\\
-\rho&-\rho&\frac{\rho^{2}}{1-\rho^{2}}\end{pmatrix}\begin{pmatrix}1&\rho^{2}&\rho(1-\rho^{2})\\\
\rho^{2}&1&\rho(1-\rho^{2})\\\
\rho(1-\rho^{2})&\rho(1-\rho^{2})&2(1-\rho^{2})^{2}\end{pmatrix}\begin{pmatrix}\sigma_{1}\\\
-\sigma_{2}\\\ 0\end{pmatrix}$
$\displaystyle=\frac{1}{4}\left(\begin{array}[]{cc}\sigma_{1}&-\sigma_{2}\end{array}\right)\boldsymbol{1}_{2}\boldsymbol{1}_{2}^{T}\begin{pmatrix}\sigma_{1}\\\
-\sigma_{2}\end{pmatrix}=\frac{1}{4}(\sigma_{1}-\sigma_{2})^{2}.$

Finally, denoting
$b_{\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})=\left(\alpha+1\right)^{2}\beta_{\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})$,
we get (46).

Case 3 (Fixing a value of the correlation coefficient of two dependent
populations with normal distributions).
If we consider the function

$m\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)=\rho-\rho_{0},$

the null hypothesis can be given by
$m\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)=0$. In this case
$\boldsymbol{M}^{T}(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho)=\left(\begin{array}[]{ccccc}0&0&0&0&1\end{array}\right)$
and we have

$\left(\boldsymbol{M}^{T}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\boldsymbol{V}_{\alpha}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\boldsymbol{M}(\widehat{\boldsymbol{\theta}}_{R,\alpha})\right)^{-1}=\frac{\left(2\alpha+1\right)^{3}}{\left(\alpha+1\right)^{6}}\frac{1}{(1-\widehat{\rho}_{R,\alpha}^{2})^{2}}.$
(87)

Therefore, we get (49).

Case 4 (Comparing means and variances of two dependent populations with normal
distribution).
If we consider the function

$\boldsymbol{m}^{T}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)=\left(\mu_{1}-\mu_{2},\sigma_{1}-\sigma_{2}\right),$

the null hypothesis can be written by
$\boldsymbol{m}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)=\boldsymbol{0}_{2}$.
In this case,

$\boldsymbol{M}^{T}(\widehat{\boldsymbol{\theta}}_{R,\alpha})=\left(\begin{array}[]{ccccc}1&-1&0&0&0\\\
0&0&1&-1&0\end{array}\right)$

and

$\boldsymbol{M}^{T}\left(\boldsymbol{\theta}\right)\boldsymbol{V}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{M}\left(\boldsymbol{\theta}\right)=\frac{\left(\alpha+1\right)^{4}}{\left(2\alpha+1\right)^{2}}\left(\begin{array}[]{cc}\sigma_{1}^{2}-2\rho\sigma_{1}\sigma_{2}+\sigma_{2}^{2}&0\\\
0&\frac{\left(\alpha+1\right)^{2}}{2\alpha+1}\beta_{\alpha}(\boldsymbol{\theta})\end{array}\right),$

with $\beta_{\alpha}(\boldsymbol{\theta})$ given in (86). Therefore, we get
(52).

Case 5 (Fixing a value for covariance of two normal populations).
If we consider the function

$m\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)=\sigma_{1}\sigma_{2}\rho-\sigma_{12,0},$

the null hypothesis can be written as
$m\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)=0$ and

$\displaystyle\boldsymbol{M}^{T}\left(\boldsymbol{\theta}\right)\boldsymbol{V}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{M}\left(\boldsymbol{\theta}\right)$
$\displaystyle=$
$\displaystyle\sigma_{1}^{2}\sigma_{2}^{2}\left(\begin{array}[]{ccc}\rho&\rho&1\end{array}\right)\boldsymbol{D}_{1,\sigma_{1},\sigma_{2}}\boldsymbol{V}_{2,\alpha}(\rho)\boldsymbol{D}_{1,\sigma_{1},\sigma_{2}}\left(\begin{array}[]{c}\rho\\\
\rho\\\ 1\end{array}\right)$ $\displaystyle=$
$\displaystyle\sigma_{1}^{2}\sigma_{2}^{2}\left[\left(\alpha+1\right)^{2}(\widehat{\rho}_{R,\alpha}^{2}+1)+\frac{\alpha^{2}}{2}\widehat{\rho}_{R,\alpha}^{2}\right],$

where $\boldsymbol{V}_{2,\alpha}(\rho)$ is given by (35).Therefore, we get
(54).

Case 6 (Fixing values for means of two dependent populations with normal
distribution).
If we consider the function

$\boldsymbol{m}^{T}(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho)=\left(\mu_{1}-\mu_{1,0},\mu_{2}-\mu_{2,0}\right),$

the null hypothesis can be written by
$m\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)=0$. It is clear that

$\boldsymbol{M}^{T}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)=\left(\begin{array}[]{ccccc}1&0&0&0&0\\\
0&1&0&0&0\end{array}\right).$

Therefore, we get (56) since

$\displaystyle\left(\boldsymbol{M}^{T}\left(\boldsymbol{\theta}\right)\boldsymbol{V}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{M}\left(\boldsymbol{\theta}\right)\right)^{-1}$
$\displaystyle=\boldsymbol{V}_{1,\alpha}^{-1}\boldsymbol{\left({\boldsymbol{\theta}}\right)}$
$\displaystyle=\left(\frac{\left(\alpha+1\right)^{4}}{\left(2\alpha+1\right)^{2}}\left(\begin{array}[]{cc}\sigma_{1}^{2}&\rho\sigma_{1}\sigma_{2}\\\
\rho\sigma_{1}\sigma_{2}&\sigma_{2}^{2}\end{array}\right)\right)^{-1}$
$\displaystyle=\frac{\left(2\alpha+1\right)^{2}}{(1-\rho^{2})\left(\alpha+1\right)^{4}}\left(\begin{array}[]{cc}\frac{1}{\sigma_{1}^{2}}&-\frac{\rho}{\sigma_{1}\sigma_{2}}\\\
-\frac{\rho}{\sigma_{1}\sigma_{2}}&\frac{1}{\sigma_{2}^{2}}\end{array}\right),$
$\displaystyle\boldsymbol{m}^{T}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\text{
}\right)\left(\boldsymbol{M}^{T}\left(\boldsymbol{\theta}\right)\boldsymbol{V}_{\alpha}\left(\boldsymbol{\theta}\right)\boldsymbol{M}\left(\boldsymbol{\theta}\right)\right)^{-1}\boldsymbol{m}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\text{
}\right)$
$\displaystyle=\frac{\left(2\alpha+1\right)^{2}}{\left(\alpha+1\right)^{4}}\bigg{[}\frac{(\widehat{\mu}_{1,R,\alpha}-\mu_{1,0})^{2}\widehat{\sigma}_{2,R,\alpha}^{2}}{\widehat{\sigma}_{1,R,\alpha}^{2}\widehat{\sigma}_{2,R,\alpha}^{2}(1-\widehat{\rho}_{R,\alpha}^{2})}$
$\displaystyle\hskip
14.22636pt+\frac{-2\widehat{\rho}_{R,\alpha}(\widehat{\mu}_{1,R,\alpha}-\mu_{1,0})(\widehat{\mu}_{2,R,\alpha}-\mu_{2,0})\widehat{\sigma}_{1,R,\alpha}\widehat{\sigma}_{2,R,\alpha}+(\widehat{\mu}_{2,R,\alpha}-\mu_{2,0})^{2}\widehat{\sigma}_{1,R,\alpha}^{2}}{\widehat{\sigma}_{1,R,\alpha}^{2}\widehat{\sigma}_{2,R,\alpha}^{2}(1-\widehat{\rho}_{R,\alpha}^{2})}\bigg{]}$
$\displaystyle=\frac{\left(2\alpha+1\right)^{2}}{\left(\alpha+1\right)^{4}}\Bigg{[}\frac{\left(\frac{\widehat{\mu}_{1,R,\alpha}-\mu_{1,0}}{\widehat{\sigma}_{1,R,\alpha}}\right)^{2}-2\widehat{\rho}_{R,\alpha}\left(\frac{\widehat{\mu}_{1,R,\alpha}-\mu_{1,0}}{\widehat{\sigma}_{1,R,\alpha}}\right)\left(\frac{\widehat{\mu}_{2,R,\alpha}-\mu_{2,0}}{\widehat{\sigma}_{2,R,\alpha}}\right)}{1-\widehat{\rho}_{R,\alpha}^{2}}$
$\displaystyle\hskip
65.44142pt+\frac{\left(\dfrac{\widehat{\mu}_{2,R,\alpha}-\mu_{2,0}}{\widehat{\sigma}_{2,R,\alpha}}\right)^{2}}{1-\widehat{\rho}_{R,\alpha}^{2}}\Bigg{]}$
$\displaystyle=\dfrac{\left(2\alpha+1\right)^{2}}{\left(\alpha+1\right)^{4}}\bigg{[}\frac{\left(\dfrac{\widehat{\mu}_{1,R,\alpha}-\mu_{1,0}}{\widehat{\sigma}_{1,R,\alpha}}-\dfrac{\widehat{\mu}_{2,R,\alpha}-\mu_{2,0}}{\widehat{\sigma}_{2,R,\alpha}}\right)^{2}}{1-\widehat{\rho}_{R,\alpha}^{2}}$
$\displaystyle\hskip