# Maximum $\log_{q}$ Likelihood Estimation for Parameters of Weibull
Distribution and Properties: Monte Carlo Simulation

Mehmet Niyazi Çankayaa1,a2 Roberto Vilaa3 a1 Faculty of Applied Sciences,
Department of International Trading and Finance, Uşak University,  a2 Faculty
of Art and Sciences, Department of Statistics, Uşak University, Uşak, Turkey
a3 Departamento de Estatística, Universidade de Brasília, Brazil

###### Abstract

Abstract

The maximum ${\log}_{q}$ likelihood estimation method is a generalization of
the known maximum $\log$ likelihood method to overcome the problem for
modeling non-identical observations (inliers and outliers). The parameter $q$
is a tuning constant to manage the modeling capability. Weibull is a flexible
and popular distribution for problems in engineering. In this study, this
method is used to estimate the parameters of Weibull distribution when non-
identical observations exist. Since the main idea is based on modeling
capability of objective function
$\rho(x;\boldsymbol{\theta})=\log_{q}\big{[}f(x;\boldsymbol{\theta})\big{]}$,
we observe that the finiteness of score functions cannot play a role in the
robust estimation for inliers. The properties of Weibull distribution are
examined. In the numerical experiment, the parameters of Weibull distribution
are estimated by $\log_{q}$ and its special form, $\log$, likelihood methods
if the different designs of contamination into underlying Weibull distribution
are applied. The optimization is performed via genetic algorithm. The modeling
competence of $\rho(x;\boldsymbol{\theta})$ and insensitiveness to non-
identical observations are observed by Monte Carlo simulation. The value of
$q$ can be chosen by use of the mean squared error in simulation and the
$p$-value of Kolmogorov-Smirnov test statistic used for evaluation of fitting
competence. Thus, we can overcome the problem about determining of the value
of $q$ for real data sets.

Mathematics Subject Classification. 62C05; 62E10; 62F10.
Keywords. Weibull distribution; inference; $q$-deformed logarithm; robustness.

##  I Introduction

After the study of a real-world phenomenon or the realization of an
experiment, it may be desirable to model the experimental data by means of a
proposed parametric model $f(x;\boldsymbol{\theta})$. In other words, the
experimental data set is assumed to be a member of a parametric model.
However, it cannot be a realistic assumption for the real world which will be
modeled only with certain values of parameters in a model. The observations
can be mixed with a different parameter of the same distribution or a
different distribution. That is, a contamination exists into majority of the
distribution, which leads to have non-identical observations. The type of
contamination is defined as deviant observations. In other words, one or more
observations are made to behave differently from what it is present when
creating a deviant observation. Deviant observations can be divided into
inlier and outlier deviations. The data set may have both inward/inlier and
outward/outlier deviations at the same time. Inward deviations are generated
by short-tailed distributions and outward deviations by thick-tailed
distributions. For inward deviations, it can also be realized by generating
random numbers from uniform distribution in the closed range $[a,b]$. In fact,
deviant observations in the data set indicate that an assumption trusting on
identically distributed random variables will not be realistic to model a
phenomenia (tikuinlier ; LehmannCas98 ). If the assumption showing that the
data set includes identically distributed random observations is violated,
robust estimation method for the parameters of model
$f(x;\boldsymbol{\theta})$ have been applied by use of the different objective
functions. Robust methods trust the used objective function (Hampeletal86 ).
Deformed algebras such as Tsallis and Kaniakadis statistics are important to
derive a neighborhood of a parametric model in order to overcome the problem
which will occur when the assumption for ideniticality is violated
(Wadatwopara ; Bercher10 ; Bercher12a ).

The origin of robust estimation method was started by biologist at 18th
century (Hampeletal86 ). The main working principle is based on the estimating
equations (EEs) (God60 ; GodTh78 ). EEs can be derived by use of maximum
composite likelihood estimation method. Tsallis $q$-entropy is creator of
deformed logarithm ($\log_{q}$). In the direction of estimation method,
$\log_{q}$ from Tsallis $q$-entropy has been studied recently. For this aim,
maximum $\log_{q}$ likelihood estimation (MLqE) method are studied for outward
observations. MLqE which is a generalization of the maximum likelihood (MLE)
method LehmannCas98  is used to obtain robust and also efficient estimators
(Tsallisbook09 ; FerrariYang10 ). Different deformed logarithms can be
obtained from entropy functions (Wadatwopara ). However, the deformed or the
generalized logarithms should map the probability density function
$f(x_{i};\boldsymbol{\theta})$ as one-to-one and overlay (Lindsay94 ; CanKor18
). Kaniakadis’ deformed logarithm (${\log}_{\kappa{}}$) can have same property
with ${\log}_{q}$ due to fact that $(\alpha,\beta)$-difference operator in
fractional calculus (FC) is used to generate entropies. The other genaralized
entropies can have same role in the estimation procedure (Wadatwopara ;
Jansent ). Note that the performance of efficiency can be managed by using the
generalized entropies and generalized logarithms from FC. MLqE is simple and
its computational implementation is not heavy. No extra condition is requried
to apply MLqE for estimation when it is compared with divergences
(alphabetadivergences ). The parameters of Gamma distribution are estimated by
MLqE method (MLqEGamma ). As an another approach for the robust estimation,
divergences have been used. Minimization of entropies and divergences are
equivalent to maximization of the generalized maximum likelihood estimation
method (Lindsay94 ; PardoSD ).

FC has been started to play an important role in the estimations of parameters
of a probability density (p.d.) function $f(x;\boldsymbol{\theta}).$ Tsallis
distributions ($q$-distributions) have encountered a large success because of
their remarkable agreement with experimental data. The parameter $q$ behaves
as a microscope and generates neigborhoods of p.d. function
$f(x;\boldsymbol{\theta})$. Thus, the different behaviour of p.d. function
$f(x;\boldsymbol{\theta})$ can be explored by use of ${\log}_{q}$ (see
Bercher10 ; Bercher12a  and references therein). After we use $\log_{q}$ from
Tsallis $q$-entropy to estimate robustly the parameters $\boldsymbol{\theta}$
of underlying distribution $f(x;\boldsymbol{\theta})$, that is, we will have
estimators which are not affected by inliers and outliers in a data set, the
optimization of
$\sum_{i=1}^{n}\log_{q}\big{[}f(x_{i};\boldsymbol{\theta})\big{]}$ according
to the parameters $\boldsymbol{\theta}$ is an another challeging problem for
the estimation process if we have a nonlinear function. The optimization is
performed by means of genetic algorithm (GA) which is not attached to local
points. Thus, we will have estimators. GA mimicing evolutionary biology as a
stochasticity is a derivative-independent method. We use hybrid method in ‘ga’
module in MATLAB2013a to decrease the computational error as well. In hybrid
method, GA and derivative-based methods work simultenaously. Further, GA has
an effective attitude to reach the global point in the fuction
$\rho{}(x;\boldsymbol{\theta})=\Lambda[f(x;\boldsymbol{\theta})]$ (introga ).

The aim is to estimate robustly shape and scale parameters of Weibull
distribution when the different types of contamination into artifical data set
are added. The properties of Weibull distribution are examined extensively.
The roles of $\Lambda{}$ from the deformed algebras and DPD, etc. as an
objective function $\rho{}$, are observed easily by using the illustrative
representations (see Fig. 1), which is a key to analyze their roles on p.d.
function $f(x;\boldsymbol{\theta})$. Outliers and inliers are added into the
artificial data sets at the simulation and the different types of
contaminations to real data sets are applied to observe the robustness and the
modeling capability of
$\rho(x;\boldsymbol{\theta})=\log_{q}\big{[}f(x;\boldsymbol{\theta})\big{]}$.
The value of parameter $q$ is determined according to the $p$-value of KS test
statistic of estimates of parameters for Weibull distribution. Fisher
information based on ${\log}_{q}$ is used to evaluate the variance-covariance
matrix of estimators derived by MLqE.

The organization of study is as follows. Section II includes Weibull
distribution and its properties which indicate that Weibull can be used for
the modeling fruitfully. We provide the essential tools to pass the modeling
sketch of estimation procedure such as convexity, (concavity), entropies, etc
Haberman  in the M-estimation (Hub81 ). We also provide main tools to get the
elements of Fisher information matrix. Section III introduces estimation
methods varincomposite  and estimating equations (God60 ; GodTh78 ). Section
IV introduces the tools and robustness to outliers and also we propose a tool
based on score function for robustness to inliers. Section V provides the tool
used for optimization and numerical experiments. The last section VI is given
for conclusions.

##  II Weibull Distribution

Weibull distribution is chosen because it has many applications in the field
of the applied science. Further, it has many features such as existence of
cumulative distribution (c.d.) function, moments and entropies, etc
(Malikcondition ; Cankaya2018 ). We say that a non-negative random variable
$X$ has a Weibull distribution with vector parameter
$\boldsymbol{\theta}=(\alpha,\beta)$, denoted by
$X\sim\text{Weibull}(\boldsymbol{\theta})$, if its probability density
function is given by

$\displaystyle
f(x;\boldsymbol{\theta})={\alpha\over\beta}\,\bigg{(}{x\over\beta}\bigg{)}^{\alpha-1}\exp\Bigg{[}-\bigg{(}{x\over\beta}\bigg{)}^{\alpha}\Bigg{]},\quad
x\geqslant 0;\ \alpha,\beta>0,$ (1)

where $\alpha$ is the shape parameter and $\beta$ is the scale parameter
(Weibullref ).

###  II.1 Properties

If $X\sim\text{Weibull}(\boldsymbol{\theta})$ then the following properties
are satisfied (see reviewwei  for review of modified Weibull distributions).

  * 1)

Asymptotic behavior of $f(x;\boldsymbol{\theta})$. The behavior of
$f(x;\boldsymbol{\theta})$ with $x\to 0$ or $x\to\infty$ is as follows:

$\displaystyle\lim_{x\to
0}f(x;\boldsymbol{\theta})=\begin{cases}\infty&\text{for}\quad 0<\alpha<1,\\\
\dfrac{1}{\beta}&\text{for}\quad\alpha=1,\\\
0&\text{for}\quad\alpha>1,\end{cases}$
$\displaystyle\lim_{x\to\infty}f(x;\boldsymbol{\theta})=0\quad\forall\alpha>0.$

  * 2)

Monotonicity, unimodality, concavity and convexity of
$f(x;\boldsymbol{\theta})$. The point $x$ is a mode of the Weibull density, if
and only if it is the solution of the following equation

$\displaystyle{{\rm d}f(x;\boldsymbol{\theta})\over{\rm
d}x}=\displaystyle{f(x;\boldsymbol{\theta})\over
x}\,\Bigg{[}(\alpha-1)-\alpha\bigg{(}{x\over\beta}\bigg{)}^{\alpha}\Bigg{]}=0.$

Solving this equation, we get the following critical point

$\displaystyle
x_{0}=\begin{cases}\displaystyle\beta\bigg{(}{\alpha-1\over\alpha}\bigg{)}^{1/\alpha}&\text{for}\
\alpha>1,\\\\[11.38092pt] 0&\text{for}\ \alpha\leqslant 1.\end{cases}$

A simple calculation shows that

$\displaystyle{{\rm d}^{2}f(x;\boldsymbol{\theta})\over{\rm
d}x^{2}}=\displaystyle{f(x;\boldsymbol{\theta})\over
x^{2}}\,\Bigg{[}\alpha^{2}\bigg{(}{x\over\beta}\bigg{)}^{2\alpha}-3\alpha(\alpha-1)\bigg{(}{x\over\beta}\bigg{)}^{\alpha}+(\alpha-1)(\alpha-2)\Bigg{]}.$

Note that

$\displaystyle{{\rm d}^{2}f(x_{0};\boldsymbol{\theta})\over{\rm
d}x^{2}}=-\alpha(\alpha-1)<0,\quad\alpha>1,$

and

$\displaystyle{{\rm d}^{2}f(x;\boldsymbol{\theta})\over{\rm
d}x^{2}}=0\quad\iff\quad
x=x_{\pm}=\beta\Bigg{[}\dfrac{3(\alpha-1)\pm\sqrt{(\alpha-1)(5\alpha-1)}}{2\alpha}\Bigg{]}^{1/\alpha}.$

Therefore, the following properties follow immediately:

For $\alpha>1$,

    * $\bullet$

$f(x;\boldsymbol{\theta})$ increases as $x\to x_{0}$ and decreases thereafter.

    * $\bullet$

The point $x_{0}$ is the unique mode for the Weibull density.

    * $\bullet$

For $\alpha\neq 2$, the inflection points $x_{\pm}$ satisfy inequality
$x_{-}<x_{+}$. Furthermore, $f(x_{0};\boldsymbol{\theta})$ is convex on
$(0,x_{-})\cup(x_{+},\infty)$ and is concave on $(x_{-},x_{+})$.

    * $\bullet$

For $\alpha=2$, $x_{-}=0$ and $x_{+}=6\beta/4$. Then
$f(x_{0};\boldsymbol{\theta})$ is concave on $(0,x_{+})$ and is convex on the
interval $(x_{+},\infty)$.

For $\alpha\leqslant 1$,

    * $\bullet$

$f(x_{0};\boldsymbol{\theta})$ decreases monotonically and is convex.

    * $\bullet$

The mode is non-existent.

  * 3)

Reliability. If $F(t;\boldsymbol{\theta})$ denotes the cumulative distribution
function of $X$, then the reliability function is written as

$\displaystyle\displaystyle
R(t;\boldsymbol{\theta})=1-F(t;\boldsymbol{\theta})=\exp\Bigg{[}-\bigg{(}{t\over\beta}\bigg{)}^{\alpha}\Bigg{]}.$

  * 4)

Hazard rate. The hazard rate is given by

$\displaystyle
H(t;\boldsymbol{\theta})=\dfrac{f(t;\boldsymbol{\theta})}{R(t;\boldsymbol{\theta})}=\displaystyle{\alpha\over\beta}\,\bigg{(}{t\over\beta}\bigg{)}^{\alpha-1}.$

The function $H(t;\boldsymbol{\theta})$ is increasing when $\alpha>1$,
decreasing when $\alpha<1$ and constant when $\alpha=1$.

  * 5)

Truncated moment. Integration by parts gives

$\displaystyle\displaystyle\mathbb{E}\big{[}{1}_{\\{X\geqslant
t\\}}X^{s}\big{]}=t^{s}\exp\Bigg{[}-\bigg{(}{t\over\beta}\bigg{)}^{\alpha}\Bigg{]}+{s\beta^{s}\over\alpha}\,\Gamma\bigg{(}{s\over\alpha},{t^{\alpha}\over\beta^{\alpha}}\bigg{)},\quad
s\geqslant 0,$

where $\Gamma(s,x)$, $s>0$, is the upper incomplete gamma function, and where
we adopt the notation $\Gamma(0,x)=0$.

  * 6)

Moment of the residual life. For each $t\geqslant 0$, we have

$\displaystyle\displaystyle\phi_{n}(t;\boldsymbol{\theta})$
$\displaystyle=\mathbb{E}\big{[}(X-t)^{n}|X\geqslant t\big{]}$
$\displaystyle=\sum_{k=1}^{n}\binom{n}{k}(-t)^{n-k}\Bigg{\\{}t^{k}+\beta^{k}\Gamma\bigg{(}1+\dfrac{k}{\alpha},\dfrac{t^{\alpha}}{\beta^{\alpha}}\bigg{)}\exp\Bigg{[}\bigg{(}{t\over\beta}\bigg{)}^{\alpha}\Bigg{]}\Bigg{\\}}+(-t)^{n}.$

The proof of this identity is immediate since

$\phi_{n}(t;\boldsymbol{\theta})={1\over
R(t;\boldsymbol{\theta})}\,\sum_{k=1}^{n}\binom{n}{k}(-t)^{s-k}\mathbb{E}\big{[}{1}_{\\{X\geqslant
t\\}}X^{n}\big{]}+(-t)^{n},$

where $R(t;\boldsymbol{\theta})$ and $\mathbb{E}\big{[}{1}_{\\{X\geqslant
t\\}}X^{n}\big{]}$ are given in Items 3) and 5), respectively.

  * 7)

First main tool. By using the formula of Item (6) in Cankaya2018 , we have

$\displaystyle\displaystyle\mathbb{E}\big{[}X^{s}f^{r-1}(X;\boldsymbol{\theta})\big{]}=\dfrac{\alpha^{r-1}\beta^{s-r+1}}{r^{r+(s-r+1)/\alpha}}\,\beta^{s}\Gamma\bigg{(}r+{s-r+1\over\alpha}\bigg{)},\quad
s-r+1>-\alpha r.$

    * $\bullet$

Real moments. By taking $r=1$ in the first main tool, we reach

$\displaystyle\displaystyle\mathbb{E}(X^{s})=\beta^{s}\Gamma\bigg{(}1+{s\over\alpha}\bigg{)},\quad
s>-\alpha,$

where $\Gamma(z)$ is the complete gamma function. Taking $t\to 0$ in the
moment of the residual life $\phi_{n}(t;\boldsymbol{\theta})$ we see that the
entire moments are justified by the above identity.

    * $\bullet$

Tsallis entropy. As an immediate application of the first main tool, we have
that if $q(\alpha-1)>-1$ then the Tsallis entropy Tsallis1988  is given by

$\displaystyle\displaystyle S_{q}(X)$
$\displaystyle=\dfrac{1}{q-1}\,\Bigg{[}1-\int_{0}^{\infty}f^{q}(x;\boldsymbol{\theta})\,{\rm
d}x\Bigg{]}$
$\displaystyle=\dfrac{1}{q-1}\,\Bigg{[}1-\bigg{(}\dfrac{\alpha}{\beta}\bigg{)}^{q-1}\dfrac{1}{q^{q+(1-q)/\alpha}}\,\Gamma\bigg{(}q+{1-q\over\alpha}\bigg{)}\Bigg{]},\quad
q\in\mathbb{R}\backslash\\{1\\}.$

      * *

Quadratic entropy. It is followed directly by Tsallis entropy that, if
$\alpha>1/2$ then the quadratic entropy can be written as

$\displaystyle\displaystyle H_{2}(X)$
$\displaystyle=-\log\Bigg{[}\int_{0}^{\infty}f^{2}(x;\boldsymbol{\theta})\,{\rm
d}x\Bigg{]}$
$\displaystyle=\log(\beta)-\log(\alpha)+\bigg{(}2-\dfrac{1}{\alpha}\bigg{)}\log(2)-\log\Bigg{[}{\displaystyle\Gamma\bigg{(}2-{1\over\alpha}\bigg{)}}\Bigg{]}.$

      * *

Shannon entropy. By combining the Tsallis entropy with the well known relation
$H_{1}(X)=\lim_{q\to 1}S_{q}(X),$ we have that the Shannon entropy can be
written as

$\displaystyle\displaystyle
H_{1}(X)=-\int_{0}^{\infty}f(x;\boldsymbol{\theta})\log\big{[}f(x;\boldsymbol{\theta})\big{]}\,{\rm
d}x=\displaystyle
1+\log\bigg{(}\dfrac{\beta}{\alpha}\bigg{)}+\bigg{(}1-\dfrac{1}{\alpha}\bigg{)}\gamma,$

where $\gamma=-{{\rm d}\Gamma(x)\over{\rm d}x}\big{|}_{x=1}\approx 0.57721$ is
the Euler-Mascheroni constant.

  * 8)

Moment generating function. By applying Fubini’s Theorem we have that the
moment generating function, $M_{X}(t)=\mathbb{E}[\exp({tX})]$, can be
expressed as follows

$\displaystyle\displaystyle
M_{X}(t)=\begin{cases}\displaystyle\dfrac{1}{1-\beta t}&\text{for}\ \alpha=1\
\text{and}\ |t|<1/\beta,\\\\[11.38092pt]
\displaystyle\sum_{n=0}^{\infty}{(\beta t)^{n}\over
n!}\,\Gamma\bigg{(}1+{n\over\alpha}\bigg{)}&\text{for}\ \alpha>1\ \text{and}\
t\in\mathbb{R}.\end{cases}$

  * 9)

Light-tailed distribution. From Item 8) it follows that, if $\alpha\geqslant
1$ then there exists $t_{0}>0$ such that
$\mathbb{P}(X>x)\leqslant\exp(-t_{0}x)$ for $x$ large enough.

  * 10)

Second main tool. A simple change of variable shows that

$\displaystyle\displaystyle\mathbb{E}\big{[}X^{s}\log(X)f^{r-1}(X;\boldsymbol{\theta})\big{]}$
$\displaystyle={\beta^{s-r+1}\alpha^{r-1}\over
r^{1+\zeta/\alpha}}\,\Gamma\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\Bigg{[}\log\bigg{(}{\beta\over
r^{1/\alpha}}\bigg{)}+{1\over\alpha}\,\Psi^{(0)}\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\Bigg{]},\quad\zeta>-\alpha,$

where $\zeta=s+(r-1)(\alpha-1)$, $r\in\mathbb{R}$, and $\Psi^{(m)}(z)={{\rm
d}^{m+1}\log[\Gamma(z)]\over{\rm d}z^{m+1}}$ is the polygamma function of
order $m$.

Indeed, by taking the change of variable
$w=r\big{(}{x\over\beta}\big{)}^{\alpha}$ we have, for each $\zeta>-\alpha$,

$\displaystyle\mathbb{E}\big{[}X^{s}\log(X)f^{r-1}(X;\boldsymbol{\theta})\big{]}$
$\displaystyle=\displaystyle\bigg{(}{\alpha\over\beta}\bigg{)}^{r}\,\int_{0}^{\infty}x^{s}\log(y)\,\bigg{(}{x\over\beta}\bigg{)}^{r(\alpha-1)}\exp\Bigg{[}-r\bigg{(}{x\over\beta}\bigg{)}^{\alpha}\Bigg{]}\,{\rm
d}x$ $\displaystyle={\beta^{s-r+1}\alpha^{r-1}\over
r^{1+\zeta/\alpha}}\,\int_{0}^{\infty}w^{\zeta\over\alpha}\Bigg{[}\log\bigg{(}{\beta\over
r^{1/\alpha}}\bigg{)}+{1\over\alpha}\,\log(w)\Bigg{]}\exp(-w)\,{\rm d}w.$ (2)

By combining the following known formulas

$\displaystyle\int_{0}^{\infty}w^{\zeta\over\alpha}\exp(-w)\,{\rm
d}w=\Gamma\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)},$ (3)
$\displaystyle\int_{0}^{\infty}w^{\zeta\over\alpha}\log(w)\exp(-w)\,{\rm
d}w=\Gamma\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\Psi^{(0)}\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)},$
(4)

with the identity (2), the formula for the expectation
$\mathbb{E}\big{[}X^{s}\log(X)f^{r-1}(X;\boldsymbol{\theta})\big{]}$ follows.

    * $\bullet$

In the particular case when $r=1$ we have

$\displaystyle\displaystyle\mathbb{E}[X^{s}\log(X)]=\beta^{s}\Gamma\bigg{(}1+\dfrac{s}{\alpha}\bigg{)}\Bigg{[}\log(\beta)+{1\over\alpha}\,\Psi^{(0)}\bigg{(}1+\dfrac{s}{\alpha}\bigg{)}\Bigg{]},\quad
s>-\alpha.$

  * 11)

Third main tool. Analogously to the proof of Item 10), a simple change of
variable shows that, for each $s>-\alpha$,

$\displaystyle\mathbb{E}\big{[}X^{s}\log^{2}(X)f^{r-1}(X;\boldsymbol{\theta})\big{]}$
$\displaystyle={2\beta^{s-r+1}\alpha^{r-2}\over
r^{1+\zeta/\alpha}}\,\Gamma\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\log\bigg{(}{\beta\over
r^{1/\alpha}}\bigg{)}\Psi^{(0)}\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\displaystyle$
$\displaystyle+{\beta^{s-r+1}\alpha^{r-1}\over
r^{1+\zeta/\alpha}}\,\Gamma\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\Bigg{\\{}\log^{2}\bigg{(}{\beta\over
r^{1/\alpha}}\bigg{)}+{1\over\alpha^{2}}\,\bigg{[}\Psi^{(0)}\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}^{2}+\Psi^{(1)}\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\bigg{]}\Bigg{\\}},$

where $\zeta=s+(r-1)(\alpha-1)$ and $r\in\mathbb{R}$ and $\zeta>-\alpha$.

The proof of this formula follows by taking the change of variable
$w=r\big{(}{x\over\beta}\big{)}^{\alpha}$ and by combining the formulas (3)
and (4) with the following formula, for each $\zeta>-\alpha$,

$\displaystyle\int_{0}^{\infty}w^{\zeta\over\alpha}\log^{2}(w)\exp(-w)\,{\rm
d}w=\Gamma\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\Bigg{[}\Psi^{(0)}\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}^{2}+\Psi^{(1)}\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\Bigg{]}.$

    * $\bullet$

In the particular case when $r=1$ we have, for $s>-\alpha$,

$\displaystyle\displaystyle\mathbb{E}[X^{s}\log^{2}(X)]=\beta^{s}\Gamma\bigg{(}1+\dfrac{s}{\alpha}\bigg{)}\Bigg{\\{}\log^{2}(\beta)+{1\over\alpha^{2}}\,\bigg{[}\Psi^{(0)}\bigg{(}1+\dfrac{s}{\alpha}\bigg{)}^{2}+\Psi^{(1)}\bigg{(}1+\dfrac{s}{\alpha}\bigg{)}\bigg{]}\displaystyle+{2\log(\beta)\over\alpha}\,\Psi^{(0)}\bigg{(}1+\dfrac{s}{\alpha}\bigg{)}\Bigg{\\}}.$

##  III Inference: Estimation Methods and Fisher information

###  III.1 Maximum likelihood estimation method

Maximum likelihood is the standard approach in parametric estimation, mainly
due to the desirable asymptotic properties of consistency, efficiency and
asymptotic normality under some regularity conditions (LehmannCas98 ).

  * •

The log-likelihood function for $\boldsymbol{\theta}$ is given by

$\displaystyle
l(\boldsymbol{\theta};\boldsymbol{x})=n\log\Big{(}\frac{\alpha}{\beta}\Big{)}+\sum_{i=1}^{n}\left\\{(\alpha-1)\log\Big{(}\frac{x_{i}}{\beta}\Big{)}-\Big{(}{x_{i}\over\beta}\Big{)}^{\alpha}\right\\}.$

A standard calculation shows that the first-order partial derivatives of
$l(\boldsymbol{\theta};\boldsymbol{x})$ are

$\displaystyle{\partial
l(\boldsymbol{\theta};\boldsymbol{x})\over\partial\alpha}$
$\displaystyle={n\over\alpha}-n\log(\beta)+\sum_{i=1}^{n}\log(x_{i})-{1\over\beta^{\alpha}}\sum_{i=1}^{n}x_{i}^{\alpha}\log(x_{i}),$
(5) $\displaystyle{\partial
l(\boldsymbol{\theta};\boldsymbol{x})\over\partial\beta}$
$\displaystyle=-{n\alpha\over\beta}+{\alpha\over\beta^{\alpha+1}}\sum_{i=1}^{n}x_{i}^{\alpha}.$
(6)

The second-order partial derivatives of
$l(\boldsymbol{\theta};\boldsymbol{x})$ can be written as

$\displaystyle{\partial^{2}l(\boldsymbol{\theta};\boldsymbol{x})\over\partial\alpha^{2}}$
$\displaystyle=-{n\over\alpha^{2}}+{\log(\beta)\over\beta^{\alpha}}\sum_{i=1}^{n}x_{i}^{\alpha}\log(x_{i})-{1\over\beta^{\alpha}}\sum_{i=1}^{n}x_{i}^{\alpha}\log^{2}(x_{i}),$
$\displaystyle{\partial^{2}l(\boldsymbol{\theta};\boldsymbol{x})\over\partial\beta^{2}}$
$\displaystyle={n\alpha\over\beta^{2}}-{\alpha(\alpha+1)\over\beta^{\alpha+2}}\sum_{i=1}^{n}x_{i}^{\alpha},$
$\displaystyle{\partial^{2}l(\boldsymbol{\theta};\boldsymbol{x})\over\partial\alpha\partial\beta}$
$\displaystyle={\partial^{2}l(\boldsymbol{\theta};\boldsymbol{x})\over\partial\beta\partial\alpha}=-{n\over\beta}+{\alpha\over\beta^{\alpha+1}}\sum_{i=1}^{n}x_{i}^{\alpha}\log(x_{i}).$

  * •

The $\log_{q}$-likelihood function for $\boldsymbol{\theta}$ is given by

$\displaystyle
l_{q}(\boldsymbol{\theta};\boldsymbol{x})=\sum_{i=1}^{n}{f^{1-q}(x_{i};\boldsymbol{\theta})-1\over
1-q}.$

Note that the first-order partial derivatives of
$l_{q}(\boldsymbol{\theta};\boldsymbol{x})$ are

$\displaystyle{\partial
l_{q}(\boldsymbol{\theta};\boldsymbol{x})\over\partial\alpha}$
$\displaystyle=\sum_{i=1}^{n}f^{-q}(x_{i};\boldsymbol{\theta})\,{\partial
f(x_{i};\boldsymbol{\theta})\over\partial\alpha}=\sum_{i=1}^{n}f^{1-q}(x_{i};\boldsymbol{\theta})\,\Bigg{\\{}\dfrac{1}{\alpha}+\Bigg{[}1-\bigg{(}\dfrac{x_{i}}{\beta}\bigg{)}^{\alpha}\Bigg{]}\log\bigg{(}{x_{i}\over\beta}\bigg{)}\Bigg{\\}},$
$\displaystyle{\partial
l_{q}(\boldsymbol{\theta};\boldsymbol{x})\over\partial\beta}$
$\displaystyle=\sum_{i=1}^{n}f^{-q}(x_{i};\boldsymbol{\theta})\,{\partial
f(x_{i};\boldsymbol{\theta})\over\partial\beta}=\dfrac{\alpha}{\beta}\sum_{i=1}^{n}f^{1-q}(x_{i};\boldsymbol{\theta})\,\Bigg{[}\bigg{(}\dfrac{x_{i}}{\beta}\bigg{)}^{\alpha}-1\Bigg{]}.$

###  III.2 Fisher information

The Fisher information matrix is defined by

$\displaystyle F=n\begin{bmatrix}E_{\alpha\alpha}&E_{\alpha\beta}\\\
E_{\beta\alpha}&E_{\beta\beta}\end{bmatrix},$ (7)

where $n$ is sample size. $E$ is integral for partial derivatives of $\log(L)$
according to parameters and it is taken by probability density function
$f(x;\boldsymbol{\theta})$. The subscript in $E$ represents second-order
partial derivatives of $\log(L)$ according to parameters $\alpha$ and $\beta$.
In other words, if $X\sim\text{Weibull}(\boldsymbol{\theta})$, by using Items
7), 10) and 11) of Subsection II.1, we have

$\displaystyle\begin{array}[]{lllll}E_{\alpha\alpha}&=\displaystyle-{1\over\alpha^{2}}+{\log(\beta)\over\beta^{\alpha}}\,\mathbb{E}\big{[}X^{\alpha}\log(X)\big{]}-{1\over\beta^{\alpha}}\,\mathbb{E}\big{[}X^{\alpha}\log^{2}(X)\big{]}\\\\[8.5359pt]
&=\displaystyle-{1\over\alpha^{2}}\,\big{[}1+\Psi^{(1)}(2)+\Psi^{(0)}(2)^{2}+{\alpha\log(\beta)}\,\Psi^{(0)}(2)\big{]},\\\\[11.38092pt]
E_{\beta\beta}&=\displaystyle{\alpha\over\beta^{2}}-{\alpha(\alpha+1)\over\beta^{\alpha+2}}\,\mathbb{E}\big{(}X^{\alpha}\big{)}=-{\alpha^{2}\over\beta^{2}},\\\\[11.38092pt]
E_{\alpha\beta}&=\displaystyle
E_{\beta\alpha}=-{1\over\beta}+{\alpha\over\beta^{\alpha+1}}\,\mathbb{E}\big{[}X^{\alpha}\log(X)\big{]}=-{1\over\beta}+{\alpha\over\beta}\,\bigg{[}\log(\beta)+{1\over\alpha}\,\Psi^{(0)}(2)\bigg{]}.\end{array}$

Since, $\Psi^{(0)}(2)={{\rm d}\log[\Gamma(x)]\over{\rm
d}x}\big{|}_{x=2}=1-\gamma$ and $\Psi^{(1)}(2)={{\rm
d^{2}}\log[\Gamma(x)]\over{\rm d}x^{2}}\big{|}_{x=2}={\pi^{2}\over 6}-1$,
where $\gamma=-{{\rm d}\Gamma(x)\over{\rm d}x}\big{|}_{x=1}\approx 0.57721$ is
the Euler-Mascheroni constant. Then the matrix (7) is given by

$\displaystyle
F=n\begin{bmatrix}\displaystyle-{1\over\alpha^{2}}\,\bigg{\\{}{\pi^{2}\over
6}+(1-\gamma)\big{[}{\alpha\log(\beta)}+(1-\gamma)\big{]}\bigg{\\}}&&&\displaystyle{\alpha\over\beta}\,\bigg{\\{}\log(\beta)-{\gamma\over\alpha}\bigg{\\}}\\\\[14.22636pt]
\displaystyle{\alpha\over\beta}\,\bigg{\\{}\log(\beta)-{\gamma\over\alpha}\bigg{\\}}&&&\displaystyle-{\alpha^{2}\over\beta^{2}}\end{bmatrix}.$

######  Theorem III.1.

Let
$\Theta=\\{\boldsymbol{\theta}=(\alpha,\beta)\in\mathbb{R}^{+}\times\mathbb{R}^{+}:\alpha\
\text{is known}\ \text{and}\ \beta>1\\}$ be the parameter space. Then, with
probability approaching $1$, as $n\to\infty,$ the likelihood equation ${{\rm
d}\,l(\boldsymbol{\theta};\boldsymbol{x})\over{\rm d}\beta}=0$ has a
consistent solution, denoted by $\widehat{\beta}$.

###### Proof.

If $X\sim\text{Weibull}(\boldsymbol{\theta})$, a simple calculation shows that

  1. 1.

$\mathbb{E}\big{[}{{\rm d}\log f(X;\boldsymbol{\theta})\over{\rm
d}\beta}\big{]}=-{\alpha\over\beta}+{\alpha\over\beta^{\alpha+1}}\,\mathbb{E}(X^{\alpha})=0$
for all $\boldsymbol{\theta}\in\Theta$;

  2. 2.

$-\infty<\mathbb{E}\big{[}{{\rm d}^{2}\log f(X;\boldsymbol{\theta})\over{\rm
d}\beta^{2}}\big{]}=-{\alpha^{2}\over\beta^{2}}<0$ for all $\beta\in\Theta$;

  3. 3.

There exits a function $H(x)$ such that for all
$\boldsymbol{\theta}\in\Theta$,

$\displaystyle\biggl{|}{{\rm d}^{3}\log f(x;\boldsymbol{\theta})\over{\rm
d}\beta^{3}}\biggr{|}$
$\displaystyle=\biggl{|}-{2\alpha\over\beta^{3}}-{\alpha(\alpha+1)(\alpha+2)\over\beta^{2\alpha+4}}\,\beta^{\alpha+1}x^{\alpha}\biggl{|}$
$\displaystyle<2\alpha+\alpha(\alpha+1)(\alpha+2)x^{\alpha}=H(x),$

because $\beta>1$, and

$\displaystyle\mathbb{E}\big{[}H(X)\big{]}$
$\displaystyle=2\alpha+\alpha(\alpha+1)(\alpha+2)\mathbb{E}(X^{\alpha})$
$\displaystyle=2\alpha+\alpha(\alpha+1)(\alpha+2)\beta^{\alpha}=M(\beta)<\infty.$

Hence, by Cramer46  the proof follows. ∎

###  III.3 Estimating equations derived by objective functions in the
composite likelihood

The maximum composite likelihood estimation (MCLE) is a generalization of
maximum likelihood estimation (MLE), MCLE is given by

$L_{C}\big{[}f(\boldsymbol{x};\boldsymbol{\theta})\big{]}=\prod_{i=1}^{n}f(x_{i};\boldsymbol{\theta})^{w_{i}}$
(8)

$w_{i}\in\mathbb{R}$ is a weight function,
$\boldsymbol{x}=(x_{1},x_{2},\ldots,x_{n})$ and $f(x;\boldsymbol{\theta})$ is
a p.d. function. MCLE can cover density power divergence and its generalized
forms (varincomposite ; alphabetadivergences ). The M-estimators are obtained
by optimizing the objective function:

$\sum_{i=1}^{n}\rho(x_{i};\boldsymbol{\theta}):=\sum_{i=1}^{n}w_{i}\log\big{[}f(x_{i};\boldsymbol{\theta})\big{]}=\sum_{i=1}^{n}\Lambda\big{[}f(x_{i};\boldsymbol{\theta})\big{]}.$
(9)

When $w_{i}=1$ and $\log$ is replaced by $\log_{q}$, we have MLqE.
$\log_{q}(f)={f^{1-q}-1\over 1-q},~{}\lim_{q\to 1}\log_{q}(f)=\log(f)$.
$\log_{\kappa}(f)={f^{\kappa}-f^{-\kappa}\over 2\kappa},~{}\lim_{\kappa\to
0}\log_{\kappa}(f)=\log(f)$ and $\log_{q}$ and $\log_{\kappa}$ are deformed
logarithm of $\log$. $q\in\mathbb{R}\backslash\\{1\\},\kappa\in\mathbb{R}$ and
$\gamma\geq 0$ are tuning constants used to adjust robustness and also
efficiency (FerrariYang10 ; Wadatwopara ; Vajda86 ). The concavity property of
$\Lambda$ is examined by CanKor18 ; Jan2  and references therein to use
$\Lambda$ for the estimation process accurately.

The density power divergence (DPD) as an objective function between
$f(x;\boldsymbol{\theta})$ and $g(x)$ which is free from parameter was
proposed and the reorganized form of DPD is given by Basuetal98 :

$\displaystyle\sum_{i=1}^{n}\rho(x_{i};\boldsymbol{\theta}):=\sum_{i=1}^{n}\Lambda\big{[}f(x_{i};\boldsymbol{\theta})\big{]}=\int
f(x;\boldsymbol{\theta})^{1+\gamma}\mathrm{d}x-\bigg{(}1+\dfrac{1}{\gamma}\bigg{)}{1\over
n}\sum_{i=1}^{n}f(x_{i};\boldsymbol{\theta})^{\gamma}.$ (10)

Let us try to get $\log_{q}$ from DPD after algebraic rearrangement and
$\gamma=1-q>0$. Since $q<1$, DPD puts a restriction for the values of tuning
constant when we compare with $\log_{q}$. If equation (10) is rewritten for
$\gamma=1-q$, then we have the following expression:

${2-q\over
n}\sum_{i=1}^{n}\log_{q}\big{[}f(x_{i};\boldsymbol{\theta})\big{]}+\frac{n}{1-q}+\int
f(x;\boldsymbol{\theta})^{2-q}\mathrm{d}x=\sigma\sum_{i=1}^{n}\log_{q}\big{[}f(x_{i};\boldsymbol{\theta})\big{]}+\mu,$
(11)

where $\sigma$ and $\mu$ represent scale and location for
$\sum_{i=1}^{n}\log_{q}\big{[}f(x_{i};\boldsymbol{\theta})\big{]}$. When we
consider to apply the optimization for DPD and maximum $\log_{q}$-likelihood,
$\sigma$ and $\mu$ will change where the optimized region is. In other words,
there is an equivalence between DPD and $\log_{q}$ if they are optimized
according to parameters $\boldsymbol{\theta}$. Further, the integral value of
$I=\int f(x;\boldsymbol{\theta})\mathrm{d}x$ in DPD depends on Gamma function
if $f$ is Weibull (see Cankaya2018 ). The arguman of Gamma function, i.e.
$\Gamma,r>0$, has to be a positive and requires that the values of parameters
remain within certain values, which is disadvantegous for a case in
estimation. In addition, the computation time Muhammedbadwei  for numerical
integration can be high according to the used p.d. function which is not
tractable to calculate and get an expression for the result of integral. MCLE
method is useful to get robust estimators for parameters. As a generalized
form of MCLE, we can use MLqE which is for robust estimation, because MLqE is
a simple method and does not have extra conditions. If $I$ does not exist, it
is mandatory to show that $I$ is finite for the values of parameters in a p.d.
function by use of tools in (Malikcondition ). Otherwise, $f$ in $I$ does not
have a finite value, which shows that we will not have
$\hat{\boldsymbol{\theta}}$. DPD for estimation of parameters of Weibull
distribution does not work properly, as introduced by (Muhammedbadwei ).

Let us derive the estimating equations (EEs) to examine the role of objective
functions. EEs are obtained after taking derivatives of objective functions,
$\rho=\Lambda(f)$, according to parameters $\boldsymbol{\theta}$. For all of
parameters, we have a system of EEs. The rearrangement forms of EEs are as
follows:

$\sum_{i=1}^{n}{\partial\log_{q}\big{[}f(x_{i};\boldsymbol{\theta})\big{]}\over\partial\boldsymbol{\theta}}=\sum_{i=1}^{n}f(x;\boldsymbol{\theta})^{1-q}\mathcal{Z}(x_{i};\boldsymbol{\theta})=\sum_{i=1}^{n}w_{q}(x_{i};\boldsymbol{\theta})\mathcal{Z}(x_{i};\boldsymbol{\theta})=\boldsymbol{0},$
(12)
$\sum_{i=1}^{n}{\partial\log_{\kappa}\big{[}f(x_{i};\boldsymbol{\theta})\big{]}\over\partial\boldsymbol{\theta}}=\sum_{i=1}^{n}\big{[}f(x_{i};\boldsymbol{\theta})^{\kappa}+f(x_{i};\boldsymbol{\theta})^{-\kappa}\big{]}\,{\mathcal{Z}(x_{i};\boldsymbol{\theta})\over
2}=\sum_{i=1}^{n}w_{\kappa}(x_{i};\boldsymbol{\theta})\mathcal{Z}(x_{i};\boldsymbol{\theta})=\boldsymbol{0},$
(13)
$\sum_{i=1}^{n}{\partial\log\big{[}\propto+f(x_{i};\boldsymbol{\theta})\big{]}\over\partial\boldsymbol{\theta}}=\sum_{i=1}^{n}{f(x_{i};\boldsymbol{\theta})\over\propto+f(x_{i};\boldsymbol{\theta})}\,\mathcal{Z}(x_{i};\boldsymbol{\theta})=\sum_{i=1}^{n}w_{\propto}(x_{i};\boldsymbol{\theta})\mathcal{Z}(x_{i};\boldsymbol{\theta})=\boldsymbol{0},$
(14) ${1\over
n}\sum_{i=1}^{n}\mathcal{Z}(x_{i};\boldsymbol{\theta})f(x_{i};\boldsymbol{\theta})^{\gamma}-I=\sum_{i=1}^{n}w_{\gamma}(x_{i};\boldsymbol{\theta})\mathcal{Z}(x_{i};\boldsymbol{\theta})-I=\boldsymbol{0},$
(15)

where
$I=\int\mathcal{Z}(x_{i};\boldsymbol{\theta})f(x_{i};\boldsymbol{\theta})^{1+\gamma}\mathrm{d}x$.
Eqs. (12)-(15) are the weighted score function with
$w_{q},w_{\kappa},w_{\propto}$ and $w_{\gamma}$.
$\mathcal{Z}(x_{i};\boldsymbol{\theta})={{\partial
f(x_{i};\boldsymbol{\theta})\over\partial\boldsymbol{\theta}}/f(x_{i};\boldsymbol{\theta})}$
is a score function. Note that DPD has a weighting $0\leq w_{\gamma}\leq 1$
which is disadvantegous because of the poorness in modeling capability
(Basuetal98 ). Let us rewrite the system of estimating equations (EE) given by
the following form:

$\sum_{i=1}^{n}w(x_{i};\boldsymbol{\theta})\mathcal{Z}(x_{i};\boldsymbol{\theta})=\boldsymbol{0}.$
(16)

Note that $w_{q}$ is bigger than 1 if $q>1$, which shows an advantage for us
when we compare with $0\leq w_{\propto},w_{\gamma}\leq 1$ for
$\propto,\gamma\geq 0$. If $0\leq w\leq 1$, then a partial form of
$\mathcal{Z}$ is produced. If $w>1$, then $(1+v)\mathcal{Z}$ as an extended
form of $\mathcal{Z}$ is produced, which shows us that the information gained
from the joint role of $w$ and $\mathcal{Z}$ at same time is managed by not
only $\mathcal{Z}$ from $f^{{}^{\prime}}/f$ but also the different values of
$w$ from $f^{1-q}$. For this reason, $\log_{q}(f)$ as an objective function is
used to perform an efficient fitting on a data set. Thus, it is seen that $w$
can manage to produce the efficient estimators $\hat{\boldsymbol{\theta}}$
from equation (16). Note that the chosen $w$ affects the estimations of
parameters which are shape and scale. After solving the systems of the
estimating equations according to parameters, the estimators
$\hat{\boldsymbol{\theta}}$ are also obtained instead of optimizing the
objective function (Udristegeo ; Amari16 ).

###  III.4 Investigation for behaviour of objective functions in the composite
likelihood

Estimation is performed when we assume that the empirical data sets are a
member of an objective function $\rho$. For this reason, we can remove
integral or summation from relative entropy or divergence to have function
part of analytical expression. Since $f$ is on the closed inverval $[0,1]$, we
have advantage to display the behaviour of objective functions $\rho$ derived
by $\Lambda(f)$ and the values of tuning parameters are chosen as the interval
$[0,1]$. $\Lambda$ can be chosen as $\log_{q}$, $\log_{\kappa}$,
$\log(\propto+f)$, etc (Lindsay94 ). Let us display their behaviours via Fig.
1:

Figure 1: $f\in[0,1]$ and $q,\kappa,\propto,\gamma\in[0,1]$

Figs. 1 and 1 have big range for the values of functions when compared with
the Figs. 1 and 1. As it seen from plots in Fig. 1, the concavity property of
Figs. 1 and 1 can be better than Figs. 1 and 1. When we look at plots in Fig.
1, we can observe that $\log_{q}$ is better than other functions from
$\Lambda$ (BrondivergencesBas ; Vajda86 ). Thus, we test the role of $\Lambda$
on p.d. function $f$.

###  III.5 $q$-Fisher information

Let us remind the definition of Fisher information based on $\log_{q}$
(CanKor18 ).

${}_{q}\mathbb{E}\left\\{{\partial\log
f(X;\boldsymbol{\theta})\over\partial\boldsymbol{\theta}}\cdot\left[{\partial\log
f(X;\boldsymbol{\theta})\over\partial\boldsymbol{\theta}}\right]^{\intercal}f^{1-q}(X;\boldsymbol{\theta})\right\\}$
$\displaystyle=\int_{0}^{\infty}{\partial\log
f(x;\boldsymbol{\theta})\over\partial\boldsymbol{\theta}}\cdot\left[{\partial\log
f(x;\boldsymbol{\theta})\over\partial\boldsymbol{\theta}}\right]^{\intercal}f^{2-q}(x;\boldsymbol{\theta})\,\mathrm{d}x.$
(17)

Let us rewrite the Equation (III.5) for the parameters $\alpha$ and $\beta$ as
follows

${}_{q}F=n\begin{bmatrix}{}_{q}E_{\alpha\alpha}&{}_{q}E_{\alpha\beta}\\\
{}_{q}E_{\beta\alpha}&{}_{q}E_{\beta\beta}\end{bmatrix}.$

The elements of Fisher Information (FI) matrix based on $\log_{q}$ (${}_{q}F$)
can be written as the following form.

  * •

By taking $n=1$ in (5), we get

${}_{q}E_{\alpha\alpha}$
$\displaystyle=_{q}\mathbb{E}\left\\{\left[{\partial\log
f(X;\boldsymbol{\theta})\over\partial{\alpha}}\right]^{2}f^{1-q}(X;\boldsymbol{\theta})\right\\}$
$\displaystyle=\bigg{[}{1\over\alpha}-\log(\beta)\bigg{]}^{2}\mathbb{E}\big{[}f^{1-q}(X;\boldsymbol{\theta})\big{]}$
$\displaystyle+\sum_{\ell\in\\{0,\alpha\\}}a_{\ell}\,\mathbb{E}\big{[}X^{\ell}\log(X)f^{1-q}(X;\boldsymbol{\theta})\big{]}+\sum_{\ell\in\\{0,\alpha,2\alpha\\}}\widetilde{a}_{\ell}\,\mathbb{E}\big{[}X^{\ell}\log^{2}(X)f^{1-q}(X;\boldsymbol{\theta})\big{]},$

where $a_{0}=2\big{[}{1\over\alpha}-\log(\beta)\big{]}$,
$a_{\alpha}=-{2\over\beta^{\alpha}}\,\big{[}{1\over\alpha}-\log(\beta)\big{]}$,
$\widetilde{a}_{0}=1$, $\widetilde{a}_{\alpha}=-{2\over\beta^{\alpha}}$ and
$\widetilde{a}_{2\alpha}={1\over\beta^{2\alpha}}$. By using Items 7), 10) and
11) of Subsection II.1, we have

${}_{q}E_{\alpha\alpha}$
$\displaystyle=\bigg{[}{1\over\alpha}-\log(\beta)\bigg{]}^{2}\bigg{(}\dfrac{\alpha}{\beta}\bigg{)}^{1-q}\dfrac{1}{q^{2-q+(q-1)/\alpha}}\,\Gamma\bigg{(}2-q+{q-1\over\alpha}\bigg{)}$
$\displaystyle+\sum_{\ell\in\\{0,\alpha\\}}a_{\ell}\,{\beta^{\ell+q-1}\alpha^{1-q}\over(2-q)^{1+\zeta/\alpha}}\,\Gamma\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\Bigg{[}\log\bigg{(}{\beta\over\\{2-q\\}^{1/\alpha}}\bigg{)}+{1\over\alpha}\,\Psi^{(0)}\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\Bigg{]}$
$\displaystyle+\sum_{\ell\in\\{0,\alpha,2\alpha\\}}\widetilde{a}_{\ell}\,{\beta^{\ell+q-1}\alpha^{1-q}\over(2-q)^{1+\zeta/\alpha}}\,\Gamma\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\Bigg{\\{}\log^{2}\bigg{(}{\beta\over\\{2-q\\}^{1/\alpha}}\bigg{)}+\displaystyle$
$\displaystyle+{1\over\alpha^{2}}\,\bigg{[}\Psi^{(0)}\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}^{2}+\Psi^{(1)}\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\bigg{]}+{2\over\alpha}\,\log\bigg{(}{\beta\over\\{2-q\\}^{1/\alpha}}\bigg{)}\Psi^{(0)}\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\Bigg{\\}},$

where $\zeta=\ell+(1-q)(\alpha-1)$.

  * •

Letting $n=1$ in (6), we get

${}_{q}E_{\beta\beta}$
$\displaystyle=_{q}\mathbb{E}\left\\{\left[{\partial\log
f(X;\boldsymbol{\theta})\over\partial{\beta}}\right]^{2}f^{1-q}(X;\boldsymbol{\theta})\right\\}=\sum_{\ell\in\\{0,\alpha,2\alpha\\}}b_{\ell}\,\mathbb{E}\big{[}X^{\ell}f^{1-q}(X;\boldsymbol{\theta})\big{]},$

where $b_{0}=\big{(}{\alpha\over\beta}\big{)}^{2}$,
$b_{\alpha}=-{2\alpha^{2}\over\beta^{\alpha+2}}$ and
$b_{2\alpha}={\alpha^{2}\over\beta^{2(\alpha+1)}}$. By using Item 7) of
Subsection II.1, we reach

${}_{q}E_{\beta\beta}$
$\displaystyle=\sum_{\ell\in\\{0,\alpha,2\alpha\\}}b_{\ell}\,\dfrac{\alpha^{1-q}\beta^{\ell+q-1}}{(2-q)^{r+(\ell+q-1)/\alpha}}\,\beta^{\ell}\Gamma\bigg{(}r+{\ell+q-1\over\alpha}\bigg{)}.$

  * •

Multiplying (5) and (6) with $n=1$, we obtain

${}_{q}E_{\alpha\beta}$ $\displaystyle=_{q}\mathbb{E}\left[{\partial\log
f(X;\boldsymbol{\theta})\over\partial{\alpha}}\,{\partial\log
f(X;\boldsymbol{\theta})\over\partial{\beta}}\,f^{1-q}(X;\boldsymbol{\theta})\right]$
$\displaystyle=\sum_{\ell\in\\{0,\alpha\\}}c_{\ell}\,\mathbb{E}\big{[}X^{\ell}f^{1-q}(X;\boldsymbol{\theta})\big{]}+\sum_{\ell\in\\{0,\alpha,2\alpha\\}}d_{\ell}\,\mathbb{E}\big{[}X^{\ell}\log(X)f^{1-q}(X;\boldsymbol{\theta})\big{]},$

where $c_{0}=-{\alpha\over\beta}\,\big{[}{1\over\alpha}-\log(\beta)\big{]}$,
$c_{\alpha}={\alpha\over\beta^{\alpha+1}}\,\big{[}{1\over\alpha}-\log(\beta)\big{]}$,
$d_{0}=-{\alpha\over\beta}$,
$d_{\alpha}={\alpha\over\beta}\big{(}1+{1\over\beta^{\alpha}}\big{)}$ and
$d_{2\alpha}=-{\alpha\over\beta^{2\alpha+1}}$. By using Items 7) and 10) of
Subsection II.1, we have

${}_{q}E_{\alpha\beta}$
$\displaystyle=\sum_{\ell\in\\{0,\alpha\\}}c_{\ell}\,\dfrac{\alpha^{1-q}\beta^{\ell+q-1}}{(2-q)^{r+(\ell+q-1)/\alpha}}\,\beta^{\ell}\Gamma\bigg{(}r+{\ell+q-1\over\alpha}\bigg{)}$
$\displaystyle+\sum_{\ell\in\\{0,\alpha,2\alpha\\}}d_{\ell}\,{\beta^{\ell+q-1}\alpha^{1-q}\over(2-q)^{1+\zeta/\alpha}}\,\Gamma\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\Bigg{[}\log\bigg{(}{\beta\over\\{2-q\\}^{1/\alpha}}\bigg{)}+{1\over\alpha}\,\Psi^{(0)}\bigg{(}1+\dfrac{\zeta}{\alpha}\bigg{)}\Bigg{]},$

where $\zeta=\ell+(1-q)(\alpha-1)$.

Note that arguments in $\log$ and $\Gamma$ functions should be positive.

##  IV Robustness

Influence function is a measure which is used to evaluate the robustness of
M-estimators (Hampeletal86 ). Let $\boldsymbol{\theta}$ be a vector for
parameters $\alpha$ and $\beta$, i.e.
$\boldsymbol{\theta}=\left(\alpha,\beta\right)$, and
$f(x;\boldsymbol{\theta})$ represents p. d. function of Weibull distribution.
In this case, the objective function is defined as

$\displaystyle\rho(x;\boldsymbol{\theta})=\Lambda\big{[}f(x;\boldsymbol{\theta})\big{]},\quad\psi_{\xi}\left(x\right)=\frac{\partial\rho(x;\boldsymbol{\theta})}{\partial{\xi}},\quad\xi\in\\{\alpha,\beta\\},$

where $\Psi=(\psi_{\alpha},\psi_{\beta})$ is a vector of score functions
$\psi_{\alpha}$ and $\psi_{\beta}$ and $\Lambda(z)=\log_{q}(z)$.

Robustness trusts on the finiteness of score functions from
${\partial\rho(x;\boldsymbol{\theta})\over\partial\boldsymbol{\theta}}$ when
$x$ goes to infinity, i.e.
$\lim_{x\to\infty}{\partial\rho(x;\boldsymbol{\theta})\over\partial\boldsymbol{\theta}}$.
By using same way from robustness to outliers, the finiteness of score
functions should be tested for the case in which we have $\lim_{x\to
0}{\partial\rho(x;\boldsymbol{\theta})\over\partial\boldsymbol{\theta}}$.
Thus, we will imply the robustness to inlier observations in a data set.

###  IV.1 Examination of score functions of parameters in objective function
$\log_{q}(f)$

In order to get MLqE of parameters $\alpha$ and $\beta$, $\log_{q}$ is applied
to $f(x;\boldsymbol{\theta})$. Thus, we have objective function
$\rho(x;\boldsymbol{\theta})=\log_{q}\big{[}f(x;\boldsymbol{\theta})\big{]}$.
The score functions derived by $\rho(x;\boldsymbol{\theta})$ for the
corresponding parameters are given in the following order:

$\displaystyle\psi_{\alpha}\left(x\right)$
$\displaystyle={\partial\rho(x;\boldsymbol{\theta})\over\partial\alpha}=f^{1-q}(x;\boldsymbol{\theta})\,\Bigg{\\{}\dfrac{1}{\alpha}+\Bigg{[}1-\bigg{(}\dfrac{x}{\beta}\bigg{)}^{\alpha}\Bigg{]}\log\bigg{(}{x\over\beta}\bigg{)}\Bigg{\\}},$
(18) $\displaystyle\psi_{\beta}(x)$
$\displaystyle={\partial\rho(x;\boldsymbol{\theta})\over\partial\beta}=\dfrac{\alpha}{\beta}\,f^{1-q}(x;\boldsymbol{\theta})\,\Bigg{[}\bigg{(}\dfrac{x}{\beta}\bigg{)}^{\alpha}-1\Bigg{]}.$
(19)

A simple calculation shows that

$\displaystyle\lim_{x\rightarrow
0}\psi_{\alpha}(x)=\begin{cases}+\infty&\text{for}\ \alpha\geq 1\ \text{and}\
q>1,\\\ 0&\text{for}\ \alpha\geq 1\ \text{and}\ 0<q<1,\\\ 0&\text{for}\
0<\alpha<1\ \text{and}\ q>1,\\\ +\infty&\text{for}\ 0<\alpha<1\ \text{and}\
0<q<1,\end{cases}\quad\lim_{x\rightarrow
0}\psi_{\beta}(x)=\begin{cases}-\infty&\text{for}\ \alpha\geq 1\ \text{and}\
q>1,\\\ 0&\text{for}\ \alpha\geq 1\ \text{and}\ 0<q<1,\\\ 0&\text{for}\
0<\alpha<1\ \text{and}\ q>1,\\\ -\infty&\text{for}\ 0<\alpha<1\ \text{and}\
0<q<1,\end{cases}$
$\displaystyle\lim_{x\rightarrow+\infty}\psi_{\alpha}(x)=\begin{cases}-\infty&\text{for}\
\alpha\geq 1\ \text{and}\ q>1,\\\ 0&\text{for}\ \alpha\geq 1\ \text{and}\
0<q<1,\\\ -\infty&\text{for}\ 0<\alpha<1\ \text{and}\ q>1,\\\ 0&\text{for}\
0<\alpha<1\ \text{and}\
0<q<1,\end{cases}\quad\lim_{x\rightarrow+\infty}\psi_{\beta}(x)=\begin{cases}+\infty&\text{for}\
\alpha\geq 1\ \text{and}\ q>1,\\\ 0&\text{for}\ \alpha\geq 1\ \text{and}\
0<q<1,\\\ +\infty&\text{for}\ 0<\alpha<1\ \text{and}\ q>1,\\\ 0&\text{for}\
0<\alpha<1\ \text{and}\ 0<q<1,\end{cases}$

Briefly, the limits above can be written using the following tables:

Table 1: Limit values of vector $\Psi(x)=(\psi_{\alpha}(x),\psi_{\beta}(x))$ $x\to 0$ | $q>1$ | $0<q<1$
---|---|---
$\alpha\geq 1$ | $(+\infty,-\infty)$ | $(0,0)$
$0<\alpha<1$ | $(0,0)$ | $(+\infty,-\infty)$
Table 2: Limit values of a vector $\Psi(x)=(\psi_{\alpha}(x),\psi_{\beta}(x))$ $x\to+\infty$ | $q>1$ | $0<q<1$
---|---|---
$\alpha\geq 1$ | $(-\infty,\infty)$ | $(0,0)$
$0<\alpha<1$ | $(-\infty,\infty)$ | $(0,0)$

$\Psi=(\psi_{\alpha},\psi_{\beta})$ is a vector of score functions
$\psi_{\alpha}$ and $\psi_{\beta}$. Influence function is proportional to
score function. If score functions $\psi_{\alpha}$ and $\psi_{\beta}$ are
finite, then we have finite influence function, which shows that the
estimators $\hat{\alpha}$ and $\hat{\beta}$ from MLqE and MLE will be robust
(Hampeletal86 ). When MLqE is used, the influence function of the estimators
$\hat{\alpha}$ and $\hat{\beta}$ is finite if $q\in(0,1)$ (see Table 2).
However, one can show that there are cases for $\log(f)$ in which the score
functions of parameters of $f$ are finite or infinite according to property of
$f$ (Canetal19 ). In addition, for an arbitrary $f$, we cannot get FI and the
definition of influence function also includes the inverse of Fisher. There
can be cases in which an element of FI matrix is not defined for some values
of parameters such as $\Gamma(r),r>0$, $a/b,b\neq 0$, etc and the inverse of
FI matrix cannot exist. However, we can get estimates of parameters for these
cases in which FI and its inverse does not exist. Instead of calling
robustness and trusting on the tools in robustness (M. Thompson, e-mail
communication, March 2, 2016), the main approach should be based on the
modelling competence of an arbitrary function. Further, Table 1 includes the
infinity cases. The modeling is carried out for the finite sample size. The
robustness is for the case in which we take limit at the values which are zero
and infinity. However, it is expected that the robustness should be supported
by simulation. There is an open question: Even if the score functions are
infinite for limit values at zero and infinity, can we perform a modeling
capability on a finite sample size? Yes and we can find a counter example
against the robustness theory from simulation results (see Case 4 in Table 5).

##  V Optimization and numerical experiments

###  V.1 Optimization via genetic algorithm for $\Lambda(f)$

The genetic algorithm (GA) can be applied to solve a variety of optimization
problems that are unsuitable for standard optimization algorithms which
include problems where the objective function is undifferentiated in Radon
Nikodym derivative, highly nonlinear, discontinuous, (absolutely) continuous,
non-smooth, even stochastic and random subsets from the real line. GA method
is preferred to ensure that such objective functions converge to a global
point. When $\log_{q}$ is used as the objective function, GA method has been
used by (CanKor18 ). The codes used to get estimates of MLqE are given by
Appendix Computation of parameters via GA.

###  V.2 Structure of contamination: The distributions and their parameter
values used at Monte Carlo simulation

Contamination makes a disorder in the identically distributed random variables
represented by $X_{1},X_{2},\dots,X_{n}$. If these random variables are
disrtibuted non-identically, then we express the structure of non-identicality
from mixing of two p.d. functions $f_{0}$ and $f_{1}$, as given by following
form:

$f_{\varepsilon}(x;\boldsymbol{\theta};\boldsymbol{\tau})=(1-\varepsilon)f_{0}(x;\boldsymbol{\theta})+\varepsilon
f_{1}(x;\boldsymbol{\tau})$ (20)

is the contaminated distribution. The constant $\varepsilon$ is the
contamination rate. $f_{0}$ is the underlying and $f_{1}$ is contamination.
$f_{1}$ can be same distribution with $f_{0}$, but the parameter values of
$f_{1}$ are different from $f_{0}$, i.e. $f_{0}=$
Weibull($\alpha_{0}$,$\beta_{0}$) and $f_{1}=$ Weibull
($\alpha_{1}$,$\beta_{1}$). We can select $f_{1}$ distribution with the given
values of parameters. For example, $f_{1}$ =
BurrIII($\alpha_{1}$,$\beta_{1}$). In the real world, after assuming that a
data set is a member of $f_{0}$ distribution, a contamination to $f_{0}$ by
means of $f_{1}$ can occur in an empirical distribution. We do not know how
much rate $\varepsilon$ of contamination into the data set exist. Further,
there are two types of contamination. These are inliers and outliers. A data
set can include both of contaminations. The aim is to estimate robustly
parameters
$\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2},\dots,\boldsymbol{\theta}_{p}$
of $f_{0}$ under contamination.

The distributions and parameter values selected to generate different forms of
contamination are those in which both inliers and outliers are at the same
time. The plots of functions of the selected values of parameters are given in
Figures 2-4. Thus, it can be observed the structure of underlying and
contamination distributions. We provide three blocks for Figures 2-4. Each of
plots in the blocked Figures 2-4 are given to depict the role of underlying
and contamination distributions clearly. They are depicted separately and same
length $x$-axis to avoid bad illustration which can occur the different values
of parameters for values of p.d. function at $y$-axis. Thus, we observe the
contamination structure and so that we can test the performance of MLqE for
such contaminations in all cases given by Figures 2-4. Blue and red lines
representing p. d. function abbverivated as PDF show the underlying $f_{0}$
and the contamination $f_{1}$ into underlying distribution respectively.
Simulation in section V.3 includes the robust estimations of parameters of
blue lines. As it is proven theoretically by II.1, Weibull distribution has
one unimodal for $\alpha>1$. Note that the mode of Weibull does not exist and
makes asymptotic to $y$-axis for $\alpha\leq 1$. The uniform and BurrIII
Canetal19  distributions have one mode as well.

For the estimation process, the modality of p.d. function
$f(x;\boldsymbol{\theta})$ and the concavity of $\Lambda$ are taken in account
to get $\Lambda(f)$ and apply for the estimation. Thus, the cooperation
between $f(x;\boldsymbol{\theta})$ and $\Lambda$ for conducting an accurate
modeling on a data set can be performed successfully. For example, the
smoothness property of objective function
$\rho(x;\boldsymbol{\theta})=\Lambda\big{[}f(x;\boldsymbol{\theta})\big{]}$ is
important to apply for estimation if a data set does not have several jumpings
on an interval on the real line, i.e. the smoothness of data set should be
provided. The structure of inliers in Monte Carlo simulation will not be
strictly existing. In other words, the frequencies of artificial data sets for
a narrow interval on the real line are not extremely high degree. One can
observe the schema of underlying and contamination distributions in Figures
2-4.

(a) Case 1

(b) Case 2

(c) Case 3

(d) Case 4

(e) Case 5

(f) Case 6

(g) Case 7

(h) Case 8

Figure 2: The underlying and contamination distributions are Weibull (color
online)

(a) Case 1

(b) Case 2

(c) Case 3

(d) Case 4

Figure 3: The underlying is Weibull and contamination is Uniform distributions
(color online)

(a) Case 1

(b) Case 2

(c) Case 3

(d) Case 4

Figure 4: The underlying is Weibull and contamination is BurrIII distributions
(color online)

###  V.3 Monte Carlo simulation: The design of artificial random numbers and
results of simulation

A comprehensive simulation study is performed to make a comparison between
MLqE and MLE methods. There are three contamination structures in the
simulation. Contamination makes a disorder in the identically distributed
random observations represented by $x_{1},x_{2},\ldots,x_{n}$. Contaminations
are as follows:

  1. 1.

$(1-\varepsilon)f_{0}(x;\alpha_{0},\beta_{0})+\varepsilon
f_{1}(x;\alpha_{1},\beta_{1})=(1-\varepsilon)$Weibull($x;\alpha_{0}$,$\beta_{0}$)+$\varepsilon$Weibull($x;\alpha_{1}$,$\beta_{1}$),

  2. 2.

$(1-\varepsilon)f_{0}(x;\alpha_{0},\beta_{0})+\varepsilon
f_{1}(x;\alpha_{1},\beta_{1})=(1-\varepsilon)$Weibull($x;\alpha_{0}$,$\beta_{0}$)+$\varepsilon$Uniform($x;\alpha_{1}$,$\beta_{1}$),

  3. 3.

$(1-\varepsilon)f_{0}(x;\alpha_{0},\beta_{0})+\varepsilon
f_{1}(x;\alpha_{1},\beta_{1})=(1-\varepsilon)$Weibull($x;\alpha_{0}$,$\beta_{0}$)+$\varepsilon$BurrIII($x;\alpha_{1}$,$\beta_{1}$).

The contamination rate $\varepsilon$ is chosen as 10% and 20%. The parameters
$\alpha_{0}$ and $\beta_{0}$ of underlying distribution
Weibull($\alpha_{0}$,$\beta_{0}$) are estimated under contamination. $f_{1}$
can be chosen as Weibull($\alpha_{1},\beta_{1}$), single-peak
BurrIII($\alpha_{1},\beta_{1}$) Canetal19  and Uniform($\alpha_{1},\beta_{1}$)
with the same mode for all values of $x\in[a,b]$. Thus, inliers and outliers
can be observed through shape of function (see Figures 2-4). Four different
sample sizes $n=50,100,150$ and $n=200$ are used in the designs of simulation.
$n_{0}$ and $n_{1}$ represent the number of random numbers from $f_{0}$ and
$f_{1}$ distributions respectively. The number of replication for $n$ is
$10^{4}$.

Simulation variance $\widehat{Var}(\hat{\boldsymbol{\theta}})$ and simulation
mean error squares $\widehat{MSE}(\hat{\boldsymbol{\theta}})$ are computed by
using the following forms:

$\displaystyle\widehat{MSE}(\hat{\boldsymbol{\theta}})$ $\displaystyle=$
$\displaystyle\widehat{Var}(\hat{\boldsymbol{\theta}})+\widehat{Bias}(\hat{\boldsymbol{\theta}})^{2}$
(21) $\displaystyle\widehat{MSE}(\hat{\boldsymbol{\theta}})$ $\displaystyle=$
$\displaystyle\hat{E}[(\hat{\boldsymbol{\theta}}-\hat{E}(\hat{\boldsymbol{\theta}}))^{2}]+(\hat{E}(\hat{\boldsymbol{\theta}})-\boldsymbol{\theta})^{2}$

$\widehat{Bias}^{2}$ and $\widehat{E}$ show the bias and the expected values
obtained from the simulation, respectively. $\widehat{MSE}$ and
$\widehat{Var}$ obtained through simulation are sampling forms of theoretical
MSE and Var, respectively. If the estimators $\hat{\boldsymbol{\theta}}$
obtained by MLqE are unbiased, then
MSE($\hat{\boldsymbol{\theta}}$)=Var($\hat{\boldsymbol{\theta}}$). For
comparison between MLqE and MLE methods, Tables 3-6 give the estimates of
parameters $\alpha_{0}$ and $\beta_{0}$,
$\widehat{Var}(\hat{\boldsymbol{\theta}}$) and
$\widehat{MSE}(\hat{\boldsymbol{\theta}}$). Note that other objective
functions from $\log_{\kappa}$ and DPD did not give results which are better
than MLqE for different types of contaminations. For this reason, we do not
give them for the sake of not increasing the page numbers (see Figures 1-1 for
objective functions).

Table 3: The underlying and contamination distributions are Weibull

$\hat{\boldsymbol{\theta}}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| Case 1: $(1-\varepsilon)$W(4,2) +$\varepsilon$W(1,5), $\varepsilon$=0.1 | Case 1: $(1-\varepsilon)$W(4,2) +$\varepsilon$W(1,5), $\varepsilon$=0.2
|  $n=50,q=0.84$ |  $n=100,q=0.84$ |  $n=150,q=0.84$ |  $n=200,q=0.84$ |  $n=50,q=0.77$ |  $n=100,q=0.77$ |  $n=150,q=0.77$ |  $n=200,q=0.77$
MLqE($\boldsymbol{\theta}$) | 4.0117 | 1.9984 | 3.9449 | 1.9981 | 3.9294 | 1.9967 | 3.9238 | 1.9970 | 3.9537 | 1.9951 | 3.8851 | 1.9938 | 3.8627 | 1.9939 | 3.8454 | 1.9928
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.3883 | 0.0073 | 0.1775 | 0.0036 | 0.1161 | 0.0023 | 0.0883 | 0.0018 | 0.4907 | 0.0092 | 0.2389 | 0.0044 | 0.1516 | 0.0030 | 0.1108 | 0.0022
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.3884 | 0.0073 | 0.1806 | 0.0036 | 0.1210 | 0.0024 | 0.0941 | 0.0018 | 0.4928 | 0.0092 | 0.2522 | 0.0045 | 0.1705 | 0.0030 | 0.1347 | 0.0023
MLE($\boldsymbol{\theta}$) | 1.7602 | 2.3742 | 1.5993 | 2.3786 | 1.5511 | 2.3800 | 1.5226 | 2.3812 | 1.3897 | 2.6791 | 1.3278 | 2.6798 | 1.3082 | 2.6802 | 1.2964 | 2.6796
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.2919 | 0.0518 | 0.1015 | 0.0259 | 0.0576 | 0.0174 | 0.0372 | 0.0130 | 0.0771 | 0.0869 | 0.0289 | 0.0435 | 0.0181 | 0.0284 | 0.0124 | 0.0214
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 5.3086 | 0.1919 | 5.8647 | 0.1692 | 6.0546 | 0.1618 | 6.1746 | 0.1583 | 6.8909 | 0.5480 | 7.1698 | 0.5056 | 7.2639 | 0.4911 | 7.3221 | 0.4832
| Case 2: $(1-\varepsilon)$W(6,4) +$\varepsilon$W(1,5), $\varepsilon$=0.1 | Case 2: $(1-\varepsilon)$W(6,4) +$\varepsilon$W(1,5), $\varepsilon$=0.2
|  $n=50,q=0.82$ |  $n=100,q=0.82$ |  $n=150,q=0.82$ |  $n=200,q=0.82$ |  $n=50,q=0.76$ |  $n=100,q=0.76$ |  $n=150,q=0.76$ |  $n=200,q=0.76$
MLqE($\boldsymbol{\theta}$) | 6.0814 | 3.9807 | 5.9821 | 3.9823 | 5.9549 | 3.9824 | 5.9371 | 3.9828 | 5.8268 | 3.9667 | 5.7126 | 3.9646 | 5.6736 | 3.9658 | 5.6606 | 3.9650
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.8033 | 0.0126 | 0.3677 | 0.0065 | 0.2322 | 0.0043 | 0.1738 | 0.0032 | 1.0473 | 0.0159 | 0.4823 | 0.0079 | 0.3153 | 0.0054 | 0.2322 | 0.0040
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.8099 | 0.0130 | 0.3680 | 0.0068 | 0.2342 | 0.0046 | 0.1777 | 0.0035 | 1.0773 | 0.0170 | 0.5648 | 0.0092 | 0.4219 | 0.0066 | 0.3474 | 0.0053
MLE($\boldsymbol{\theta}$) | 2.9198 | 4.2649 | 2.5628 | 4.2746 | 2.4039 | 4.2797 | 2.3270 | 4.2847 | 2.1074 | 4.4416 | 1.9521 | 4.4407 | 1.8948 | 4.4488 | 1.8674 | 4.4494
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 1.0803 | 0.0952 | 0.5275 | 0.0509 | 0.2924 | 0.0334 | 0.1991 | 0.0254 | 0.3171 | 0.1437 | 0.1272 | 0.0721 | 0.0753 | 0.0490 | 0.0539 | 0.0364
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 10.5682 | 0.1654 | 12.3418 | 0.1263 | 13.2245 | 0.1116 | 13.6903 | 0.1064 | 15.4692 | 0.3388 | 16.5125 | 0.2663 | 16.9276 | 0.2504 | 17.1320 | 0.2384
| Case 3: $(1-\varepsilon)$W(5,5) +$\varepsilon$W(2,6), $\varepsilon$=0.1 | Case 3: $(1-\varepsilon)$W(5,5) +$\varepsilon$W(2,6), $\varepsilon$=0.2
|  $n=50,q=0.83$ |  $n=100,q=0.83$ |  $n=150,q=0.83$ |  $n=200,q=0.83$ |  $n=50,q=0.73$ |  $n=100,q=0.73$ |  $n=150,q=0.73$ |  $n=200,q=0.73$
MLqE($\boldsymbol{\theta}$) | 5.0053 | 5.0198 | 4.9333 | 5.0161 | 4.9117 | 5.0177 | 4.9002 | 5.0177 | 5.0820 | 5.0168 | 4.9997 | 5.0160 | 4.9665 | 5.0140 | 4.9475 | 5.0166
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.5207 | 0.0289 | 0.2328 | 0.0144 | 0.1590 | 0.0097 | 0.1163 | 0.0072 | 0.6637 | 0.0365 | 0.3162 | 0.0178 | 0.1996 | 0.0117 | 0.1496 | 0.0090
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.5207 | 0.0293 | 0.2372 | 0.0146 | 0.1667 | 0.0100 | 0.1262 | 0.0075 | 0.6704 | 0.0368 | 0.3162 | 0.0181 | 0.2008 | 0.0119 | 0.1524 | 0.0093
MLE($\boldsymbol{\theta}$) | 3.8745 | 5.1391 | 3.6735 | 5.1411 | 3.6032 | 5.1439 | 3.5537 | 5.1446 | 3.2677 | 5.2637 | 3.1351 | 5.2671 | 3.0810 | 5.2674 | 3.0652 | 5.2669
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.5877 | 0.0465 | 0.3243 | 0.0237 | 0.2239 | 0.0158 | 0.1695 | 0.0118 | 0.3543 | 0.0665 | 0.1678 | 0.0333 | 0.1046 | 0.0223 | 0.0796 | 0.0170
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 1.8545 | 0.0659 | 2.0839 | 0.0436 | 2.1750 | 0.0365 | 2.2613 | 0.0327 | 3.3551 | 0.1360 | 3.6455 | 0.1046 | 3.7871 | 0.0938 | 3.8231 | 0.0883
| Case 4: $(1-\varepsilon)$W(10,8) +$\varepsilon$W(4,10), $\varepsilon$=0.1 | Case 4: $(1-\varepsilon)$W(10,8) +$\varepsilon$W(4,10), $\varepsilon$=0.2
|  $n=50,q=0.82$ |  $n=100,q=0.82$ |  $n=150,q=0.82$ |  $n=200,q=0.82$ |  $n=50,q=0.74$ |  $n=100,q=0.73$ |  $n=150,q=0.73$ |  $n=200,q=0.73$
MLqE($\boldsymbol{\theta}$) | 10.0432 | 8.0523 | 9.9416 | 8.0521 | 9.9305 | 8.0521 | 9.8742 | 8.0522 | 10.0133 | 8.0936 | 10.0648 | 8.0798 | 10.0319 | 8.0819 | 10.0015 | 8.0829
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 2.4484 | 0.0201 | 1.1634 | 0.0096 | 0.8164 | 0.0065 | 0.6065 | 0.0051 | 3.1352 | 0.0264 | 1.6086 | 0.0126 | 1.1025 | 0.0085 | 0.8469 | 0.0067
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 2.4502 | 0.0229 | 1.1668 | 0.0123 | 0.8212 | 0.0092 | 0.6223 | 0.0078 | 3.1354 | 0.0352 | 1.6128 | 0.0190 | 1.1035 | 0.0152 | 0.8469 | 0.0135
MLE($\boldsymbol{\theta}$) | 6.3489 | 8.2799 | 5.9287 | 8.2881 | 5.8087 | 8.2890 | 5.7108 | 8.2928 | 5.2740 | 8.5174 | 5.0506 | 8.5238 | 4.9863 | 8.5280 | 4.9650 | 8.5302
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 2.2001 | 0.0424 | 0.9516 | 0.0206 | 0.5923 | 0.0145 | 0.4180 | 0.0109 | 0.9024 | 0.0577 | 0.3525 | 0.0292 | 0.2149 | 0.0190 | 0.1645 | 0.0143
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 15.5306 | 0.1207 | 17.5273 | 0.1036 | 18.1596 | 0.0980 | 18.8150 | 0.0966 | 23.2372 | 0.3253 | 24.8492 | 0.3035 | 25.3524 | 0.2977 | 25.5160 | 0.2954
| Case 5: $(1-\varepsilon)$W(3,5) +$\varepsilon$W(2,7), $\varepsilon$=0.1 | Case 5: $(1-\varepsilon)$W(3,5) +$\varepsilon$W(2,7), $\varepsilon$=0.2
|  $n=50,q=0.84$ |  $n=100,q=0.82$ |  $n=150,q=0.82$ |  $n=200,q=0.82$ |  $n=50,q=0.75$ |  $n=100,q=0.75$ |  $n=150,q=0.74$ |  $n=200,q=0.74$
MLqE($\boldsymbol{\theta}$) | 3.0231 | 5.0735 | 3.0327 | 5.0527 | 3.0191 | 5.0543 | 3.0108 | 5.0532 | 3.0520 | 5.1156 | 3.0003 | 5.1164 | 3.0117 | 5.1089 | 3.0024 | 5.1063
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.1642 | 0.0781 | 0.0782 | 0.0382 | 0.0519 | 0.0253 | 0.0381 | 0.0192 | 0.1995 | 0.0964 | 0.0931 | 0.0468 | 0.0624 | 0.0308 | 0.0456 | 0.0231
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.1647 | 0.0835 | 0.0793 | 0.0410 | 0.0523 | 0.0283 | 0.0383 | 0.0221 | 0.2022 | 0.1098 | 0.0931 | 0.0604 | 0.0625 | 0.0426 | 0.0456 | 0.0344
MLE($\boldsymbol{\theta}$) | 2.6699 | 5.2183 | 2.6007 | 5.2154 | 2.5756 | 5.2189 | 2.5600 | 5.2188 | 2.4435 | 5.4241 | 2.3859 | 5.4267 | 2.3642 | 5.4295 | 2.3518 | 5.4286
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.1476 | 0.0888 | 0.0733 | 0.0424 | 0.0513 | 0.0285 | 0.0371 | 0.0216 | 0.1148 | 0.1111 | 0.0552 | 0.0542 | 0.0348 | 0.0360 | 0.0261 | 0.0269
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.2565 | 0.1364 | 0.2327 | 0.0888 | 0.2315 | 0.0764 | 0.2307 | 0.0695 | 0.4245 | 0.2910 | 0.4324 | 0.2363 | 0.4391 | 0.2204 | 0.4462 | 0.2106

Table 4: The underlying and contamination distributions are Weibull
(continuation of the Table 3)

| Case 6: $(1-\varepsilon)$W(1,5) +$\varepsilon$W(2,8), $\varepsilon$=0.1 | Case 6: $(1-\varepsilon)$W(1,5) +$\varepsilon$W(2,8), $\varepsilon$=0.2
---|---|---
|  $n=50,q=0.97$ |  $n=100,q=0.97$ |  $n=150,q=0.97$ |  $n=200,q=0.97$ |  $n=50,q=0.87$ |  $n=100,q=0.87$ |  $n=150,q=0.87$ |  $n=200,q=0.87$
MLqE($\boldsymbol{\theta}$) | 1.0720 | 5.1854 | 1.0565 | 5.1629 | 1.0519 | 5.1597 | 1.0513 | 5.1636 | 1.1229 | 5.0455 | 1.1103 | 5.0387 | 1.1053 | 5.0389 | 1.1020 | 5.0298
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.0157 | 0.5523 | 0.0074 | 0.2762 | 0.0049 | 0.1820 | 0.0037 | 0.1358 | 0.0217 | 0.5414 | 0.0106 | 0.2750 | 0.0070 | 0.1796 | 0.0051 | 0.1326
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.0209 | 0.5866 | 0.0106 | 0.3027 | 0.0076 | 0.2075 | 0.0063 | 0.1626 | 0.0368 | 0.5435 | 0.0228 | 0.2765 | 0.0181 | 0.1811 | 0.0155 | 0.1335
MLE($\boldsymbol{\theta}$) | 1.0727 | 5.3231 | 1.0568 | 5.3022 | 1.0521 | 5.2999 | 1.0514 | 5.3035 | 1.1238 | 5.6089 | 1.1099 | 5.6057 | 1.1044 | 5.6091 | 1.1008 | 5.6013
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.0151 | 0.5622 | 0.0071 | 0.2814 | 0.0047 | 0.1853 | 0.0035 | 0.1390 | 0.0173 | 0.5579 | 0.0084 | 0.2808 | 0.0055 | 0.1824 | 0.0041 | 0.1360
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.0204 | 0.6666 | 0.0103 | 0.3727 | 0.0074 | 0.2752 | 0.0062 | 0.2311 | 0.0326 | 0.9287 | 0.0205 | 0.6477 | 0.0164 | 0.5534 | 0.0142 | 0.4976
| Case 7: $(1-\varepsilon)$W(5,1) +$\varepsilon$W(2,8), $\varepsilon$=0.1 | Case 7: $(1-\varepsilon)$W(5,1) +$\varepsilon$W(2,8), $\varepsilon$=0.2
|  $n=50,q=0.87$ |  $n=100,q=0.87$ |  $n=150,q=0.87$ |  $n=200,q=0.87$ |  $n=50,q=0.8$ |  $n=100,q=0.8$ |  $n=150,q=0.8$ |  $n=200,q=0.8$
MLqE($\boldsymbol{\theta}$) | 5.3941 | 1.0002 | 5.3295 | 1.0007 | 5.3101 | 1.0006 | 5.2996 | 1.0009 | 5.6228 | 1.0009 | 5.5442 | 1.0007 | 5.5227 | 1.0006 | 5.5089 | 1.0010
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.5300 | 0.0011 | 0.2401 | 0.0005 | 0.1615 | 0.0003 | 0.1201 | 0.0003 | 0.6812 | 0.0015 | 0.3046 | 0.0006 | 0.2071 | 0.0004 | 0.1522 | 0.0003
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.6854 | 0.0011 | 0.3487 | 0.0005 | 0.2577 | 0.0003 | 0.2098 | 0.0003 | 1.0691 | 0.0015 | 0.6008 | 0.0006 | 0.4803 | 0.0004 | 0.4113 | 0.0003
MLE($\boldsymbol{\theta}$) | 1.0943 | 1.5828 | 1.0772 | 1.5837 | 1.0698 | 1.5858 | 1.0685 | 1.5851 | 0.9719 | 2.0993 | 0.9644 | 2.1000 | 0.9620 | 2.1006 | 0.9612 | 2.1001
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.0156 | 0.0129 | 0.0062 | 0.0064 | 0.0037 | 0.0042 | 0.0027 | 0.0031 | 0.0048 | 0.0249 | 0.0021 | 0.0123 | 0.0014 | 0.0084 | 0.0011 | 0.0063
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 15.2700 | 0.3525 | 15.3944 | 0.3472 | 15.4501 | 0.3474 | 15.4596 | 0.3455 | 16.2302 | 1.2334 | 16.2880 | 1.2223 | 16.3068 | 1.2197 | 16.3127 | 1.2166
| Case 8: $(1-\varepsilon)$W(5,3) +$\varepsilon$W(2,7), $\varepsilon$=0.1 | Case 8: $(1-\varepsilon)$W(5,3) +$\varepsilon$W(2,7), $\varepsilon$=0.2
|  $n=50,q=0.84$ |  $n=100,q=0.84$ |  $n=150,q=0.84$ |  $n=200,q=0.83$ |  $n=50,q=0.77$ |  $n=100,q=0.77$ |  $n=150,q=0.76$ |  $n=200,q=0.76$
MLqE($\boldsymbol{\theta}$) | 5.0450 | 3.0315 | 4.9924 | 3.0291 | 4.9679 | 3.0253 | 5.0395 | 3.0237 | 5.0236 | 3.0612 | 4.9861 | 3.0476 | 5.0784 | 3.0395 | 5.0649 | 3.0376
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.7555 | 0.0112 | 0.3561 | 0.0053 | 0.2309 | 0.0035 | 0.1613 | 0.0026 | 1.1168 | 0.0181 | 0.4973 | 0.0076 | 0.3107 | 0.0046 | 0.2178 | 0.0033
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.7575 | 0.0121 | 0.3561 | 0.0062 | 0.2320 | 0.0042 | 0.1629 | 0.0032 | 1.1174 | 0.0218 | 0.4975 | 0.0099 | 0.3169 | 0.0061 | 0.2221 | 0.0047
MLE($\boldsymbol{\theta}$) | 2.2789 | 3.4869 | 2.1464 | 3.4979 | 2.1086 | 3.4970 | 2.0953 | 3.4989 | 1.9144 | 3.9000 | 1.8628 | 3.9065 | 1.8508 | 3.9044 | 1.8441 | 3.9029
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.2579 | 0.0414 | 0.0829 | 0.0205 | 0.0486 | 0.0141 | 0.0333 | 0.0103 | 0.0700 | 0.0636 | 0.0286 | 0.0316 | 0.0175 | 0.0201 | 0.0129 | 0.0156
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 7.6625 | 0.2785 | 8.2258 | 0.2684 | 8.4086 | 0.2611 | 8.4705 | 0.2592 | 9.5907 | 0.8736 | 9.8707 | 0.8533 | 9.9347 | 0.8381 | 9.9725 | 0.8308

Table 5: The underlying is Weibull and contamination is Uniform distributions

$\hat{\boldsymbol{\theta}}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| Case 1: $(1-\varepsilon)$W(8,10) +$\varepsilon$U(4,10), $\varepsilon$=0.1 | Case 1: $(1-\varepsilon)$W(8,10) +$\varepsilon$U(4,10), $\varepsilon$=0.2
|  $n=50,q=0.84$ |  $n=100,q=0.84$ |  $n=150,q=0.82$ |  $n=200,q=0.82$ |  $n=50,q=0.7$ |  $n=100,q=0.7$ |  $n=150,q=0.7$ |  $n=200,q=0.7$
MLqE($\boldsymbol{\theta}$) | 8.0064 | 9.8590 | 7.9204 | 9.8638 | 8.0010 | 9.8689 | 7.9822 | 9.8692 | 8.1291 | 9.7391 | 8.0101 | 9.7455 | 7.9531 | 9.7452 | 7.9363 | 9.7461
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.9125 | 0.0385 | 0.4443 | 0.0204 | 0.3111 | 0.0138 | 0.2331 | 0.0100 | 1.3036 | 0.0470 | 0.6139 | 0.0238 | 0.4175 | 0.0161 | 0.3002 | 0.0122
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.9125 | 0.0584 | 0.4507 | 0.0389 | 0.3111 | 0.0310 | 0.2334 | 0.0271 | 1.3202 | 0.1151 | 0.6140 | 0.0886 | 0.4197 | 0.0810 | 0.3043 | 0.0766
MLE($\boldsymbol{\theta}$) | 7.1776 | 9.8106 | 7.1036 | 9.8159 | 7.0731 | 9.8169 | 7.0499 | 9.8152 | 6.4031 | 9.6178 | 6.3368 | 9.6245 | 6.3101 | 9.6246 | 6.3005 | 9.6251
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.6376 | 0.0365 | 0.3069 | 0.0191 | 0.2049 | 0.0127 | 0.1524 | 0.0093 | 0.4950 | 0.0386 | 0.2386 | 0.0197 | 0.1605 | 0.0133 | 0.1140 | 0.0098
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 1.3140 | 0.0724 | 1.1105 | 0.0530 | 1.0639 | 0.0462 | 1.0552 | 0.0435 | 3.0452 | 0.1848 | 3.0048 | 0.1607 | 3.0162 | 0.1542 | 3.0022 | 0.1503
| Case 2: $(1-\varepsilon)$W(3,5) +$\varepsilon$U(5,15), $\varepsilon$=0.1 | Case 2: $(1-\varepsilon)$W(3,5) +$\varepsilon$U(5,15), $\varepsilon$=0.2
|  $n=50,q=0.75$ |  $n=100,q=0.75$ |  $n=150,q=0.74$ |  $n=200,q=0.74$ |  $n=50,q=0.65$ |  $n=100,q=0.65$ |  $n=150,q=0.65$ |  $n=200,q=0.65$
MLqE($\boldsymbol{\theta}$) | 3.1173 | 5.1332 | 3.0509 | 5.1345 | 3.0231 | 5.1392 | 3.0161 | 5.1385 | 3.2591 | 5.2930 | 3.1292 | 5.2903 | 3.0989 | 5.2890 | 3.0789 | 5.2872
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.2868 | 0.0827 | 0.1362 | 0.0412 | 0.0871 | 0.0277 | 0.0665 | 0.0209 | 0.5459 | 0.1214 | 0.2484 | 0.0618 | 0.1641 | 0.0415 | 0.1236 | 0.0311
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.3006 | 0.1004 | 0.1388 | 0.0593 | 0.0876 | 0.0471 | 0.0668 | 0.0400 | 0.6130 | 0.2072 | 0.2650 | 0.1461 | 0.1738 | 0.1250 | 0.1299 | 0.1135
MLE($\boldsymbol{\theta}$) | 2.2236 | 5.6668 | 2.1925 | 5.6698 | 2.1816 | 5.6751 | 2.1756 | 5.6775 | 2.0557 | 6.3144 | 2.0339 | 6.3136 | 2.0278 | 6.3161 | 2.0252 | 6.3139
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.0656 | 0.0843 | 0.0280 | 0.0415 | 0.0173 | 0.0274 | 0.0125 | 0.0207 | 0.0363 | 0.0990 | 0.0165 | 0.0492 | 0.0107 | 0.0336 | 0.0079 | 0.0249
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.6684 | 0.5288 | 0.6800 | 0.4901 | 0.6870 | 0.4832 | 0.6921 | 0.4797 | 0.9280 | 1.8265 | 0.9499 | 1.7746 | 0.9560 | 1.7656 | 0.9580 | 1.7513
| Case 3: $(1-\varepsilon)$W(10,8) +$\varepsilon$U(4,10), $\varepsilon$=0.1 | Case 3: $(1-\varepsilon)$W(10,8) +$\varepsilon$U(4,10), $\varepsilon$=0.2
|  $n=50,q=0.85$ |  $n=100,q=0.85$ |  $n=150,q=0.85$ |  $n=200,q=0.85$ |  $n=50,q=0.74$ |  $n=100,q=0.73$ |  $n=150,q=0.7$ |  $n=200,q=0.7$
MLqE($\boldsymbol{\theta}$) | 9.9399 | 7.9939 | 9.8051 | 7.9961 | 9.7881 | 7.9996 | 9.7611 | 7.9982 | 9.7928 | 7.9929 | 9.7810 | 7.9984 | 10.0043 | 8.0030 | 9.9722 | 8.0032
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 1.3651 | 0.0185 | 0.6472 | 0.0092 | 0.4316 | 0.0062 | 0.3394 | 0.0049 | 1.6467 | 0.0226 | 0.8393 | 0.0119 | 0.6388 | 0.0080 | 0.4839 | 0.0062
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 1.3687 | 0.0186 | 0.6852 | 0.0092 | 0.4765 | 0.0062 | 0.3964 | 0.0049 | 1.6896 | 0.0226 | 0.8873 | 0.0119 | 0.6388 | 0.0080 | 0.4847 | 0.0062
MLE($\boldsymbol{\theta}$) | 9.0213 | 7.9756 | 8.8908 | 7.9779 | 8.8674 | 7.9813 | 8.8412 | 7.9802 | 8.0909 | 7.9542 | 8.0147 | 7.9587 | 7.9808 | 7.9593 | 7.9593 | 7.9605
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 1.0064 | 0.0186 | 0.4656 | 0.0091 | 0.3062 | 0.0062 | 0.2331 | 0.0048 | 0.7783 | 0.0221 | 0.3634 | 0.0114 | 0.2341 | 0.0074 | 0.1771 | 0.0058
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 1.9643 | 0.0192 | 1.6960 | 0.0096 | 1.5889 | 0.0065 | 1.5759 | 0.0052 | 4.4230 | 0.0242 | 4.3048 | 0.0131 | 4.3113 | 0.0091 | 4.3414 | 0.0074
| Case 4: $(1-\varepsilon)$W(3,5) +$\varepsilon$U(3,5), $\varepsilon$=0.1 | Case 4: $(1-\varepsilon)$W(3,5) +$\varepsilon$U(3,5), $\varepsilon$=0.2
|  $n=50,q=1.07$ |  $n=100,q=1.09$ |  $n=150,q=1.09$ |  $n=200,q=1.09$ |  $n=50,q=1.09$ |  $n=100,q=1.1$ |  $n=150,q=1.09$ |  $n=200,q=1.1$
MLqE($\boldsymbol{\theta}$) | 3.0854 | 4.9569 | 3.0185 | 4.9681 | 2.9972 | 4.9665 | 2.9902 | 4.9669 | 3.1509 | 4.9048 | 3.0867 | 4.9119 | 3.0773 | 4.9113 | 3.0555 | 4.9142
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.1239 | 0.0569 | 0.0556 | 0.0282 | 0.0362 | 0.0191 | 0.0272 | 0.0143 | 0.1287 | 0.0536 | 0.0587 | 0.0261 | 0.0391 | 0.0179 | 0.0289 | 0.0136
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.1312 | 0.0588 | 0.0559 | 0.0292 | 0.0362 | 0.0203 | 0.0272 | 0.0154 | 0.1514 | 0.0626 | 0.0662 | 0.0338 | 0.0451 | 0.0258 | 0.0319 | 0.0210
MLE($\boldsymbol{\theta}$) | 3.1676 | 4.9365 | 3.1250 | 3.1250 | 3.1052 | 4.9384 | 3.0986 | 4.9386 | 3.2627 | 4.8741 | 3.2150 | 4.8761 | 3.1958 | 4.8784 | 3.1864 | 4.8775
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.1317 | 0.0567 | 0.0599 | 0.0279 | 0.0392 | 0.0189 | 0.0292 | 0.0141 | 0.1372 | 0.0521 | 0.0630 | 0.0251 | 0.0414 | 0.0174 | 0.0308 | 0.0131
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.1598 | 0.0607 | 0.0755 | 0.0315 | 0.0503 | 0.0227 | 0.0390 | 0.0178 | 0.2062 | 0.0680 | 0.1092 | 0.0405 | 0.0797 | 0.0322 | 0.0655 | 0.0281

Table 6: The underlying is Weibull and contamination is BurrIII distributions

$\hat{\boldsymbol{\theta}}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| Case 1: $(1-\varepsilon)$W(1,5) +$\varepsilon$B(2,20), $\varepsilon$=0.1 | Case 1: $(1-\varepsilon)$W(1,5) +$\varepsilon$B(2,20), $\varepsilon$=0.2
|  $n=50,q=0.94$ |  $n=100,q=0.94$ |  $n=150,q=0.94$ |  $n=200,q=0.94$ |  $n=50,q=0.89$ |  $n=100,q=0.89$ |  $n=150,q=0.89$ |  $n=200,q=0.89$
MLqE($\boldsymbol{\theta}$) | 1.0576 | 5.0057 | 1.0422 | 5.0046 | 1.0374 | 4.9953 | 1.0347 | 4.9909 | 1.1055 | 5.0499 | 1.0895 | 5.0407 | 1.0855 | 5.0421 | 1.0832 | 5.0372
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.0179 | 0.5367 | 0.0083 | 0.2677 | 0.0054 | 0.1777 | 0.0041 | 0.1363 | 0.0221 | 0.5177 | 0.0103 | 0.2601 | 0.0069 | 0.1736 | 0.0050 | 0.1310
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.0212 | 0.5367 | 0.0100 | 0.2677 | 0.0068 | 0.1777 | 0.0053 | 0.1363 | 0.0332 | 0.5202 | 0.0183 | 0.2617 | 0.0142 | 0.1753 | 0.0119 | 0.1324
MLE($\boldsymbol{\theta}$) | 1.0530 | 5.3062 | 1.0344 | 5.3124 | 1.0274 | 5.3053 | 1.0236 | 5.3019 | 1.0795 | 5.6179 | 1.0560 | 5.6162 | 1.0484 | 5.6207 | 1.0424 | 5.6168
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.0174 | 0.5776 | 0.0089 | 0.2903 | 0.0061 | 0.1932 | 0.0050 | 0.1495 | 0.0218 | 0.6135 | 0.0116 | 0.3085 | 0.0088 | 0.2059 | 0.0070 | 0.1576
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.0202 | 0.6714 | 0.0100 | 0.3879 | 0.0069 | 0.2864 | 0.0055 | 0.2407 | 0.0281 | 0.9953 | 0.0148 | 0.6882 | 0.0111 | 0.5911 | 0.0088 | 0.5381
$\hat{\boldsymbol{\theta}}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$
| Case 2: $(1-\varepsilon)$W(3,5) +$\varepsilon$B(2,20), $\varepsilon$=0.1 | Case 2: $(1-\varepsilon)$W(3,5) +$\varepsilon$B(2,20), $\varepsilon$=0.2
|  $n=50,q=0.89$ |  $n=100,q=0.87$ |  $n=150,q=0.87$ |  $n=200,q=0.87$ |  $n=50,q=0.81$ |  $n=100,q=0.81$ |  $n=150,q=0.81$ |  $n=200,q=0.81$
MLqE($\boldsymbol{\theta}$) | 3.0249 | 5.0598 | 3.0102 | 5.0515 | 2.9923 | 5.0422 | 2.9935 | 5.0425 | 3.0853 | 5.0898 | 3.0407 | 5.0752 | 3.0137 | 5.0756 | 3.0157 | 5.0760
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.1834 | 0.0740 | 0.0874 | 0.0351 | 0.0562 | 0.0227 | 0.0429 | 0.0183 | 0.2157 | 0.0858 | 0.0967 | 0.0417 | 0.0636 | 0.0272 | 0.0491 | 0.0209
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.1840 | 0.0776 | 0.0875 | 0.0378 | 0.0562 | 0.0245 | 0.0430 | 0.0201 | 0.2230 | 0.0939 | 0.0984 | 0.0473 | 0.0638 | 0.0329 | 0.0494 | 0.0266
MLE($\boldsymbol{\theta}$) | 2.4092 | 5.3579 | 2.1905 | 5.3762 | 2.0784 | 5.3782 | 2.0086 | 5.3820 | 2.0350 | 5.7129 | 1.8471 | 5.7146 | 1.7609 | 5.7196 | 1.7135 | 5.7233
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.4280 | 0.1964 | 0.2898 | 0.1031 | 0.2456 | 0.0725 | 0.2043 | 0.0569 | 0.3500 | 0.3050 | 0.2018 | 0.1572 | 0.1518 | 0.1118 | 0.1191 | 0.0862
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.7771 | 0.3245 | 0.9450 | 0.2446 | 1.0950 | 0.2155 | 1.1872 | 0.2028 | 1.2812 | 0.8132 | 1.5309 | 0.6679 | 1.6872 | 0.6296 | 1.7742 | 0.6094
| Case 3: $(1-\varepsilon)$W(5,1) +$\varepsilon$B(2,20), $\varepsilon$=0.1 | Case 3: $(1-\varepsilon)$W(5,1) +$\varepsilon$B(2,20), $\varepsilon$=0.2
|  $n=50,q=0.86$ |  $n=100,q=0.88$ |  $n=150,q=0.88$ |  $n=200,q=0.88$ |  $n=50,q=0.8$ |  $n=100,q=0.8$ |  $n=150,q=0.8$ |  $n=200,q=0.8$
MLqE($\boldsymbol{\theta}$) | 5.4852 | 0.9989 | 5.3394 | 1.0000 | 5.3218 | 0.9995 | 5.3078 | 0.9992 | 5.6303 | 1.0037 | 5.5981 | 0.9994 | 5.5816 | 0.9987 | 5.5781 | 0.9993
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.5120 | 0.0010 | 0.2345 | 0.0005 | 0.1585 | 0.0004 | 0.1169 | 0.0003 | 0.9365 | 0.0028 | 0.3323 | 0.0009 | 0.1932 | 0.0004 | 0.1482 | 0.0003
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 0.7474 | 0.0010 | 0.3497 | 0.0005 | 0.2621 | 0.0004 | 0.2117 | 0.0003 | 1.3338 | 0.0028 | 0.6900 | 0.0009 | 0.5315 | 0.0004 | 0.4824 | 0.0003
MLE($\boldsymbol{\theta}$) | 1.1026 | 1.5800 | 1.0635 | 1.5789 | 1.0379 | 1.5827 | 1.0278 | 1.5812 | 0.9505 | 2.1014 | 0.9278 | 2.0925 | 0.9142 | 2.0963 | 0.9075 | 2.0951
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 0.0410 | 0.0223 | 0.0243 | 0.0121 | 0.0174 | 0.0082 | 0.0134 | 0.0060 | 0.0208 | 0.0498 | 0.0110 | 0.0220 | 0.0089 | 0.0160 | 0.0071 | 0.0114
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 15.2304 | 0.3587 | 15.5201 | 0.3473 | 15.7159 | 0.3478 | 15.7914 | 0.3437 | 16.4190 | 1.2628 | 16.5937 | 1.2155 | 16.7024 | 1.2177 | 16.7557 | 1.2105
| Case 4: $(1-\varepsilon)$W(10,8) +$\varepsilon$B(2,20), $\varepsilon$=0.1 | Case 4: $(1-\varepsilon)$W(10,8) +$\varepsilon$B(2,20), $\varepsilon$=0.2
|  $n=50,q=0.82$ |  $n=100,q=0.82$ |  $n=150,q=0.82$ |  $n=200,q=0.8$ |  $n=50,q=0.7$ |  $n=100,q=0.7$ |  $n=150,q=0.69$ |  $n=200,q=0.69$
MLqE($\boldsymbol{\theta}$) | 9.8204 | 7.9524 | 9.6892 | 7.9526 | 9.6657 | 7.9558 | 9.8622 | 7.9581 | 10.0918 | 7.9249 | 9.8777 | 7.9264 | 9.9913 | 7.9367 | 9.9574 | 7.9362
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 1.7467 | 0.0190 | 0.8708 | 0.0094 | 0.5688 | 0.0064 | 0.4775 | 0.0048 | 2.8331 | 0.0239 | 1.3950 | 0.0122 | 0.9690 | 0.0082 | 0.7346 | 0.0067
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 1.7789 | 0.0212 | 0.9673 | 0.0116 | 0.6806 | 0.0084 | 0.4965 | 0.0066 | 2.8415 | 0.0295 | 1.4100 | 0.0176 | 0.9691 | 0.0123 | 0.7364 | 0.0108
MLE($\boldsymbol{\theta}$) | 5.5193 | 8.2448 | 4.4450 | 8.2947 | 3.8883 | 8.3273 | 3.5565 | 8.3300 | 3.5681 | 8.4284 | 2.9810 | 8.4351 | 2.6693 | 8.4701 | 2.5040 | 8.4876
$\widehat{Var}(\hat{\boldsymbol{\theta}})$ | 5.4952 | 0.3408 | 4.0032 | 0.2034 | 3.0931 | 0.1523 | 2.2446 | 0.1183 | 2.5827 | 0.4989 | 1.4444 | 0.2649 | 0.9102 | 0.1950 | 0.6641 | 0.1617
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ | 25.5720 | 0.4007 | 34.8611 | 0.2903 | 40.4455 | 0.2594 | 43.7628 | 0.2272 | 43.9517 | 0.6824 | 50.7104 | 0.4542 | 54.6498 | 0.4160 | 56.8545 | 0.3994

When the Tables 3-6 are examined;

  * •

It is generally observed that the modeling capability of
$\log_{q}$($f_{0}$=Weibull($\alpha_{0},\beta_{0}$)) from MLqE is better than
that of $\log$($f_{0}$=Weibull($\alpha_{0},\beta_{0}$)) from MLE.

  * •

In MLqE, it is observed especially that the estimated values of shape
parameter $\alpha$ can be significantly small for some tried designs.

  * •

As it is logical to expect, $\widehat{MSE}(\hat{\boldsymbol{\theta}})$ of MLE
cannot get the smaller values when the sample size $n$ is increased for some
designs of contamination; because $n=n_{0}+n_{1}$ increases sample size
$n_{1}$ of a contamination, which makes more contaminated data set when it is
compared with small sample sizes, such as $n=50,100,$ etc. In addition,
$\log(f)$ cannot model well. However, for some designs of contamination,
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ of $\log(f)$ can be smaller than
that of $\log_{q}(f)$, because the used objective function is an important
indicator for modeling capability. For example,
$(1-\varepsilon)$Weibull($\alpha_{0}=1$,$\beta_{0}=5$) + $\varepsilon$
Weibull($\alpha_{1}=2$,$\beta_{1}=8$).

  * •

$\widehat{MSE}(\hat{\boldsymbol{\theta}})$ were used to compare the
performance of MLqE and MLE. Depending on the structure of contamination, the
unbiasedness of estimators $\hat{\boldsymbol{\theta}}$ obtained by MLqE were
examined by simulation. However, the contamination structure, i.e. the
selected $f_{1}$ distribution and its parameter values and contamination rate
$\varepsilon$ can lead to observe the biasedness in Tables 3-6. In addition,
in the some cases of MLqE,
$\widehat{MSE}(\hat{{\alpha}})\approx\widehat{Var}(\hat{{\alpha}})$ and
$\widehat{MSE}(\hat{{\beta}})\approx\widehat{Var}(\hat{{\beta}})$ for the
shape $\alpha$ and the scale $\beta$ parameters, respectively.

  * •

When the sample size is increased, the values of
$\widehat{MSE}(\hat{{\alpha}})$ and $\widehat{MSE}(\hat{{\beta}})$ for MLqE
decreased, as expected. However, this is not observed by the results from MLE.
The structure of inliers and outliers affected partially the estimation of
shape parameter $\alpha$.

  * •

For some contamination and the values of parameters $\alpha_{0}$ and
$\beta_{0}$ for Weibull, the values of $q$ can differ for different sample
sizes. In addition, $q$ differs for different contamination structure and
Weibull($\alpha_{0},\beta_{0}$), as expected. The value of $q$ is chosen until
the smallest values of $\widehat{MSE}(\hat{\boldsymbol{\theta}})$ are
obtained. If the constant $q$ with MLqE provides the smallest values for
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$, then it is not possible to find
another value for $q$ which will give the smallest values for
$\widehat{MSE}(\hat{\boldsymbol{\theta}})$.

In overall assesment about robustness and modeling, the modeling is a
procedure applied the finite sample size. The robustness of score function is
based on the limit values at point zero and inifinity. Since the main idea is
based on the modeling capability of $\log_{q}(f)$, using finiteness of score
functions for implying robustness is not necessary (see Table 1 for $q>1$ and
$\alpha\geq 1$). When we look at the results of simulation, we have a result
in which $q>1$ and $\alpha\geq 1$. When we use p.d.
$f(x;\hat{\boldsymbol{\theta}})$ and $q>1$, the values of
$w_{q}=f(x;\hat{\boldsymbol{\theta}})^{1-q}$ are bigger than 1. $\log_{q}(f)$
gives an advantageous for us to have $w\in[0,\infty]$ and so it is flexible to
perform an efficient modeling.

###  V.4 Real data application

Real data sets are applied to test the performance of MLE and MLqE methods.
The estimates of $\hat{\boldsymbol{\theta}}$ and Kolmogorov-Smirnov (KS) test
statistics are provided by Tables 7-10. We give the plots of c.d. and p.d.
functions (CDF and PDF) in Figures 5-9 for illustrative purpose and make a
comparison among the fitting competence with estimates from MLE and MLqE. The
variance-covariance of M-estimators Hub81 ; God60 ; GodTh78  are mainly based
on the Taylor expansion of score function around the true value of parameter.
Even if M-estimation method provides a family for estimation methodology, a
tool from information geometry was provided by CanKor18  as well. The Taylor
expansion approach Hub81  is also used by MLqEGamma  to determine the value of
$q$. Instead of using variance-covariance based on Taylor expansion as a rough
approach for score function based on $\log_{q}$, the $p$-value of KS test
statistic should be preferred. The different $q$-values are tried until the
highest $p$-value of KS test statistic from c.d. function of Weibull is
reached. This approach is also supported by Figures 5-9 which depict the
fitting of c.d. and p.d. functions of Weibull distribution with $\hat{\alpha}$
and $\hat{\beta}$ (Canorder20 ).

It is difficult to know the nature of reality and also knowing modality and
bimodality in an empirical distribution or a real data set. We have to assume
a parametric model and estimate the parameters of underlying distribution as
much as we can do. We use two real data sets which are modeled by objective
functions $\log_{q}\big{[}f(x;\alpha,\beta)\big{]}$ and
$\log\big{[}f(x;\alpha,\beta)\big{]}$. Contaminations are applied into real
data sets. Thus, we will test the performance of MLE and MLqE when the
different types of contamination exist in real data sets. Three types of
contamination to the real data set were performed. These are given by the
following items:

  1. 1.

Inliers: $\alpha_{1}=5;\beta_{1}=10;n_{1}=100$ and $\alpha_{1}=\min(x)+0.5$;
$\beta_{1}=\max(x)-0.5;n_{1}=10;$ for examples 1 and 2, respectively.
$x_{1}=randuni(n_{1},\alpha_{1},\beta_{1})$; $x=[x;x_{1}$];

randuni is a function written in MATLAB2013a to generate uniform artificial
data set from the interval $[a,b]$. $\min(x)$ and $\max(x)$ represent the
minimum and maximum value of observations, respectively.

  2. 2.

Outliers: $x=[x;2\max(x);3\max(x);4\max(x);5\max(x)]$.

  3. 3.

Both of inliers and outliers:
$x=[x;2\max(x);3\max(x);4\max(x);5\max(x);x_{1}]$;

####  V.4.1 Real data application: Example 1

This section consists of the numerical example for the application of real
data set. R Version 4.0.2 and some packages such as
source(”http://bioconductor.org/biocLite.R”) biocLite(”GEOquery”) and
require(GEOquery) are used to reach the real data set. We use the real data
set from ”test$myMean”. The sample size $n$ for this data set is 6136. The
value of tuning constant is chosen until the highest $p$-value of KS test
statistc is obtained when $q$ is $0.85$, which means that the best values for
estimates of parameters can obtained.

Table 7: The estimates of parameters $\alpha$ and $\beta$ via MLE and MLqE
methods

| Without contamination |  With inliers |  With outliers |  With both
---|---|---|---|---
$\hat{\boldsymbol{\theta}}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$
MLE($\boldsymbol{\theta}$) | 3.5427(.00045) | 7.9935(.00038) | 3.5623(.00045) | 7.9937(.00038) | 2.5596(.00065) | 7.9907(.00044) | 2.5869(.00032) | 8.0862 (.00053)
MLqE($\boldsymbol{\theta}$) | 3.9137(.00065) | 7.9942(.00044) | 3.9382(.00064) | 7.9985(.00043) | 3.9133(.00033) | 7.9940(.00054) | 3.9398(.00064) | 7.9961(.00043)

Table 7 shows that MLqE can be robust to inliers and ouliers at each case. The
estimates of scale parameter $\beta$ can be similar to each other at each case
for MLE and MLqE methods. MLE and MLqE with inliers cannot be more different
than that of without the contamination case. However, when MLE and MLqE
methods are compared for the case of outliers, it is seen that the shape
parameter $\alpha$ obtained by MLE is very sensitive outlier. For the real
data set, the results show that MLqE is robust to outliers, because the score
functions derived from $\log_{q}$ for parameters $\alpha$ and $\beta$ are
finite. For this real data set, the sensivity of estimates of $\beta$ from MLE
cannot be more, however when the estimates of $\beta$ is compared with that of
MLqE, MLqE is insensitive to contamination in all of cases. The estimates of
$\alpha$ from MLqE can be insensitive to contamination in all of cases.
Especially, the estimates of $\beta$ from MLqE for both contamination are
insensitivite when compared with that of MLE.

Table 8: The $p$-value of KS test statistics computed by the estimates of
parameters $\alpha$ and $\beta$ via MLE and MLqE

| Without contamination | With inliers | With outliers | With both
---|---|---|---|---
$\hat{\boldsymbol{\theta}}$ |  $p$-value |  $p$-value |  $p$-value |  $p$-value
MLE($\boldsymbol{\theta}$) | 4.3225e-07 | 2.3326e-07 | 6.8452e-76 | 2.6225e-67
MLqE($\boldsymbol{\theta}$) | 1.2516e-04 | 1.2895e-04 | 1.2767e-04 | 1.3000e-04

Figure 5: CDF, PDF and histogram when real data set does not have inliers and
outliers for bioconductor test data(color online)

Figure 6: CDF, PDF and histogram when real data set has inliers for
bioconductor test data (color online)

Figure 7: CDF, PDF and histogram when real data set has outliers for
bioconductor test data(color online)

Figure 8: CDF, PDF and histogram when real data set has inliers and outliers
for bioconductor test data(color online)

Table 8 shows the $p$-value of KS test statistics. According to $p$-values of
KS test statistic, there does not exist enough evidence to accept the null
hypothesis $H_{0}$ which shows that the real data set is a member of Weibull
distribution. However, note that if the significance level of test statistic
is chosen to be $10^{-5}$, then Weibull with MLqE($\boldsymbol{\theta}$)
provides sufficient evidence not to reject the null hypothesis $H_{0}$. Let us
focus on the $p$-values instead of considering whether or not the data set
does really come from Weibull distribution with parameters $\alpha$ and
$\beta$ which were estimated by MLqE and MLE methods. The $p$-values of KS
test statistic obtained from two cases which are outliers and both
contaminations are very small when they are compared with that of MLqE, which
shows that adding outliers to real data set makes more far from Weibull
distribution with the estimates obtained from MLE. However, the $p$-values of
KS test statistic with estimates of MLqE($\boldsymbol{\theta}$) did not differ
much for three cases which are with inliers, outliers and both contaminations.
Let us focus on the comparison of MLE for without contamination and with
inliers cases, $p$-values from $4.3225e-07$ to $2.3326e-07$ tend to be near to
zero due to the fact that adding inliers affects the estimates of MLE. In
MLqE, such tendency being zero is not observed because of robustness property
of MLqE.

####  V.4.2 Real data application: Example 2

The data are the strengths of 1.5 cm glass fibres measured by the National
Physical Laboratory, England. The sample size is $n=63$ (glass15data ). The
distributions and references therein alphapowerglass15  are used to model the
data set. The estimates from Weibull with $\log_{q}(f)$ as an objective
function give the $p$-value which is bigger than that of distributions in
(alphapowerglass15 ). Note that the parametric model and objective functions
should be tried for the case in which we can improve the modeling competence.

Table 9: The estimates of parameters $\alpha$ and $\beta$ via MLE and MLqE
methods

| Without contamination |  With inliers |  With outliers |  With both
---|---|---|---|---
$\hat{\boldsymbol{\theta}}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{\alpha}$ | $\hat{\beta}$
MLE($\boldsymbol{\theta}$) | 5.7762(.07149) | 1.6275(.00471) | 5.8626(.06262) | 1.6164(.00398) | 1.4587(.01697) | 2.1153(.02278) | 1.4961(.01515) | 2.0564(.01879)
MLqE($\boldsymbol{\theta}$) | 7.5423(.10755) | 1.6401(.00393) | 7.6146(.09359) | 1.6325(.00334) | 7.5136(.10077) | 1.6392(.00371) | 7.5692(.08819) | 1.6231(.00317)

Table 9 shows that MLqE can be robust to inliers and ouliers at each case. For
all of scenarios of contaminations which are inliers, outliers and both of
them, the estimates $\hat{\alpha}$ and $\hat{\beta}$ from MLE are sensitive to
contamination. However, such sensivity has not been observed at MLqE. When
outliers and both contaminations are examined, it is observed that the
estimates from MLE are very sensitive to contamination schemas. However, the
estimates from MLqE can have similar values at which there are not
contaminations into data set, which shows that MLqE are robust.

Table 10: The $p$-value of KS test statistics computed by the estimates of
parameters $\alpha$ and $\beta$ via MLE and MLqE

| Without contamination | With inliers | With outliers | With both
---|---|---|---|---
$\hat{\boldsymbol{\theta}}$ |  $p$-value |  $p$-value |  $p$-value |  $p$-value
MLE($\boldsymbol{\theta}$) | 0.0936 | 0.0740 | 9.6151e-07 | 1.1024e-07
MLqE($\boldsymbol{\theta}$) | 0.7283 | 0.4501 | 0.4504 | 0.2700

Table 10 shows the $p$-value of KS test statistics. According to $p$-values of
Kolmogorov-Smirnov (KS) as a goodness of fit test, there exists enough
evidence to accept the null hypothesis $H_{0}$ which shows that the real data
set is a member of Weibull distribution. MLqE and MLE depend on $\log_{q}$ and
$\log$ functions respectively. So the importance of the used objective
function has been observed when we make a comparsion between the $p$-values
which are $0.0936$ and $0.7283$ of MLE and MLqE respectively. The value of
tuning constant is chosen until the highest $p$-value of KS test statistcs is
obtained when $q$ is $0.8$. Even though there is no strict changing of the
values of estimates from MLqE, the $p$-values of KS test statistics from MLqE
go to lower values. This is due to the definition of KS test statistic which
uses values of $x$ (see codes for the computation of $p$-value in Appendix
Computation of $p$-value of KS test statistic). Figures 9-12 illustrate that
there exist a good fitting by MLqE, i.e. CDF${}_{\text{MLqE}}$ and
PDF${}_{\text{MLqE}}$, as supported by $p$-values of KS test statistics in
Table 10.

Figure 9: CDF, PDF and histogram when real data set does not have inliers and
outliers for 1.5 cm glass fibres(color online)

Figure 10: CDF, PDF and histogram when real data set has inliers for 1.5 cm
glass fibres(color online)

Figure 11: CDF, PDF and histogram when real data set has outliers for 1.5 cm
glass fibres(color online)

Figure 12: CDF, PDF and histogram when real data set has inliers and outliers
for 1.5 cm glass fibres(color online)

The insensitivities for $\hat{\alpha}$ and $\hat{\beta}$ in Tables 7-9 to
contaminations show not only the robustness of MLqE but also we can conclude
that the modeling competence of $\log_{q}(f)$ is better than $\log(f)$.

##  VI Conclusions and discussions

If the random variables $X_{1},X_{2},\ldots,X_{n}$ are not distributed
identically, different values of parameters of Weibull distribution have been
estimated robustly by using MLqE. Numerical experiments have been applied to
assess the performance of MLqE. In the simulation, Weibull with different
values of parameters from underlying Weibull, BurrIII and uniform
distributions have been used for schema of contamination which occurs non-
identicality. The performance of MLqE was tested for the functions having one
mode property. The contaminated distributions $f_{1}$ have one mode property
as well. GA which is powerful tool to avoid the local points of the optimized
function has been used to get the estimates of parameters. Researchers can use
MLqE to estimate the parameters of Weibull distribution having contamination
from an one mode function. The $p$-value of KS test statistic should be used
to determine the value of constant $q$ for the real data set. Thus, the
evaluation of fitting performance of Weibull with the estimated parameters can
be tested easily. The score functions derived by $\log(f)$ and $\log_{q}(f)$
are infinite and finite for $0<q<1$ and $\alpha\geq 1$, respectively. The
results in real data show that if $0<q<1$, then we have estimates which can be
insensitive to the added inliers and outliers. When we consider on the results
of simulation, we can have for inlier case in which $q>1$ which shows that the
score functions of parameters are infinite for $q>1$ and $\alpha\geq 1$.
Consequently, the numerical experiments show that robustness is not enough to
imply that the best modeling is accomplished. By using the MLqE, the values of
the parameters representing the majority of the distribution were estimated
with small $\widehat{MSE}(\hat{\boldsymbol{\theta}})$ for different scenarios
of contaminations. Many results from simulation and the application of real
data sets show that MLqE is capable to model efficiently and gives an
advantage to obtain the estimates for parameters of underlying distribution.

Information geometry will be used (Amari16 ; Udristegeo ); and the adopted
goodness of fit test will be proposed for further advance to determine the
value of $q$ by means of tools in statistics. In our future works, we will try
to find counter examples from deformation family and its generalization as
theoretical results and numerical experiments for different types of
contamination will be used to test their modeling competence even if their
score function is infinite.

## Appendix

### Computation of parameters via GA

  * •

Run:

        ΨΨopt=gaoptimset(’CrossoverFcn’,{@crossoversinglepoint},’display’,’off’);
    Ψ

    1. 1.

                 lb=[0 0];ub=[10^10 10^10];
        ΨΨΨMLqE(t,:)=ga(@(p)fMLqE(p,x),2,[],[],[],[],lb,ub,[],opt);
        ΨΨ

  * •

Apply: ’fMLqE’ is a function given by

    1. 1.

                 function [sumL] = fMLqE(p,x)

    2. 2.

                ΨΨΨa=p(1);b=p(2);f=(a/b)*(x/b).^(a-1).*exp(-(x/b).^a);
        ΨΨ

    3. 3.

                 sumL=-sum((f.^(1-q)-1)./(1-q));

When $q=1$, line 3 is replaced by $\log(f)$.

### Computation of $p$-value of KS test statistic

Apply: WeiCDF is a function written in MATLAB 2013a to compute the CDF values.

  1. 1.

         function F=WeiCDF(a,s,x)

  2. 2.

         F=1-exp(-(x/s).^a);

The $p$-value is given by

  1. 1.

         F=WeiCDF(alpha,beta,x); Ψ

  2. 2.

         test_cdf=[x,F];

  3. 3.

         [p_value]=kstest(x,’CDF’,test_cdf);

## Acknowledgements

We would like to thank so much Editorial Board and anonymous referees to
provide the invaluable comments. This study was financed in part by the
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) -
Finance Code 001.

## Disclosure statement

No potential conflict of interest was reported by the author(s).

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