# Parametric Analysis of Gumbel Type-II Distribution under Step-stress Life
Test

Subhankar Dutta 1 , Farha Sultana 1 , Suchandan Kayal 2 Email address:
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(corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we focus on the parametric inference based on the Tampered
Random Variable (TRV) model for simple step-stress life testing (SSLT) using
Type-II censored data. The baseline lifetime of the experimental units under
normal stress conditions follows Gumbel Type-II distribution with $\alpha$ and
$\lambda$ being the shape and scale parameters, respectively. Maximum
likelihood estimator (MLE) and Bayes estimator of the model parameters are
derived based on Type-II censored samples. We obtain asymptotic intervals of
the unknown parameters using the observed Fisher information matrix. Bayes
estimators are obtained using Markov Chain Monte Carlo (MCMC) method under
squared error loss function and LINEX loss function. We also construct highest
posterior density (HPD) intervals of the unknown model parameters. Extensive
simulation studies are performed to investigate the finite sample properties
of the proposed estimators. Finally, the methods are illustrated with the
analysis of a real data set.

1,3Department of Mathematics, National Institute of Technology,
Rourkela-769008, India.
2 Department of Mathematics and Statistics, Indian Institute of Technology,
Kanpur-208016, India.

Keywords: Baseline lifetime, Tampering times, Tampering coefficients, Bayesian
Analysis, MCMC method, Metropolis-Hastings algorithm, Highest posterior
density credible interval.

##  1 Introduction

In reliability analysis, there are many problems with life-testing experiments
which require a long time to acquire the test data such as for highly reliable
products at the specified use condition. There may occur very few failures or
even no failures within a limited testing time under normal operating
conditions. In such problems, to induce early failures an accelerated life
test (ALT) is often used. If all test units are exposed to accelerated
conditions than usual normal conditions, then the test is called ALT. A
special class of the ALT, known as step-stress life testing (SSLT), which
allows the experimenter to gradually increase the stress levels at some pre-
fixed time points during the experiment. In such a life-testing experiment
with two stress levels $s_{1}$ and $s_{2}$, $n$ identical units are placed on
test initially under normal stress level $s_{1}$. The stress level is changed
from $s_{1}$ to $s_{2}$, at pre-fixed time $\tau$, known as the tampering
time. There may be more stress levels and corresponding to each stress change
there would be more than one number of tampering time points we call the
multiple step-stress life testing. If there are only two stress levels with
one tampering time, then it is known as the simple step-stress life-testing,
or simple SSLT. The lifetime distribution under the initial normal stress
level is termed as the baseline lifetime distribution. Some key references on
ALT model are [1], [2], [3], [4].

There are many situations in life testing and reliability experiments in which
the experimental units are lost or removed from test before failure.
Therefore, the experimenter may not obtain complete information of failure
times of all the units put on a life testing experiment. Data observed from
such experiments are called censored data. The most commonly used censoring
schemes are Type-I and Type-II censoring schemes. Briefly, these can be
described as follows. Consider $n$ units are placed under observations on a
particular experiment. In the conventional Type-I censoring scheme, the
experiment continues upto a pre-specified time $\eta$, which is pre-fixed. The
main drawback of this censoring scheme is one may get very few failures or in
worst case no failure till time $\eta$. Therefore, estimating the unknown
parameters can not be done efficiently. On the other hand, the conventional
Type-II censoring scheme requires the experiment to continue until a pre-
specified number $r$ (say) of failures ($r\leq n$) occurs. However, the Type-
II censoring always ensures $r$ number of failures during the life testing
experiment. Here, also the major drawback is the experimental time. If one
uses a highly reliable product for the life testing experiment then it will
take more time to get a pre-specified number of failures and which leads to a
cost constraint for the experimenter.

Goel [5] first introduced the tampered random variable (TRV) modeling in the
context of a simple SSLT (see also [6]), which assumes that the effect of
change of the stress level at time $\tau$ is equivalent to changing the
remaining life of the experimental unit by an unknown positive factor, say
$\beta$ (usually, less than 1). Let $T$ be the random variable representing
the baseline lifetime under normal stress condition. Then, the overall
lifetime, denoted by the random variable $T_{TRV}$, is defined as

$\displaystyle T_{TRV}=\left\\{\begin{array}[]{ll}\displaystyle
T,&\leavevmode\nobreak\ 0<T\leq\tau\\\
\displaystyle\tau+\beta(T-\tau),&\leavevmode\nobreak\
T>\tau,\end{array}\right.$

where the scale factor $\beta$, called the tampering coefficient, depends on
both the stress levels $s_{1}$ and $s_{2}$ and possibly on $\tau$ as well. The
time point $\tau$ is called the tampering time. In literature many authors
considered this model for estimating different lifetime distributions. Abdel-
Ghaly et al. [7] considered the estimation problem of the Weibull distribution
in ALT. Maximum likelihood estimators (MLEs) are obtained for the distribution
parameters and the acceleration factor in both Type-I and Type-II censored
samples. The modified quasilinearization method is used to solve the nonlinear
maximum likelihood equations. Also, the confidence intervals of the estimators
are obtained. Wang et al. [8] studied the estimation of the parameters of the
Weibull distribution in step-stress ALT under multiply censored data. The MLEs
are used to obtain the parameters of the Weibull distribution and the
acceleration factor under multiply censored data. Additionally, the confidence
intervals for the estimators are also obtained. Ismail [9] obtained the MLEs
of Weibull distribution parameters and the acceleration factor under adaptive
Type-II progressively hybrid censored data. The method has been extended for
an adaptive Type-I progressive hybrid censored data by Ismail [10].

In this paper, we consider the problem of estimation for a Gumbel Type-II
distribution based on TRV modeling under simple SSLT using Type-II censoring.
The baseline lifetime $T$ follows Gumbel Type-II distribution whose
probability density function (PDF) and the cumulative distribution function
(CDF) are, respectively,

$\displaystyle f_{T}(t)=\alpha\lambda t^{-(\alpha+1)}e^{-\lambda
t^{-\alpha}},\quad t>0,\leavevmode\nobreak\ \leavevmode\nobreak\
\alpha>0,\leavevmode\nobreak\ \leavevmode\nobreak\ \lambda>0,$ (1.2)

and

$\displaystyle F_{T}(t)=e^{-\lambda t^{-\alpha}},\quad t>0,$ (1.3)

where $\alpha$, $\lambda$ are the shape and scale parameters, respectively.
The hazard rate function of Gumbel Type-II distribution is decreasing or
upside-down bathtub (UTB) shape depends on the parameters values. Due to these
shapes of the hazard rate function, the Gumbel Type-II distribution is very
flexible to model meteorological phenomena such as floods, earthquakes, and
natural disasters, also in medical and epidemiological applications. In recent
years many authors have studied statistical properties of the estimators of
the model parameters of Gumbel Type-II distribution. Abbas et al. [11]
discussed Bayesian estimation of the model parameters of Gumbel Type-II
distribution. Then E-Bayesian estimation of the unknown model shape parameter
has been studied by Reyad and Ahmed [12]. Sindhu et al. [13] obtained Bayes
estimates and corresponding risk of the model parameters based on left-
censored data.

Due to importance of the Gumbel Type-II distribution and simple SSLT, in this
paper, we have considered the problem of estimation of the parameters of
Gumbel Type-II distribution using TRV modeling under simple SSLT. To the best
of our knowledge, this problem has not been studied yet. The rest of the
article is organized as follows. In Section 2, we introduce the TRV modeling
under simple SSLT and derive the corresponding CDF and PDF for Gumbel Type-II
baseline lifetimes. Also, the MLEs of the unknown parameters, $\alpha$,
$\lambda$, and $\beta$ are derived using Type-II censored samples. We also
construct asymptotic intervals of unknown parameters based on the observed
Fisher information matrix. Bayes estimates are obtained under the squared
error loss function as well as LINEX loss function in Section 3. We compute
these estimates using the MH-algorithm of MCMC method. The HPD credible
intervals of unknown parameters are discussed as well. Section 4 presents some
simulation studies to investigate the finite sample properties of the MLEs. We
illustrate the proposed methods through the analysis of a real life data set
in Section 5, while Section 6 ends with some concluding remarks.

##  2 Model Description and MLEs

In a simple SSLT model, let us consider $n$ number of experimental units are
placed with initial stress $s_{1}$. After a prefixed time $\tau$, the initial
stress level is changed from $s_{1}$ to $s_{2}$. The experiment will be
terminated when $r^{th}$ failure occurs, where $r$ is a pre-fixed integer.
Therefore, the time of failures $t_{1:n}<t_{2:n}<...<t_{r:n}$, are denoted as
the observed data. The following are all possible types of data we can get
from the Type-II censoring under simple SSLT :
Case-I : $t_{1:n}<t_{2:n}<\cdots<t_{r:n}<\tau$,
Case-II :
$t_{1:n}<t_{2:n}<\cdots<t_{N:n}<\tau<t_{N+1:n}<t_{N+2:n}<\cdots<t_{r:n}$,
Case-III : $\tau<t_{1:n}<t_{2:n}<\cdots<t_{r:n}$,
where N is the number of failures at stress level $s_{1}$. Note that, for
Case-I, $N=r$ and for Case-III, $N=0$. Basically Case-I and Case-III are the
special cases of Case-II thereafter we will only focus on Case-II in the
reaming part of this paper.

Let us assume that the baseline lifetime follows Gumbel Type-II distribution
with $\alpha$, $\lambda$ are shape and scale parameters, respectively. Also,
assume that the experimental units are independent and identically distributed
(i.i.d.) in the life testing experiment. Now, under the assumption of TRV
model, the CDF $F_{TRV}(.)$, of a $T_{TRV}$ is given by

$\displaystyle F_{{TRV}}(t)$ $\displaystyle=\begin{cases}F_{T}(t),&\text{if
}0\leq t<\tau\\\ F_{T}\left(\tau+\frac{t-\tau}{\beta}\right),&\text{if
}t\geq\tau\end{cases}$ $\displaystyle=\begin{cases}e^{-\lambda
t^{-\alpha}},&\text{if }0\leq t<\tau\\\
e^{-\lambda(\tau+\frac{t-\tau}{\beta})^{-\alpha}},&\text{if
}t\geq\tau.\end{cases}$ (2.1)

Therefore, the corresponding PDF is given by

$\displaystyle f_{{TRV}}(t)$ $\displaystyle=\begin{cases}f_{T}(t),&\text{if
}0\leq t<\tau\\\
\frac{1}{\beta}f_{T}\left(\tau+\frac{t-\tau}{\beta}\right),&\text{if
}t\geq\tau\end{cases}$ $\displaystyle=\begin{cases}\alpha\lambda
t^{-(\alpha+1)}e^{-\lambda t^{-\alpha}},&\text{if }0\leq t<\tau\\\
\frac{\alpha\lambda}{\beta}(\tau+\frac{t-\tau}{\beta})^{-(\alpha+1)}e^{-\lambda(\tau+\frac{t-\tau}{\beta})^{-\alpha}},&\text{if
}t\geq\tau.\end{cases}$ (2.2)

Next we will discuss about the MLEs of the unknown model parameters.

###  2.1 Maximum Likelihood Estimation

In this section we will determine MLEs for the unknown model parameters under
Type-II censored sample using TRV modeling based on simple SSLT. Let
$T_{1:n},T_{2:n},\cdots,T_{n:n}$ be the Type-II censored random sample of size
$n$ from the Gumbel Type-II distribution described in (2.1) with the unknown
model parameters $\alpha$, $\lambda$, and $\beta$. Therefore, the likelihood
function for TRV modeling under Type-II censoring can be written as

$\displaystyle
L(\alpha,\lambda,\beta|data)\propto\prod_{i=1}^{N}f_{T}(t_{i:n})\prod_{i=N+1}^{r}f_{T}\left(\tau+\frac{t_{i:n}-\tau}{\beta}\right)\left[1-F_{T}\left(\tau+\frac{t_{r:n}-\tau}{\beta}\right)\right]^{n-r}.$

Thus, the likelihood function for Gumbel Type-II distribution can be written
as

$\displaystyle L(\alpha,\lambda,\beta|data)\propto\leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle{\alpha}^{r}{\lambda}^{r}{\beta}^{r-N}\prod_{i=1}^{N}{t_{i:n}}^{-(\alpha+1)}e^{-\lambda{t_{i}}^{-\alpha}}\prod_{i=1}^{r-N}\bigg{(}\tau+\frac{t_{i:n}-\tau}{\beta}\bigg{)}^{-(\alpha+1)}$
$\displaystyle\times
e^{-\lambda\sum_{i=1}^{r-N}\bigg{(}\tau+\frac{t_{i}-\tau}{\beta}\bigg{)}^{-\alpha}}\bigg{[}1-e^{-\lambda\left(\tau+\frac{t_{r:n}-\tau}{\beta}\right)^{-\alpha}}\bigg{]}^{n-r}.$
(2.3)

Therefore, the log-likelihood function can be written as

$\displaystyle l(\alpha,\lambda,\beta)=\leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle
r\log\alpha+r\log\lambda+(r-N)\log\beta-(\alpha+1)\bigg{(}\sum_{i=1}^{N}\log
t_{i}+\sum_{i=1}^{r-N}\log z_{i}\bigg{)}$
$\displaystyle-\lambda\bigg{(}\sum_{i=1}^{N}{t_{i}}^{-\alpha}+\sum_{i=1}^{r-N}{z_{i}}^{-\alpha}\bigg{)}+(n-r)\log\bigg{(}1-e^{-\lambda{z_{r}}^{-\alpha}}\bigg{)},$
(2.4)

where, $z_{i}=\leavevmode\nobreak\ \tau+\frac{t_{i}-\tau}{\beta}$ and
$z_{r}=\leavevmode\nobreak\ \tau+\frac{t_{r}-\tau}{\beta}$. Note that, for
notational simplicity, rest of the paper we write $t_{i:n}$ and $t_{r:n}$ as
$t_{i}$ and $t_{r},$ respectively.

Now, taking partial derivatives of $l(\alpha,\lambda,\beta)$ with respect to
unknown parameters we get likelihood equations as given below

$\displaystyle\frac{\partial l}{\partial\alpha}=\leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\frac{r}{\alpha}-\bigg{(}\sum_{i=1}^{N}\log
t_{i}+\sum_{i=1}^{r-N}\log
z_{i}\bigg{)}+\lambda\bigg{(}\sum_{i=1}^{N}{t_{i}}^{-\alpha}\log
t_{i}+\sum_{i=1}^{r-N}{z_{i}}^{-\alpha}\log z_{i}\bigg{)}$
$\displaystyle-(n-r)\frac{\lambda{z_{r}}^{-\alpha}\log
z_{r}}{(1-e^{-\lambda{z_{r}}^{-\alpha}})}=0,$ (2.5)
$\displaystyle\frac{\partial l}{\partial\lambda}=\leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\frac{r}{\lambda}-\sum_{i=1}^{N}{t_{i}}^{-\alpha}-\sum_{i=1}^{r-N}{z_{i}}^{-\alpha}+(n-r)\frac{{z_{r}}^{-\alpha}}{(e^{\lambda{z_{r}}^{-\alpha}}-1)}=0,$
(2.6) and $\displaystyle\frac{\partial l}{\partial\beta}=\leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\frac{r-N}{\beta}-(\alpha+1)\sum_{i=1}^{r-N}\frac{{z_{i}}^{{}^{\prime}}}{z_{i}}-\alpha\lambda\sum_{i=1}^{r-N}{z_{i}}^{-(\alpha+1)}{z_{i}}^{{}^{\prime}}$
$\displaystyle-(n-r)\frac{\alpha\lambda{z_{r}}^{-(\alpha+1)}{z_{r}}^{{}^{\prime}}}{(e^{\lambda{z_{r}}^{-\alpha}}-1)}=0,$
(2.7)

where ${z_{i}}^{{}^{\prime}}=\frac{\tau-t_{i}}{{\beta}^{2}}$ and
${z_{r}}^{{}^{\prime}}=\frac{\tau-t_{r}}{{\beta}^{2}}$.
As the likelihood equations are in implicit form of the unknown parameters,
thus we cannot solve it explicitly to determine the MLEs of $\alpha$,
$\lambda$, and $\beta$ as $\widehat{\alpha}$, $\widehat{\lambda}$, and
$\widehat{\beta}$. Therefore solve the equations (2.5), (2.6) and (2.7)
numerically by using some numerical method, such as Newton-Raphson.

###  2.2 Approximate Confidence Intervals

In this section we want to construct the asymptotic confidence intervals for
the unknown model parameters. To obtain $100(1-\gamma)\%$ confidence interval
of the unknown parameters of Gumbel Type-II distribution under simple SSLT, we
have to calculate the asymptotic variance-covariance matrix. Using asymptotic
normality properties of MLEs of the parameters, an asymptotic variance-
covariance matrix can be obtained. In doing so, the variance of
$\widehat{\alpha}$, $\widehat{\lambda}$, and $\widehat{\beta}$ are required.
These can be obtained from the diagonal elements of the inverse of the
observed Fisher information matrix,
$\widehat{I}^{-1}(\widehat{\alpha},\widehat{\lambda},\widehat{\beta})$, where

$\displaystyle\widehat{I}(\widehat{\alpha},\widehat{\lambda},\widehat{\beta})={\begin{bmatrix}-l_{11}&-l_{12}&-l_{13}\\\
-l_{12}&-l_{22}&-l_{23}\\\ -l_{13}&-l_{23}&-l_{33}\\\
\end{bmatrix}}_{({\alpha},\leavevmode\nobreak\ {\lambda},\leavevmode\nobreak\
{\beta})=(\widehat{\alpha},\leavevmode\nobreak\
\widehat{\lambda},\leavevmode\nobreak\ \widehat{\beta})},$ (2.8)

and $l_{ij}=\frac{\partial^{2}logL}{\partial\theta_{i}\partial\theta_{j}}$ for
$i,j=1,2,3,$ where,
$\Theta=(\theta_{1},\theta_{2},\theta_{3})=(\alpha,\lambda,\beta)$. So here,

$\displaystyle l_{11}=$
$\displaystyle-\frac{r}{{\alpha}^{2}}-\lambda\sum_{i=1}^{N}\frac{(\log{t_{i}})^{2}}{{t_{i}}^{\alpha}}-\lambda\sum_{i=1}^{r-N}\frac{(\log{z_{i}})^{2}}{{z_{i}}^{\alpha}}-(n-r)\frac{\lambda[{z_{r}}^{\alpha}+e^{\lambda{z_{r}}^{-\alpha}(\lambda-{z_{r}}^{\alpha})}]}{{z_{r}}^{2\alpha}(e^{\lambda{z_{r}}^{-\alpha}-1)^{2}}(\log
z_{r})^{-2}}.$ $\displaystyle l_{12}=l_{21}=$
$\displaystyle\sum_{i=1}^{N}\frac{\log{t_{i}}}{{t_{i}}^{\alpha}}+\sum_{i=1}^{N}\frac{\log{z_{i}}}{{z_{i}}^{\alpha}}-(n-r)\frac{\bigg{(}e^{\lambda{z_{r}}^{-\alpha}}({z_{r}}^{\alpha}-\lambda)\log{z_{r}}\bigg{)}}{{z_{r}}^{2\alpha}(e^{\lambda{z_{r}}^{-\alpha}}-1)^{2}}.$
$\displaystyle l_{13}=l_{31}=$
$\displaystyle-(n-r)\frac{\lambda{z_{r}}^{{}^{\prime}}\bigg{(}{z_{r}}^{\alpha}(e^{\lambda{z_{r}}^{-\alpha}}-1)+\alpha\bigg{(}{z_{r}}^{\alpha}+e^{\lambda{z_{r}}^{-\alpha}}(\lambda-{z_{r}}^{\alpha})\bigg{)}\log
z_{r}\bigg{)}}{{z_{r}}^{2\alpha+1}(e^{\lambda{z_{r}}^{-\alpha}}-1)^{2}}$
$\displaystyle-\sum_{i=1}^{r-N}\frac{{z_{i}}^{{}^{\prime}}}{z_{i}}-\lambda\sum_{i=1}^{r-N}{z_{i}}^{-\alpha}\frac{{z_{i}}^{{}^{\prime}}}{z_{i}}(\alpha\log
z_{i}-1).$ $\displaystyle l_{22}=$
$\displaystyle-\frac{r}{{\lambda}^{2}}-(n-r)\frac{{z_{r}}^{-2\alpha}e^{\lambda{z_{r}}^{-\alpha}}}{(e^{\lambda{z_{r}}^{-\alpha}}-1)^{2}}.$
$\displaystyle l_{23}=l_{32}=$
$\displaystyle\sum_{i=1}^{r-N}\alpha{z_{i}}^{-\alpha}\frac{{z_{i}}^{{}^{\prime}}}{z_{i}}-(n-r)\frac{{z_{r}}^{{}^{\prime}}}{z_{r}}\frac{{z_{r}}^{-2\alpha}}{(e^{\lambda{z_{r}}^{-\alpha}}-1)}\bigg{(}e^{\lambda{z_{r}}^{-\alpha}}({z_{r}}^{\alpha}-\lambda)-{z_{r}}^{\alpha}\bigg{)}.$
$\displaystyle l_{33}=$
$\displaystyle-\frac{r-N}{{\beta}^{2}}+(\alpha+1)\sum_{i=1}^{r-N}\bigg{(}\frac{{z_{i}}^{{}^{\prime}}}{z_{i}}\bigg{[}\frac{{z_{i}}^{{}^{\prime}}}{z_{i}}+\frac{2}{\beta}\bigg{]}\bigg{)}-\alpha\lambda\sum_{i=1}^{r-N}\bigg{(}{z_{i}}^{-\alpha}\frac{{z_{i}}^{{}^{\prime}}}{z_{i}}\bigg{[}(\alpha+1)\frac{{z_{i}}^{{}^{\prime}}}{z_{i}}-\frac{2}{\beta}\bigg{]}\bigg{)}$
$\displaystyle-\frac{\alpha\lambda}{{\beta}^{2}}\bigg{(}\frac{{z_{r}}^{{}^{\prime}}}{z_{r}}\bigg{)}^{2}\frac{{z_{r}}^{-2\alpha}}{(e^{\lambda{z_{r}}^{-\alpha}}-1)^{2}}\bigg{[}(\alpha-1)t_{r}{z_{r}}^{\alpha}-(2\beta+\alpha-1)\tau{z_{r}}^{\alpha}+\tau
e^{\lambda{z_{r}}^{-\alpha}}\bigg{(}(2\beta-1){z_{r}}^{\alpha}$
$\displaystyle+\alpha({z_{r}}^{\alpha}-\lambda)\bigg{)}+t_{r}e^{\lambda{z_{r}}^{-\alpha}}\bigg{(}{z_{r}}^{\alpha}+\alpha(\lambda-{z_{r}}^{\alpha})\bigg{)}\bigg{]}.$

Then the $100(1-\gamma)\%$ approximate confidence intervals for $\alpha$,
$\lambda$, and $\beta$ are given by,

$\displaystyle\bigg{(}\widehat{\alpha}\leavevmode\nobreak\
\underline{+}\leavevmode\nobreak\
z_{\frac{\gamma}{2}}\sqrt{Var(\widehat{\alpha})}\bigg{)},\leavevmode\nobreak\
\leavevmode\nobreak\ \bigg{(}\widehat{\lambda}\leavevmode\nobreak\
\underline{+}\leavevmode\nobreak\
z_{\frac{\gamma}{2}}\sqrt{Var(\widehat{\lambda})}\bigg{)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\bigg{(}\widehat{\beta}\leavevmode\nobreak\ \underline{+}\leavevmode\nobreak\
z_{\frac{\gamma}{2}}\sqrt{Var(\widehat{\beta})}\bigg{)},$

where $z_{\frac{\gamma}{2}}$ is the upper $\frac{\gamma}{2}$-th percentile of
a standard normal distribution.

##  3 Bayesian Estimation

In this section, we will determine the Bayes estimates of the unknown
parameters $\alpha$, $\lambda$, and $\beta$ based on Type-II censored data
from Gumble Type-II distribution. The most commonly used symmetric loss
function is squared error loss (SEL) function and an asymmetric loss function
is LINEX loss (LL) function. These loss functions are defined as

$\displaystyle
L_{SE}\big{(}h(\theta),\widehat{h}(\theta)\big{)}=\leavevmode\nobreak\ $
$\displaystyle\big{(}h(\theta)-\widehat{h}(\theta)\big{)}^{2},$ and
$\displaystyle
L_{LI}\big{(}h(\theta),\widehat{h}(\theta)\big{)}=\leavevmode\nobreak\ $
$\displaystyle
e^{u\big{(}\widehat{h}(\theta)-h(\theta)\big{)}}-u\big{(}\widehat{h}(\theta)-h(\theta)\big{)}-1,\leavevmode\nobreak\
u\neq 0,$

where $\widehat{h}(\theta)$ is an estimate of a parametric function
$h(\theta)$ and $u$ is a real number. All the parameters $\alpha$, $\lambda$,
and $\beta$ are unknown. In this case, there doesn’t exist any natural joint
conjugate prior distribution. Thus, according to Kundu and Pradhan [14] we
assume independent priors for $\alpha$, $\lambda$, and $\beta$ as Gamma(a,b),
Gamma(c,d), and Beta(p,q) distributions, respectively. We recall that
$X\sim\mbox{Gamma}(a,b)$, if its PDF is given by

$\displaystyle f_{1}(x)\propto x^{a-1}e^{-bx},\leavevmode\nobreak\
x>0,\leavevmode\nobreak\ a,\leavevmode\nobreak\ b>0.$ (3.1)

Further, if $X\sim\mbox{Beta}(p,q)$, then its PDF is

$\displaystyle f_{2}(x)\propto x^{p-1}(1-x)^{q-1},\leavevmode\nobreak\
0<x<1,\leavevmode\nobreak\ p,\leavevmode\nobreak\ q>0.$ (3.2)

Now, the joint prior distribution of the unknown parameters is obtained as

$\displaystyle{\pi}^{*}(\alpha,\lambda,\beta)\propto{\alpha}^{a-1}e^{-b\alpha}{\lambda}^{c-1}e^{-d\lambda}{\beta}^{p-1}(1-\beta)^{q-1},\leavevmode\nobreak\
\leavevmode\nobreak\ \alpha>0,\leavevmode\nobreak\
\lambda>0,\leavevmode\nobreak\ 0<\beta<1,$ (3.3)

where, $a,\leavevmode\nobreak\ b,\leavevmode\nobreak\ c,\leavevmode\nobreak\
d,\leavevmode\nobreak\ p,$ and $q$ are the hyper parameters. Note that the
hyper parameters reflect the prior knowledge about the unknown parameters, and
can take the value from any positive real numbers. After some calculations,
the joint posterior PDF of the unknown parameters $\alpha$, $\lambda$, and
$\beta$ can be obtained as

$\displaystyle\pi(\alpha,\lambda,\beta|data)=$
$\displaystyle\leavevmode\nobreak\
k\frac{{\alpha}^{r+a-1}{\lambda}^{r+c-1}}{{\beta}^{N+1-r-p}}(1-\beta)^{q-1}e^{-(b\alpha+d\lambda)}e^{-\big{[}(\alpha+1)\sum_{i=1}^{N}\log(t_{i})+\lambda\sum_{i=1}^{N}{t_{i}}^{-\alpha}\big{]}}$
$\displaystyle\times
e^{-\big{[}(\alpha+1)\sum_{i=1}^{r-N}\log(\tau+\frac{t_{i}-\tau}{\beta})+\lambda\sum_{i=1}^{r-N}(\tau+\frac{t_{i}-\tau}{\beta})^{-\alpha}\big{]}}\big{[}1-e^{-\lambda{(\tau+\frac{t_{r}-\tau}{\beta})}^{-\alpha}}\big{]}^{(n-r)},$
(3.4)

where

$\displaystyle k^{-1}=$
$\displaystyle\int_{0}^{1}\int_{0}^{\infty}\int_{0}^{\infty}\bigg{[}\frac{{\alpha}^{r+a-1}{\lambda}^{r+c-1}}{{\beta}^{N+1-r-p}}(1-\beta)^{q-1}e^{-(b\alpha+d\lambda)}e^{-\big{[}(\alpha+1)\sum_{i=1}^{N}\log(t_{i})+\lambda\sum_{i=1}^{N}{t_{i}}^{-\alpha}\big{]}}$
$\displaystyle\times
e^{-\big{[}(\alpha+1)\sum_{i=1}^{r-N}\log(\tau+\frac{t_{i}-\tau}{\beta})+\lambda\sum_{i=1}^{r-N}(\tau+\frac{t_{i}-\tau}{\beta})^{-\alpha}\big{]}}\big{[}1-e^{-\lambda{(\tau+\frac{t_{r}-\tau}{\beta})}^{-\alpha}}\big{]}^{(n-r)}\bigg{]}d\alpha\leavevmode\nobreak\
d\lambda\leavevmode\nobreak\ d\beta.$

Under the loss functions SEL and LL, the Bayes estimates of
$h(\alpha,\lambda,\beta)$ can be written as
$\widehat{h}_{SE}(\alpha,\lambda,\beta)$ and
$\widehat{h}_{LI}(\alpha,\lambda,\beta)$, respectively, where

$\displaystyle\widehat{h}_{SE}(\alpha,\lambda,\beta)=$
$\displaystyle\int_{0}^{1}\int_{0}^{\infty}\int_{0}^{\infty}h(\alpha,\lambda,\beta)\pi(\alpha,\lambda,\beta|data)\leavevmode\nobreak\
d\alpha\leavevmode\nobreak\ d\lambda\leavevmode\nobreak\ d\beta,$ (3.5) and
$\displaystyle\widehat{h}_{LI}(\alpha,\lambda,\beta)=$
$\displaystyle-\bigg{(}\frac{1}{u}\bigg{)}\log\bigg{[}\int_{0}^{1}\int_{0}^{\infty}\int_{0}^{\infty}e^{-uh(\alpha,\lambda,\beta)}\pi(\alpha,\lambda,\beta|data)\leavevmode\nobreak\
d\alpha\leavevmode\nobreak\ d\lambda\leavevmode\nobreak\ d\beta\bigg{]}.$
(3.6)

Since $(\ref{3.3})$ and $(\ref{3.4})$ can not be solved explicitly, hence we
use a numerical method to solve these equations.

###  3.1 MCMC Method

In this subsection, Markov Chain Monte Carlo (MCMC) method is adopted to
enumerate the Bayes estimates of unknown parameters $\alpha$, $\lambda$, and
$\beta$ under both the loss functions SEL and LL. In addition, HPD intervals
are also composed by using the generated MCMC samples. From the posterior
density function given by $(\ref{3.2})$, the conditional posterior densities
can be written as

$\displaystyle\pi_{1}(\alpha|\lambda,\beta,data)\propto$
$\displaystyle\leavevmode\nobreak\
{\alpha}^{r+a-1}e^{-b\alpha}e^{-\big{[}(\alpha+1)\sum_{i=1}^{N}\log(t_{i})+\lambda\sum_{i=1}^{N}{t_{i}}^{-\alpha}\big{]}}\big{[}1-e^{-\lambda{(\tau+\frac{t_{r}-\tau}{\beta})}^{-\alpha}}\big{]}^{(n-r)}$
$\displaystyle\times
e^{-\big{[}(\alpha+1)\sum_{i=1}^{r-N}\log(\tau+\frac{t_{i}-\tau}{\beta})+\lambda\sum_{i=1}^{r-N}(\tau+\frac{t_{i}-\tau}{\beta})^{-\alpha}\big{]}},$
(3.7) $\displaystyle\pi_{2}(\lambda|\alpha,\beta,data)\propto$
$\displaystyle\leavevmode\nobreak\
{\lambda}^{r+c-1}e^{-d\lambda}e^{-\lambda\big{[}\sum_{i=1}^{N}{t_{i}}^{-\alpha}+\sum_{i=1}^{r-N}(\tau+\frac{t_{i}-\tau}{\beta})^{-\alpha}\big{]}}\big{[}1-e^{-\lambda{(\tau+\frac{t_{r}-\tau}{\beta})}^{-\alpha}}\big{]}^{(n-r)},$
(3.8) and $\displaystyle\pi_{3}(\beta|\alpha,\lambda,data)\propto$
$\displaystyle\leavevmode\nobreak\
\frac{(1-\beta)^{q-1}}{{\beta}^{N+1-r-p}}\leavevmode\nobreak\
e^{-\big{[}(\alpha+1)\sum_{i=1}^{r-N}\log(\tau+\frac{t_{i}-\tau}{\beta})+\lambda\sum_{i=1}^{r-N}(\tau+\frac{t_{i}-\tau}{\beta})^{-\alpha}\big{]}}$
$\displaystyle\times\big{[}1-e^{-\lambda{(\tau+\frac{t_{r}-\tau}{\beta})}^{-\alpha}}\big{]}^{(n-r)}.$
(3.9)

The above density functions $\pi_{1}(\alpha|\lambda,\beta,data)$,
$\pi_{2}(\lambda|\alpha,\beta,data)$, and $\pi_{3}(\beta|\alpha,\lambda,data)$
can not be written in the form of any well known distributions. Therefore, the
MCMC samples can not be generated from these densities given in $(\ref{3.5})$,
$(\ref{3.6})$, and $(\ref{3.7})$ directly. So, the Metropolis-Hastings
algorithm is used to generate MCMC samples from the conditional densities.
Then the Bayes estimates can be obtained by using the following steps:

Step 1: Choose initial values as ${\alpha}^{(1)}=\widehat{\alpha}$,
${\lambda}^{(1)}=\widehat{\lambda}$ and ${\beta}^{(1)}=\widehat{\beta}$ and
set $i=1$.
Step 2: Generate ${\alpha}^{(i)}$, ${\lambda}^{(i)}$, and ${\beta}^{(i)}$ with
normal distribution as ${\alpha}^{(i)}\sim
N\big{(}{\alpha}^{(i-1)},var(\widehat{\alpha})\big{)}$, ${\lambda}^{(i)}\sim
N\big{(}{\lambda}^{(i-1)},var(\widehat{\lambda})\big{)}$, and
${\beta}^{(i)}\sim N\big{(}{\beta}^{(i-1)},var(\widehat{\beta})\big{)}$.
Step 3: Compute
${\Omega}_{\alpha}=min\bigg{(}1,\frac{\pi_{1}\big{(}{\alpha}^{(i)}|{\lambda}^{(i-1)},{\beta}^{(i-1)},data\big{)}}{\pi_{1}\big{(}{\alpha}^{(i-1)}|{\lambda}^{(i-1)},{\beta}^{(i-1)},data\big{)}}\bigg{)}$,
${\Omega}_{\lambda}=min\bigg{(}1,\frac{\pi_{2}\big{(}{\lambda}^{(i)}|{\alpha}^{(i-1)},{\beta}^{(i-1)},data\big{)}}{\pi_{2}\big{(}{\lambda}^{(i-1)}|{\alpha}^{(i-1)},{\beta}^{(i-1)},data\big{)}}\bigg{)}$,
and
${\Omega}_{\beta}=min\bigg{(}1,\frac{\pi_{3}\big{(}{\beta}^{(i)}|{\alpha}^{(i-1)},{\lambda}^{(i-1)},data\big{)}}{\pi_{3}\big{(}{\beta}^{(i-1)}|{\alpha}^{(i-1)},{\lambda}^{(i-1)},data\big{)}}\bigg{)}$.
Step 4: Generate samples for $\tau_{1}$, $\tau_{2}$, and $\tau_{3}$, where
$\tau_{1}\sim Uniform(0,1)$, $\tau_{2}\sim Uniform(0,1)$, and $\tau_{3}\sim
Uniform(0,1)$.
Step 5: Set

$\left\\{\begin{array}[]{lr}\alpha={\alpha}^{(i)},&\mbox{if
$\tau_{1}\leq{\Omega}_{\alpha}$; otherwise $\alpha={\alpha}^{(i-1)}$},\\\
\lambda={\lambda}^{(i)},&\mbox{if $\tau_{2}\leq{\Omega}_{\lambda}$; otherwise
$\lambda={\lambda}^{(i-1)}$},\\\ \beta={\beta}^{(i)},&\mbox{if
$\tau_{3}\leq{\Omega}_{\beta}$; otherwise
$\beta={\beta}^{(i-1)}$}.\end{array}\right.$

Step 6: Set $i=i+1$.
Step 7: Repeat steps $1$ to $6$, M times to get
${\alpha}^{(1)},\cdots,{\alpha}^{(M)}$;
${\lambda}^{(1)},\cdots,{\lambda}^{(M)}$, and
${\beta}^{(1)},\cdots,{\beta}^{(M)}$.
Then, the Bayes estimates of $\alpha$, $\lambda$, and $\beta$ under SEL
function are given as

$\displaystyle\widehat{\alpha}_{SE}=\frac{1}{M}\sum_{i=1}^{M}{\alpha}^{(i)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\widehat{\lambda}_{SE}=\frac{1}{M}\sum_{i=1}^{M}{\lambda}^{(i)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\widehat{\beta}_{SE}=\frac{1}{M}\sum_{i=1}^{M}{\beta}^{(i)}.$

Further, the Bayes estimates of $\alpha$, $\lambda$, and $\beta$ under LL
function are given as

$\displaystyle\widehat{\alpha}_{LI}=$
$\displaystyle-\frac{1}{u}\log\bigg{(}\frac{1}{M}\sum_{i=1}^{M}e^{-u{\alpha}^{(i)}}\bigg{)},$
$\displaystyle\widehat{\lambda}_{LI}=$
$\displaystyle-\frac{1}{u}\log\bigg{(}\frac{1}{M}\sum_{i=1}^{M}e^{-u{\lambda}^{(i)}}\bigg{)},$
and $\displaystyle\widehat{\beta}_{LI}=$
$\displaystyle-\frac{1}{u}\log\bigg{(}\frac{1}{M}\sum_{i=1}^{M}e^{-u{\beta}^{(i)}}\bigg{)}.$

Furthermore, to construct $100(1-\gamma)\%$ HPD credible intervals for
$\alpha$, $\lambda$, and $\beta$, we use the method given by [15]. According
to the method, the samples are re-arranged in increasing order and these are
obtained as ${\alpha}^{[1]},\cdots,{\alpha}^{[M]}$;
${\lambda}^{[1]},\cdots,{\lambda}^{[M]}$ and
${\beta}^{[1]},\cdots,{\beta}^{[M]}$. Then, the $100(1-\gamma)\%$ HPD credible
intervals are obtained as

$\displaystyle\bigg{(}{\alpha}^{[M\gamma/2]},{\alpha}^{[M(1-\gamma/2)]}\bigg{)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\bigg{(}{\lambda}^{[M\gamma/2]},{\lambda}^{[M(1-\gamma/2)]}\bigg{)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\bigg{(}{\beta}^{[M\gamma/2]},{\beta}^{[M(1-\gamma/2)]}\bigg{)},$

where $\gamma$ is the nominal significance level.

##  4 Simulation Study

In this section, a simulation study is carried out to compare the performance
of different estimates of parameters for Gumbel Type-II distribution under
simple SSLT based on Type-II censoring. The performance of estimates are
compared on the basis of the average estimate (AE) values and mean squared
error (MSE). To observe the changes in the values of the parameters, the
simulation study is constructed based on $10,000$ Type-II censored samples
under simple SSLT. Three different choices of the sample size n= 50, 150, and
250 are taken to study the behavior of the estimates with change in the sample
size. Also, corresponding to each sample size we choose some moderate value
for $r$. We have also considered the Gumbel Type-II baseline lifetime
distribution with two choices of the shape parameter $\alpha=1,1.5$ and the
common scale parameter $0.75$. In the case of LL function, we consider the
values of $u$ as $-0.05$ and $1$. In Table 1 and Table 2, the AE, MSEs, and
average length (AL) of the corresponding asymptotic confidence intervals with
coverage probabilities (CP) based on classical estimation are presented with
$\tau=0.6$ and $0.75$, respectively. In Table 3 and Table 4, the average
values of Bayesian estimates under SEL and LL function with corresponding
MSEs, and AL of confidence intervals with corresponding CP of HPD credible
intervals are presented when $\alpha=1$ along with $\tau=0.6$ and $0.75$,
respectively. In Table 5 and Table 6, the Bayesian estimates under SEL and LL
function are presented when $\alpha=1.5$ along with $\tau=0.6$ and $0.75$,
respectively. From Table 1 and Table 2, the following conclusions have been
made:

  * •

For fixed values of $\tau$, $\beta$, $\alpha$, and $n$, when values of $r$
increase then in most of the cases the values of MSEs, average length of
asymptotic confidence intervals and corresponding coverage probabilities
decrease.

  * •

For fixed values of $\tau$, $\beta$, $\alpha$, when values of $n$ increase
then the average values of estimates, MSEs,and average length of asymptotic
confidence intervals decrease.

  * •

For fixed values of $\tau$, $\beta$, $n$, and $r$, when values of $\alpha$
increases then the average value of estimates for $\alpha$ and $\beta$
increase but the average values of estimates of $\lambda$ decrease.

  * •

For fixed values of $\tau$, $\alpha$, $n$, and $r$, when values of $\beta$
increase then the average values of estimates, MSEs and average length of
asymptotic confidence intervals decrease but the corresponding coverage
probabilities increase.

  * •

For fixed values of $\beta$, $\alpha$, $n$, and $r$, when values of $\tau$
increase then in most of the cases the MSEs, average length of asymptotic
confidence intervals and corresponding coverage probabilities decrease.

  * •

For fixed values of $\beta$, $\alpha$, $n$, and $r$, when values of $\tau$
increase then in most of the cases the average values of the estimates of
$\alpha$ and $\beta$ decrease but the average values of estimates of $\lambda$
increase.

Now, from Tables 3, 4, 5 and 6, the following conclusions can been made:

  * •

For fixed values of $\tau$, $\beta$, $\alpha$, $n$, and $r$, LL function under
$u=1$ performs better than other estimates. Also in that case, HPD intervals
perform better than asymptotic confidence intervals.

  * •

For fixed values of $\tau$, $\beta$, $\alpha$, and $n$, when values of $r$
increase then in most of the cases the values of MSEs and the average length
of HPD intervals decrease but the corresponding coverage probability
increases.

  * •

For fixed values of $\tau$, $\beta$, and $\alpha$, when values of $n$ increase
then MSEs and the average length of HPD intervals decrease.

  * •

For fixed values of $\tau$, $\beta$, $n$ and $r$, when values of $\alpha$
increase then the average values of the Bayes estimates, MSEs and the average
length of HPD intervals increase.

  * •

For fixed values of $\tau$, $\alpha$, $n$, and $r$, when values of $\beta$
increase then the average values of Bayes estimates, MSEs and average length
of HPD credible intervals decrease.

  * •

For fixed values of $\beta$, $\alpha$, $n$, and $r$, when values of $\tau$
increase then in most of the cases the MSEs, average length of HPD credible
intervals and corresponding coverage probabilities decrease.

Table 1: Simulation results for the Gumbel Type-II baseline lifetime
distribution with scale parameter $\lambda=0.75$ and shape parameter
$\alpha=1.5$ for different choices of $\beta$ when $\tau=0.6$ under simple
SSLT.

| $\alpha=1$ | $\alpha=1.5$ |
---|---|---|---
$\tau$ | $\beta$ | $n$ | $r$ | Parameters | AE | MSE | AL | CP | AE | MSE | AL | CP |
0.6 | 0.35 | 50 | 30 | $\alpha$ | 1.143 | 0.072 | 0.851 | 0.943 | 1.806 | 0.301 | 1.637 | 0.944 |
|  |  |  | $\lambda$ | 0.684 | 0.032 | 0.678 | 0.936 | 0.646 | 0.053 | 0.833 | 0.901 |
|  |  |  | $\beta$ | 0.486 | 0.061 | 0.839 | 0.978 | 0.495 | 0.063 | 0.829 | 0.981 |
|  |  | 40 | $\alpha$ | 1.143 | 0.068 | 0.816 | 0.940 | 1.788 | 0.270 | 1.570 | 0.943 |
|  |  |  | $\lambda$ | 0.682 | 0.031 | 0.660 | 0.937 | 0.651 | 0.051 | 0.817 | 0.903 |
|  |  |  | $\beta$ | 0.496 | 0.065 | 0.837 | 0.981 | 0.496 | 0.064 | 0.829 | 0.976 |
|  | 150 | 60 | $\alpha$ | 1.042 | 0.017 | 0.468 | 0.944 | 1.593 | 0.061 | 0.870 | 0.949 |
|  |  |  | $\lambda$ | 0.729 | 0.011 | 0.412 | 0.933 | 0.716 | 0.018 | 0.515 | 0.923 |
|  |  |  | $\beta$ | 0.406 | 0.024 | 0.543 | 0.961 | 0.395 | 0.014 | 0.435 | 0.966 |
|  |  | 120 | $\alpha$ | 1.046 | 0.015 | 0.438 | 0.944 | 1.590 | 0.054 | 0.812 | 0.944 |
|  |  |  | $\lambda$ | 0.727 | 0.010 | 0.396 | 0.926 | 0.716 | 0.017 | 0.495 | 0.914 |
|  |  |  | $\beta$ | 0.397 | 0.013 | 0.406 | 0.961 | 0.395 | 0.013 | 0.399 | 0.956 |
|  | 250 | 100 | $\alpha$ | 1.025 | 0.009 | 0.360 | 0.947 | 1.557 | 0.033 | 0.663 | 0.948 |
|  |  |  | $\lambda$ | 0.738 | 0.006 | 0.322 | 0.939 | 0.729 | 0.010 | 0.403 | 0.930 |
|  |  |  | $\beta$ | 0.381 | 0.011 | 0.377 | 0.956 | 0.378 | 0.007 | 0.322 | 0.959 |
|  |  | 200 | $\alpha$ | 1.027 | 0.008 | 0.334 | 0.943 | 1.555 | 0.029 | 0.618 | 0.945 |
|  |  |  | $\lambda$ | 0.735 | 0.006 | 0.309 | 0.934 | 0.729 | 0.010 | 0.387 | 0.928 |
|  |  |  | $\beta$ | 0.378 | 0.007 | 0.303 | 0.957 | 0.377 | 0.007 | 0.298 | 0.953 |
0.6 | 0.7 | 50 | 30 | $\alpha$ | 1.097 | 0.044 | 0.825 | 0.975 | 1.694 | 0.144 | 1.557 | 0.986 |
|  |  |  | $\lambda$ | 0.714 | 0.025 | 0.693 | 0.946 | 0.691 | 0.034 | 0.866 | 0.949 |
|  |  |  | $\beta$ | 0.820 | 0.056 | 1.381 | 0.978 | 0.831 | 0.056 | 1.383 | 0.983 |
|  |  | 40 | $\alpha$ | 1.085 | 0.035 | 0.773 | 0.980 | 1.665 | 0.112 | 1.457 | 0.987 |
|  |  |  | $\lambda$ | 0.720 | 0.023 | 0.674 | 0.953 | 0.699 | 0.031 | 0.842 | 0.960 |
|  |  |  | $\beta$ | 0.826 | 0.056 | 1.377 | 0.978 | 0.832 | 0.057 | 1.384 | 0.980 |
|  | 150 | 60 | $\alpha$ | 1.036 | 0.014 | 0.469 | 0.956 | 1.576 | 0.050 | 0.864 | 0.966 |
|  |  |  | $\lambda$ | 0.734 | 0.010 | 0.469 | 0.942 | 0.724 | 0.015 | 0.518 | 0.948 |
|  |  |  | $\beta$ | 0.769 | 0.040 | 0.966 | 0.961 | 0.763 | 0.033 | 0.829 | 0.972 |
|  |  | 120 | $\alpha$ | 1.038 | 0.012 | 0.435 | 0.966 | 1.574 | 0.040 | 0.804 | 0.972 |
|  |  |  | $\lambda$ | 0.732 | 0.009 | 0.397 | 0.951 | 0.723 | 0.014 | 0.497 | 0.954 |
|  |  |  | $\beta$ | 0.769 | 0.032 | 0.785 | 0.974 | 0.771 | 0.032 | 0.779 | 0.977 |
|  | 250 | 100 | $\alpha$ | 1.022 | 0.008 | 0.359 | 0.955 | 1.551 | 0.029 | 0.661 | 0.958 |
|  |  |  | $\lambda$ | 0.740 | 0.006 | 0.323 | 0.953 | 0.731 | 0.010 | 0.403 | 0.944 |
|  |  |  | $\beta$ | 0.745 | 0.027 | 0.726 | 0.964 | 0.745 | 0.023 | 0.632 | 0.971 |
|  |  | 200 | $\alpha$ | 1.025 | 0.008 | 0.334 | 0.958 | 1.549 | 0.025 | 0.615 | 0.962 |
|  |  |  | $\lambda$ | 0.737 | 0.006 | 0.309 | 0.946 | 0.731 | 0.009 | 0.387 | 0.947 |
|  |  |  | $\beta$ | 0.748 | 0.022 | 0.599 | 0.976 | 0.747 | 0\. 021 | 0.591 | 0.973 |

Table 2: Simulation results for the Gumbel Type-II baseline lifetime
distribution with scale parameter $\lambda=0.75$ and shape parameter
$\alpha=1.5$ for different choices of $\beta$ when $\tau=0.75$ under simple
SSLT.

| $\alpha=1$ | $\alpha=1.5$ |
---|---|---|---
$\tau$ | $\beta$ | $n$ | $r$ | Parameters | AE | MSE | AL | CP | AE | MSE | AL | CP |
0.75 | 0.35 | 50 | 30 | $\alpha$ | 1.107 | 0.047 | 0.734 | 0.947 | 1.683 | 0.137 | 1.215 | 0.942 |
|  |  |  | $\lambda$ | 0.705 | 0.024 | 0.605 | 0.918 | 0.699 | 0.029 | 0.659 | 0.908 |
|  |  |  | $\beta$ | 0.478 | 0.061 | 0.869 | 0.971 | 0.458 | 0.046 | 0.742 | 0.972 |
|  |  | 40 | $\alpha$ | 1.108 | 0.046 | 0.712 | 0.942 | 1.694 | 0.139 | 1.197 | 0.938 |
|  |  |  | $\lambda$ | 0.705 | 0.024 | 0.596 | 0.912 | 0.694 | 0.028 | 0.649 | 0.903 |
|  |  |  | $\beta$ | 0.476 | 0.055 | 0.783 | 0.977 | 0.464 | 0.045 | 0.699 | 0.979 |
|  | 150 | 60 | $\alpha$ | 1.036 | 0.014 | 0.408 | 0.945 | 1.564 | 0.038 | 0.663 | 0.941 |
|  |  |  | $\lambda$ | 0.731 | 0.009 | 0.360 | 0.936 | 0.726 | 0.033 | 0.388 | 0.932 |
|  |  |  | $\beta$ | 0.407 | 0.028 | 0.599 | 0.953 | 0.409 | 0.033 | 0.664 | 0.942 |
|  |  | 120 | $\alpha$ | 1.040 | 0.012 | 0.393 | 0.940 | 1.571 | 0.033 | 0.654 | 0.942 |
|  |  |  | $\lambda$ | 0.728 | 0.008 | 0.352 | 0.928 | 0.724 | 0.010 | 0.385 | 0.925 |
|  |  |  | $\beta$ | 0.387 | 0.011 | 0.381 | 0.958 | 0.382 | 0.009 | 0.344 | 0.958 |
|  | 250 | 100 | $\alpha$ | 1.024 | 0.009 | 0.311 | 0.941 | 1.538 | 0.020 | 0.512 | 0.944 |
|  |  |  | $\lambda$ | 0.736 | 0.007 | 0.279 | 0.938 | 0.735 | 0.006 | 0.304 | 0.936 |
|  |  |  | $\beta$ | 0.388 | 0.007 | 0.511 | 0.944 | 0.383 | 0.015 | 0.427 | 0.954 |
|  |  | 200 | $\alpha$ | 1.024 | 0.007 | 0.302 | 0.943 | 1.538 | 0.020 | 0.304 | 0.942 |
|  |  |  | $\lambda$ | 0.736 | 0.005 | 0.275 | 0.935 | 0.735 | 0.006 | 0.304 | 0.936 |
|  |  |  | $\beta$ | 0.372 | 0.006 | 0.287 | 0.956 | 0.384 | 0.015 | 0.433 | 0.953 |
0.75 | 0.7 | 50 | 30 | $\alpha$ | 1.079 | 0.035 | 0.721 | 0.966 | 1.635 | 0.096 | 1.190 | 0.968 |
|  |  |  | $\lambda$ | 0.726 | 0.021 | 0.614 | 0.943 | 0.717 | 0.024 | 0.666 | 0.942 |
|  |  |  | $\beta$ | 0.807 | 0.055 | 1.382 | 0.971 | 0.801 | 0.051 | 1.250 | 0.975 |
|  |  | 40 | $\alpha$ | 1.074 | 0.030 | 0.692 | 1.311 | 1.630 | 0.083 | 1.160 | 0.976 |
|  |  |  | $\lambda$ | 0.726 | 0.020 | 0.630 | 0.945 | 0.720 | 0.023 | 0.661 | 0.948 |
|  |  |  | $\beta$ | 0.814 | 0.035 | 1.311 | 0.975 | 0.809 | 0.050 | 1.206 | 0.978 |
|  | 150 | 60 | $\alpha$ | 1.030 | 0.011 | 0.407 | 0.955 | 1.554 | 0.030 | 0.669 | 0.956 |
|  |  |  | $\lambda$ | 0.735 | 0.008 | 0.361 | 0.941 | 0.732 | 0.009 | 0.393 | 0.944 |
|  |  |  | $\beta$ | 0.754 | 0.035 | 0.892 | 0.961 | 0.749 | 0.032 | 0.813 | 0.961 |
|  |  | 120 | $\alpha$ | 1.038 | 0.010 | 0.393 | 0.955 | 1.562 | 0.029 | 0.650 | 0.961 |
|  |  |  | $\lambda$ | 0.731 | 0.007 | 0.353 | 0.942 | 0.727 | 0.009 | 0.386 | 0.934 |
|  |  |  | $\beta$ | 0.756 | 0.030 | 0.742 | 0.968 | 0.753 | 0.026 | 0.678 | 0.972 |
|  | 250 | 100 | $\alpha$ | 1.020 | 0.007 | 0.314 | 0.952 | 1.534 | 0.018 | 0.513 | 0.952 |
|  |  |  | $\lambda$ | 0.740 | 0.005 | 0.282 | 0.947 | 0.738 | 0.006 | 0.305 | 0.945 |
|  |  |  | $\beta$ | 0.742 | 0.027 | 0.712 | 0.958 | 0.737 | 0.024 | 0.660 | 0.962 |
|  |  | 200 | $\alpha$ | 1.025 | 0.006 | 0.302 | 0.953 | 1.539 | 0.017 | 0.499 | 0.951 |
|  |  |  | $\lambda$ | 0.736 | 0.004 | 0.275 | 0.942 | 0.735 | 0.006 | 0.301 | 0.942 |
|  |  |  | $\beta$ | 0.740 | 0.020 | 0.568 | 0.970 | 0.733 | 0.017 | 0.517 | 0.969 |

Table 3: Simulation results of Bayesian estimates for the Gumbel Type-II
baseline lifetime distribution with scale parameter $\lambda=0.75$ and shape
parameter $\alpha=1$ for different choices of $\beta$ when $\tau=0.6$ under
simple SSLT.

| SEL |  LL ($u=-0.05$) |  LL ($u=1$) | HPD |
---|---|---|---|---|---
$\tau$ | $\beta$ | $n$ | $r$ | Parameters | AE | MSE | AE | MSE | AE | MSE | AL | CP |
0.6 | 0.35 | 50 | 30 | $\alpha$ | 1.021 | 0.015 | 1.022 | 0.015 | 1.015 | 0.014 | 0.443 | 0.962 |
|  |  |  | $\lambda$ | 0.737 | 0.008 | 0.736 | 0.008 | 0.734 | 0.008 | 0.338 | 0.956 |
|  |  |  | $\beta$ | 0.393 | 0.002 | 0.393 | 0.002 | 0.392 | 0.002 | 0.133 | 0.986 |
|  |  | 40 | $\alpha$ | 1.027 | 0.014 | 1.028 | 0.014 | 1.022 | 0.013 | 0.358 | 0.968 |
|  |  |  | $\lambda$ | 0.734 | 0.007 | 0.734 | 0.007 | 0.732 | 0.008 | 0.220 | 0.962 |
|  |  |  | $\beta$ | 0.392 | 0.002 | 0.392 | 0.002 | 0.391 | 0.002 | 0.124 | 0.988 |
|  | 150 | 60 | $\alpha$ | 1.014 | 0.008 | 1.015 | 0.008 | 1.010 | 0.007 | 0.293 | 0.974 |
|  |  |  | $\lambda$ | 0.739 | 0.006 | 0.739 | 0.006 | 0.737 | 0.005 | 0.202 | 0.932 |
|  |  |  | $\beta$ | 0.391 | 0.002 | 0.391 | 0.002 | 0.389 | 0.001 | 0.129 | 0.983 |
|  |  | 120 | $\alpha$ | 1.042 | 0.007 | 1.042 | 0.007 | 1.039 | 0.006 | 0.279 | 0.983 |
|  |  |  | $\lambda$ | 0.727 | 0.005 | 0.726 | 0.005 | 0.724 | 0.005 | 0.193 | 0.957 |
|  |  |  | $\beta$ | 0.388 | 0.001 | 0.388 | 0.001 | 0.0386 | 0.001 | 0.114 | 0.989 |
|  | 250 | 100 | $\alpha$ | 1.029 | 0.006 | 1.029 | 0.006 | 1.026 | 0.005 | 0.365 | 0.945 |
|  |  |  | $\lambda$ | 0.732 | 0.004 | 0.732 | 0.004 | 0.730 | 0.004 | 0.186 | 0.944 |
|  |  |  | $\beta$ | 0.394 | 0.002 | 0.394 | 0.002 | 0.392 | 0.002 | 0.117 | 0.973 |
|  |  | 200 | $\alpha$ | 1.052 | 0.006 | 1.052 | 0.006 | 1.049 | 0.005 | 0.259 | 0.947 |
|  |  |  | $\lambda$ | 0.721 | 0.004 | 0.721 | 0.004 | 0.719 | 0.004 | 0.188 | 0.949 |
|  |  |  | $\beta$ | 0.435 | 0.007 | 0.435 | 0.007 | 0.433 | 0.007 | 0.208 | 0.994 |
0.6 | 0.7 | 50 | 30 | $\alpha$ | 1.014 | 0.014 | 1.015 | 0.014 | 1.008 | 0.013 | 0.368 | 0.942 |
|  |  |  | $\lambda$ | 0.743 | 0.007 | 0.744 | 0.007 | 0.741 | 0.008 | 0.221 | 0.931 |
|  |  |  | $\beta$ | 0.793 | 0.009 | 0.793 | 0.009 | 0.789 | 0.008 | 0.257 | 0.967 |
|  |  | 40 | $\alpha$ | 1.020 | 0.012 | 1.020 | 0.012 | 1.014 | 0.011 | 0.356 | 0.944 |
|  |  |  | $\lambda$ | 0.739 | 0.008 | 0.739 | 0.008 | 0.736 | 0.008 | 0.225 | 0.939 |
|  |  |  | $\beta$ | 0.792 | 0.009 | 0.793 | 0.009 | 0.789 | 0.008 | 0.256 | 0.977 |
|  | 150 | 60 | $\alpha$ | 1.013 | 0.008 | 1.013 | 0.008 | 1.009 | 0.007 | 0.292 | 0.952 |
|  |  |  | $\lambda$ | 0.743 | 0.005 | 0.743 | 0.005 | 0.741 | 0.005 | 0.201 | 0.931 |
|  |  |  | $\beta$ | 0.788 | 0.008 | 0.788 | 0.008 | 0.785 | 0.007 | 0.249 | 0.966 |
|  |  | 120 | $\alpha$ | 1.2024 | 0.006 | 1.023 | 0.006 | 1.020 | 0.005 | 0.276 | 0.941 |
|  |  |  | $\lambda$ | 0.737 | 0.005 | 0.738 | 0.005 | 0.735 | 0.005 | 0.195 | 0.956 |
|  |  |  | $\beta$ | 0.787 | 0.008 | 0.787 | 0.008 | 0.785 | 0.007 | 0.246 | 0.973 |
|  | 250 | 100 | $\alpha$ | 1.020 | 0.006 | 1.020 | 0.006 | 1.017 | 0.005 | 0.255 | 0.953 |
|  |  |  | $\lambda$ | 0.738 | 0.004 | 0.738 | 0.004 | 0.736 | 0.004 | 0.185 | 0.946 |
|  |  |  | $\beta$ | 0.782 | 0.007 | 0.783 | 0.007 | 0.779 | 0.006 | 0.238 | 0.983 |
|  |  | 200 | $\alpha$ | 1.029 | 0.004 | 1.029 | 0.004 | 1.027 | 0.003 | 0.224 | 0.963 |
|  |  |  | $\lambda$ | 0.732 | 0.003 | 0.732 | 0.003 | 0.730 | 0.003 | 0.176 | 0.927 |
|  |  |  | $\beta$ | 0.782 | 0.007 | 0.782 | 0.007 | 0.779 | 0.006 | 0.237 | 0.991 |

Table 4: Simulation results of Bayesian estimates for the Gumbel Type-II
baseline lifetime distribution with scale parameter $\lambda=0.75$ and shape
parameter $\alpha=1$ for different choices of $\beta$ when $\tau=0.75$ under
simple SSLT.

| SEL |  LL ($u=-0.05$) |  LL ($u=1$) | HPD |
---|---|---|---|---|---
$\tau$ | $\beta$ | $n$ | $r$ | Parameters | AE | MSE | AE | MSE | AE | MSE | AL | CP |
0.75 | 0.35 | 50 | 30 | $\alpha$ | 1.009 | 0.014 | 1.010 | 0.014 | 1.004 | 0.013 | 0.366 | 0.938 |
|  |  |  | $\lambda$ | 0.743 | 0.007 | 0.743 | 0.007 | 0.740 | 0.007 | 0.222 | 0.949 |
|  |  |  | $\beta$ | 0.383 | 0.001 | 0.383 | 0.001 | 0.383 | 0.001 | 0.134 | 0.943 |
|  |  | 40 | $\alpha$ | 1.013 | 0.013 | 1.014 | 0.013 | 1.007 | 0.012 | 0.359 | 0.946 |
|  |  |  | $\lambda$ | 0.744 | 0.008 | 0.744 | 0.008 | 0.741 | 0.008 | 0.220 | 0.955 |
|  |  |  | $\beta$ | 0.384 | 0.001 | 0.384 | 0.001 | 0.383 | 0.001 | 0.134 | 0.962 |
|  | 150 | 60 | $\alpha$ | 1.027 | 0.012 | 1.028 | 0.012 | 1.023 | 0.011 | 0.310 | 0.941 |
|  |  |  | $\lambda$ | 0.732 | 0.009 | 0.732 | 0.009 | 0.730 | 0.009 | 0.212 | 0.934 |
|  |  |  | $\beta$ | 0.381 | 0.001 | 0.381 | 0.001 | 0.379 | 0.001 | 0.128 | 0.953 |
|  |  | 120 | $\alpha$ | 1.017 | 0.005 | 1.018 | 0.005 | 1.014 | 0.004 | 0.265 | 0.963 |
|  |  |  | $\lambda$ | 0.742 | 0.005 | 0.742 | 0.005 | 0.740 | 0.005 | 0.194 | 0.930 |
|  |  |  | $\beta$ | 0.380 | 0.001 | 0.380 | 0.001 | 0.379 | 0.001 | 0.127 | 0.964 |
|  | 250 | 100 | $\alpha$ | 1.020 | 0.008 | 1.021 | 0.008 | 1.017 | 0.007 | 0.262 | 0.939 |
|  |  |  | $\lambda$ | 0.736 | 0.006 | 0.736 | 0.006 | 0.734 | 0.006 | 0.193 | 0.926 |
|  |  |  | $\beta$ | 0.378 | 0.001 | 0.378 | 0.001 | 0.377 | 0.001 | 0.124 | 0.947 |
|  |  | 200 | $\alpha$ | 1.014 | 0.003 | 1.014 | 0.003 | 1.012 | 0.003 | 0.220 | 0.946 |
|  |  |  | $\lambda$ | 0.741 | 0.003 | 0.741 | 0.003 | 0.739 | 0.003 | 0.177 | 0.957 |
|  |  |  | $\beta$ | 0.377 | 0.001 | 0.377 | 0.001 | 0.376 | 0.001 | 0.122 | 0.962 |
0.75 | 0.7 | 50 | 30 | $\alpha$ | 1.023 | 0.014 | 1.024 | 0.014 | 1.017 | 0.013 | 0.367 | 0.922 |
|  |  |  | $\lambda$ | 0.738 | 0.008 | 0.738 | 0.008 | 0.735 | 0.008 | 0.227 | 0.935 |
|  |  |  | $\beta$ | 0.756 | 0.003 | 0.756 | 0.003 | 0.754 | 0.003 | 0.175 | 0.957 |
|  |  | 40 | $\alpha$ | 1.016 | 0.012 | 1.016 | 0.012 | 1.010 | 0.011 | 0.352 | 0.936 |
|  |  |  | $\lambda$ | 0.742 | 0.008 | 0.742 | 0.008 | 0.739 | 0.007 | 0.223 | 0.942 |
|  |  |  | $\beta$ | 0.755 | 0.003 | 0.756 | 0.003 | 0.754 | 0.003 | 0.174 | 0.962 |
|  | 150 | 60 | $\alpha$ | 1.020 | 0.010 | 1.021 | 0.010 | 1.010 | 0.009 | 0.303 | 0.940 |
|  |  |  | $\lambda$ | 0.737 | 0.007 | 0.737 | 0.007 | 0.735 | 0.007 | 0.211 | 0.932 |
|  |  |  | $\beta$ | 0.750 | 0.003 | 0.750 | 0.003 | 0.747 | 0.002 | 0.165 | 0.956 |
|  |  | 120 | $\alpha$ | 1.023 | 0.006 | 1.023 | 0.006 | 1.019 | 0.005 | 0.264 | 0.958 |
|  |  |  | $\lambda$ | 0.742 | 0.005 | 0.743 | 0.005 | 0.741 | 0.004 | 0.193 | 0.957 |
|  |  |  | $\beta$ | 0.751 | 0.003 | 0.751 | 0.003 | 0.750 | 0.003 | 0.166 | 0.961 |
|  | 250 | 100 | $\alpha$ | 1.015 | 0.007 | 1.014 | 0.007 | 1.012 | 0.006 | 0.254 | 0.931 |
|  |  |  | $\lambda$ | 0.744 | 0.005 | 0.744 | 0.005 | 0.742 | 0.005 | 0.192 | 0.945 |
|  |  |  | $\beta$ | 0.745 | 0.002 | 0.745 | 0.002 | 0.744 | 0.002 | 0.165 | 0.974 |
|  |  | 200 | $\alpha$ | 1.017 | 0.004 | 1.017 | 0.004 | 1.015 | 0.003 | 0.208 | 0.934 |
|  |  |  | $\lambda$ | 0.740 | 0.003 | 0.740 | 0.003 | 0.738 | 0.003 | 0.175 | 0.966 |
|  |  |  | $\beta$ | 0.746 | 0.002 | 0.746 | 0.002 | 0.745 | 0.002 | 0.155 | 0.983 |

Table 5: Simulation results of Bayesian estimates for the Gumbel Type-II
baseline lifetime distribution with scale parameter $\lambda=0.75$ and shape
parameter $\alpha=1.5$ for different choices of $\beta$ when $\tau=0.6$ under
simple SSLT.

| SEL |  LL($u=-0.05$) |  LL($u=0.5$) | HPD |
---|---|---|---|---|---
$\tau$ | $\beta$ | $n$ | $r$ | Parameters | AE | MSE | AE | MSE | AE | MSE | AL | CP |
0.6 | 0.35 | 50 | 30 | $\alpha$ | 1.505 | 0.026 | 1.506 | 0.026 | 1.497 | 0.025 | 0.422 | 0.958 |
|  |  |  | $\lambda$ | 0.737 | 0.007 | 0.737 | 0.007 | 0.735 | 0.007 | 0.225 | 0.963 |
|  |  |  | $\beta$ | 0.394 | 0.002 | 0.394 | 0.002 | 0.393 | 0.002 | 0.125 | 0.975 |
|  |  | 40 | $\alpha$ | 1.511 | 0.022 | 1.512 | 0.022 | 1.503 | 0.021 | 0.415 | 0.972 |
|  |  |  | $\lambda$ | 0.742 | 0.007 | 0.743 | 0.007 | 0.741 | 0.007 | 0.219 | 0.975 |
|  |  |  | $\beta$ | 0.394 | 0.002 | 0.394 | 0.002 | 0.393 | 0.002 | 0.114 | 0.990 |
|  | 150 | 60 | $\alpha$ | 1.519 | 0.017 | 1.520 | 0.017 | 1.513 | 0.016 | 0.378 | 0.947 |
|  |  |  | $\lambda$ | 0.736 | 0.005 | 0.736 | 0.005 | 0.733 | 0.006 | 0.204 | 0.954 |
|  |  |  | $\beta$ | 0.392 | 0.002 | 0.392 | 0.002 | 0.391 | 0.002 | 0.131 | 0.987 |
|  |  | 120 | $\alpha$ | 1.538 | 0.012 | 1.539 | 0.012 | 1.533 | 0.011 | 0.352 | 0.962 |
|  |  |  | $\lambda$ | 0.735 | 0.005 | 0.735 | 0.004 | 0.733 | 0.005 | 0.196 | 0.943 |
|  |  |  | $\beta$ | 0.392 | 0.002 | 0.392 | 0.002 | 0.391 | 0.002 | 0.130 | 0.989 |
|  | 250 | 100 | $\alpha$ | 1.522 | 0.012 | 1.523 | 0.012 | 1.517 | 0.011 | 0.334 | 0.965 |
|  |  |  | $\lambda$ | 0.740 | 0.004 | 0.741 | 0.004 | 0.738 | 0.004 | 0.185 | 0.946 |
|  |  |  | $\beta$ | 0.381 | 0.001 | 0.381 | 0.001 | 0.380 | 0.001 | 0.129 | 0.981 |
|  |  | 200 | $\alpha$ | 1.531 | 0.008 | 1.531 | 0.008 | 1.526 | 0.007 | 0.325 | 0.957 |
|  |  |  | $\lambda$ | 0.736 | 0.003 | 0.736 | 0.003 | 0.734 | 0.003 | 0.189 | 0.936 |
|  |  |  | $\beta$ | 0.381 | 0.001 | 0.381 | 0.001 | 0.379 | 0.001 | 0.128 | 0.983 |
0.6 | 0.7 | 50 | 30 | $\alpha$ | 1.511 | 0.025 | 1.511 | 0.025 | 1.503 | 0.024 | 0.434 | 0.935 |
|  |  |  | $\lambda$ | 0.733 | 0.008 | 0.733 | 0.008 | 0.731 | 0.008 | 0.229 | 0.947 |
|  |  |  | $\beta$ | 0.793 | 0.009 | 0.793 | 0.009 | 0.789 | 0.008 | 0.258 | 0.953 |
|  |  | 40 | $\alpha$ | 1.516 | 0.021 | 1.516 | 0.021 | 1.507 | 0.019 | 0.419 | 0.942 |
|  |  |  | $\lambda$ | 0.732 | 0.007 | 0.732 | 0.007 | 0.729 | 0.008 | 0.221 | 0.939 |
|  |  |  | $\beta$ | 0.794 | 0.009 | 0.794 | 0.009 | 0.790 | 0.008 | 0.253 | 0.974 |
|  | 150 | 60 | $\alpha$ | 1.522 | 0.016 | 1.522 | 0.016 | 1.516 | 0.015 | 0.367 | 0.948 |
|  |  |  | $\lambda$ | 0.736 | 0.005 | 0.737 | 0.005 | 0.735 | 0.005 | 0.169 | 0.947 |
|  |  |  | $\beta$ | 0.789 | 0.008 | 0.789 | 0.008 | 0.786 | 0.007 | 0.251 | 0.958 |
|  |  | 120 | $\alpha$ | 1.544 | 0.013 | 1.543 | 0.013 | 1.538 | 0.012 | 0.362 | 0.953 |
|  |  |  | $\lambda$ | 0.728 | 0.005 | 0.727 | 0.005 | 0.725 | 0.005 | 0.201 | 0.929 |
|  |  |  | $\beta$ | 0.790 | 0.008 | 0.790 | 0.008 | 0.787 | 0.007 | 0.251 | 0.965 |
|  | 250 | 100 | $\alpha$ | 1.529 | 0.011 | 1.529 | 0.011 | 1.524 | 0.010 | 0.336 | 0.937 |
|  |  |  | $\lambda$ | 0.733 | 0.004 | 0.733 | 0.004 | 0.731 | 0.004 | 0.188 | 0.933 |
|  |  |  | $\beta$ | 0.787 | 0.008 | 0.788 | 0.008 | 0.784 | 0.007 | 0.246 | 0.971 |
|  |  | 200 | $\alpha$ | 1.549 | 0.009 | 1.549 | 0.009 | 1.544 | 0.008 | 0.332 | 0.940 |
|  |  |  | $\lambda$ | 0.724 | 0.003 | 0.724 | 0.003 | 0.721 | 0.004 | 0.186 | 0.947 |
|  |  |  | $\beta$ | 0.786 | 0.008 | 0.786 | 0.008 | 0.783 | 0.007 | 0.245 | 0.984 |

Table 6: Simulation results of Bayesian estimates for the Gumbel Type-II
baseline lifetime distribution with scale parameter $\lambda=0.75$ and shape
parameter $\alpha=1.5$ for different choices of $\beta$ when $\tau=0.75$ under
simple SSLT.

| SEL |  LL ($u=-0.05$) |  LL ($u=1$) | HPD |
---|---|---|---|---|---
$\tau$ | $\beta$ | $n$ | $r$ | Parameters | AE | MSE | AE | MSE | AE | MSE | AL | CP |
0.75 | 0.35 | 50 | 30 | $\alpha$ | 1.510 | 0.025 | 1.511 | 0.025 | 1.502 | 0.024 | 0.489 | 0.956 |
|  |  |  | $\lambda$ | 0.739 | 0.007 | 0.739 | 0.007 | 0.736 | 0.008 | 0.219 | 0.916 |
|  |  |  | $\beta$ | 0.384 | 0.001 | 0.384 | 0.001 | 0.383 | 0.001 | 0.135 | 0.948 |
|  |  | 40 | $\alpha$ | 1.513 | 0.021 | 1.514 | 0.022 | 1.505 | 0.021 | 0.413 | 0.969 |
|  |  |  | $\lambda$ | 0.739 | 0.008 | 0.740 | 0.008 | 0.737 | 0.008 | 0.224 | 0.939 |
|  |  |  | $\beta$ | 0.382 | 0.001 | 0.382 | 0.001 | 0.383 | 0.001 | 0.133 | 0.967 |
|  | 150 | 60 | $\alpha$ | 1.513 | 0.016 | 1.514 | 0.016 | 1.508 | 0.015 | 0.365 | 0.943 |
|  |  |  | $\lambda$ | 0.746 | 0.005 | 0.746 | 0.005 | 0.744 | 0.005 | 0.199 | 0.932 |
|  |  |  | $\beta$ | 0.382 | 0.001 | 0.382 | 0.001 | 0.381 | 0.001 | 0.130 | 0.948 |
|  |  | 120 | $\alpha$ | 1.528 | 0.012 | 1.529 | 0.012 | 1.523 | 0.011 | 0.351 | 0.952 |
|  |  |  | $\lambda$ | 0.738 | 0.005 | 0.738 | 0.005 | 0.736 | 0.005 | 0.197 | 0.958 |
|  |  |  | $\beta$ | 0.381 | 0.001 | 0.381 | 0.001 | 0.380 | 0.001 | 0.129 | 0.959 |
|  | 250 | 100 | $\alpha$ | 1.509 | 0.012 | 1.510 | 0.012 | 1.505 | 0.011 | 0.328 | 0.955 |
|  |  |  | $\lambda$ | 0.746 | 0.004 | 0.747 | 0.004 | 0.744 | 0.004 | 0.184 | 0.933 |
|  |  |  | $\beta$ | 0.379 | 0.001 | 0.379 | 0.001 | 0.378 | 0.001 | 0.124 | 0.948 |
|  |  | 200 | $\alpha$ | 1.526 | 0.007 | 1.526 | 0.007 | 1.522 | 0.007 | 0.311 | 0.964 |
|  |  |  | $\lambda$ | 0.740 | 0.003 | 0.740 | 0.003 | 0.738 | 0.003 | 0.179 | 0.942 |
|  |  |  | $\beta$ | 0.378 | 0.001 | 0.378 | 0.001 | 0.377 | 0.001 | 0.121 | 0.987 |
0.75 | 0.7 | 50 | 30 | $\alpha$ | 1.507 | 0.027 | 1.507 | 0.027 | 1.499 | 0.026 | 0.428 | 0.939 |
|  |  |  | $\lambda$ | 0.744 | 0.008 | 0.744 | 0.008 | 0.741 | 0.007 | 0.223 | 0.935 |
|  |  |  | $\beta$ | 0.755 | 0.003 | 0.755 | 0.003 | 0.753 | 0.003 | 0.175 | 0.958 |
|  |  | 40 | $\alpha$ | 1.516 | 0.022 | 1.517 | 0.022 | 1.508 | 0.021 | 0.402 | 0.941 |
|  |  |  | $\lambda$ | 0.741 | 0.008 | 0.741 | 0.008 | 0.738 | 0.008 | 0.221 | 0.946 |
|  |  |  | $\beta$ | 0.756 | 0.003 | 0.756 | 0.003 | 0.754 | 0.003 | 0.173 | 0.966 |
|  | 150 | 60 | $\alpha$ | 1.525 | 0.016 | 1.526 | 0.016 | 1.520 | 0.015 | 0.360 | 0.962 |
|  |  |  | $\lambda$ | 0.742 | 0.005 | 0.742 | 0.005 | 0.740 | 0.005 | 0.198 | 0.938 |
|  |  |  | $\beta$ | 0.752 | 0.003 | 0.752 | 0.003 | 0.750 | 0.003 | 0.167 | 0.969 |
|  |  | 120 | $\alpha$ | 1.531 | 0.013 | 1.532 | 0.013 | 1.526 | 0.012 | 0.325 | 0.934 |
|  |  |  | $\lambda$ | 0.734 | 0.005 | 0.734 | 0.005 | 0.732 | 0.005 | 0.196 | 0.955 |
|  |  |  | $\beta$ | 0.752 | 0.003 | 0.752 | 0.003 | 0.750 | 0.003 | 0.163 | 0.981 |
|  | 250 | 100 | $\alpha$ | 1.518 | 0.012 | 1.518 | 0.012 | 1.513 | 0.011 | 0.342 | 0.933 |
|  |  |  | $\lambda$ | 0.743 | 0.004 | 0.743 | 0.004 | 0.741 | 0.004 | 0.183 | 0.941 |
|  |  |  | $\beta$ | 0.748 | 0.002 | 0.748 | 0.003 | 0.746 | 0.002 | 0.161 | 0.968 |
|  |  | 200 | $\alpha$ | 1.529 | 0.007 | 1.530 | 0.007 | 1.526 | 0.007 | 0.296 | 0.944 |
|  |  |  | $\lambda$ | 0.739 | 0.003 | 0.740 | 0.003 | 0.738 | 0.003 | 0.174 | 0.964 |
|  |  |  | $\beta$ | 0.748 | 0.003 | 0.748 | 0.003 | 0.746 | 0.002 | 0.160 | 0.981 |

##  5 Real Data Analysis

In this section, we consider a real data set from Lee and Wang [16] to
illustrate the estimation methods developed in this paper. The data set
contains remission time (in months) of a random sample of $124$ bladder cancer
patients and the corresponding failure times are given in Table 7. To check
the goodness-of-fit of this data to the Gumbel Type-II distribution, K-S test
has been employed. From this test, we observe the K-S distance is $0.14243$
and the corresponding $p$-value is $0.01306$. Also, for the purpose of
goodness-of-fit test, different plots are considered in Figure $1$ and Figure
$2$. Figure $1$ represents the comparison between the theoretical CDF of
Gumbel Type-II distribution and the empirical CDF and QQ-plot of the given
real data set. If $(X_{1},\cdots,X_{n})$ are $n$ number of i.i.d. random
variables with CDF $F(t)$, then the empirical CDF (ECDF) is given as
$F_{n}(t)=\frac{1}{n}\sum_{i=1}^{n}I_{X_{i}\leq t}$, where $I_{W}$ denotes the
indicator of the event $W$. Here Q-Q plot represents the points
$(F^{-1}(i/(n+1)),x_{(i)})$, where $x_{(i)}$ denotes the ordered data for
$i=1,\cdots,n$. Then Figure $2$ represents the comparison between the
theoretical density of Gumbel Type-II distribution and the histogram and box-
plot of given real data. From the box-plot, it can be concluded that the given
distribution is right-skewed.

Table 7: Lifetimes of the failure units are presented for Real Data. $\tau$ | $r$ | Stress level | Failure time
---|---|---|---
2.5 | 50 | 1 | 0.080 0.200 0.400 0.500 0.510 0.810 0.900 1.050 1.190 1.260 1.350 1.400 1.460
|  |  | 1.760 2.020 2.020 2.090 2.220 2.260 2.460
|  | 2 | 2.520 2.560 2.570 2.595 2.595 2.615 2.625 2.665 2.685 2.760 2.875 2.905 2.930
|  |  | 2.990 3.010 3.035 3.070 3.100 3.160 3.190 3.340 3.365 3.380 3.415 3.420 3.450
|  |  | 3.500 3.505 3.685 3.740
| 60 | 1 | 0.080 0.200 0.400 0.500 0.510 0.810 0.900 1.050 1.190 1.260 1.350 1.400 1.460
|  |  | 1.760 2.020 2.020 2.090 2.220 2.260 2.460
|  | 2 | 2.520 2.560 2.570 2.595 2.595 2.615 2.625 2.665 2.685 2.760 2.875 2.905 2.930
|  |  | 2.990 3.010 3.035 3.070 3.100 3.160 3.190 3.340 3.365 3.380 3.415 3.420 3.450
|  |  | 3.500 3.505 3.685 3.740 3.780 3.795 3.835 3.910 3.910 3.920 3.955 3.955 3.995
|  |  | 4.060
4 | 50 | 1 | 0.080 0.200 0.400 0.500 0.510 0.810 0.900 1.050 1.190 1.260 1.350 1.400 1.460
|  |  | 1.760 2.020 2.020 2.090 2.220 2.260 2.460 2.540 2.620 2.640 2.690 2.690 2.730
|  |  | 2.750 2.830 2.870 3.020 3.250 3.310 3.360 3.480 3.520 3.570 3.640 3.700 3.820
|  |  | 3.880
|  | 2 | 4.090 4.115 4.130 4.165 4.170 4.200 4.250 4.255 4.435 4.490.
| 60 | 1 | 0.080 0.200 0.400 0.500 0.510 0.810 0.900 1.050 1.190 1.260 1.350 1.400 1.460
|  |  | 1.760 2.020 2.020 2.090 2.220 2.260 2.460 2.540 2.620 2.640 2.690 2.690 2.730
|  |  | 2.750 2.830 2.870 3.020 3.250 3.310 3.360 3.480 3.520 3.570 3.640 3.700 3.820
|  |  | 3.880
|  | 2 | 4.090 4.115 4.130 4.165 4.170 4.200 4.250 4.255 4.435 4.490 4.530 4.545 4.585
|  |  | 4.660 4.660 4.670 4.705 4.705 4.745 4.810.

Different simple step-stress samples based Type-II censoring scheme are
considered by using different values of $\tau$, $\beta$ and $r$ when
$\alpha=0.75$ and $\lambda=2.5$. In Table 8 computed values of the MLEs, Bayes
estimates based on SEL and LL functions, average length of asymptotic
confidence intervals and HPD credible intervals based on the real data set are
tabulated. It is observed that the Bayes estimates perform better than MLEs.
Further it has been noticed that the HPD credible interval performs better
than asymptotic confidence intervals.

Figure 1: (a) ECDF and CDF comparison and (b) QQ-Plot for the Gumbel Type-II
distribution fitted to given real data set.

Figure 2: (a) Histogram and theoretical density comparison and (b) Box-plot
for Gumbel Type-II distribution fitted to given real data set.  Table 8:
Simulation results of Classical and Bayesian estimates and length of
asymptotic confidence intervals (ACI) and highest posterior density (HPD)
confidence intervals for the Gumbel Type-II baseline lifetime distribution
under simple SSLT based on given real data.

$\tau$ | $\beta$ | $r$ | Para. | MLE | SEL |  LL($u=-0.05$) |  LL($u=1$) | ACI | HPD
---|---|---|---|---|---|---|---|---|---
2.5 | 0.5 | 50 | $\alpha$ | 0.401 | 0.566 | 0.566 | 0.564 | 0.229 | 0.195
|  |  | $\lambda$ | 2.623 | 2.497 | 2.497 | 2.495 | 1.041 | 0.229
|  |  | $\beta$ | 0.118 | 0.518 | 0.518 | 0.517 | 0.201 | 0.054
|  | 60 | $\alpha$ | 0.441 | 0.605 | 0.605 | 0.604 | 0.222 | 0.179
|  |  | $\lambda$ | 2.633 | 2.414 | 2.415 | 2.413 | 1.039 | 0.242
|  |  | $\beta$ | 0.123 | 0.533 | 0.533 | 0.533 | 0.188 | 0.056
| 0.75 | 50 | $\alpha$ | 0.401 | 0.562 | 0.562 | 0.560 | 0.229 | 0.214
|  |  | $\lambda$ | 2.624 | 2.630 | 2.631 | 2.628 | 1.041 | 0.234
|  |  | $\beta$ | 0.177 | 0.780 | 0.780 | 0.779 | 0.302 | 0.067
|  | 60 | $\alpha$ | 0.441 | 0.618 | 0.618 | 0.616 | 0.222 | 0.237
|  |  | $\lambda$ | 2.633 | 2.533 | 2.533 | 2.530 | 1.039 | 0.262
|  |  | $\beta$ | 0.185 | 0.769 | 0769 | 0.768 | 0.283 | 0.060
4 | 0.5 | 50 | $\alpha$ | 0.505 | 0.570 | 0.570 | 0.568 | 0.219 | 0.214
|  |  | $\lambda$ | 2.452 | 2.537 | 2.537 | 2.536 | 0.921 | 0.117
|  |  | $\beta$ | 0.254 | 0.578 | 0.578 | 0.578 | 0.415 | 0.140
|  | 60 | $\alpha$ | 0.525 | 0.658 | 0.658 | 0.655 | 0.214 | 0.204
|  |  | $\lambda$ | 2.454 | 2.479 | 2.479 | 2.478 | 0.923 | 0.117
|  |  | $\beta$ | 0.181 | 0.565 | 0.565 | 0.564 | 0.242 | 0.118
| 0.75 | 50 | $\alpha$ | 0.514 | 0.699 | 0.699 | 0.697 | 0.219 | 0.189
|  |  | $\lambda$ | 2.446 | 2.550 | 2.550 | 2.549 | 0.921 | 0.125
|  |  | $\beta$ | 0.381 | 0.784 | 0.784 | 0.781 | 0.623 | 0.073
|  | 60 | $\alpha$ | 0.525 | 0.681 | 0.681 | 0.680 | 0.214 | 0.120
|  |  | $\lambda$ | 2.454 | 2.528 | 2.528 | 2.526 | 0.923 | 0.167
|  |  | $\beta$ | 0.271 | 0.789 | 0.789 | 0.789 | 0.363 | 0.088

##  6 Conclusion

In this paper, we obtained estimates of the unknown model parameters of the
Gumbel Type-II distribution under both classical and the Bayesian approaches
using TRV modeling for simple SSLT. It has been observed that the MLEs can not
be obtained explicitly for both the unknown parameters. Therefore, we used the
Newton-Raphson iterative method to compute MLEs by using $R$ software. Also,
we obtained the Bayes estimates based on the symmetric and asymmetric loss
functions under the assumption of independent priors. A Monte Carlo simulation
study is performed to compare the performance of the estimates in terms of the
average values and MSEs. It has been noticed that the Bayes estimates under LL
function perform better than the other estimates. The asymptotic confidence
intervals and HPD intervals are also obtained. It is noticed that the HPD
credible intervals perform better than asymptotic confidence intervals, as
expected. Further, a real data set on bladder cancer is considered for
illustrative purposes.

The authors declare that they do not have any conflict of interests.

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