# Some Unified Results on Isotonic Regression Estimators of Order Restricted
Parameters of a General Bivariate Location/Scale Model

Naresh Garg and Neeraj Misra
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Kanpur-208016, Uttar Pradesh, India

## Abstract

We consider component-wise estimation of order restricted location/scale
parameters $\theta_{1}$ and $\theta_{2}$ ($\theta_{1}\leq\theta_{2}$) of a
general bivariate distribution under the squared error loss function. To find
improvements over the best location/scale equivariant estimators (BLEE/BSEE)
of $\theta_{1}$ and $\theta_{2}$, we study isotonic regression of suitable
location/scale equivariant estimators (LEE/SEE) of $\theta_{1}$ and
$\theta_{2}$ with general weights. Let $\mathcal{D}_{1,\nu}$ and
$\mathcal{D}_{2,\beta}$ denote suitable classes of isotonic regression
estimators of $\theta_{1}$ and $\theta_{2}$, respectively. Under the squared
error loss function, we characterize admissible estimators within classes
$\mathcal{D}_{1,\nu}$ and $\mathcal{D}_{2,\beta}$, and identify estimators
that dominate the BLEE/BSEE of $\theta_{1}$ and $\theta_{2}$. Our study
unifies and extends several studies reported in the literature for specific
probability distributions having independent marginals. Additionally, some new
and interesting results are obtained. A simulation study is also considered to
compare the risk performances of various estimators.

Keywords: Admissible estimators; BLEE; BSEE; Inadmissible estimators; Isotonic
regression; Location/Scale model; Mixed estimators Scaled squared error loss
(SSEL); Squared error loss (SEL).

##  1\. Introduction

Let $\underline{X}=(X_{1},X_{2})$ be a random vector having the Lebesgue
probability density function (pdf) $f_{\underline{\theta}}(x_{1},x_{2})$,
$(x_{1},x_{2})\in\Re^{2}$, where
$\underline{\theta}=(\theta_{1},\theta_{2})\in\Theta\subseteq\Re^{2}$ is a
vector of unknown parameters and $\Theta$ is the parameter space (to be called
unrestricted parameter space); here $\Re$ denotes the real line and
$\Re^{2}=\Re\times\Re$. For a completely specified pdf $f(\cdot,\cdot)$, we
assume one of the following models for $f_{\underline{\theta}}(x_{1},x_{2})$:

(1.1) $\displaystyle f_{\underline{\theta}}(x_{1},x_{2})$
$\displaystyle=f(x_{1}-\theta_{1},x_{2}-\theta_{2}),\;(x_{1},x_{2})\in\Re^{2},\;\underline{\theta}=(\theta_{1},\theta_{2})\in\Theta=\Re^{2},$
(1.2) $\displaystyle\text{or}\quad f_{\underline{\theta}}(x_{1},x_{2})$
$\displaystyle=\frac{1}{\theta_{1}\theta_{2}}f\\!\left(\frac{x_{1}}{\theta_{1}},\frac{x_{2}}{\theta_{2}}\right),\;(x_{1},x_{2})\in\Re^{2},\;\underline{\theta}=(\theta_{1},\theta_{2})\in\Theta=\Re_{++}^{2},\;\;\,$

where $\Re_{++}=(0,\infty)$, $\Re_{++}^{2}=\Re_{++}\times\Re_{++}$, and
$\Theta$ denotes the parameter space. The models (1.1) and (1.2) will be
referred to as the location and scale probability models, respectively, and
the corresponding parameters $\theta_{1}$ and $\theta_{2}$ will be called the
location and the scale parameters, receptively. Generally,
$\underline{X}=(X_{1},X_{2})$ would be a minimal sufficient statistic based on
a random sample from a bivariate distribution or a minimal sufficient
statistic based on independent random samples. Under the scale model (1.2), we
will assume, throughout, that the distributional support of
$\underline{X}=(X_{1},X_{2})$ is a subset of $\Re_{++}^{2}$.

Consider estimation of $\theta_{i}$ under the loss function
$L_{i}(\underline{\theta},a),\;\underline{\theta}\in\Theta,\;a\in\mathcal{A},\;i=1,2,$
where $\mathcal{A}$ denotes the action space. While working under the location
model (1.1), we assume that

(1.3)
$L_{i}(\underline{\theta},a)=(a-\theta_{i})^{2},\;\;\;\underline{\theta}\in\Theta,\;a\in\mathcal{A}=\Re,\;i=1,2,$

and, under the scale model (1.2), we take

(1.4)
$L_{i}(\underline{\theta},a)=\left(\frac{a}{\theta_{i}}-1\right)^{2},\;\;\;\underline{\theta}\in\Theta,\;a\in\mathcal{A}=\Re_{++},\;i=1,2.$

The loss functions (1.3) and (1.4) will be referred to as the squared error
loss (SEL) function and the scaled squared error loss (SSEL) function,
respectively.

The problem of estimating location parameter $\theta_{i}$, under the location
model (1.1) and loss function (1.3), is invariant under the additive group of
transformations
$\mathcal{G}_{L}=\\{g_{c_{1},c_{2}}:(c_{1},c_{2})\in\Re^{2}\\}$, where
$g_{c_{1},c_{2}}(x_{1},x_{2})=(x_{1}+c_{1},x_{2}+c_{2})$,
$(x_{1},x_{2})\in\Re^{2},\;(c_{1},c_{2})\in\Re^{2}$. Any location equivariant
estimator (LEE) of $\theta_{i}$ has the form
$\delta_{i,c}(\underline{X})=X_{i}-c,\;c\in\Re,\;i=1,2$. The best location
equivariant estimator (BLEE) of $\theta_{i}$ is
$\delta_{i,c_{0,i}}(\underline{X})=X_{i}-c_{0,i}$, where

(1.5)
$c_{0,i}\\!=E[Z_{i}]=\\!\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\\!\\!z_{i}f(z_{1},z_{2})dz_{1}dz_{2},$

for $Z_{i}=X_{i}-\theta_{i},\;i=1,2$. Similarly, estimation of $\theta_{i}$,
under the scale model (1.2) and the SSEL function (1.4), is invariant under
the multiplicative group of transformations
$\mathcal{G}_{S}=\\{g_{c_{1},c_{2}}:(c_{1},c_{2})\in\Re_{++}^{2}\\}$, where
$g_{c_{1},c_{2}}(x_{1},x_{2})=(c_{1}x_{1},c_{2}x_{2})$,
$(x_{1},x_{2})\in\Re^{2},\;(c_{1},c_{2})\in\Re_{++}^{2}$. Any scale
equivariant estimator (SEE) of $\theta_{i}$ is of the form
$\delta_{i,c}(\underline{X})=cX_{i},\;c>0,\;i=1,2$. The best scale equivariant
estimator (BSEE) of $\theta_{i}$ is
$\delta_{i,c_{0,i}}(\underline{X})=c_{0,i}X_{i}$, where, for
$Z_{i}=\frac{X_{i}}{\theta_{i}}$,

(1.6)
$c_{0,i}\\!=\frac{E[Z_{i}]}{E[Z_{i}^{2}]}=\\!\frac{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}z_{i}f(z_{1},z_{2})dz_{1}dz_{2}}{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}z_{i}^{2}f(z_{1},z_{2})dz_{1}dz_{2}},\;i=1,2.$

Under the unrestricted parameter space $\Theta$, the BLEE and BSEE of
$\theta_{i}$ are known to possess various optimality properties.

In many situations, it may be known apriori that $\theta_{1}\leq\theta_{2}$ so
that the parameter space of interest is
$\Theta_{0}=\\{(x_{1},x_{2})\in\Theta:x_{1}\leq x_{2}\\}$, called the
restricted parameter space. In such situations, it may be of interest to
estimate $\theta_{1}$ and $\theta_{2}$. For descriptions of various real-life
situations where such an estimation problem may be of interest, one may refer
to Barlow et. al. (1972), Robertson et. al. (1988) and Kumar and Sharma
(1988). In such problems, a natural question that arises is whether the prior
information $\theta_{1}\leq\theta_{2}$ can be exploited to find better
alternatives to the BLEE/BSEE. This issue has been extensively investigated in
the literature for specific probability distributions, having independent
marginals (see, for example, Kumar and Sharma (1988), Kushary and Cohen
(1989), Pal and Kushary (1992), Kubokawa and Saleh (1994), Misra and Singh
(1994) and Chang et al. (2017)). An account of these studies can be found in
van Eeden (2006).

To find improvements over the BLEE/BSEE, we will consider isotonic regression
of suitable pair of LEE/SEE of $\underline{\theta}=(\theta_{1},\theta_{2})$.
Let $\omega_{1}$ and $\omega_{2}$ be given non-negative weights, such that
$\omega_{1}+\omega_{2}\neq 0$, and let
$(\delta_{1,\beta}(\underline{X}),\delta_{2,\nu}(\underline{X}))$ be a given
pair of unrestricted LEE/SEE of $\underline{\theta}=(\theta_{1},\theta_{2})$
($\delta_{1,\beta}(\underline{X})\leq\delta_{2,\nu}(\underline{X})$, almost
surely, does not hold). The isotonic regression of
$(\delta_{1,\beta}(\underline{X}),\delta_{2,\nu}(\underline{X}))$, with
weights $(\omega_{1},\omega_{2})$, is defined as the pair
$(\delta^{*}_{1,\beta}(\underline{X}),\delta^{*}_{2,\nu}(\underline{X}))$ that
minimizes, for each sample point $\underline{x},$

$Q=\omega_{1}(\delta_{1,\beta}(\underline{x})-y_{1}(\underline{x}))^{2}+\omega_{2}(\delta_{2,\nu}(\underline{x})-y_{2}(\underline{x}))^{2},$

among all $y_{1}(\underline{x})$ and $y_{2}(\underline{x})$ such that
$y_{1}(\underline{x})\leq y_{1}(\underline{x})$. For
$\alpha=\frac{\omega_{1}}{\omega_{1}+\omega_{2}}$, it turns out that (see
Barlow et al. (1972))

$\displaystyle\delta^{*}_{1,\beta}(\underline{X})$
$\displaystyle=\min\\{\delta_{1,\beta}(\underline{X}),\alpha\delta_{1,\beta}(\underline{X})+(1-\alpha)\delta_{2,\nu}(\underline{X})\\}$
$\displaystyle=\begin{cases}\delta_{1,\beta}(\underline{X}),&\text{
if}\quad\delta_{1,\beta}(\underline{X})\leq\delta_{2,\nu}(\underline{X})\\\
\alpha\delta_{1,\beta}(\underline{X})+(1-\alpha)\delta_{2,\nu}(\underline{X}),&\text{
if}\quad\delta_{1,\beta}(\underline{X})>\delta_{2,\nu}(\underline{X})\end{cases}$

and

$\displaystyle\delta^{*}_{2,\nu}(\underline{X})$
$\displaystyle=\max\\{\delta_{2,\nu}(\underline{X}),\alpha\delta_{1,\beta}(\underline{X})+(1-\alpha)\delta_{2,\nu}(\underline{X})\\}$
$\displaystyle=\begin{cases}\delta_{2,\nu}(\underline{X}),&\text{
if}\quad\delta_{1,\beta}(\underline{X})\leq\delta_{2,\nu}(\underline{X})\\\
\alpha\delta_{1,\beta}(\underline{X})+(1-\alpha)\delta_{2,\nu}(\underline{X}),&\text{
if}\quad\delta_{1,\beta}(\underline{X})>\delta_{2,\nu}(\underline{X})\end{cases}.$

We call the above estimators as isotonic regression estimators based on
$(\delta_{1,\beta},\delta_{2,\nu})$. For a detailed treatment of the notion of
isotonic regression, readers are referred to Barlow et. al. (1972) and
Robertson et. al. (1988).

Let $c_{0,1}$ and $c_{0,2}$ be as defined by (1.5)/(1.6), so that
$\delta_{i,c_{0,i}}$ is the BLEE/BSEE of $\theta_{i},\;i=1,2$. To find
improvements over the BLEE/BSEE $\delta_{1,c_{0,1}}$ of $\theta_{1}$, under
the prior information $\underline{\theta}\in\Theta_{0}$, we propose to
consider isotonic regression
$(\delta_{1,c_{0,1},\alpha}^{(\nu)},\delta_{2,c_{0,1},\alpha}^{(\nu)})$ of
$(\delta_{1,c_{0,1}},\delta_{2,\nu})$, for appropriately chosen $\nu$ and
weight $\alpha$, and take $\delta_{1,c_{0,1},\alpha}^{(\nu)}$ as a competitor
of the BLEE/BSEE $\delta_{1,c_{0,1}}$. Similarly, for estimation of
$\theta_{2}$, under the prior information $\underline{\theta}\in\Theta_{0}$,
we propose to consider isotonic regression
$(\delta_{1,c_{0,2},\alpha}^{(\beta)},\delta_{2,c_{0,2},\alpha}^{(\beta)})$ of
$(\delta_{1,\beta},\delta_{2,c_{0,2}})$, for appropriately chosen $\beta$ and
weight $\alpha$, and take $\delta_{2,c_{0,2},\alpha}^{(\beta)}$ as a
competitor of the BLEE/BSEE $\delta_{2,c_{0,2}}$. Here, one may argue that
natural choices for $\nu$ and $\beta$ are $c_{0,2}$ and $c_{0,1}$ (the one
corresponding to the BLEE/BSEE of $\theta_{2}$ and $\theta_{1}$,
respectively). As we will observe later that such a choice of $\nu$ or $\beta$
may not always be successful in providing improvements over the BLEE/BSEE
$\delta_{1,c_{0,1}}$ or $\delta_{2,c_{0,2}}$, as desired (see examples
provided in Sections 3.3 and 4.3). For this reason, to begin with, we take
$\nu$ or $\beta$ to be a fixed constant and, depending on the application at
hand, we choose them appropriately to achieve our goal of improving the
BLEE/BSEE. The guiding principle for choosing $\nu$ or $\beta$ in any
application is that $\delta_{2,\nu}$ or $\delta_{1,\beta}$ are close to
$\delta_{2,c_{0,2}}$ or $\delta_{1,c_{0,1}}$ and the required conditions (as
derived in various results proved in the paper) for improving on the BLEE/BSEE
are satisfied. This leads to studying isotonic regression estimators

(1.7)
$\displaystyle\delta_{1,c_{0,1},\alpha}^{(\nu)}(\underline{X})=\begin{cases}\delta_{1,c_{0,1}}(\underline{X}),&\text{if}\;\delta_{1,c_{0,1}}(\underline{X})\leq\delta_{2,\nu}(\underline{X})\\\
\alpha\delta_{1,c_{0,1}}(\underline{X})+(1-\alpha)\delta_{2,\nu}(\underline{X}),&\text{if}\;\delta_{1,c_{0,1}}(\underline{X})>\delta_{2,\nu}(\underline{X})\end{cases},\alpha\in[0,1],$

for estimating $\theta_{1}$, and isotonic regression estimators

(1.8)
$\displaystyle\delta_{2,c_{0,2},\alpha}^{(\beta)}(\underline{X})=\begin{cases}\delta_{2,c_{0,2}}(\underline{X}),&\text{if}\;\delta_{1,\beta}(\underline{X})\leq\delta_{2,c_{0,2}}(\underline{X})\\\
\alpha\delta_{1,\beta}(\underline{X})+(1-\alpha)\delta_{2,c_{0,2}}(\underline{X}),&\text{if}\;\delta_{1,\beta}(\underline{X})>\delta_{2,c_{0,2}}(\underline{X})\end{cases},\;\;\alpha\in[0,1],$

By allowing weight $\alpha$ to vary on $\Re$, we consider general class of
isotonic regression estimators

(1.9) $\displaystyle\mathcal{D}_{1,\nu}$
$\displaystyle=\\{\delta_{1,c_{0,1},\alpha}^{(\nu)}:-\infty<\alpha<\infty\\}$
(1.10) $\displaystyle\text{and}\quad\mathcal{D}_{2,\beta}$
$\displaystyle=\\{\delta_{2,c_{0,2},\alpha}^{(\beta)}:-\infty<\alpha<\infty\\},$

for estimation of $\theta_{1}$ and $\theta_{2}$, respectively, when it is
known apriori that $\underline{\theta}\in\Theta_{0}$; here $\nu$ and $\beta$
are fixed constants that are to be chosen appropriately based on the
application at hand, as described above. In the literature estimators
$\delta_{1,c_{0,1},\alpha}^{(\nu)}$ and $\delta_{2,c_{0,2},\alpha}^{(\beta)}$,
defined by (1.7) and (1.8), are also called mixed estimators of $\theta_{1}$
and $\theta_{2}$, respectively. Isotonic regression estimators have been
studied by many researchers for specific probability models having independent
marginals. It may be worth mentioning here that, for many probability models
(e.g. independent Gaussian, independent gamma, etc.), the restricted maximum
likelihood estimators (under restriction $\underline{\theta}\in\Theta_{0}$)
are the same as isotonic regression estimators based on some LEE/SEE.

Kumar and Sharma (1988) dealt with simultaneous estimation of ordered means of
two normal distributions having known variances and characterized admissible
estimators among isotonic regression estimators based on BLEEs of $\theta_{1}$
and $\theta_{2}$. For component-wise estimation of order restricted means of
$k$ ($\geq 2$) normal distributions having known variances, Lee (1981)
established component-wise dominance of restricted maximum likelihood
estimators over the unrestricted maximum likelihood estimators under the
squared error loss function. Kelly (1989) extended the results of Lee (1981)
under the criterion of stochastic dominance. Hwang and Peddada (1994) further
extended the results of Lee (1981) and Kelly (1989) to general settings and
proved certain stochastic dominance results. Kushary and Cohen (1989) and
Misra and Singh (1994) studied component-wise estimation of order restricted
location parameters of two exponential distributions, having known scale
parameters, under the squared error loss function. Patra and Kumar (2017)
considered simultaneous estimation of ordered means of a bivariate normal
distribution under the squared error loss function and discussed admissibility
of isotonic regression estimators based on BLEEs of $\theta_{1}$ and
$\theta_{2}$. Chang et. al. (2017) compared different isotonic regression
estimators of the order restricted means of the bivariate normal distribution
having a known correlation matrix under the criterion of stochastic dominance.

In this paper we aim to unify some of the above results proved for specific
probability models by considering a general bivariate location or scale model,
defined by (1.1) or (1.2). The loss function considered is the squared error
loss function, defined by (1.3) or (1.4). For estimation of $\theta_{1}$
($\theta_{2}$), under the restricted parameter space $\Theta_{0}$ and squared
error loss functions (1.3) or (1.4), we consider the class
$\mathcal{D}_{1,\nu}$ ($\mathcal{D}_{2,\beta}$) of isotonic regression
estimators defined in (1.9) ((1.10)), where $\nu$ $(\beta)$ are suitably
chosen, as described above. We will characterize admissible estimators in
$\mathcal{D}_{1,\nu}$ ($\mathcal{D}_{2,\beta}$) and obtain subclass of
estimators that dominate the BLEE/BSEE $\delta_{1,c_{0,1}}(\underline{X})$
($\delta_{2,c_{0,2}}(\underline{X})$). The rest of the paper is organized as
follows. In Section 2, we introduce some useful notations, definitions, and
results that are used later in the paper. Sections 3.1 and 3.2 (4.1 and 4.2),
respectively, deal with estimation of the smaller and larger location (scale)
parameters and under the loss functions $L_{1}$ and $L_{2}$, defined by (1.3)
and (1.4), respectively. In Section 3.3 (4.3), we present a simulation study
to demonstrate that an isotonic regression estimator based on BLEEs (BSEEs)
may not uniformly dominate the BLEE (BSEE).

##  2\. Some Useful Notations, Definitions and Results

The following notations will be used throughout the paper:

  * $\ast$

$\Re=(-\infty,\infty);\,\Re_{++}=(0,\infty),\,\Re^{2}=\Re\times\Re,\,\Re_{++}^{2}=\Re_{++}\times\Re_{++}$;

  * $\ast$

For any positive integer $m$, $\Re^{m}$ will denote the $m$-dimensional
Euclidean space;

  * $\ast$

For any set $A\subseteq\Re$, $I_{A}(\cdot)$ will denote its indicator
function;

  * $\ast$

$N(\mu,\sigma^{2}):$ Normal distribution with mean $\mu\in\Re$ and standard
deviation $\sigma>0$;

  * $\ast$

$N_{2}(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},\rho):$ Bivariate normal
distribution with mean vector $\begin{pmatrix}\mu_{1}\\\
\mu_{2}\end{pmatrix}\in\Re^{2}$ and positive definite variance-covariance
matrix $\Sigma=\begin{pmatrix}\sigma_{1}^{2}&\rho\sigma_{1}\sigma_{2}\\\
\rho\sigma_{1}\sigma_{2}&\sigma_{2}^{2}\end{pmatrix}$;

  * $\ast$

$\Phi(z):$ distribution function of $N(0,1)$;

  * $\ast$

$\phi(z):$ probability density function of $N(0,1)$;

  * $\ast$

$=^{d}$ would mean equality in distribution.

We extend the usual orders ”$\leq$” and ”$<$” on $\Re$ to
$\Re\cup\\{-\infty,\infty\\}$ with the convention that, for
$a>(<)\,0,\;\frac{a}{0}=\infty\,(-\infty),\;\frac{a}{\infty}=0,\;\frac{0}{\infty}=0$
and that, for any $b\in\Re$, $b>-\infty$ and $b<\infty$.

Whenever, we say that an estimator $\delta_{1}$ dominates (or improves on)
another estimator $\delta_{2}$ it is in the non-strict sense, i.e., it simply
means that the risk of $\delta_{1}$ is never larger than that of $\delta_{2}$,
for any configuration of parametric values. Also, when we say that a function
$k:A\rightarrow\Re$, where $A\subseteq\Re$, is increasing (decreasing) it
means that it is non-decreasing (non-increasing).

We first introduce definitions of some stochastic orders relevant to our study
(see Shaked and Shanthikumar (2007)).

Definition 2.1 Let $X$ and $Y$ be random variables with the Lebesgue pdfs
$f(\cdot)$ and $g(\cdot)$, respectively, dfs $F(\cdot)$ and $G(\cdot)$,
respectively, and distributional supports $S_{X}$ and $S_{Y}$, respectively.
We say that the random variable (r.v.) $X$ is smaller than the r.v. $Y$ in
the:

(i) usual stochastic order (written as $X\underset{\mathrm{st}}{\leq}Y$) if
$F(x)\geq G(x)$, $\forall\;x\in\Re$;

(ii) likelihood ratio order (written as $X\underset{\mathrm{lr}}{\leq}Y$) if

$f(x)g(y)\geq f(y)g(x),\;\;\text{whenever}\;-\infty<x<y<\infty,$

or equivalently if $S_{X}\subseteq S_{Y}$ and $\frac{g(x)}{f(x)}$ is
increasing in $x\in S_{Y}$;

It is well known that (see Shaked and Shanthikumar (2007))
$X\underset{\mathrm{lr}}{\leq}Y\Rightarrow X\underset{\mathrm{st}}{\leq}Y\iff
E[h(X)]\leq E[h(Y)]$, for any increasing function $h:S_{X}\cup
S_{Y}\rightarrow\Re$ for which expectations exists.

Now we will discuss some properties of log-concave and log-convex functions
(see Pecaric et. al. (1992)), as they are relevant to our study. We begin with
the definitions of log-concave and log-convex functions.

Definition 2.2 Let $D$ be a convex subset of $\Re^{m}$. A function
$h:D\rightarrow[0,\infty)$ is said to be log-concave (log-convex) if, for all
$\underline{x},\,\underline{y}\in D$ and all $\alpha\in[0,1]$,

$h(\alpha\underline{x}+(1-\alpha)\underline{y})\geq(\leq)(h(\underline{x}))^{\alpha}(h(\underline{y}))^{1-\alpha}.$

The following properties are well known.

P1. If $g:\Re\rightarrow[0,\infty)$ is a pdf with interior of support $(a,b)$
$(-\infty\leq a<b\leq\infty)$, then $g$ is log-concave (log-convex) on $(a,b)$
if, and only if,

(2.1) $g(x_{1})\,g(x_{2}-\delta)\geq(\leq)g(x_{1}-\delta)\,g(x_{2}),$

for all $0<\delta<b-a$ and for all $a+\delta<x_{1}<x_{2}<b$ (see Pecaric et.
al. (1992)).

P2. Let $h:\Re^{m}\times\Re^{n}\rightarrow[0,\infty)$ be a log-concave
function. Then the function

$g(\underline{x})=\int_{\Re^{n}}h(\underline{x},\underline{y})\,d\underline{y}$

is log-concave on $\Re^{m}$ (see Prekopa (1971) and Pecaric et. al. (1992)).

P3. Let $-\infty\leq a<b\leq\infty$ and $-\infty\leq c<d\leq\infty$. Let
$h:(a,b)\times(c,d)\rightarrow[0,\infty)$ be such that, for each $y\in(c,d)$,
$h(x,y)$ is log-convex in $x\in(a,b)$. Then

$g(x)=\int_{a}^{b}h(x,y)dy$

is log-convex on $(a,b)$ (Artin (1931) and Marshall and Olkin (1979)).

The following result taken from Misra and van der Meulen (2003) will be used
in proving the main results of the paper.

Proposition 2.1 Let $T_{1}$ and $T_{2}$ be random variables having
distributional supports $B_{1}$ and $B_{2}$, respectively. Let
$k_{i}:B_{1}\cup B_{2}\rightarrow\Re,\;i=1,2,$ be given functions and let
$\beta_{i}=E[k_{i}(T_{i})],\;i=1,2$. Suppose that $T_{1}\leq_{st}T_{2}$
($T_{2}\leq_{st}T_{1}$). Then
(i) $\beta_{1}\leq\beta_{2}$, provided $k_{1}(x)\leq k_{2}(x)$, $\forall\;x\in
B_{1}\cup B_{2}$, and $k_{1}(x)$ or $k_{2}(x)$ is a increasing (decreasing)
function of $x\in B_{1}\cup B_{2}$;
(ii) $\beta_{2}\leq\beta_{1}$, provided $k_{2}(x)\leq k_{1}(x)$,
$\forall\;x\in B_{1}\cup B_{2}$, and $k_{1}(x)$ or $k_{2}(x)$ is a decreasing
(increasing) function of $x\in B_{1}\cup B_{2}$.

The following result, famously known as Chebyshev’s inequality, will also be
used in our study (see Marshall and Olkin (2007)).

Proposition 2.2 Let $S$ be r.v. and let $k_{1}(\cdot)$ and $k_{2}(\cdot)$ be
real-valued monotonic functions defined on the distributional support of the
r.v. $S$. If $k_{1}(\cdot)$ and $k_{2}(\cdot)$ are monotonic functions of the
same (opposite) type, then

$E[k_{1}(S)k_{2}(S)]\geq(\leq)E[k_{1}(S)]E[k_{2}(S)],$

provided the above expectations exist.

In the following section we consider isotonic regression estimators of order
restricted location parameters $\theta_{1}$ and $\theta_{2}$
($\theta_{1}\leq\theta_{2}$).

##  3\. Isotonic Regression Estimators For Component-wise Estimation of Order
Restricted Location Parameters

Consider estimation of $\theta_{i}\;(i=1,2)$ under the location model (1.1)
and the SEL function (1.3), when it is known apriori that
$\underline{\theta}\in\Theta_{0}=\\{(x_{1},x_{2})\in\Re^{2}:x_{1}\leq
x_{2}\\}$. The risk function of an estimator $\delta_{i}$ of $\theta_{i}$ is
given by

$R_{i}(\underline{\theta},\delta_{i})=E_{\theta}[(\delta_{i}(\underline{X})-\theta_{i})^{2}],\;\underline{\theta}\in\Theta_{0},\;i=1,2.$

Let $\delta_{i,c_{0,i}}(\underline{X})=X_{i}-c_{0,i},\;i=1,2,$ be the BLEE of
$\theta_{i},$ where $c_{0,i}$ is defined by (1.5), $i=1,2$. For suitably
chosen $\nu\in\Re$ and $\beta\in\Re$, let $\mathcal{D}_{1,\nu}$ and
$\mathcal{D}_{2,\beta}$ be the classes of isotonic regression estimators of
$\theta_{1}$ and $\theta_{2}$ defined by (1.7)-(1.10), where
$\delta_{1,\beta}(\underline{X})=X_{1}-\beta$ and
$\delta_{2,\nu}(\underline{X})=X_{2}-\nu$. The choices of $\nu$ and $\beta$
are to be such that LEEs $\delta_{2,\nu}(\underline{X})=X_{2}-\nu$ and
$\delta_{1,\beta}(\underline{X})=X_{1}-\beta$ are close to BLEEs
$\delta_{2,c_{0,2}}(\underline{X})=X_{2}-c_{0,2}$ and
$\delta_{1,c_{0,1}}(\underline{X})=X_{1}-c_{0,1}$, respectively, and dominance
of $\delta_{1,c_{0,1,\alpha}}^{(\nu)}(\underline{X})$ and
$\delta_{2,c_{0,2},\alpha}^{(\beta)}(\underline{X})$ over the BLEEs
$\delta_{1,c_{0,1}}$ and $\delta_{2,c_{0,2}}$, respectively, is ensured for
some $\alpha$’s. Our goal is to characterize admissible estimators within
classes $\mathcal{D}_{1,\nu}$ and $\mathcal{D}_{2,\beta}$ of isotonic
regression estimators of $\theta_{1}$ and $\theta_{2}$, respectively, and to
find estimators in these classes that dominate the BLEEs
$\delta_{1,c_{0,1}}(\underline{X})=X_{1}-c_{0,1}$ and
$\delta_{2,c_{0,2}}(\underline{X})=X_{2}-c_{0,2}$, respectively. The following
subsection deals with estimation of the location parameter $\theta_{1}$.

###  3.1. Isotonic Regression Estimators of the smaller Location Parameter
$\theta_{1}$

Let $\nu$ be a fixed real constant. Consider the class $\mathcal{D}_{1,\nu}$
of isotonic regression estimators defined by (1.7) and (1.9), where
$\delta_{1,c_{0,1}}(\underline{X})=X_{1}-c_{0,1}$ and
$\delta_{2,\nu}(\underline{x})=X_{2}-\nu$. Note that the BLEE
$\delta_{1,c_{0,1}}(\underline{X})=\delta_{1,c_{0,1},1}^{(\nu)}(\underline{X})\in\mathcal{D}_{1,\nu}$.
Define $D=X_{2}-X_{1}$, $\lambda=\theta_{2}-\theta_{1}$ ($\lambda\geq 0$),
$Z_{1}=X_{1}-\theta_{1}$, $Z_{2}=X_{2}-\theta_{2}$, $Z=Z_{2}-Z_{1}$ and
$\underline{Z}=(Z_{1},Z_{2})$. For estimating $\theta_{1}$ under the SEL
function (1.3), the risk function of the estimator
$\delta_{1,c_{0,1},\alpha}^{(\nu)}\in\mathcal{D}_{1,\nu}$,
$-\infty<\alpha<\infty$, is given by

(3.1)
$R_{1}(\underline{\theta},\delta_{1,c_{0,1},\alpha}^{(\nu)})=E_{\underline{\theta}}[(Z_{1}-c_{0,1})^{2}I_{(\nu-
c_{0,1}-\lambda,\infty)}(Z)]\\\
+E_{\underline{\theta}}[((Z_{2}+\lambda-\nu)-\alpha(Z+\lambda+c_{0,1}-\nu))^{2}I_{(-\infty,\nu-
c_{0,1}-\lambda)}(Z)].$

The risk function
$R_{1}(\underline{\theta},\delta_{1,c_{0,1},\alpha}^{(\nu)})$ depends on
$\underline{\theta}\in\Theta_{0}$ only through
$\lambda=\theta_{2}-\theta_{1}\;(\geq 0)$. Let $f_{Z}(\cdot)$ denote the pdf
of $Z$, so that
$f_{Z}(z)=\int_{-\infty}^{\infty}f(s,s+z)\,ds,\;-\infty<z<\infty$. Clearly,
for any fixed $\underline{\theta}\in\Theta_{0}$ (or $\lambda\geq 0)$,
$R_{1}(\underline{\theta},\delta_{1,c_{0,1},\alpha}^{(\nu)})$ is minimized at
$\alpha=\alpha_{1}(\lambda)$, where, for $\lambda\geq 0$,

(3.2)
$\alpha_{1}(\lambda)=\frac{E_{\underline{\theta}}[(Z_{2}+\lambda-\nu)(Z+\lambda+c_{0,1}-\nu)I_{(-\infty,\nu-
c_{0,1}-\lambda)}(Z)]}{E_{\underline{\theta}}[(Z+\lambda+c_{0,1}-\nu)^{2}I_{(-\infty,\nu-
c_{0,1}-\lambda)}(Z)]}=1+\alpha_{1}^{*}(\lambda)$ (3.3)
$\text{and}\;\;\;\alpha_{1}^{*}(\lambda)=\frac{E_{\underline{\theta}}[(Z_{1}-c_{0,1})(Z+\lambda+c_{0,1}-\nu)I_{(-\infty,\nu-
c_{0,1}-\lambda)}(Z)]}{E_{\underline{\theta}}[(Z+\lambda+c_{0,1}-\nu)^{2}I_{(-\infty,\nu-
c_{0,1}-\lambda)}(Z)]},\;\;\lambda\geq 0.$

Let $S_{Z}$ denote the distribution support of $Z$ and

$\psi_{1}(z)=E[Z_{1}|Z=z]=\frac{\int_{-\infty}^{\infty}s\,f(s,s+z)ds}{\int_{-\infty}^{\infty}f(s,s+z)ds},\;z\in
S_{Z}.$

Then, for $\lambda\geq 0$,

$\displaystyle\alpha_{1}^{*}(\lambda)=\frac{\int_{-\infty}^{0}(\psi_{1}(z+\nu-
c_{0,1}-\lambda)-c_{0,1})z\,f_{Z}(z+\nu-
c_{0,1}-\lambda)dz}{\int_{-\infty}^{0}z^{2}\,f_{Z}(z+\nu-
c_{0,1}-\lambda)dz}=E_{\lambda}[k_{1}(S_{1,\lambda},\lambda)],$

where $k_{1}(z,\lambda)=\frac{\psi_{1}(z+\nu-
c_{0,1}-\lambda)-c_{0,1}}{z},\;z<0,\;\lambda\geq 0,$ and $S_{1,\lambda}$ is a
r.v. having the pdf

(3.4) $h_{1,\lambda}(z)=\begin{cases}\frac{z^{2}f_{Z}(z+\nu-
c_{0,1}-\lambda)}{\int_{-\infty}^{0}s^{2}f_{Z}(s+\nu-
c_{0,1}-\lambda)\,ds},&\text{if}\;\;z<0\\\
0,&\text{otherwise}\end{cases},\;\;\lambda\geq 0.$

Using (2.1), it is easy to check if $f_{Z}(\cdot)$ is log-concave (log-convex)
on $(-\infty,\nu-c_{0,1})$, then $S_{1,\lambda_{1}}\leq_{lr}S_{1,\lambda_{2}}$
($S_{1,\lambda_{2}}\leq_{lr}S_{1,\lambda_{1}}$) and, consequently,
$S_{1,\lambda_{1}}\leq_{st}S_{1,\lambda_{2}}$
($S_{1,\lambda_{2}}\leq_{st}S_{1,\lambda_{1}}$), whenever
$0\leq\lambda_{1}<\lambda_{2}<\infty$.

The following lemma will be useful in proving the main result of this
subsection.

Lemma 3.1.1 (a) Suppose that $f_{Z}(\cdot)$ is log-concave (log-convex) on
$(-\infty,\nu-c_{0,1})$, $\psi_{1}(z)=E[Z_{1}|Z=z]$ is increasing on
$(-\infty,\nu-c_{0,1})$ and, for every $\lambda\geq 0$,
$k_{1}(z,\lambda)=\frac{\psi_{1}(z+\nu-c_{0,1}-\lambda)-c_{0,1}}{z}$ is
increasing (decreasing) in $z\in(-\infty,0)$. Then $\alpha_{1}^{*}(\lambda)$
(and hence $\alpha_{1}(\lambda)$), defined by (3.3) ((3.2)), is an increasing
function of $\lambda\in[0,\infty)$, $\inf_{\lambda\geq
0}\alpha_{1}(\lambda)=1+\alpha_{1}^{*}(0)=\alpha_{1,0},$ say, and
$\sup_{\lambda\geq
0}\alpha_{1}(\lambda)=1+\lim_{\lambda\to\infty}\alpha_{1}^{*}(\lambda)=\alpha_{1,\infty},\text{
say}.$

(b) Suppose that $f_{Z}(\cdot)$ is log-concave (log-convex) on $(-\infty,\nu-
c_{0,1})$, $\psi_{1}(z)$ is decreasing on $(-\infty,\nu-c_{0,1})$ and, for
every $\lambda\geq 0$, $k_{1}(z,\lambda)=\frac{\psi_{1}(z+\nu-
c_{0,1}-\lambda)-c_{0,1}}{z}$ is decreasing (increasing) in $z\in(-\infty,0)$.
Then $\alpha_{1}^{*}(\lambda)$ (and hence $\alpha_{1}(\lambda)$) is a
decreasing function of $\lambda\in[0,\infty)$, $\inf_{\lambda\geq
0}\alpha_{1}(\lambda)=1+\lim_{\lambda\to\infty}\alpha_{1}^{*}(\lambda)=\alpha_{1,\infty},\text{
say, and }\sup_{\lambda\geq
0}\alpha_{1}(\lambda)=1+\alpha_{1}^{*}(0)=\alpha_{1,0},\text{ say}.$

###### Proof.

(a) Let $0\leq\lambda_{1}<\lambda_{2}<\infty$,
$T_{i}=S_{1,\lambda_{i}},\;i=1,2,$ and
$m_{1,i}(z)=k_{1}(z,\lambda_{i})=\frac{\psi_{1}(z+\nu-
c_{0,1}-\lambda_{i})-c_{0,1}}{z},\;z<0,\;i=1,2,$ where $S_{1,\lambda}$ is a
r.v. having pdf given by (3.4). Then
$\alpha_{1}^{*}(\lambda_{i})=E_{\lambda_{i}}[m_{1,i}(T_{i})],\;i=1,2$. The
hypothesis that $f_{Z}(\cdot)$ is log-concave on $(-\infty,\nu-c_{0,1})$
implies that $T_{1}\leq_{lr}T_{2}$ ($T_{2}\leq_{lr}T_{1}$) and, consequently,
$T_{1}\leq_{st}T_{2}$ ($T_{2}\leq_{st}T_{1}$). Also, under the hypothesis of
(a), $m_{1,1}(t)\leq(\geq)\,m_{1,2}(t),\;\forall\;t<0$, and $m_{1,i}(t)$ is an
increasing (decreasing) function of $t\in(-\infty,0)$. Now, using Proposition
2.1 (a), it follows that

$\alpha_{1}^{*}(\lambda_{1})=E_{\lambda_{1}}[m_{1,1}(T_{1})]\leq
E_{\lambda_{2}}[m_{1,2}(T_{2})]=\alpha_{1}^{*}(\lambda_{2}).$

(b) Using Proposition 2.1 (ii), the assertion follows on the lines of the
proof of assertion (a) above. ∎

Now we prove the main result of this subsection

Theorem 3.1.1 (a) Suppose that assumptions of Lemma 3.1.1 (a) hold. Then the
estimators that are admissible within the class $\mathcal{D}_{1,\nu}$ are
$\\{\delta_{1,c_{0,1},\alpha}^{(\nu)}:\alpha\in[\alpha_{1,0},\alpha_{1,\infty}]\\}$.
Moreover, for $-\infty<\alpha_{1}<\alpha_{2}\leq\alpha_{1,0}$ or
$\alpha_{1,\infty}\leq\alpha_{2}<\alpha_{1}$, the estimator
$\delta_{1,c_{0,1},\alpha_{2}}^{(\nu)}$ dominates the estimator
$\delta_{1,c_{0,1},\alpha_{1}}^{(\nu)}$, for any
$\underline{\theta}\in\Theta_{0}$.

(b) Suppose that assumptions of Lemma 3.1.1 (b) hold. Then the estimators that
are admissible within the class $\mathcal{D}_{1,\nu}$ are
$\\{\delta_{1,c_{0,1},\alpha}^{(\nu)}:\alpha\in[\alpha_{1,\infty},\alpha_{1,0}]\\}$.
Moreover, for $-\infty<\alpha_{1}<\alpha_{2}\leq\alpha_{1,\infty}$ or
$\alpha_{1,0}\leq\alpha_{2}<\alpha_{1}<\infty$, the estimator
$\delta_{1,c_{0,1},\alpha_{2}}^{(\nu)}$ dominates the estimator
$\delta_{1,c_{0,1},\alpha_{1}}^{(\nu)}$, for any
$\underline{\theta}\in\Theta_{0}$.

###### Proof.

(a) Let $\alpha_{1}(\lambda)$ be as defined by (3.2). Note that, for any fixed
$\underline{\theta}\in\Theta_{0}$ (or fixed $\lambda\geq 0$), the risk
function $R_{1}(\underline{\theta},\delta_{1,c_{0,1},\alpha}^{(\nu)})$, given
by (3.1), is uniquely minimized at $\alpha=\alpha_{1}(\lambda)$, it is a
strictly decreasing function of $\alpha$ on $(-\infty,\alpha_{1}(\lambda))$
and strictly increasing function of $\alpha$ on
$(\alpha_{1}(\lambda),\infty)$. Since, for any $\lambda\geq 0$,
$\alpha_{1}(\lambda)$ is a continuous function of $\lambda\in[0,\infty)$, it
assumes all values between $\alpha_{1,0}=\inf_{\lambda\geq
0}\alpha_{1}(\lambda)$ and $\alpha_{1,\infty}=\sup_{\lambda\geq
0}\alpha_{1}(\lambda)$ as $\lambda$ varies on $[0,\infty)$. It follows that,
each $\alpha\in[\alpha_{1,0},\alpha_{1,\infty}]$ uniquely minimizes the risk
function $R_{1}(\underline{\theta},\delta_{1,c_{0,1},\alpha}^{(\nu)})$ at some
$\underline{\theta}\in\Theta_{0}$ (or at some $\lambda\geq 0$). This proves
that the estimators
$\\{\delta_{1,c_{0,1},\alpha}^{(\nu)}:\alpha\in[\alpha_{1,0},\alpha_{1,\infty}]\\}$
are admissible among the estimators in the class $\mathcal{D}_{1,\nu}$. Since
$\alpha_{1,0}\leq\alpha_{1}(\lambda)\leq\alpha_{1,\infty},\;\forall\;\lambda\geq
0$, the foregoing discussion implies that, for any
$\underline{\theta}\in\Theta_{0}$ (or for any $\lambda\geq 0$),
$R_{1}(\underline{\theta},\delta_{1,c_{0,1},\alpha}^{(\nu)})$ is a decreasing
function of $\alpha$ on $(-\infty,\alpha_{1,0}]$ and, for any
$\underline{\theta}\in\Theta_{0}$, it is an increasing function of $\alpha$ on
$[\alpha_{1,\infty},\infty)$. This establishes the second assertion of (a).

(b) Similar to the proof of (a), and hence omitted. ∎

In many applications verifying validity of assumptions of the above theorem
will be straightforward. In this direction, the following points are
noteworthy:

  * $\ast$

If $f:\Re^{2}\rightarrow[0,\infty)$ be log-concave on $\Re^{2}$, then
$f_{Z}(z)=\int_{-\infty}^{\infty}f(s,s+z)ds,\;z\in\Re,$ is log-concave on
$\Re$ (see property P2);

  * $\ast$

If, for every fixed $s\in\Re$, $f(s,z)$ is a log-convex function of
$z\in(-\infty,\nu-c_{0,1})$, then
$f_{Z}(z)=\int_{-\infty}^{\infty}f(s,s+z)ds,\;\;\;z\in\Re,$ is log-convex on
$(-\infty,\nu-c_{0,1})$ (see property P3);

  * $\ast$

Let $X_{1}$ and $X_{2}$ be independent r.v.s with pdfs $f_{1}(\cdot)$ and
$f_{2}(\cdot)$, respectively, so that
$f(z_{1},z_{2})=f_{1}(z_{1})f_{2}(z_{2}),\;-\infty<z_{1},z_{2}<\infty$. If
$f_{1}$ and $f_{2}$ are log-concave on $\Re$, then
$f_{Z}(z)=\int_{-\infty}^{\infty}f(s,s+z)ds,\,z\in\Re,$ is log-concave on
$\Re$. Similarly, if for any fixed $s\in\Re$, $f_{2}(s+z)$ is log-convex on
(-$\infty,\nu-c_{0,1}$), then
$f_{Z}(z)=\int_{-\infty}^{\infty}f(s,s+z)ds,\;z\in\Re,$ is log-convex on
$(-\infty,\nu-c_{0,1})$ (follows from the above two points);

  * $\ast$

Suppose that $\frac{f(s,s+z_{2})}{f(s,s+z_{1})}$ is increasing (decreasing) in
$s\in\Re$, whenever $-\infty<z_{1}<z_{2}\leq\nu-c_{0,1}$. Then
$\psi_{1}(z)=E[Z_{1}|Z=z]$ is an increasing (decreasing) function of
$z\in(-\infty,\nu-c_{0,1})$.

To establish the last point, note that

$\psi_{1}(z)=\frac{\int_{-\infty}^{\infty}sf(s,s+z)ds}{\int_{-\infty}^{\infty}f(s,s+z)ds}=E[Y_{z}],\;\;z\in(-\infty,\nu-
c_{0,1}),$

where $Y_{z},\;z\in(-\infty,\nu-c_{0,1})$, is a r.v. having the pdf

$h_{z}(s)=\frac{f(s,s+z)}{\int_{-\infty}^{\infty}f(s,s+z)ds},\;-\infty<s<\infty,\;z\in(-\infty,\nu-
c_{0,1}).$

Let $-\infty<z_{1}<z_{2}\leq\nu-c_{0,1}$, $T_{i}=Y_{z_{i}},\;i=1,2,$ and
$k_{i}(t)=t,\;-\infty<t<\infty,\;i=1,2$. Then the hypotheses of assertion
implies that $T_{1}\leq_{lr}\,(\geq_{lr}T_{2})$. Now, using Proposition 2.1,
it follows that $\psi_{1}(z_{1})=E[T_{1}]\leq E[T_{2}]=\psi_{1}(z_{2})$.

Note that
$\delta_{1,c_{0,1}}(\underline{X})=\delta_{1,c_{0,1},1}^{(\nu)}(\underline{x})=X_{1}-c_{0,1}$
is the BLEE of $\theta_{1}$. Using Theorem 3.1.1 (a) ((b)), one can obtain a
class of estimators that dominate the BLEE $\delta_{1,c_{0,1}}^{(\nu)}$,
provided $\alpha_{1,0}>(<)\,1$. Note that $\alpha_{1,0}>(<)\,1$ if, and only
if,
$E_{\underline{\theta}}[(\psi_{1}(Z)-c_{0,1})(Z+c_{0,1}-\nu)I_{(-\infty,\nu-
c_{0,1})}(Z)]>(<)\,0$. Let $k_{1}(s)=\psi_{1}(s)-c_{0,1}$ and
$k_{2}(s)=(s+c_{0,1}-\nu)I_{(-\infty,\nu-c_{0,1})}(s)$. Then $k_{2}(s)$ is an
increasing function of $s\in\Re$. Using Proposition 2.2, it follows that

$E[(\psi_{1}(Z)-c_{0,1})(Z+c_{0,1}-\nu)I_{(-\infty,\nu-c_{0,1})}(Z)]\\\
\geq(\leq)E[(\psi_{1}(Z)-c_{0,1})]E[(Z+c_{0,1}-\nu)I_{(-\infty,\nu-
c_{0,1})}(Z)]=0,$

provided $\psi_{1}(z)$ is an increasing (decreasing) function of
$z\in(-\infty,\nu-c_{0,1})$. Thus, we have the following proposition.

Proposition 3.1.1 Suppose that $\psi_{1}(z)=E[Z_{1}|Z=z]$ is increasing
(decreasing) in $z\in(-\infty,\nu-c_{0,1})$. Then,
$\alpha_{1,0}\geq(\leq)\;1$.

The above proposition suggests that, under the assumptions of Theorem 3.1.1
(a) (or (b)), the BLEE $\delta_{1,c_{0,1}}(\underline{X})=X_{1}-c_{0,1}$ is
inadmissible for estimating $\theta_{1}$ and dominating estimators are
$\delta_{1,c_{0,1},\alpha_{1,0}}^{(\nu)}$, for any fixed $\nu$, provided, for
any fixed $z\in S_{Z}$, $\psi_{1}(z)=E[Z_{1}|Z=z]$ is an increasing
(decreasing) function of $z\in S_{Z}$.

Now we illustrate some applications of Theorem 3.1.1.

Example 3.1.1 Let $\underline{X}=(X_{1},X_{2})$ have the Lebesgue pdf
belonging to location family (1.1), where $\underline{\theta}\in\Theta_{0}$.
Assume that $f(z_{1},z_{2})=f(z_{2},z_{1})$,
$\forall\;(z_{1},z_{2})\in\Re^{2}$ and
$f(z_{1},z_{2})=f(-z_{1},-z_{2})\;\forall\;(z_{1},z_{2})\in\Re^{2}$. Further
suppose that $f(z_{1},z_{2})$ is log-concave on $\Re^{2}$. Then, by property
P2, $f_{Z}(z)$ is log-concave on $\Re$. Consider estimation of $\theta_{1}$
under the SEL function $L_{1}$, defined by (1.3), when it is known apriori
that $\underline{\theta}\in\Theta_{0}$. Then $Z_{i}=^{d}-Z_{i},\;i=1,2,$ and
$(Z_{1},Z_{2})=^{d}(-Z_{2},-Z_{1})$. Consequently $c_{0,1}=c_{0,2}=0$ and, for
$-\infty<z<\infty$,

$\displaystyle\psi_{1}(z)\\!=\\!E_{\underline{\theta}}[Z_{1}|Z_{2}\\!-\\!Z_{1}\\!=\\!z]\\!=\\!-z\\!+\\!E_{\underline{\theta}}[Z_{2}|Z_{2}\\!-\\!Z_{1}\\!=\\!z]\\!=\\!-z\\!-\\!\psi_{1}(z)\;\;((Z_{1},Z_{2})=^{d}(-Z_{2},-Z_{1})),$

i.e., $\psi_{1}(z)=-\frac{z}{2},\;\;-\infty<z<\infty$. Thus, $\psi_{1}(z)$ is
a decreasing function of $z\in\Re$. Also, for any $\lambda\geq 0$,

$k_{1}(z,\lambda)=\frac{\psi_{1}(z+\nu-
c_{0,1}-\lambda)-c_{0,1}}{z}=-\frac{1}{2}+\frac{\lambda-\nu}{2z}$

is a decreasing function of $z\in(-\infty,0)$ for the choice $\nu\leq 0$. As
$c_{0,2}=0$, here, an appropriate choice of $\nu$ is 0, which amounts to
considering isotonic regression of $(X_{1},X_{2})$. For $\nu=0$, we have
$\alpha_{1,0}=\alpha_{1}(0)=\frac{1}{2}$. Using Theorem 3.1.1 (b), it follows
that the estimators
$\\{\delta_{1,0,\alpha}^{(0)}:\alpha_{1,\infty}\leq\alpha\leq\frac{1}{2}\\}$
are admissible among the isotonic regression estimators in the class
$\mathcal{D}_{1,0}=\\{\delta_{1,0,\alpha}^{(0)}:-\infty<\alpha<\infty\\}$,
where

$\delta_{1,0,\alpha}^{(0)}(\underline{X})=\begin{cases}X_{1},&\text{
if}\;\;\;X_{1}\leq X_{2}\\\ \alpha X_{1}+(1-\alpha)X_{2},&\text{
if}\;\;\;X_{1}>X_{2}\end{cases},\;\;-\infty<\alpha<\infty.$

Also, for any $\alpha_{1}>\alpha_{2}\geq\frac{1}{2}$, the estimator
$\delta_{1,0,\alpha_{2}}^{(0)}(\underline{X})$ dominates the estimator
$\delta_{1,0,\alpha_{1}}^{(0)}(\underline{X})$. In particular the estimator

$\delta_{1,0,0.5}^{(0)}(\underline{X})=\begin{cases}X_{1},&\text{
if}\;\;X_{1}\leq X_{2}\\\ \frac{X_{1}+X_{2}}{2},&\text{
if}\;\;X_{1}>X_{2}\end{cases}=\min\bigg{\\{}X_{1},\frac{X_{1}+X_{2}}{2}\bigg{\\}}$

dominates the BLEE $\delta_{1,0}(\underline{X})=\delta_{1,0,1}^{(\nu)}=X_{1}$.

Examples of log-concave densities $f(z_{1},z_{2})$ satisfying the assumptions
of this example are:

  * •

$f(z_{1},z_{2})=\frac{1}{2\pi\sigma^{2}\sqrt{1-\rho^{2}}}e^{-\frac{1}{2(1-\rho^{2})\sigma^{2}}\left[z_{1}^{2}-2\rho\,z_{1}z_{2}+z_{2}^{2}\right]},\;(z_{1},z_{2})\in\Re^{2},$
where $\sigma>0$ and $-1<\rho<1$.

  * •

$f(z_{1},z_{2})=\frac{1}{\sigma^{2}}\frac{e^{-\frac{z_{1}}{\sigma}}}{\left(1+e^{-\frac{z_{1}}{\sigma}}\right)^{2}}\frac{e^{-\frac{z_{2}}{\sigma}}}{\left(1+e^{-\frac{z_{2}}{\sigma}}\right)^{2}},\;(z_{1},z_{2})\in\Re^{2},$
where $\sigma>0$.

  * •

$f(z_{1},z_{2})=\frac{1}{4\sigma^{2}}e^{-\frac{|z_{1}|}{\sigma}}e^{-\frac{|z_{2}|}{\sigma}},\;(z_{1},z_{2})\in\Re^{2},$
where $\sigma>0$.

Example 3.1.2 Let $\underline{X}=(X_{1},X_{2})\sim
N_{2}(\theta_{1},\theta_{2},\sigma_{1}^{2},\sigma_{2}^{2},\rho)$, where
$-\infty<\theta_{1}\leq\theta_{2}<\infty$, $\theta_{1}$ and $\theta_{2}$ are
unknown, and $\sigma_{i}>0,\;i=1,2,$ and $\rho\in(-1,1)$ are known. Consider
estimation of $\theta_{1}$ under the squared error loss $L_{1}$, given by
(1.3). Here $Z_{i}\sim N(0,\sigma_{i}^{2}),\;i=1,2,\;c_{0,i}=0,\;i=1,2,$
$Z_{1}|Z=z\sim
N\left(\frac{\sigma_{1}(\rho\sigma_{2}-\sigma_{1})z}{\tau^{2}},\frac{(1-\rho^{2})\sigma_{1}^{2}\sigma_{2}^{2}}{\tau^{2}}\right),\;-\infty<z<\infty,$
$Z\sim N(0,\tau^{2})$ and
$\psi_{1}(z)=E[Z_{1}|Z=z]=\frac{\sigma_{1}(\rho\sigma_{2}-\sigma_{1})z}{\tau^{2}},\;-\infty<z<\infty,$
where $\tau^{2}=\sigma_{1}^{2}+\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}$.
Moreover, $f_{Z}(z)$ is log-concave on $\Re$. Clearly $\psi_{1}(z)$ is
decreasing (increasing) in $z\in(-\infty,0)$ if $\rho\sigma_{2}<\sigma_{1}$
($\rho\sigma_{2}>\sigma_{1}$). Moreover, for any fixed $\lambda\geq 0$,
$k_{1}(z,\lambda)=\frac{\sigma_{1}(\rho\sigma_{2}-\sigma_{1})}{\tau^{2}}\frac{z+\nu-\lambda}{z}$
is decreasing (increasing) in $z\in(-\infty,0)$ provided
$\rho\sigma_{2}<\sigma_{1}$ ($\rho\sigma_{2}>\sigma_{1}$) and $\nu\leq 0$.
Since $c_{0,2}=0$, an appropriate choice of $\nu$ in this case is $\nu=0$. For
$\nu=0$.

$\alpha_{1,0}\\!=\\!1-\alpha_{1}^{*}(0)\\!=\\!1+\frac{E[\psi_{1}(Z)Z\,I_{(-\infty,0)}(Z)]}{E[Z^{2}\,I_{(-\infty,0)}(Z)]}\\!=\\!1+\frac{\sigma_{1}(\rho\sigma_{2}-\sigma_{1})}{\tau^{2}}\\!=\\!\frac{\sigma_{2}(\sigma_{2}\\!-\\!\rho\sigma_{1})}{\tau^{2}}\\!=\\!\beta_{0},\;(\text{say})$

and

$\alpha_{1,\infty}\\!=\\!\lim_{\lambda\to\infty}\\!(1+\alpha_{1}^{*}(\lambda))\\!=\\!1+\frac{\sigma_{1}(\rho\sigma_{2}\\!-\\!\sigma_{1})}{\tau^{2}}\lim_{\lambda\to\infty}\\!\frac{\int_{-\infty}^{0}(z-\lambda)z\frac{1}{\tau}\phi\left(\frac{z-\lambda}{\tau}\right)dz}{\int_{-\infty}^{0}z^{2}\frac{1}{\tau}\phi\left(\frac{z-\lambda}{\tau}\right)dz}\\!=\\!\begin{cases}-\infty,\\!\\!\\!&\text{if}\;\rho\sigma_{2}<\sigma_{1}\\\
1,\\!\\!\\!&\text{if}\;\rho\sigma_{2}=\sigma_{1}\\\
\infty,\\!\\!\\!&\text{if}\;\rho\sigma_{2}>\sigma_{1}\end{cases}.$

Here $\delta_{1,0,\beta_{0}}^{(0)}(\underline{X})$ is the restricted MLE
(RMLE) of $\theta_{1}$ (see Patra and Kumar (2017)). Consider the following
cases.

Case I: $\frac{\rho\sigma_{2}}{\sigma_{1}}<1$

In this case
$\alpha_{1,0}=\beta_{0}=\frac{\sigma_{2}(\sigma_{2}-\rho\sigma_{1})}{\tau^{2}}<1,\;\alpha_{1,\infty}=-\infty,\;\psi_{1}(z)=\frac{\sigma_{1}(\rho\sigma_{2}-\sigma_{1})}{\tau^{2}}z$
is decreasing in $z\in(-\infty,0)$ and, for every fixed $\lambda\geq 0$,
$k_{1}(z,\lambda)=\frac{\psi_{1}(z-\lambda)}{z}=\frac{\sigma_{1}(\rho\sigma_{2}-\sigma_{1})}{\tau^{2}}\frac{z-\lambda}{z}$
is decreasing in $z\in(-\infty,0)$. Using Theorem 3.1.1 (b), it follows that
the estimators $\\{\delta_{1,0,\alpha}^{(0)}:\alpha\in(-\infty,\beta_{0}]\\}$
are admissible within the class
$\mathcal{D}_{1,0}=\\{\delta_{1,0,\alpha}^{(0)}:-\infty<\alpha<\infty\\}$ of
isotonic regression estimators of $\theta_{1}$. Moreover the estimators
$\\{\delta_{1,0,\alpha}^{(0)}:\alpha>\beta_{0}\\}$ are inadmissible and, for
$\beta_{0}\leq\alpha_{2}<\alpha_{1}$, the estimator
$\delta_{1,0,\alpha_{2}}^{(0)}$ dominates the estimator
$\delta_{1,0,\alpha_{1}}^{(0)}$. In particular the RMLE
$\delta_{1,0,\beta_{0}}^{(0)}$ is admissible within the class
$\mathcal{D}_{1,0}$ of isotonic regression estimators of $\theta_{1}$ and it
dominates the BLEE $\delta_{1,0}(\underline{X})=\delta_{1,0,1}^{(0)}=X_{1}$.
For the special case $\rho=0$, the dominance of RMLE over the BLEE is also
shown in Kubokawa and Saleh (1994).

Case II: $\frac{\rho\sigma_{2}}{\sigma_{1}}>1$

In this case
$\alpha_{1,0}=\beta_{0}=\frac{\sigma_{2}(\sigma_{2}-\rho\sigma_{1})}{\tau^{2}}>1,\;\alpha_{1,\infty}=\infty,\;\psi_{1}(z)=\frac{\sigma_{1}(\rho\sigma_{2}-\sigma_{1})}{\tau^{2}}z$
is increasing in $z\in(-\infty,0)$ and, for every fixed $\lambda\geq 0$,
$k_{1}(z,\lambda)=\frac{\psi_{1}(z-\lambda)}{z}=\frac{\sigma_{1}(\rho\sigma_{2}-\sigma_{1})}{\tau^{2}}\frac{z-\lambda}{z}$
is increasing in $z\in(-\infty,0)$. Using Theorem 3.1.1 (a), we conclude that
the estimators $\\{\delta_{1,0,\alpha}^{(0)}:\alpha\in[\beta_{0},\infty)\\}$
are admissible within the class
$\mathcal{D}_{1,0}=\\{\delta_{1,0,\alpha}^{(0)}:-\infty<\alpha<\infty\\}$ of
isotonic regression estimators of $\theta_{1}$. Moreover, the estimators
$\\{\delta_{1,0,\alpha}^{(0)}:\alpha<\beta_{0}\\}$ are inadmissible for
estimating $\theta_{1}$ and, for $\infty<\alpha_{1}<\alpha_{2}\leq\beta_{0}$,
the estimator $\delta_{1,0,\alpha_{2}}^{(0)}$ dominates the estimator
$\delta_{1,0,\alpha_{1}}^{(0)}$. In particular the RMLE
$\delta_{1,0,\beta_{0}}^{(0)}$ is admissible within the class
$\mathcal{D}_{1,0}$ of isotonic regression estimators of $\theta_{1}$ and it
dominates the BLEE $\delta_{1,0}(\underline{X})=\delta_{1,0,1}^{(0)}=X_{1}$.

Case III: $\frac{\rho\sigma_{2}}{\sigma_{1}}=1$

In this case $\alpha_{1,0}=\beta_{0}=\alpha_{1,\infty}=1$ and the BLEE
$\delta_{1,0}(\underline{X})=\delta_{1,0,1}^{(0)}(\underline{X})=X_{1}$ is
also the RMLE. Also,
$\psi_{1}(z)=0=k(z,\lambda),\;z\in(-\infty,0),\;\lambda\geq 0$. Using Theorem
3.1.1, it follows that the BLEE/RMLE
$\delta_{1,0}(\underline{X})=\delta_{1,0,1}^{(0)}(\underline{X})=X_{1}$ is the
only admissible estimator within the class $\mathcal{D}_{1,0}$ of isotonic
regression estimators of $\theta_{1}$. Any other isotonic regression estimator
in $\mathcal{D}_{1,0}$ is dominated by the BLEE/RMLE
$\delta_{1,0}(\underline{X})=\delta_{1,0,1}^{(0)}(\underline{X})=X_{1}$.

Example 3.1.3. Let
$f(z_{1},z_{2})=f_{1}(z_{1})f_{2}(z_{2}),\,\underline{z}=(z_{1},z_{2})\in\Re^{2},$
where, for known positive constants $\sigma_{1}>0$ and $\sigma_{2}>0$,
$f_{i}(z)=\frac{1}{\sigma_{i}}e^{-\frac{z}{\sigma_{i}}},\text{
if}\;z>0,i=1,2.$ Here $c_{0,i}=\sigma_{i},\;i=1,2$, and the BLEE of
$\theta_{1}$ is
$\delta_{1,\sigma_{1}}(\underline{X})=\delta_{1,\sigma_{1},1}^{(\nu)}(\underline{X})=X_{1}-\sigma_{1}$.
Moreover,

$f_{Z}(z)=\begin{cases}\frac{1}{\sigma_{1}+\sigma_{2}}e^{\frac{z}{\sigma_{1}}},&\text{
if}\;\;z<0\\\ \frac{1}{\sigma_{1}+\sigma_{2}}e^{-\frac{z}{\sigma_{2}}},&\text{
if}\;\;z\geq 0\end{cases}$

is log-concave on $\Re$. The conditional pdf of $Z_{1}$ given
$Z=z\;(-\infty<z<\infty)$ is
$f_{Z_{1}|Z}(z_{1}|z)=\frac{\sigma_{1}+\sigma_{2}}{\sigma_{1}\sigma_{2}}\,e^{-\left(\frac{1}{\sigma_{1}}+\frac{1}{\sigma_{2}}\right)(z_{1}-\max\\{-z,0\\})},\text{
if}\;z_{1}\geq\max\\{-z,0\\},$ and

$\psi_{1}(z)=E[Z_{1}|Z=z]=\max\\{-z,0\\}+\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}=\begin{cases}-z+\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}},&\text{
if}\;\;z<0\\\ \frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}},&\text{
if}\;\;z\geq 0\end{cases}$

is decreasing in $z\in(-\infty,\nu-\sigma_{1})$. For any fixed $\lambda\geq
0$,

$k_{1}(z,\lambda)=\frac{\psi_{1}(z+\nu-\sigma_{1}-\lambda)-\sigma_{1}}{z}=\begin{cases}\left(\lambda+\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}-\nu\right)\frac{1}{z}-1,&\text{
if}\;\;z<\lambda+\sigma_{1}-\nu\\\
\frac{-\sigma_{1}^{2}}{(\sigma_{1}+\sigma_{2})z},&\text{
if}\;\;z\geq\lambda+\sigma_{1}-\nu\end{cases}$

is decreasing in $z\in(-\infty,0)$, provided
$\nu\leq\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}$. Since
$c_{0,2}=\sigma_{2}$, an appropriate choice of $\nu$ in this case is
$\nu=\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}$. For
$\nu=\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}$, it is easy to verify
that $\alpha_{1,0}=\alpha_{1}(0)=0$ and
$\alpha_{1,\infty}=\lim_{\lambda\to\infty}\alpha_{1}(\lambda)=-\infty$. Using
Theorem 3.1.1 (b), we conclude that the estimators
$\\{\delta_{1,\sigma_{1},\alpha}^{(\nu)}:\alpha\in(-\infty,0]\\}$ are
admissible within the class
$\\{\delta_{1,\sigma_{1},\alpha}^{(\nu)}:-\infty<\alpha<\infty\\}$ of isotonic
regression estimators of $\theta_{1}$; here
$\nu=\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}$. Moreover, for
$0\leq\alpha_{2}<\alpha_{1}<\infty$, the isotonic regression estimator
$\delta_{1,\sigma_{1},\alpha_{2}}^{(\nu)}$ dominates the estimator
$\delta_{1,\sigma_{1},\alpha_{1}}^{(\nu)}$. In particular the BLEE
$\delta_{1,\sigma_{1}}(\underline{X})=\delta_{1,\sigma_{1},1}^{(\nu)}(\underline{X})=X_{1}-\sigma_{1}$
is inadmissible for estimating $\theta_{1}$ and is dominated by the isotonic
regression estimator

$\delta_{1,\sigma_{1},0}^{(\nu)}(\underline{X})=\begin{cases}X_{1}-\sigma_{1},&\text{
if}\;\;X_{1}-\sigma_{1}\leq
X_{2}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\\\
X_{2}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}},&\text{
if}\;\;X_{1}-\sigma_{1}>X_{2}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\end{cases}.$

The above estimator is a member of a class of dominating estimators proposed
in Pal and Kushary (1992). Note that, here, isotonic regression estimators
based on BLEE $(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$ may not be able to provide
improvements over the BLEE
$\delta_{1,\sigma_{1}}(\underline{X})=X_{1}-\sigma_{1}$, rather an isotonic
regression estimator based on
$(X_{1}-\sigma_{1},X_{2}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}})$
provides an improvement over the BLEE $\delta_{1,\sigma_{1}}(\underline{X})$
(see Section (3.3)).

###  3.2. Isotonic Regression Estimators of the Larger Location Parameter
$\theta_{2}$

In this section, we deal with estimation of the larger location parameter
$\theta_{2}$ under the location model (1.1) and the SEL function $L_{2}$,
defined by (1.3), when it is known apriori that
$\underline{\theta}\in\Theta_{0}$. We continue to follow the notations of
Section 3.1. Let $\beta$ be a fixed real constant, to be suitably chosen as
described in Section 1. Consider the class $\mathcal{D}_{2,\beta}$ of isotonic
regression estimators of $\theta_{2}$, defined by (1.8) and (1.10); here
$\delta_{2,c_{0,2}}(\underline{X})=X_{2}-c_{0,2}$ and
$\delta_{1,\beta}(\underline{X})=X_{1}-\beta$. Note that the BLEE
$\delta_{2,c_{0,2}}(\underline{X})=\delta_{2,c_{0,2},0}^{(\nu)}(\underline{X})=X_{2}-c_{0,2}$
is a member of the class $\mathcal{D}_{2,\beta}$ for any fixed $\beta$. Under
the notations of Section 3.1, for any fixed $\underline{\theta}\in\Theta_{0}$
(or $\lambda\geq 0)$,

(3.5)
$R_{2}(\underline{\theta},\delta_{2,c_{0,2},\alpha}^{(\beta)})=E_{\underline{\theta}}[(Z_{2}-c_{0,2})^{2}I_{(c_{0,2}-\beta-\lambda,\infty)}(Z)]\\\
\;\;\quad+E_{\underline{\theta}}[((Z_{2}-c_{0,2})-\alpha(Z+\lambda+\beta-
c_{0,2}))^{2}I_{(-\infty,c_{0,2}-\beta-\lambda)}(Z)].$

is minimized at $\alpha=\alpha_{2}(\lambda)$, where

(3.6)
$\alpha_{2}(\lambda)=\frac{E_{\underline{\theta}}[(Z_{2}-c_{0,2})(Z+\lambda+\beta-
c_{0,2})I_{(-\infty,c_{0,2}-\beta-\lambda)}(Z)]}{E_{\underline{\theta}}[(Z+\lambda+\beta-
c_{0,2})^{2}I_{(-\infty,c_{0,2}-\beta-\lambda)}(Z)]},\;\;\lambda\geq 0.$

Let
$\psi_{2}(z)=E[Z_{2}|Z=z]=\frac{\int_{-\infty}^{\infty}s\,f(s-z,s)ds}{\int_{-\infty}^{\infty}f(s-z,s)ds},\;z\in
S_{Z},$ so that $\psi_{2}(z)=z+\psi_{1}(z),\;z\in S_{Z}$, where
$\psi_{1}(z)=E[Z_{1}|Z=z],\;z\in S_{Z}$. Then, for $\lambda\geq 0$,

(3.7)
$\displaystyle\alpha_{2}(\lambda)\\!=\\!\frac{\int_{-\infty}^{0}(\psi_{2}(z+c_{0,2}-\beta-\lambda)-c_{0,2})z\,f_{Z}(z+c_{0,2}-\beta-\lambda)dz}{\int_{-\infty}^{0}z^{2}\,f_{Z}(z+c_{0,2}-\beta-\lambda)dz}\\!=\\!E_{\lambda}[k_{2}(S_{2,\lambda},\lambda)],$

where
$k_{2}(z,\lambda)=\frac{\psi_{2}(z+c_{0,2}-\beta-\lambda)-c_{0,2}}{z},\;z<0,\;\lambda\geq
0,$ and $S_{2,\lambda}$ is a r.v. having the pdf

(3.8)
$h_{2,\lambda}(z)=\begin{cases}\frac{z^{2}f_{Z}(z+c_{0,2}-\beta-\lambda)}{\int_{-\infty}^{0}s^{2}f_{Z}(s+c_{0,2}-\beta-\lambda)\,ds},&\text{if}\;\;z<0\\\
0,&\text{otherwise}\end{cases},\;\;\lambda\geq 0.$

Using property P1, it is easy to check if $f_{Z}(\cdot)$ is log-concave (log-
convex) on $(-\infty,c_{0,2}-\beta)$, then
$S_{2,\lambda_{1}}\leq_{lr}S_{2,\lambda_{2}}$
($S_{2,\lambda_{2}}\leq_{lr}S_{2,\lambda_{1}}$) and, consequently,
$S_{2,\lambda_{1}}\leq_{st}S_{2,\lambda_{2}}$
($S_{2,\lambda_{2}}\leq_{st}S_{2,\lambda_{1}}$), whenever
$0\leq\lambda_{1}<\lambda_{2}<\infty$.

The following lemma, whose proof is similar to that of Lemma 3.1.1 (a), will
be instrumental in proving the main result of this subsection.

Lemma 3.2.1 (a) Suppose that $f_{Z}(\cdot)$ is log-concave (log-convex) on
$(-\infty,c_{0,2}-\beta)$, $\psi_{2}(z)=E[Z_{2}|Z=z]$ is increasing on
$(-\infty,c_{0,2}-\beta)$ and, for every $\lambda\geq 0$,
$k_{2}(z,\lambda)=\frac{\psi_{2}(z+c_{0,2}-\beta-\lambda)-c_{0,2}}{z}$ is
increasing (decreasing) in $z\in(-\infty,0)$. Then $\alpha_{2}(\lambda)$,
defined by (3.7), is an increasing function of $\lambda\in[0,\infty)$,
$\inf_{\lambda\geq 0}\alpha_{2}(\lambda)=\alpha_{2}(0)=\alpha_{2,0},\text{
say, and }\sup_{\lambda\geq
0}\alpha_{2}(\lambda)=\lim_{\lambda\to\infty}\alpha_{2}(\lambda)=\alpha_{2,\infty},\text{
say}.$

(b) Suppose that $f_{Z}(\cdot)$ is log-concave (log-convex) on
$(-\infty,c_{0,2}-\beta)$, $\psi_{2}(z)$ is decreasing on
$(-\infty,c_{0,2}-\beta)$ and, for every $\lambda\geq 0$,
$k_{2}(z,\lambda)=\frac{\psi_{2}(z+c_{0,2}-\beta-\lambda)-c_{0,2}}{z}$ is
decreasing (increasing) in $z\in(-\infty,0)$. Then $\alpha_{2}(\lambda)$ is a
decreasing function of $\lambda\in[0,\infty)$, $\inf_{\lambda\geq
0}\alpha_{2}(\lambda)=\lim_{\lambda\to\infty}\alpha_{2}(\lambda)=\alpha_{2,\infty},\text{
say, and }\sup_{\lambda\geq
0}\alpha_{2}(\lambda)=\alpha_{2}(0)=\alpha_{2,0},\text{ say}.$

Now we have the main result of this subsection. The proof of the theorem,
being similar to the proof of Theorem 3.1.1, is omitted.

Theorem 3.2.1 (a) Suppose that assumptions of Lemma 3.2.1 (a) hold. Then, the
estimators that are admissible in the class $\mathcal{D}_{2,\beta}$ are
$\\{\delta_{2,c_{0,2},\alpha}^{(\beta)}:\alpha\in[\alpha_{2,0},\alpha_{2,\infty}]\\}$.
Moreover, for $-\infty<\alpha_{1}<\alpha_{2}\leq\alpha_{2,0}$ or
$\alpha_{2,\infty}\leq\alpha_{2}<\alpha_{1}$, the estimator
$\delta_{2,c_{0,2},\alpha_{2}}^{(\beta)}$ dominates the estimator
$\delta_{2,c_{0,2},\alpha_{1}}^{(\beta)}$, for any
$\underline{\theta}\in\Theta_{0}$.

(b) Suppose that assumptions of Lemma 3.2.1 (b) hold. Then, the estimators
that are admissible in the class $\mathcal{D}_{2,\beta}$ are
$\\{\delta_{2,c_{0,2},\alpha}^{(\beta)}:\alpha\in[\alpha_{2,\infty},\alpha_{2,0}]\\}$.
Moreover, for $-\infty<\alpha_{1}<\alpha_{2}\leq\alpha_{2,\infty}$ or
$\alpha_{2,0}\leq\alpha_{2}<\alpha_{1}<\infty$, the estimator
$\delta_{2,c_{0,2},\alpha_{2}}^{(\beta)}$ dominates the estimator
$\delta_{2,c_{0,2},\alpha_{1}}^{(\beta)}$, for any
$\underline{\theta}\in\Theta_{0}$.

Note that
$\delta_{2,c_{0,2}}(\underline{X})=\delta_{2,c_{0,2},0}^{(\beta)}(\underline{X})=X_{2}-c_{0,2}$
is the BLEE of $\theta_{2}$. Using Theorem 3.2.1 (a) ((b)), one can obtain a
class of estimators that dominate the BLEE $\delta_{2,c_{0,2}}$, provided
$\alpha_{2,0}>(<)\,0$. Note that $\alpha_{2,0}>(<)\,0$ if, and only if,
$E_{\underline{\theta}}[(\psi_{2}(Z)-c_{0,2})(Z+\beta-
c_{0,2})I_{(-\infty,c_{0,2}-\beta)}(Z)]>(<)\,0$. Let
$k_{1}(s)=\psi_{2}(s)-c_{0,2}$ and $k_{2}(s)=(s+\beta-
c_{0,2})I_{(-\infty,c_{0,2}-\beta)}(s)$. Then $k_{2}(s)$ is an increasing
function of $s\in\Re$. Using Proposition 2.2, it follows that

$E[(\psi_{2}(Z)-c_{0,2})(Z+\beta-c_{0,2})I_{(-\infty,c_{0,2}-\beta)}(Z)]\\\
\geq(\leq)E[(\psi_{2}(Z)-c_{0,2})]E[(Z+\beta-
c_{0,2})I_{(-\infty,c_{0,2}-\beta)}(Z)]=0,$

provided $\psi_{2}(z)$ is an increasing (decreasing) function of
$z\in(-\infty,c_{0,2}-\beta)$. Thus, we have the following proposition.

Proposition 3.2.2 Suppose that $\psi_{2}(z)=E[Z_{2}|Z=z]$ is increasing
(decreasing) in $z\in(-\infty,c_{0,2}-\beta)$. Then,
$\alpha_{2,0}\geq(\leq)\;0$.

Now we illustrate some applications of Theorem 3.2.1.

Example 3.2.1 Under the set-up of Example 3.1.1, consider estimation of
$\theta_{2}$ under the SEL function $L_{2}$, defined by (1.3), when it is
known apriori that $\underline{\theta}\in\Theta_{0}$. Then $c_{0,1}=c_{0,2}=0$
and $\psi_{2}(z)=z+\psi_{1}(z)=\frac{z}{2},\;\;-\infty<z<\infty$. Thus,
$\psi_{2}(z)$ is an increasing function of $z\in\Re$. Also, for any
$\lambda\geq 0$,

$k_{2}(z,\lambda)=\frac{\psi_{2}(z+c_{0,2}-\beta-\lambda)-c_{0,2}}{z}=\frac{1}{2}-\frac{\beta+\lambda}{2z}$

is an increasing function of $z\in(-\infty,0)$, for any $\beta\geq 0$. As
$c_{0,1}=0$, here the suitable choice of $\beta$ is 0. That amounts to
considering isotonic regression of $(X_{1},X_{2})$. One can verify that, for
$\beta=0$,
$\alpha_{2,0}=\alpha_{2}(0)=\frac{1}{2}\;\text{and}\;\alpha_{2,\infty}=\lim_{\lambda\to\infty}\alpha_{2,\infty}.$

Now using Theorem 3.2.1 (a), it follows that the estimators
$\\{\delta_{2,0,\alpha}^{(0)}:\frac{1}{2}\leq\alpha\leq\alpha_{2,\infty}\\}$
are admissible among the isotonic regression estimators in the class
$\mathcal{D}_{2,0}=\\{\delta_{2,0,\alpha}^{(0)}:-\infty<\alpha<\infty\\}$,
where

$\delta_{2,0,\alpha}^{(0)}(\underline{X})=\begin{cases}X_{2},&\text{
if}\;\;\;X_{1}\leq X_{2}\\\ \alpha X_{1}+(1-\alpha)X_{2},&\text{
if}\;\;\;X_{1}>X_{2}\end{cases},\;\;-\infty<\alpha<\infty.$

Also, for $-\infty<\alpha_{1}<\alpha_{2}\leq\frac{1}{2}$, the estimator
$\delta_{2,0,\alpha_{2}}^{(0)}(\underline{X})$ dominates the estimator
$\delta_{2,0,\alpha_{1}}^{(0)}(\underline{X})$. In particular the estimator

$\delta_{2,0,0.5}^{(0)}(\underline{X})=\begin{cases}X_{2},&\text{
if}\;\;X_{1}\leq X_{2}\\\ \frac{X_{1}+X_{2}}{2},&\text{
if}\;\;X_{1}>X_{2}\end{cases}=\max\bigg{\\{}X_{2},\frac{X_{1}+X_{2}}{2}\bigg{\\}}$

dominates the BLEE $\delta_{2,0}(\underline{X})=\delta_{2,0,0}^{(0)}=X_{2}$.

Example 3.2.2. Let $\underline{X}=(X_{1},X_{2})$ follow the bivariate normal
distribution described in Example 3.1.2. Consider estimation of $\theta_{2}$
under the squared error loss $L_{2}$, given by (1.3), when it is known that
$\underline{\theta}\in\Theta_{0}$. Here, for any fixed $\lambda\geq 0$,
$k_{2}(z,\lambda)=\frac{\sigma_{2}(\sigma_{2}-\rho\sigma_{1})}{\tau^{2}}\frac{z-\beta-\lambda}{z}$
is decreasing (increasing) in $z\in(-\infty,0)$, provided
$\rho\sigma_{1}>\sigma_{2}$ ($\rho\sigma_{1}<\sigma_{2}$) and $\beta\geq 0$.
Since $c_{0,1}=0$, an appropriate choice of $\beta$ in this case is $\beta=0$.
For $\beta=0$, we have

$\alpha_{2,0}\\!=\\!\alpha_{2}(0)\\!=\\!\frac{E[\psi_{2}(Z)Z\,I_{(-\infty,0)}(Z)]}{E[Z^{2}\,I_{(-\infty,0)}(Z)]}\\!=\\!1\\!+\\!\frac{E[\psi_{1}(Z)Z\,I_{(-\infty,0)}(Z)]}{E[Z^{2}\,I_{(-\infty,0)}(Z)]}\\!=\\!\frac{\sigma_{2}(\sigma_{2}-\rho\sigma_{1})}{\tau^{2}}\\!=\\!\beta_{0}$

and

$\alpha_{2,\infty}\\!=\\!\lim_{\lambda\to\infty}\alpha_{2}(\lambda)\\!=\\!\frac{\sigma_{2}(\sigma_{2}-\rho\sigma_{1})}{\tau^{2}}\lim_{\lambda\to\infty}\frac{\int_{-\infty}^{0}(z-\lambda)z\frac{1}{\tau}\phi\left(\frac{z-\lambda}{\tau}\right)dz}{\int_{-\infty}^{0}z^{2}\frac{1}{\tau}\phi\left(\frac{z-\lambda}{\tau}\right)dz}\\!=\\!\begin{cases}-\infty,\\!\\!\\!&\text{if}\,\rho\sigma_{1}>\sigma_{2}\\\
0,\\!\\!\\!&\text{if}\,\rho\sigma_{1}=\sigma_{2}\\\
\infty,\\!\\!\\!&\text{if}\,\rho\sigma_{1}<\sigma_{2}\end{cases}.$

Here $\delta_{2,0,\beta_{0}}^{(0)}(\underline{X})$ is the restricted MLE
(RMLE) of $\theta_{2}$. Consider the following cases.

Case I: $\rho\sigma_{1}>\sigma_{2}$

In this case
$\alpha_{2,0}=\beta_{0}<0,\;\alpha_{2,\infty}=-\infty,\;\psi_{2}(z)=\frac{\sigma_{2}(\sigma_{2}-\rho\sigma_{1})}{\tau^{2}}z$
is decreasing in $z\in(-\infty,0)$ and, for every fixed $\lambda\geq 0$,
$k_{2}(z,\lambda)=\frac{\psi_{2}(z-\lambda)}{z}=\frac{\sigma_{2}(\sigma_{2}-\rho\sigma_{1})}{\tau^{2}}\frac{z-\lambda}{z}$
is decreasing in $z\in(-\infty,0)$. Using Theorem 3.2.1 (b), it follows that
the estimators $\\{\delta_{2,0,\alpha}^{(0)}:\alpha\in(-\infty,\beta_{0}]\\}$
are admissible within the class
$\mathcal{D}_{2,0}=\\{\delta_{2,0,\alpha}^{(0)}:-\infty<\alpha<\infty\\}$ of
isotonic regression estimators of $\theta_{2}$. Moreover, the estimators
$\\{\delta_{2,0,\alpha}^{(0)}:\alpha>\beta_{0}\\}$ are inadmissible and, for
$\beta_{0}\leq\alpha_{2}<\alpha_{1}$, the estimator
$\delta_{2,0,\alpha_{2}}^{(0)}$ dominates the estimator
$\delta_{2,0,\alpha_{1}}^{(0)}$. In particular the RMLE
$\delta_{2,0,\beta_{0}}^{(0)}$ is admissible within the class
$\mathcal{D}_{2,0}$ of isotonic regression estimators of $\theta_{2}$ and it
dominates the BLEE $\delta_{2,0}(\underline{X})=\delta_{2,0,0}^{(0)}=X_{2}$.

Case II: $\rho\sigma_{1}<\sigma_{2}$

In this case
$\alpha_{2,0}=\beta_{0}>0,\;\alpha_{2,\infty}=\infty,\;\psi_{2}(z)=\frac{\sigma_{2}(\sigma_{2}-\rho\sigma_{1})}{\tau^{2}}z$
is increasing in $z\in(-\infty,0)$ and, for every fixed $\lambda\geq 0$,
$k_{2}(z,\lambda)=\frac{\sigma_{2}(\sigma_{2}-\rho\sigma_{1})}{\tau^{2}}\frac{z-\lambda}{z}$
is increasing in $z\in(-\infty,0)$. Using Theorem 3.2.1 (a), we conclude that
the estimators $\\{\delta_{2,0,\alpha}^{(0)}:\alpha\in[\beta_{0},\infty)\\}$
are admissible within the class
$\mathcal{D}_{2,0}=\\{\delta_{2,0,\alpha}^{(0)}:-\infty<\alpha<\infty\\}$ of
isotonic regression estimators of $\theta_{2}$. Moreover, the estimators
$\\{\delta_{2,0,\alpha}^{(0)}:\alpha<\beta_{0}\\}$ are inadmissible for
estimating $\theta_{2}$ and, for $\infty<\alpha_{1}<\alpha_{2}\leq\beta_{0}$,
the estimator $\delta_{2,0,\alpha_{2}}^{(0)}$ dominates the estimator
$\delta_{2,0,\alpha_{1}}^{(0)}$. In particular the RMLE
$\delta_{2,0,\beta_{0}}^{(0)}$ is admissible within the class
$\mathcal{D}_{2,0}$ of isotonic regression estimators of $\theta_{2}$ and it
dominates the BLEE $\delta_{2,0}(\underline{X})=\delta_{2,0,0}^{(0)}=X_{2}$.
For $\rho=0$, this result is also reported in Kubokawa and Saleh (1994).

Case III: $\rho\sigma_{1}=\sigma_{2}$

In this case $\alpha_{2,0}=\beta_{0}=\alpha_{2,\infty}=1$ and the BLEE
$\delta_{2,0}(\underline{X})=\delta_{2,0,0}^{(0)}(\underline{X})=X_{2}$ is
also the RMLE. Also,
$\psi_{2}(z)=0=k_{2}(z,\lambda),\;z\in(-\infty,0),\;\lambda\geq 0$. Using
Theorem 3.2.1, it follows that the BLEE/RMLE
$\delta_{2,0}(\underline{X})=\delta_{2,0,0}^{(0)}(\underline{X})=X_{2}$ is the
only admissible estimator within the class $\mathcal{D}_{2,0}$ of isotonic
regression estimators of $\theta_{2}$. Any other isotonic regression estimator
in $\mathcal{D}_{2,0}$ is dominated by the BLEE/RMLE
$\delta_{2,0}(\underline{X})=\delta_{2,0,0}^{(0)}(\underline{X})=X_{2}$.

Example 3.2.3. For the exponential probability model considered in Example
3.1.3, the BLEE of $\theta_{2}$ is
$\delta_{2,0}(\underline{X})=\delta_{2,\sigma_{2},0}^{(\beta)}(\underline{X})=X_{2}-\sigma_{2}$.
Moreover

$f_{Z}(z)=\frac{e^{\frac{z}{\sigma_{1}}}\,e^{-\left(\frac{1}{\sigma_{1}}+\frac{1}{\sigma_{2}}\right)\max\\{0,z\\}}}{\sigma_{1}+\sigma_{2}}=\begin{cases}\frac{e^{\frac{z}{\sigma_{1}}}}{\sigma_{1}+\sigma_{2}},&\text{
if }z<0\\\ \leavevmode\nobreak\ \\\
\frac{e^{\frac{-z}{\sigma_{2}}}}{\sigma_{1}+\sigma_{2}},&\text{ if }z\geq
0\end{cases}$

is log-concave on $\Re$,
$\psi_{2}(z)=E[Z_{2}|Z=z]=z+\psi_{1}(z)=\max\\{0,z\\}+\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}$
is increasing on $\Re$ and, for any $\lambda\geq 0$,

$k_{2}(z,\lambda)=\frac{\psi_{2}(z+\sigma_{2}-\beta-\lambda)-\sigma_{2}}{z}=\begin{cases}\frac{-\sigma_{2}^{2}}{(\sigma_{1}+\sigma_{2})z},&\text{
if}\;\;z<\lambda+\beta-\sigma_{2}\\\
1-\left(\lambda+\beta-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\right)\frac{1}{z},&\text{
if}\;\;z\geq\lambda+\beta-\sigma_{2}\end{cases}$

is increasing in $z\in(-\infty,0)$, provided
$\beta\geq\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}$. Since
$c_{0,1}=\sigma_{1}$, the suitable choice of $\beta$ in this case is
$\beta=\sigma_{1}$. For $\beta=\sigma_{1}$, after some straightforward
algebra, one gets

$\displaystyle\alpha_{2,0}$
$\displaystyle=\frac{\int_{-\infty}^{0}(\psi_{2}(z+\sigma_{2}-\sigma_{1})-\sigma_{2})\,z\,f_{Z}(z+\sigma_{2}-\sigma_{1})dz}{\int_{-\infty}^{0}z^{2}\,f_{Z}(z+\sigma_{2}-\sigma_{1})dz}$
$\displaystyle=\begin{cases}\frac{\frac{\sigma_{2}^{2}(\sigma_{1}^{2}-2\sigma_{1}\sigma_{2}-2\sigma_{2}^{2})}{\sigma_{1}+\sigma_{2}}e^{\frac{\sigma_{1}}{\sigma_{2}}}+\sigma_{2}^{2}(\sigma_{1}+\sigma_{2})e}{(\sigma_{1}+\sigma_{2})(\sigma_{1}^{2}+\sigma_{2}^{2})e-2\sigma_{2}^{3}e^{\frac{\sigma_{1}}{\sigma_{2}}}},&\text{if
}\sigma_{1}\leq\sigma_{2}\\\ \leavevmode\nobreak\ \\\
\frac{\sigma_{2}^{2}}{2\sigma_{1}(\sigma_{1}+\sigma_{2})},&\text{if
}\sigma_{1}>\sigma_{2}\end{cases}$
$\displaystyle\text{and}\qquad\alpha_{2,\infty}$
$\displaystyle=\lim_{\lambda\to\infty}\frac{\int_{-\infty}^{0}(\psi_{2}(z+\sigma_{2}-\sigma_{1}-\lambda)-\sigma_{2})\,z\,f_{Z}(z+\sigma_{2}-\sigma_{1}-\lambda)dz}{\int_{-\infty}^{0}z^{2}f_{Z}(z+\sigma_{2}-\sigma_{1}-\lambda)dz}$
$\displaystyle=\frac{\sigma_{2}^{2}}{2\sigma_{1}(\sigma_{1}+\sigma_{2})}.$

Note that, for $\sigma_{1}>\sigma_{2}$,

$\alpha_{2}(\lambda)=\alpha_{2,0}=\alpha_{2,\infty}=\frac{\sigma_{2}^{2}}{2\sigma_{1}(\sigma_{1}+\sigma_{2})},\;\forall\,\lambda\geq
0.$

It is easy to verify that $\alpha_{2,0}>0$, for any $\sigma_{1}>0$ and
$\sigma_{2}>0$. Using Theorem 3.2.1 (a), it follows that the estimator
estimators
$\\{\delta_{2,\sigma_{2},\alpha}^{(\sigma_{1})}:\alpha\in[\alpha_{2,0},\alpha_{2,\infty}]\\}$
are admissible within the class
$\mathcal{D}_{2,\sigma_{1}}=\\{\delta_{2,\sigma_{2},\alpha}^{(\sigma_{1})}:-\infty<\alpha<\infty\\}$
of isotonic regression estimators of $\theta_{2}$; here $\beta=\sigma_{1}$.
Moreover, for any $-\infty<\alpha_{1}<\alpha_{2}\leq\alpha_{2,0}$ or
$\alpha_{2,\infty}\leq\alpha_{2}<\alpha_{1}<\infty$, the estimator
$\delta_{2,\sigma_{2},\alpha_{2}}^{(\sigma_{1})}$ dominates the estimator
$\delta_{2,\sigma_{2},\alpha_{1}}^{(\sigma_{1})}$. In particular the BLEE
$\delta_{2,\sigma_{2}}(\underline{X})=\delta_{2,\sigma_{2},0}^{(\sigma_{1})}(\underline{X})=X_{2}-\sigma_{2}$
is inadmissible for estimating $\theta_{2}$ and is dominated by the isotonic
regression estimator

$\displaystyle\delta_{2,\sigma_{2},\alpha_{2,0}}^{(\sigma_{1})}(\underline{X})$
$\displaystyle=\begin{cases}X_{2}-\sigma_{2},&\text{
if}\;\;X_{1}-\sigma_{1}\leq X_{2}-\sigma_{2}\\\
\alpha_{2,0}(X_{1}-\sigma_{1})+(1-\alpha_{2,0})(X_{2}-\sigma_{2}),&\text{
if}\;\;X_{1}-\sigma_{1}>X_{2}-\sigma_{2}\end{cases}$
$\displaystyle=\max\\{X_{2}-\sigma_{2},\alpha_{2,0}(X_{1}-\sigma_{1})+(1-\alpha_{2,0})(X_{2}-\sigma_{2})\\}.$

Interestingly, for $\sigma_{1}>\sigma_{2}$, the only admissible estimator
within the class $\mathcal{D}_{2,\sigma_{1}}$ of isotonic regression
estimators is $\delta_{2,\sigma_{2},\alpha_{0}}^{(\sigma_{1})}$, where
$\alpha_{0}=\frac{\sigma_{2}^{2}}{2\sigma_{1}(\sigma_{1}+\sigma_{2})}$, and it
dominates the BLEE $\delta_{2,\sigma_{2}}(\underline{X})=X_{2}-\sigma_{2}$.
Similar results have also been obtained by Misra and Singh (1994).

###  3.3. Simulation Study For Estimation of Location Parameter $\theta_{1}$

Under the squared error loss function, in Example 3.1.3, we considered
estimation of the smaller location parameter $\theta_{1}$ of two independent
exponential distributions having unknown order restricted location parameters
(i.e., $\theta_{1}\leq\theta_{2}$) and known scale parameters ($\sigma_{1}>0$
and $\sigma_{2}>0$). We have shown that the isotonic regression estimator
(IRE)
$\min\\{X_{1}-\sigma_{1},X_{2}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\\}$,
based on
$(X_{1}-\sigma_{1},X_{2}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}})$
with $\alpha=0$, dominates the BLEE
$\delta_{0,1}(\underline{X})=X_{1}-\sigma_{1}$. To further evaluate the
performances of various estimators under the squared error loss function, in
this section, we compare the risk performances of the BLEE
$\delta_{0,1}(\underline{X})=X_{1}-\sigma_{1}$, the Restricted MLE
$\min\\{X_{1},X_{2}\\}$, the IRE
$\min\\{X_{1}-\sigma_{1},X_{2}-\sigma_{2}\\}$, based on the BLEEs
$(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$ with $\alpha=0$, and the IRE
$\min\\{X_{1}-\sigma_{1},X_{2}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\\}$,
based on
$(X_{1}-\sigma_{1},X_{2}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}})$
with $\alpha=0$, numerically, through Monte Carlo simulations. For
simulations, we generated 10000 samples of size 1 each from relevant
exponential distributions and computed the simulated risks of the BLEE, the
Restricted MLE, the Isotonic regression Estimator (IRE) based on
$(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$ with $\alpha=0$, and the IRE based on
$(X_{1}-\sigma_{1},X_{2}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}})$
with $\alpha=0$.

The simulated values of risks of various estimators are plotted in Figure 1.
The following observations are evident from Figure 1:

(i) The risk function values of the IRE based on
$(X_{1}-\sigma_{1},X_{2}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}})$
with $\alpha=0$ estimator is nowhere larger than the risk function values of
the BLEE, which is in conformity with theoretical findings of Example 3.1.3.

(ii) Interestingly, here IRE based on the BLEEs
$(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$ with $\alpha=0$ does not always
dominates the BLEE $X_{1}-\sigma_{1}$.

(iii) There is no clear cut winner between various estimators. But the IRE
based on
$(X_{1}-\sigma_{1},X_{2}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}})$
with $\alpha=0$, performs reasonably well.

(a) $\sigma_{1}=1$ and $\sigma_{2}=0.5$.

(b) $\sigma_{1}=0.5$ and $\sigma_{2}=1.5$.

(c) $\sigma_{1}=3$ and $\sigma_{2}=2$.

(d) $\sigma_{1}=5$ and $\sigma_{2}=10$.

(e) $\sigma_{1}=10$ and $\sigma_{2}=15$.

(f) $\sigma_{1}=15$ and $\sigma_{2}=12$.

Figure 1. Risk plots of the BLEE $X_{1}-\sigma_{1}$, the Restricted MLE
$\min\\{X_{1},X_{2}\\}$, the IRE based on
$(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$ with $\alpha=0$, and the IRE based on
$(X_{1}-\sigma_{1},X_{2}-\nu)$ with $\alpha=0$, where
$\nu=\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}$, against the values
of, $\lambda=\theta_{2}-\theta_{1}$.

##  4\. Isotonic Regression Estimators For Component-wise Estimation of Order
Restricted Scale Parameters

Let $\underline{X}=(X_{1},X_{2})$ be a random vector having the Lebesgue pdf
belonging to the scale family (1.2). We assume that the distributional support
of $\underline{X}=(X_{1},X_{2})$ is a subset of $\Re_{++}^{2}$. Consider
estimation of $\theta_{i}$ under the scaled squared error loss (SSEL) function
(1.4), when it is known apriori that
$\underline{\theta}\in\Theta_{0}=\\{(x_{1},x_{2})\in\Re_{++}:x_{1}\leq
x_{2}\\}$. Let $c_{0,i}$ be as defined in (1.6), so that
$\delta_{i,c_{0,i}}(\underline{X})=c_{0,i}X_{i}$ is the BSEE of
$\theta_{i},\;i=1,2$. For suitably chosen $\nu>0$ and $\beta>0$, let
$\mathcal{D}_{1,\nu}$ and $\mathcal{D}_{2,\beta}$ be the classes of isotonic
regression estimators of $\theta_{1}$ and $\theta_{2}$ defined by (1.9) and
(1.10), respectively, where $\delta_{2,\nu}(\underline{X})=\nu X_{2}$ and
$\delta_{1,\beta}(\underline{X})=\beta X_{1}$. Here $\nu$ and $\beta$ will be
chosen in a manner such that SEEs $\delta_{2,\nu}(\underline{X})=\nu X_{2}$
and $\delta_{1,\beta}(\underline{X})=\beta X_{1}$ are close to the BSEEs
$\delta_{2,c_{0,2}}(\underline{X})=c_{0,2}X_{2}$ and
$\delta_{1,c_{0,1}}(\underline{X})=c_{0,1}X_{1}$, respectively, and dominance
of $\delta_{1,c_{0,1,\alpha}}^{(\nu)}(\underline{X})$ and
$\delta_{2,c_{0,2},\alpha}^{(\beta)}(\underline{X})$ over the BSEEs
$\delta_{1,c_{0,1}}$ and $\delta_{2,c_{0,2}}$, respectively, is ensured for
some $\alpha$’s. Thus, to begin with, we take $\nu$ and $\beta$ to be fixed
positive numbers and consider classes $\mathcal{D}_{1,\nu}$ and
$\mathcal{D}_{2,\beta}$ of isotonic regression estimators of $\theta_{1}$ and
$\theta_{2}$, respectively. We aim to characterize admissible estimators
within classes $\mathcal{D}_{1,\nu}$ and $\mathcal{D}_{2,\beta}$ of isotonic
regression estimators of $\theta_{1}$ and $\theta_{2}$, respectively, and to
find estimators in these classes that dominate the BSEEs
$\delta_{1,c_{0,1}}(\underline{X})=c_{0,1}X_{1}$ and
$\delta_{1,c_{0,2}}(\underline{X})=c_{0,2}X_{2}$. The following subsection
deals with estimation of the smaller scale parameter $\theta_{1}$.

###  4.1. Isotonic Regression Estimators of the smaller Scale Parameter
$\theta_{1}$

Let $\nu$ be a fixed positive constant, to be suitably chosen as described
above. Consider the class $\mathcal{D}_{1,\nu}$ of isotonic regression
estimators defined by (1.7) and (1.9), where
$\delta_{1,c_{0,1}}(\underline{X})=c_{0,1}X_{1}$ and
$\delta_{2,\nu}(\underline{X})=\nu X_{2}$. Define $D=\frac{X_{2}}{X_{1}}$,
$\lambda=\frac{\theta_{2}}{\theta_{1}}$ ($\lambda\geq 1$),
$Z_{1}=\frac{X_{1}}{\theta_{1}}$, $Z_{2}=\frac{X_{2}}{\theta_{2}}$,
$Z=\frac{Z_{2}}{Z_{1}}$ and $\underline{Z}=(Z_{1},Z_{2})$. For estimating
$\theta_{1}$ under the SEL function $L_{2}$, defined by (1.4), the risk
function of the estimator
$\delta_{1,c_{0,1},\alpha}^{(\nu)}\in\mathcal{D}_{1,\nu}$,
$-\infty<\alpha<\infty$, is given by

$\displaystyle R_{1}(\underline{\theta},\delta_{1,c_{0,1},\alpha}^{(\nu)})$
$\displaystyle=E_{\underline{\theta}}\left[\left(\frac{\delta_{1,c_{0,1},\alpha}^{(\nu)}(\underline{X})}{\theta_{1}}-1\right)^{2}\right]$
$\displaystyle=E_{\underline{\theta}}[(c_{0,1}Z_{1}-1)^{2}I_{(\frac{c_{0,1}}{\lambda\nu},\infty)}(Z)]$
$\displaystyle\qquad\qquad+E_{\underline{\theta}}[(\alpha
c_{0,1}Z_{1}+(1-\alpha)\nu\lambda
Z_{2}-1)^{2}I_{(0,\frac{c_{0,1}}{\lambda\nu})}(Z)].$

For any fixed $\underline{\theta}\in\Theta_{0}$ (or $\lambda\geq 1)$,
$R_{1}(\underline{\theta},\delta_{1,c_{0,1},\alpha}^{(\nu)})$ is minimized at
$\alpha=\alpha_{1}(\lambda)$, where

(4.1) $\displaystyle\alpha_{1}(\lambda)=$
$\displaystyle\frac{E_{\underline{\theta}}[(1-\nu\lambda
Z_{2})(c_{0,1}Z_{1}-\nu\lambda
Z_{2})I_{(0,\frac{c_{0,1}}{\lambda\nu})}(Z)]}{E_{\underline{\theta}}[(c_{0,1}Z_{1}-\nu\lambda
Z_{2})^{2}I_{(0,\frac{c_{0,1}}{\lambda\nu})}(Z)]}=1+\alpha_{1}^{*}(\lambda),\;\;\lambda\geq
1,$ $\displaystyle\text{and}\quad\alpha_{1}^{*}(\lambda)=$
$\displaystyle\frac{E_{\underline{\theta}}[(c_{0,1}Z_{1}-\nu\lambda
Z_{2})(1-c_{0,1}Z_{1})I_{(0,\frac{c_{0,1}}{\lambda\nu})}(Z)]}{E_{\underline{\theta}}[(c_{0,1}Z_{1}-\nu\lambda
Z_{2})^{2}I_{(0,\frac{c_{0,1}}{\lambda\nu})}(Z)]}$ $\displaystyle=$
$\displaystyle\frac{1}{c_{0,1}}\frac{E_{\underline{\theta}}[(1-\nu\frac{\lambda
Z}{c_{0,1}})(\psi_{1}(Z)-c_{0,1}\psi_{2}(Z))I_{(0,\frac{c_{0,1}}{\lambda\nu})}(Z)]}{E_{\underline{\theta}}[(1-\frac{\nu\lambda
Z}{c_{0,1}})^{2}I_{(0,\frac{c_{0,1}}{\lambda\nu})}(Z)]}$ (4.2)
$\displaystyle=$
$\displaystyle\frac{1}{c_{0,1}}\frac{\int_{0}^{1}(1-z)(\psi\left(\frac{c_{0,1}z}{\nu\lambda}\right)-c_{0,1})\,\psi_{2}\left(\frac{c_{0,1}z}{\nu\lambda}\right)\,f_{Z}\left(\frac{c_{0,1}z}{\nu\lambda}\right)dz}{\int_{0}^{1}(1-z)^{2}\,\psi_{2}\left(\frac{c_{0,1}z}{\nu\lambda}\right)\,f_{Z}\left(\frac{c_{0,1}z}{\nu\lambda}\right)dz},\;\;\lambda\geq
1;$

here $f_{Z}(z)=\int_{0}^{\infty}sf(s,sz)ds,\;0<z<\infty$ is the pdf of $Z$,
$\psi_{r}(z)=E\left[Z^{{}^{r}}_{1}|Z=z\right]=\frac{\int_{0}^{\infty}s^{r+1}f(s,sz)ds}{\int_{0}^{\infty}sf(s,sz)ds},\;z\in
S_{Z},\;r=1,2$, and $\psi(z)=\frac{\psi_{1}(z)}{\psi_{2}(z)},\;z\in S_{Z}$.

Then,
$\alpha_{1}^{*}(\lambda)=\frac{1}{c_{0,1}}E_{\lambda}[k_{1}(S_{1,\lambda},\lambda)],\,\lambda\geq
1,$ where
$k_{1}(z,\lambda)=\frac{\psi\left(\frac{c_{0,1}z}{\nu\lambda}\right)-c_{0,1}}{1-z},\;0<z<1,\;\lambda\geq
1,$ and $S_{1,\lambda}$ is a r.v. having the pdf

$h_{1,\lambda}(t)=\begin{cases}\frac{(1-t)^{2}\,\psi_{2}\left(\frac{c_{0,1}t}{\nu\lambda}\right)\,f_{Z}\left(\frac{c_{0,1}t}{\nu\lambda}\right)}{\int_{0}^{1}(1-s)^{2}\,\psi_{2}\left(\frac{c_{0,1}s}{\nu\lambda}\right)\,f_{Z}\left(\frac{c_{0,1}s}{\nu\lambda}\right)ds},&\text{if}\;\;0<t<1\\\
0,&\text{otherwise}\end{cases},\;\;\lambda\geq 1.$

Clearly if, for every fixed $\theta\in(0,1)$, $\frac{\psi_{2}(\theta
t)}{\psi_{2}(t)}$ and $\frac{f_{Z}(\theta t)}{f_{Z}(t)}$ are increasing
(decreasing) functions of t on $(0,\frac{c_{0,1}}{\nu})$, then
$S_{1,\lambda_{1}}\leq_{lr}S_{1,\lambda_{2}}$
($S_{1,\lambda_{2}}\leq_{lr}S_{1,\lambda_{1}}$) and, consequently,
$S_{1,\lambda_{1}}\leq_{st}S_{1,\lambda_{2}}$
($S_{1,\lambda_{2}}\leq_{st}S_{1,\lambda_{1}}$), whenever
$1\leq\lambda_{1}<\lambda_{2}<\infty$.

The following lemma, whose proof is immediate from Proposition 2.1, will be
useful in arriving at the main result of this subsection.

Lemma 4.1.1 (a) Suppose that, for every fixed $\theta\in(0,1)$,
$\frac{\psi_{2}(\theta t)}{\psi_{2}(t)}$ and $\frac{f_{Z}(\theta
t)}{f_{Z}(t)}$ are increasing (decreasing) functions of t on
$(0,\frac{c_{0,1}}{\nu})$. Also, suppose that $\psi(z)$ is decreasing on
$(0,\frac{c_{0,1}}{\nu})$ and, for every $\lambda\geq 1$,
$k_{1}(z,\lambda)=\frac{\psi\left(\frac{c_{0,1}z}{\nu\lambda}\right)-c_{0,1}}{1-z}$
is increasing (decreasing) in $z\in(0,1)$. Then $\alpha_{1}^{*}(\lambda)$ (and
hence $\alpha_{1}(\lambda)$), defined by (4.2) ((4.1)), is an increasing
function of $\lambda\in[1,\infty)$, $\inf_{\lambda\geq
1}\alpha_{1}(\lambda)=1+\alpha_{1}^{*}(1)=\alpha_{1,1},\text{ say, and
}\sup_{\lambda\geq
1}\alpha_{1}(\lambda)=1+\lim_{\lambda\to\infty}\alpha_{1}^{*}(\lambda)=\alpha_{1,\infty}\text{,
say}.$

(b) Suppose that, for every fixed $\theta\in(0,1)$, $\frac{\psi_{2}(\theta
t)}{\psi_{2}(t)}$ and $\frac{f_{Z}(\theta t)}{f_{Z}(t)}$ are increasing
(decreasing) functions of t on $(0,\frac{c_{0,1}}{\nu})$. Also, suppose that
$\psi(z)$ is increasing on $(0,\frac{c_{0,1}}{\nu})$ and, for every
$\lambda\geq 1$,
$k_{1}(z,\lambda)=\frac{\psi\left(\frac{c_{0,1}z}{\nu\lambda}\right)-c_{0,1}}{1-z}$
is decreasing (increasing) in $z\in(0,1)$. Then $\alpha_{1}^{*}(\lambda)$ (and
hence $\alpha_{1}(\lambda)$) is a decreasing function of
$\lambda\in[1,\infty)$, $\inf_{\lambda\geq
1}\alpha_{1}(\lambda)=1+\lim_{\lambda\to\infty}\alpha_{1}^{*}(\lambda)=\alpha_{1,\infty},\text{
say, and }\sup_{\lambda\geq
1}\alpha_{1}(\lambda)=1+\alpha_{1}^{*}(1)=\alpha_{1,1}\text{, say}.$

Now we state the main result of this subsection. The proof of the theorem,
being similar to the proof of Theorem 3.1.1, is omitted.

Theorem 4.1.1 (a) Suppose that assumptions of Lemma 4.1.1 (a) hold. Then the
estimators that are admissible in the class $\mathcal{D}_{1,\nu}$ are
$\\{\delta_{1,c_{0,1},\alpha}^{(\nu)}:\alpha\in[\alpha_{1,1},\alpha_{1,\infty}]\\}$.
Moreover, for $-\infty<\alpha_{1}<\alpha_{2}\leq\alpha_{1,1}$ or
$\alpha_{1,\infty}\leq\alpha_{2}<\alpha_{1}$, the estimator
$\delta_{1,c_{0,1},\alpha_{2}}^{(\nu)}$ dominates the estimator
$\delta_{1,c_{0,1},\alpha_{1}}^{(\nu)}$, for any
$\underline{\theta}\in\Theta_{0}$.

(b) Suppose that assumptions of Lemma 4.1.1 (b) hold. Then the estimators that
are admissible in the class $\mathcal{D}_{1,\nu}$ are
$\\{\delta_{1,c_{0,1},\alpha}^{(\nu)}:\alpha\in[\alpha_{1,\infty},\alpha_{1,1}]\\}$.
Moreover, for $-\infty<\alpha_{1}<\alpha_{2}\leq\alpha_{1,\infty}$ or
$\alpha_{1,1}\leq\alpha_{2}<\alpha_{1}<\infty$, the estimator
$\delta_{1,c_{0,1},\alpha_{2}}^{(\nu)}$ dominates the estimator
$\delta_{1,c_{0,1},\alpha_{1}}^{(\nu)}$, for any
$\underline{\theta}\in\Theta_{0}$.

Now we illustrate some applications of Theorem 4.1.1.

Example 4.1.1. Let
$f(z_{1},z_{2})=f_{1}(z_{1})f_{2}(z_{2}),\;\;\underline{z}=(z_{1},z_{2})\in\Re^{2},$
where, for positive constants $\alpha_{1}$ and $\alpha_{2}$,

$f_{i}(z)=\begin{cases}\frac{e^{-z}z^{\alpha_{i}-1}}{\Gamma{\alpha_{i}}},&\text{
if}\;\;z>0\\\ 0,&\text{ otherwise}\end{cases},\;\;i=1,2.$

Here $c_{0,i}=\frac{E[Z_{i}]}{E[Z_{i}^{2}]}=\frac{1}{\alpha_{i}+1},\;i=1,2,$
and the BSEE of $\theta_{1}$ is
$\delta_{1,0}(\underline{X})=\delta_{1,0,1}^{(\nu)}(\underline{X})=\frac{X_{1}}{\alpha_{1}+1}$.
The pdf of $Z=\frac{Z_{2}}{Z_{1}}$ is

$f_{Z}(z)=\begin{cases}\frac{\Gamma{(\alpha_{1}+\alpha_{2})}}{\Gamma{\alpha_{1}}\Gamma{\alpha_{2}}}\frac{z^{\alpha_{2}-1}}{(1+z)^{\alpha_{1}+\alpha_{2}}},&\text{
if}\;\;0<z<\infty\\\ 0,&\text{ otherwise}\end{cases}.$

One can easily check that, for every fixed $\theta\in(0,1)$,
$\frac{f_{Z}(\theta t)}{f_{Z}(t)}$ is increasing in $t\in(0,\infty)$, the
conditional pdf of $Z_{1}$ given $Z=z$ ($0<z<\infty$) is

$f_{Z_{1}|Z}(z_{1}|z)=\begin{cases}\frac{(1+z)^{\alpha_{1}+\alpha_{2}}\,z_{1}^{\alpha_{1}+\alpha_{2}-1}\,e^{-(1+z)z_{1}}}{\Gamma{(\alpha_{1}+\alpha_{2})}},&\text{
if}\;\;0<z_{1}<\infty\\\ 0,&\text{ otherwise}\end{cases},$

$\psi_{1}(z)=E[Z_{1}|Z=z]=\frac{\alpha_{1}+\alpha_{2}}{1+z},\;z>0,$ and
$\psi_{2}(z)=E[Z_{1}^{2}|Z=z]=\frac{(\alpha_{1}+\alpha_{2}+1)(\alpha_{1}+\alpha_{2})}{(1+z)^{2}},\;z>0.$
Clearly, for every fixed $\theta\in(0,1)$, $\frac{\psi_{2}(\theta
z)}{\psi_{2}(z)}=\frac{(1+z)^{2}}{(1+\theta z)^{2}}$ is increasing in
$z\in(0,\infty)$, and
$\psi(z)=\frac{\psi_{1}(z)}{\psi_{2}(z)}=\frac{1+z}{\alpha_{1}+\alpha_{2}+1}$
is increasing in $z\in(0,\infty)$. For any fixed $\lambda\geq 1$,

$k_{1}(z,\lambda)=\frac{\psi\left(\frac{c_{0,1}z}{\nu\lambda}\right)-c_{0,1}}{1-z}=\frac{z-\nu\lambda\alpha_{0}}{(\alpha_{1}+1)(\alpha_{1}+\alpha_{2}+1)\nu\lambda(1-z)},\;z>0,$

is decreasing in $z\in(0,1)$, provided $\nu\geq\frac{1}{\alpha_{2}}$. Since
$c_{0,2}=\frac{1}{\alpha_{2}+1}$, an appropriate choice of $\nu$ in this case
is $\nu=\frac{1}{\alpha_{2}}$. For $\nu=\frac{1}{\alpha_{2}}$, it is easy to
verify that

$\displaystyle\alpha_{1,1}$ $\displaystyle=1+\alpha_{1}^{*}(1)$
$\displaystyle=1+\frac{(\alpha_{1}+1)\int_{0}^{1}(1-t)\left(\psi\left(\frac{\alpha_{2}t}{\alpha_{1}+1}\right)-c_{0,1}\right)\psi_{2}\left(\frac{\alpha_{2}t}{\alpha_{1}+1}\right)f_{Z}\left(\frac{\alpha_{2}t}{\alpha_{1}+1}\right)dt}{\int_{0}^{1}(1-t)^{2}\psi_{2}\left(\frac{\alpha_{2}t}{\alpha_{1}+1}\right)f_{Z}\left(\frac{\alpha_{2}t}{\alpha_{1}+1}\right)dt}$
$\displaystyle=\frac{\alpha_{1}+1}{\alpha_{1}+\alpha_{2}+1}=\beta_{0},\;\;(say),$
$\displaystyle\text{and}\quad\alpha_{1,\infty}$
$\displaystyle=1+\lim_{\lambda\to\infty}\alpha_{1}(\lambda)$
$\displaystyle=1+\lim_{\lambda\to\infty}\frac{(\alpha_{1}+1)\int_{0}^{1}(1-t)\left(\psi\left(\frac{\alpha_{2}t}{\lambda(\alpha_{1}+1)}\right)-c_{0,1}\right)\psi_{2}\left(\frac{\alpha_{2}t}{\lambda(\alpha_{1}+1)}\right)f_{Z}\left(\frac{\alpha_{2}t}{\lambda(\alpha_{1}+1)}\right)dt}{\int_{0}^{1}(1-t)^{2}\,\psi_{2}\left(\frac{\alpha_{2}t}{\lambda(\alpha_{1}+1)}\right)f_{Z}\left(\frac{\alpha_{2}t}{\lambda(\alpha_{1}+1)}\right)dt}$
$\displaystyle=1-\frac{\alpha_{2}(\alpha_{2}+2)}{2(\alpha_{1}+\alpha_{2}+1)}=\alpha_{0},\;\;(say).$

Using Theorem 4.1.1. (b), it follows that the estimators
$\\{\delta_{1,c_{0,1},\alpha}^{(\nu)}:\alpha\in\left[\alpha_{0},\beta_{0}\right]\\}$
are admissible within the class
$\mathcal{D}_{1,\nu}=\\{\delta_{1,c_{0,1},\alpha}^{(\nu)}:-\infty<\alpha<\infty\\}$
of isotonic regression estimators of $\theta_{1}$; here
$c_{0,1}=\frac{1}{\alpha_{1}+1}$ and $\nu=\frac{1}{\alpha_{2}}$. Moreover, for
$-\infty<\alpha_{1}<\alpha_{2}\leq\alpha_{0}$ or
$\beta_{0}\leq\alpha_{2}<\alpha_{1}<\infty$, the estimator
$\delta_{1,c_{0,1},\alpha_{2}}^{(\nu)}$ dominates the estimator
$\delta_{1,c_{0,1},\alpha_{1}}^{(\nu)}$. In particular, the BSEE
$\delta_{1,c_{0,1}}(\underline{X})=\delta_{1,c_{0,1},1}^{(\nu)}(\underline{X})=\frac{X_{1}}{\alpha_{1}+1}$
is inadmissible for estimating $\theta_{1}$ and is dominated by the isotonic
regression estimator

$\displaystyle\delta_{1,c_{0,1},\beta_{0}}^{(\nu)}(\underline{X})=\begin{cases}\frac{X_{1}}{\alpha_{1}+1},&\text{
if }\frac{X_{1}}{\alpha_{1}+1}\leq\frac{X_{2}}{\alpha_{2}}\\\
\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}+1},&\text{ if
}\frac{X_{1}}{\alpha_{1}+1}>\frac{X_{2}}{\alpha_{2}}\end{cases}=\min\bigg{\\{}\frac{X_{1}}{\alpha_{1}+1},\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}+1}\bigg{\\}}.$

The above estimator is also obtained in Vijayasree et al. (1995). Note that
here using isotonic regression estimator based on
$(\frac{X_{1}}{\alpha_{1}+1},\frac{X_{2}}{\alpha_{2}})$, and not
$(\frac{X_{1}}{\alpha_{1}+1},\frac{X_{2}}{\alpha_{2}+1})$, is successful in
providing improvement over the BSEE
$\delta_{1,c_{0,1}}(\underline{X})=\frac{X_{1}}{\alpha_{1}+1}$.

Example 4.1.2. Let
$f(z_{1},z_{2})=f_{1}(z_{1})f_{2}(z_{2}),\;\underline{z}=(z_{1},z_{2})\in\Re_{++}^{2},$
where, for positive constants $\alpha_{1}$ and $\alpha_{2}$,

$f_{i}(z)=\begin{cases}\alpha_{i}z^{\alpha_{i}-1},&\text{ if}\;\;0<z<1\\\
0,&\text{ otherwise}\end{cases},\;\;i=1,2.$

Here
$c_{0,i}=\frac{E[Z_{i}]}{E[Z_{i}^{2}]}=\frac{\alpha_{i}+2}{\alpha_{i}+1},\;i=1,2,$
and the BSEE of $\theta_{1}$ is
$\delta_{1,c_{0,1}}(\underline{X})=\delta_{1,0,1}^{(\nu)}(\underline{X})=\frac{\alpha_{1}+2}{\alpha_{1}+1}X_{1}$.
The pdf of $Z=\frac{Z_{2}}{Z_{1}}$ is

$f_{Z}(z)=\frac{\alpha_{1}\alpha_{2}}{\alpha_{1}+\alpha_{2}}z^{\alpha_{2}-1}\left(\min\bigg{\\{}1,\frac{1}{z}\bigg{\\}}\right)^{\alpha_{1}+\alpha_{2}},\;\;z>0,$

the conditional pdf of $Z_{1}$ given $Z=z$ ($0<z<\infty$) is

$f_{Z_{1}|Z}(z_{1}|z)=\begin{cases}\frac{(\alpha_{1}+\alpha_{2})\,z_{1}^{\alpha_{1}+\alpha_{2}-1}}{\left(\min\big{\\{}1,\frac{1}{z}\big{\\}}\right)^{\alpha_{1}+\alpha_{2}}},&\text{
if}\;\;0<z_{1}<\min\big{\\{}1,\frac{1}{z}\big{\\}}\\\ 0,&\text{
otherwise}\end{cases},$

$\psi_{1}(z)=E[Z_{1}|Z=z]=\frac{\alpha_{1}+\alpha_{2}}{\alpha_{1}+\alpha_{2}+1}\min\big{\\{}1,\frac{1}{z}\big{\\}},\,z>0,$
and $\psi_{2}(z)=E[Z_{1}^{2}|Z=z]=$
$\frac{\alpha_{1}+\alpha_{2}}{\alpha_{1}+\alpha_{2}+2}(\min\big{\\{}1,\frac{1}{z}\big{\\}})^{2},\,z>0.$
Clearly, for $0<\theta<1$,

$\displaystyle\frac{f_{Z}(\theta
t)}{f_{Z}(t)}=\begin{cases}\theta^{\alpha_{2}-1},&\text{if }\;0<t<1\\\
\theta^{\alpha_{2}-1}t,&\text{if }\;1\leq t<\frac{1}{\theta}\\\
\frac{1}{\theta^{\alpha_{1}+1}},&\text{if }\;\frac{1}{\theta}\leq
t<\infty\end{cases}\;\;\text{ and }\;\;\frac{\psi_{2}(\theta
z)}{\psi_{2}(z)}=\begin{cases}1,&\text{if }\;0<z<1\\\ z,&\text{if }\;1\leq
z<\frac{1}{\theta}\\\ \frac{1}{\theta},&\text{if }\;\frac{1}{\theta}\leq
z<\infty\end{cases}$

are increasing in $z\in[0,\infty)$ and
$\psi(z)=\frac{\psi_{1}(z)}{\psi_{2}(z)}=\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}\,\frac{1}{\min\big{\\{}1,\frac{1}{z}\big{\\}}}$
is also increasing in $z\in[0,\infty)$. For any fixed $\lambda\geq 1$,

$\displaystyle
k_{1}(z,\lambda)=\frac{\psi\left(\frac{c_{0,1}z}{\nu\lambda}\right)-c_{0,1}}{1-z}=\begin{cases}\frac{-\alpha_{2}}{(\alpha_{1}+1)(\alpha_{1}+\alpha_{2}+1)(1-z)},&\text{if
}0<z<\frac{\nu\lambda}{c_{0,1}}\\\ \leavevmode\nobreak\ \\\
\frac{c_{0,1}\left(\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}\frac{z}{\lambda\,\nu}-1\right)}{1-z},&\text{if
}\frac{\nu\lambda}{c_{0,1}}<z<\infty\end{cases}$

is decreasing in $z\in(0,\infty)$, provided
$\nu\geq\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}$. Since
$c_{0,2}=\frac{\alpha_{2}+2}{\alpha_{2}+1}>\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}$,
a suitable choice of $\nu$ in this case is
$\nu=\frac{\alpha_{2}+2}{\alpha_{2}+1}$. To avoid some tedious algebra, we
only consider the case $\alpha_{1}\geq\alpha_{2}$. For
$\alpha_{1}\geq\alpha_{2}$ and $\nu=\frac{\alpha_{2}+2}{\alpha_{2}+1}$, it is
easy to verify that

$\alpha_{1}(\lambda)=\alpha_{1,1}=\alpha_{1,\infty}=1-\frac{\alpha_{2}(\alpha_{2}+2)}{2(\alpha_{1}+2)(\alpha_{1}+\alpha_{2}+1)}=\alpha_{0},\;\;(say).$

Using Theorem 4.1.1 (b), we conclude that among the isotonic regression
estimators in the class
$\mathcal{D}_{1,\nu}=\\{\delta_{1,c_{0,1},\alpha}^{(\nu)}:-\infty<\alpha<\infty\\}$
the only admissible estimator is

$\displaystyle\delta_{1,c_{0,1},\alpha_{0}}^{(\nu)}(\underline{X})=\begin{cases}\frac{\alpha_{1}+2}{\alpha_{1}+1}X_{1},&\text{
if
}\frac{\alpha_{1}+2}{\alpha_{1}+1}X_{1}\leq\frac{\alpha_{2}+2}{\alpha_{2}+1}X_{2}\\\
\alpha_{0}\,\frac{\alpha_{1}+2}{\alpha_{1}+1}X_{1}+(1-\alpha_{0})\frac{\alpha_{2}+2}{\alpha_{2}+1}X_{2},&\text{
if
}\frac{\alpha_{1}+2}{\alpha_{1}+1}X_{1}>\frac{\alpha_{2}+2}{\alpha_{2}+1}X_{2}\end{cases};$

here
$\alpha_{0}=1-\frac{\alpha_{2}(\alpha_{2}+2)}{2(\alpha_{1}+2)(\alpha_{1}+\alpha_{2}+1)}$,
$\nu=\frac{\alpha_{2}+2}{\alpha_{2}+1}$ and
$c_{0,1}=\frac{\alpha_{1}+2}{\alpha_{1}+1}$. All other estimators in the class
$\mathcal{D}_{1,\nu}$, including the BSEE
$\delta_{1,c_{0,1}}(\underline{X})=\delta_{1,c_{0,1},1}^{(\nu)}(\underline{X})=\frac{\alpha_{1}+2}{\alpha_{1}+1}X_{1}$,
are inadmissible for estimating $\theta_{1}$ and are dominated by the isotonic
regression estimator $\delta_{1,c_{0,1},\alpha_{0}}^{(\nu)}(\underline{X})$,
defined above.

###  4.2. Isotonic Regression Estimators of Scale Parameter $\theta_{2}$

Let $\beta$ be a fixed real constant, to be suitably chosen as described
earlier. Under the notations of Section 4.1, consider the class
$\mathcal{D}_{2,\beta}=\\{\delta_{2,c_{0,2},\alpha}^{(\beta)}:-\infty<\alpha<\infty\\}$
of isotonic regression estimators defined by (1.8) and (1.10); here
$\delta_{2,c_{0,2}}(\underline{X})=c_{0,2}X_{2}$ and
$\delta_{1,\beta}(\underline{X})=\beta X_{1}$. For $z\in S_{Z}$, let
$\psi_{3}(z)=E[Z_{2}|Z=z]=z\psi_{1}(z),$
$\psi_{4}(z)=E[Z_{2}^{{}^{2}}|Z=z]=z^{2}\psi_{2}(z)$ and
$\psi^{*}(z)=\frac{\psi_{3}(z)}{\psi_{4}(z)}$. For any fixed
$\underline{\theta}\in\Theta_{0}$ (or $\lambda\geq 1$), the risk function

$\displaystyle R_{2}(\underline{\theta},\delta_{2,c_{0,2},\alpha}^{(\beta)})$
$\displaystyle=E_{\underline{\theta}}\left[\left(\frac{\delta_{2,c_{0,2},\alpha}^{(\beta)}(\underline{X})}{\theta_{2}}-1\right)^{2}\right]$
$\displaystyle=E_{\underline{\theta}}[(c_{0,2}Z_{2}-1)^{2}I_{(\frac{\beta}{\lambda
c_{0,2}},\infty)}(Z)]$
$\displaystyle\qquad+E_{\underline{\theta}}\left[\left(\frac{\alpha\beta}{\lambda}Z_{1}+(1-\alpha)c_{0,2}Z_{2}-1\right)^{2}I_{(0,\frac{\beta}{\lambda
c_{0,2}})}(Z)\right]$

is minimized at $\alpha=\alpha_{2}(\lambda)$, where

$\displaystyle\alpha_{2}(\lambda)$
$\displaystyle=\frac{1}{c_{0,2}}\frac{E\left[\left(\frac{\beta}{\lambda\,c_{0,2}Z}-1\right)(\psi^{*}(Z)-c_{0,2})\,\psi_{4}(Z)I_{(0,\frac{\beta}{\lambda
c_{0,2}})}(Z)\right]}{E\left[\left(\frac{\beta}{\lambda\,c_{0,2}Z}-1\right)^{2}\psi_{4}(Z)I_{(0,\frac{\beta}{\lambda
c_{0,2}})}(Z)\right]}$ (4.3) $\displaystyle=$
$\displaystyle\frac{1}{c_{0,2}}\frac{\int_{0}^{1}\left(\frac{1}{t}-1\right)(\psi^{*}\left(\frac{\beta
t}{\lambda c_{0,2}}\right)-c_{0,2})\,\psi_{4}\left(\frac{\beta t}{\lambda
c_{0,2}}\right)\,f_{Z}\left(\frac{\beta t}{\lambda
c_{0,2}}\right)dt}{\int_{0}^{1}\left(\frac{1}{t}-1\right)^{2}\,\psi_{4}\left(\frac{\beta
t}{\lambda c_{0,2}}\right)\,f_{Z}\left(\frac{\beta t}{\lambda
c_{0,2}}\right)dt},\;\;\lambda\geq 1,$ $\displaystyle=$
$\displaystyle\frac{1}{c_{0,2}}E_{\lambda}[k_{2}(S_{2,\lambda},\lambda)],\;\;\lambda\geq
1,$

where $k_{2}(z,\lambda)=\frac{\psi^{*}\left(\frac{\beta
z}{\lambda\,c_{0,2}}\right)-c_{0,2}}{\frac{1}{z}-1},\;0<z<1,\;\lambda\geq 1,$
and $S_{2,\lambda}$ is a r.v. having the pdf

$h_{2,\lambda}(t)=\begin{cases}\frac{\left(\frac{1}{t}-1\right)^{2}\,\psi_{4}\left(\frac{\beta
t}{\lambda c_{0,2}}\right)\,f_{Z}\left(\frac{\beta t}{\lambda
c_{0,2}}\right)}{\int_{0}^{1}\left(\frac{1}{s}-1\right)^{2}\,\psi_{4}\left(\frac{\beta
s}{\lambda c_{0,2}}\right)\,f_{Z}\left(\frac{\beta s}{\lambda
c_{0,2}}\right)ds},&\text{if}\;\;0<t<1\\\
0,&\text{otherwise}\end{cases},\;\;\lambda\geq 1.$

It is easy to verify that if, for every fixed $\theta\in(0,1)$,
$\frac{\psi_{4}(\theta t)}{\psi_{4}(t)}$ and $\frac{f_{Z}(\theta
t)}{f_{Z}(t)}$ are increasing (decreasing) functions of t on
$(0,\frac{\beta}{c_{0,2}})$, then
$S_{2,\lambda_{1}}\leq_{lr}S_{2,\lambda_{2}}$
($S_{2,\lambda_{2}}\leq_{lr}S_{2,\lambda_{1}}$) and, consequently,
$S_{2,\lambda_{1}}\leq_{st}S_{2,\lambda_{2}}$
($S_{2,\lambda_{2}}\leq_{st}S_{2,\lambda_{1}}$), whenever
$1\leq\lambda_{1}<\lambda_{2}<\infty$.

On the lines of Lemma 3.1.1, we have the following result.

Lemma 4.2.1 (a) Suppose that, for every fixed $\theta\in(0,1)$,
$\frac{\psi_{4}(\theta t)}{\psi_{4}(t)}$ and $\frac{f_{Z}(\theta
t)}{f_{Z}(t)}$ are increasing (decreasing) functions of $t$ on
$(0,\frac{\beta}{c_{0,2}})$. Also, suppose that $\psi^{*}(z)$ is decreasing on
$(0,\frac{\beta}{c_{0,2}})$ and, for every fixed $\lambda\geq 1$,
$k_{2}(z,\lambda)=\frac{\psi^{*}\left(\frac{\beta
z}{\lambda\,c_{0,2}}\right)-c_{0,2}}{\frac{1}{z}-1}$ is increasing
(decreasing) in $z\in(0,1)$. Then $\alpha_{2}(\lambda)$, defined by (4.3), is
an increasing function of $\lambda\in[1,\infty)$, $\inf_{\lambda\geq
1}\alpha_{2}(\lambda)=\alpha_{2}(1)=\alpha_{2,1},\text{ say, and
}\sup_{\lambda\geq
1}\alpha_{2}(\lambda)=\lim_{\lambda\to\infty}\alpha_{2}(\lambda)=\alpha_{2,\infty},\text{
say}.$

(b) Suppose that, for every fixed $\theta\in(0,1)$, $\frac{\psi_{4}(\theta
t)}{\psi_{4}(t)}$ and $\frac{f_{Z}(\theta t)}{f_{Z}(t)}$ are increasing
(decreasing) functions of $t$ on $(0,\frac{\beta}{c_{0,2}})$. Also, suppose
that $\psi^{*}(z)$ is increasing on $(0,\frac{\beta}{c_{0,2}})$ and, for every
fixed $\lambda\geq 1$, $k_{2}(z,\lambda)=\frac{\psi^{*}\left(\frac{\beta
z}{\lambda\,c_{0,2}}\right)-c_{0,2}}{\frac{1}{z}-1}$ is decreasing
(increasing) in $z\in(0,1)$. Then $\alpha_{2}(\lambda)$ is a decreasing
function of $\lambda\in[1,\infty)$, $\inf_{\lambda\geq
1}\alpha_{2}(\lambda)=\lim_{\lambda\to\infty}\alpha_{2}(\lambda)=\alpha_{2,\infty},\text{
say, and }\sup_{\lambda\geq
1}\alpha_{2}(\lambda)=\alpha_{2}(1)=\alpha_{2,1}\text{, say}.$

Now we present the main result of this subsection, whose proof is similar to
the proof of Theorem 3.1.1.

Theorem 4.2.1 (a) Suppose that assumptions of Lemma 4.2.1 (a) hold. Then the
estimators that are admissible in the class $\mathcal{D}_{2,\beta}$ are
$\\{\delta_{2,c_{0,2},\alpha}^{(\beta)}:\alpha\in[\alpha_{2,1},\alpha_{2,\infty}]\\}$.
Moreover, for $-\infty<\alpha_{1}<\alpha_{2}\leq\alpha_{2,1}$ or
$\alpha_{2,\infty}\leq\alpha_{2}<\alpha_{1}$, the estimator
$\delta_{2,c_{0,2},\alpha_{2}}^{(\beta)}$ dominates the estimator
$\delta_{2,c_{0,2},\alpha_{1}}^{(\beta)}$, for any
$\underline{\theta}\in\Theta_{0}$.

(b) Suppose that assumptions of Lemma 4.2.1 (b) hold. Then the estimators that
are admissible in the class $\mathcal{D}_{2,\beta}$ are
$\\{\delta_{2,c_{0,2},\alpha}^{(\beta)}:\alpha\in[\alpha_{2,\infty},\alpha_{2,1}]\\}$.
Moreover, for $-\infty<\alpha_{1}<\alpha_{2}\leq\alpha_{2,\infty}$ or
$\alpha_{2,1}\leq\alpha_{2}<\alpha_{1}<\infty$, the estimator
$\delta_{2,c_{0,2},\alpha_{2}}^{(\beta)}$ dominates the estimator
$\delta_{2,c_{0,2},\alpha_{1}}^{(\beta)}$, for any
$\underline{\theta}\in\Theta_{0}$.

Now we illustrate some applications of Theorem 4.2.1.

Example 4.2.1. Let $(X_{1},X_{2})$ be a random vector as defined in Example
4.1.1. Here
$c_{0,i}=\frac{E[Z_{i}]}{E[Z_{i}^{2}]}=\frac{1}{\alpha_{i}+1},\;i=1,2,$ the
BSEE of $\theta_{2}$ is
$\delta_{2,0}(\underline{X})=\delta_{2,c_{0,2},0}^{(\beta)}(\underline{X})=\frac{X_{2}}{\alpha_{2}+1}$,
$\psi_{3}(z)=E[Z_{2}|Z=z]=z\,E[Z_{1}|Z=z]=(\alpha_{1}+\alpha_{2})\frac{z}{1+z},\;z>0,\;\psi_{4}(z)=E[Z_{2}^{{}^{2}}|Z=z]=z^{2}\,E[Z_{1}^{{}^{2}}|Z=z]=(\alpha_{1}+\alpha_{2}+1)(\alpha_{1}+\alpha_{2})\frac{z^{2}}{(1+z)^{2}},\;z>0,$
and
$\psi^{*}(z)=\frac{\psi_{3}(z)}{\psi_{4}(z)}=\frac{1+z}{(\alpha_{1}+\alpha_{2}+1)z}$.
Clearly, for $0<\theta<1$, $\frac{f_{Z}(\theta z)}{f_{Z}(z)}$ and
$\frac{\psi_{2}(\theta z)}{\psi_{2}(z)}=\frac{\theta^{2}(1+z)^{2}}{(1+\theta
z)^{2}}$ are increasing in $z\in\left[0,\frac{\beta}{c_{0,2}}\right]$ and
$\psi^{*}(z)=\frac{\psi_{3}(z)}{\psi_{4}(z)}=\frac{1+z}{(\alpha_{1}+\alpha_{2}+1)z}$
is decreasing in $z\in\left[0,\frac{\beta}{c_{0,2}}\right]$. For any fixed
$\lambda\geq 1$, $k_{2}(z,\lambda)=\frac{\psi^{*}\left(\frac{\beta
z}{\lambda\,c_{0,2}}\right)-c_{0,2}}{\frac{1}{z}-1}=\frac{\lambda-\alpha_{1}\beta
z}{(\alpha_{1}+\alpha_{2}+1)(\alpha_{2}+1)\beta(1-z)},\;z>0$ is increasing in
$z\in[0,1]$, provided $\beta\leq\frac{1}{\alpha_{1}}$. Since
$c_{0,1}=\frac{1}{\alpha_{1}+1}$, a suitable choice of $\beta$ in this case is
$\beta=\frac{1}{\alpha_{1}+1}$. For $\beta=\frac{1}{\alpha_{1}+1}$, it is easy
to verify that

$\displaystyle\alpha_{2,1}$ $\displaystyle=\alpha_{2}(1)$
$\displaystyle=\frac{1}{c_{0,2}}\frac{\int_{0}^{1}\left(\frac{1}{t}-1\right)\left(\psi^{*}\left(\frac{\beta
t}{c_{0,2}}\right)-c_{0,2}\right)\psi_{4}\left(\frac{\beta
t}{c_{0,2}}\right)f_{Z}\left(\frac{\beta
t}{c_{0,2}}\right)dt}{\int_{0}^{1}\left(\frac{1}{t}-1\right)^{2}\,\psi_{4}\left(\frac{\beta
t}{c_{0,2}}\right)f_{Z}\left(\frac{\beta t}{c_{0,2}}\right)dt}$
$\displaystyle=\frac{1}{\alpha_{1}+\alpha_{2}+1}\left(\alpha_{1}+1+\frac{\int_{0}^{1}\left(\frac{1}{t}-1\right)\frac{1}{\left(1+\frac{\alpha_{1}+1}{(\alpha_{2}+1)t}\right)^{\alpha_{1}+\alpha_{2}+2}}\frac{1}{t^{\alpha_{1}+1}}dt}{\int_{0}^{1}\left(\frac{1}{t}-1\right)^{2}\frac{1}{\left(1+\frac{\alpha_{1}+1}{(\alpha_{2}+1)t}\right)^{\alpha_{1}+\alpha_{2}+2}}\frac{1}{t^{\alpha_{1}+1}}dt}\right)$
$\displaystyle\text{and}\qquad\alpha_{2,\infty}=\lim_{\lambda\to\infty}\alpha_{2}(\lambda)=\infty.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

Using Theorem 4.2.1. (b), it follows that the estimators
$\\{\delta_{2,c_{0,2},\alpha}^{(\beta)}:\alpha\in[\alpha_{2,1},\infty)\\}$ are
admissible within the class
$\mathcal{D}_{2,\beta}=\\{\delta_{2,c_{0,2},\alpha}^{(\beta)}:-\infty<\alpha<\infty\\}$
of isotonic regression estimators of $\theta_{2}$; here
$\beta=\frac{1}{\alpha_{1}+1}$ and $c_{0,2}=\frac{1}{\alpha_{2}+1}$. Moreover,
for $-\infty<\alpha_{1}<\alpha_{2}\leq\alpha_{2,1}$, the estimator
$\delta_{2,c_{0,2},\alpha_{2}}^{(\beta)}$ dominates the estimator
$\delta_{2,c_{0,2},\alpha_{1}}^{(\beta)}$. Since
$\alpha_{2,1}>\frac{\alpha_{1}+1}{\alpha_{1}+\alpha_{2}+1}>\frac{\alpha_{1}}{\alpha_{1}+\alpha_{2}}$,
it follows that the BSEE
$\delta_{2,c_{0,2}}(\underline{X})=\delta_{2,c_{0,2},0}^{(\beta)}(\underline{X})=\frac{X_{2}}{\alpha_{2}+1}$
is inadmissible for estimating $\theta_{2}$ and is dominated by

$\displaystyle\delta_{2,c_{0,2},\beta_{0}}^{(\beta)}(\underline{X})$
$\displaystyle=\begin{cases}\frac{X_{2}}{\alpha_{2}+1},&\text{ if
}\frac{X_{1}}{\alpha_{1}+1}\leq\frac{X_{2}}{\alpha_{2}+1}\\\
\alpha_{2,1}\frac{X_{1}}{\alpha_{1}+1}+(1-\alpha_{2,1})\frac{X_{2}}{\alpha_{2}+1},&\text{
if }\frac{X_{1}}{\alpha_{1}+1}>\frac{X_{2}}{\alpha_{2}+1}\end{cases}$
$\displaystyle=\max\bigg{\\{}\frac{X_{2}}{\alpha_{2}+1},\alpha_{2,1}\frac{X_{1}}{\alpha_{1}+1}+(1-\alpha_{2,1})\frac{X_{2}}{\alpha_{2}+1}\bigg{\\}}.$

Example 4.2.2. Let $X_{1}$ and $X_{2}$ be independent random variables, as
defined in Example 4.1.2. Here
$c_{0,i}=\frac{E[Z_{i}]}{E[Z_{i}^{2}]}=\frac{\alpha_{i}+2}{\alpha_{i}+1},\;i=1,2,$
and, for any fixed $\beta>0$, the BSEE of $\theta_{2}$ is
$\delta_{2,0}(\underline{X})=\delta_{2,c_{0,2},0}^{(\beta)}(\underline{X})=\frac{\alpha_{2}+2}{\alpha_{2}+1}X_{2}$.
$\text{Moreover,
}\psi_{3}(z)=E[Z_{2}|Z=z]=zE[Z_{1}|Z=z]=\frac{\alpha_{1}+\alpha_{2}}{\alpha_{1}+\alpha_{2}+1}\min\\{z,1\\},\,z>0,\;\psi_{4}(z)=E[Z_{2}^{2}|Z=z]=z^{2}E[Z_{1}^{2}|Z=z]=\frac{\alpha_{1}+\alpha_{2}}{\alpha_{1}+\alpha_{2}+2}\left(\min\\{z,1\\}\right)^{2},\,z>0,\text{
and, for }0<\theta<1$,

$\frac{\psi_{4}(\theta z)}{\psi_{4}(z)}=\frac{\left(\min\\{\theta
z,1\\}\right)^{2}}{\left(\min\\{z,1\\}\right)^{2}}=\begin{cases}\theta^{2},&\text{if
}\;0<z<1\\\ \theta^{2}z^{2},&\text{if }\;1\leq z<\frac{1}{\theta}\\\
1,&\text{if }\;\frac{1}{\theta}\leq z<\infty\end{cases}$

is increasing in $z\in\left[0,\frac{\beta}{c_{0,2}}\right]$, where $\beta>0$
is fixed. Also,
$\psi^{*}(z)=\frac{\psi_{3}(z)}{\psi_{4}(z)}=\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}\,\frac{1}{\min\\{z,1\\}}$
is decreasing in $z\in\left[0,\frac{\beta}{c_{0,2}}\right]$. For any fixed
$\lambda\geq 1$,

$\displaystyle k_{2}(z,\lambda)$ $\displaystyle=\frac{\psi\left(\frac{\beta
z}{c_{0,2}\,\lambda}\right)-c_{0,2}}{\frac{1}{z}-1}=\begin{cases}\frac{\alpha_{2}+2}{\alpha_{2}+1}\frac{\left(\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}\frac{\lambda}{\beta}-z\right)}{1-z},&\text{if
}0<z<\frac{\lambda\,c_{0,2}}{\beta}\\\ \leavevmode\nobreak\ \\\
\frac{-\alpha_{1}}{(\alpha_{2}+1)(\alpha_{1}+\alpha_{2}+1)\left(\frac{1}{z}-1\right)},&\text{if
}\frac{\lambda\,c_{0,2}}{\beta}<z<\infty\end{cases}$

is increasing in $z\in[0,1]$, provided
$\beta\leq\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}$. Since
$c_{0,1}=\frac{\alpha_{1}+2}{\alpha_{1}+1}>\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}$,
an appropriate choice of $\beta$ in this case is
$\beta=\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}$. For
$\beta=\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}$, it is easy to
verify that

$\alpha_{2}(\lambda)=\frac{\int_{0}^{1}(1-t)(\lambda-t)t^{\alpha_{2}-1}dt}{\int_{0}^{1}(1-t)^{2}\,t^{\alpha_{2}-1}dt},\;\lambda\geq
0,$
$\alpha_{2,1}=\alpha_{2}(1)=1\qquad\text{and}\qquad\alpha_{2,\infty}=\lim_{\lambda\to\infty}\alpha_{2}(\lambda)=\infty.$

Using Theorem 4.2.1. (a), it follows that the estimators
$\\{\delta_{2,c_{0,2},\alpha}^{(\beta)}:\alpha\in[1,\infty)\\}$ are admissible
within the class
$\mathcal{D}_{2,\beta}=\\{\delta_{2,c_{0,2},\alpha}^{(\beta)}:-\infty<\alpha<\infty\\}$
of isotonic regression estimators of $\theta_{2}$; here
$\beta=\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}$ and
$c_{0,2}=\frac{\alpha_{2}+2}{\alpha_{2}+1}$. Moreover, for
$-\infty<\alpha_{1}<\alpha_{2}\leq 1$, the estimator
$\delta_{2,c_{0,2},\alpha_{2}}^{(\beta)}$ dominates the estimator
$\delta_{2,c_{0,2},\alpha_{1}}^{(\beta)}$. In particular the BSEE
$\delta_{2,c_{0,2}}(\underline{X})=\delta_{2,c_{0,2},0}^{(\beta)}(\underline{X})=\frac{\alpha_{2}+2}{\alpha_{2}+1}X_{2}$
is inadmissible for estimating $\theta_{2}$ and is dominated by the isotonic
regression estimator

$\displaystyle\delta_{2,c_{0,2},1}^{(\beta)}(\underline{X})$
$\displaystyle=\begin{cases}\frac{\alpha_{2}+2}{\alpha_{2}+1}X_{2},&\text{ if
}\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}X_{1}\leq\frac{\alpha_{2}+2}{\alpha_{2}+1}X_{2}\\\
\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}X_{1},&\text{ if
}\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}X_{1}>\frac{\alpha_{2}+2}{\alpha_{2}+1}X_{2}\end{cases}=\max\bigg{\\{}\frac{\alpha_{2}+2}{\alpha_{2}+1}X_{2},\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}X_{1}\bigg{\\}}.$

This is another situation where the isotonic regression estimator based on the
BSEE
$(\frac{\alpha_{1}+2}{\alpha_{1}+1}X_{1},\frac{\alpha_{2}+2}{\alpha_{2}+1}X_{2})$
may not be able to provide improvement over the BSEE
$\delta_{2,c_{0,2}}(\underline{X})=\frac{\alpha_{2}+2}{\alpha_{2}+1}X_{2}$.
Rather an isotonic regression estimator based on
$(\frac{\alpha_{1}+\alpha_{2}+2}{\alpha_{1}+\alpha_{2}+1}X_{1},\frac{\alpha_{2}+2}{\alpha_{2}+1}X_{2})$
provides improvement over the BSEE $\delta_{2,c_{0,2}}(\underline{X})$.

###  4.3. Simulation Study For Estimation of Scale Parameter $\theta_{1}$

In Example 4.1.1, we have considered two independent gamma distributions with
unknown order restricted scale parameters (i.e., $\theta_{1}\leq\theta_{2}$)
and known shape parameters ($\alpha_{1}>0$ and $\alpha_{2}>0$). For estimation
of smaller scale parameter $\theta_{1}$, under the scaled squared error loss
function, we showed that the IRE
$\min\\{\frac{X_{1}}{\alpha_{1}},\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}+1}\\}$,
based on ($\frac{X_{1}}{\alpha_{1}+1},\frac{X_{2}}{\alpha_{2}}$) with
$\alpha=\beta_{0}=\frac{\alpha_{1}+1}{\alpha_{1}+\alpha_{2}+1}$, dominates the
BLEE $\frac{X_{1}}{\alpha_{1}+1}$. To further evaluate the performances of
various estimators, under the scaled squared error loss function, in this
section, we compare the risk performances of the BSEE
$\frac{X_{1}}{\alpha_{1}+1}$, the IRE based on the BSEEs
($\frac{X_{1}}{\alpha_{1}+1},\frac{X_{2}}{\alpha_{2}+1}$) with
$\alpha=\frac{\alpha_{1}}{\alpha_{1}+\alpha_{2}}$, and the IRE based on
($\frac{X_{1}}{\alpha_{1}+1},\frac{X_{2}}{\alpha_{2}}$) with
$\alpha=\frac{\alpha_{1}+1}{\alpha_{1}+\alpha_{2}+1}$, numerically, through
Monte Carlo simulations. For simulations, we have generated 10000 sample of
size 1 each from relevant gamma distributions, for different values of known
shape parameters ($\alpha_{1}$ and $\alpha_{2}$), and computed the simulated
risks of the BSEE, the IRE based on
($\frac{X_{1}}{\alpha_{1}+1},\frac{X_{2}}{\alpha_{2}+1}$) with
$\alpha=\frac{\alpha_{1}}{\alpha_{1}+\alpha_{2}}$, and the IRE based on
($\frac{X_{1}}{\alpha_{1}+1},\frac{X_{2}}{\alpha_{2}}$) with
$\alpha=\frac{\alpha_{1}+1}{\alpha_{1}+\alpha_{2}+1}$. The simulated values of
risks of various estimators are plotted in Figure 2. The following
observations are evident from Figure 2:

(i) The risk function values of the IRE based on
($\frac{X_{1}}{\alpha_{1}+1},\frac{X_{2}}{\alpha_{2}}$) with
$\alpha=\frac{\alpha_{1}+1}{\alpha_{1}+\alpha_{2}+1}$ is nowhere larger than
the risk function values of the BSEE $\frac{X_{1}}{\alpha+1}$, which is in
conformity with theoretical findings of Example 4.1.3.
(ii) As we mentioned in the Example 4.1.1, here isotonic regression estimators
(IREs) based on the BSEEs
($\frac{X_{1}}{\alpha_{1}+1},\frac{X_{2}}{\alpha_{2}+1}$) does not always
provide improvements over the BSEE $\frac{X_{1}}{\alpha_{1}+1}$.
(iii) There is no clear cut winner between the the IRE based on
($\frac{X_{1}}{\alpha_{1}+1},\frac{X_{2}}{\alpha_{2}+1}$) with
$\alpha=\frac{\alpha_{1}}{\alpha_{1}+\alpha_{2}}$ and the IRE based on
($\frac{X_{1}}{\alpha_{1}+1},\frac{X_{2}}{\alpha_{2}}$) with
$\alpha=\frac{\alpha_{1}+1}{\alpha_{1}+\alpha_{2}+1}$. But the IRE based on
($\frac{X_{1}}{\alpha_{1}+1},\frac{X_{2}}{\alpha_{2}}$) with
$\alpha=\frac{\alpha_{1}+1}{\alpha_{1}+\alpha_{2}+1}$ performs reasonably well
in comparison to other estimators.

(a) $\alpha_{1}=0.1$ and $\alpha_{2}=0.2$.

(b) $\alpha_{1}=0.8$ and $\alpha_{2}=0.5$.

(c) $\alpha_{1}=1$ and $\alpha_{2}=1$.

(d) $\alpha_{1}=2$ and $\alpha_{2}=1$.

Figure 2. Risk plots of the BSEE $c_{0,1}X_{1}$, the IRE based on
$(c_{0,1}X_{1},c_{0,2}X_{2})$ with
$\alpha=\frac{\alpha_{1}}{\alpha_{1}+\alpha_{2}}$, and the IRE based on
$(c_{0,1}X_{1},\nu X_{2})$ with $\alpha=\beta_{0}$, where
$\nu=\frac{1}{\alpha_{2}}$ and $c_{0,i}=\frac{1}{\alpha_{i}+1},\;i=1,2$,
against the values of $\lambda=\frac{\theta_{2}}{\theta_{1}}$.

## Funding

This work was supported by the [Council of Scientific and Industrial Research
(CSIR)] under Grant [number 09/092(0986)/2018].

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