# Statistical inference for dependent competing risks data under adaptive
Type-II progressive hybrid censoring

Subhankar Dutta and Suchandan Kayal  Subhankar Dutta (Corresponding author :
<EMAIL_ADDRESS>
Department of Mathematics, National Institute of Technology Rourkela,
Rourkela-769008, India

Abstract

In this article, we consider statistical inference based on dependent
competing risks data from Marshall-Olkin bivariate Weibull distribution. The
maximum likelihood estimates of the unknown model parameters have been
computed by using Newton-Raphson method under adaptive Type II progressive
hybrid censoring with partially observed failure causes. Existence and
uniqueness of maximum likelihood estimates are derived. Approximate confidence
intervals have been constructed via the observed Fisher information matrix
using asymptotic normality property of the maximum likelihood estimates. Bayes
estimates and highest posterior density credible intervals have been
calculated under gamma-Dirichlet prior distribution by using Markov chain
Monte Carlo technique. Convergence of Markov chain Monte Carlo samples is
tested. In addition, a Monte Carlo simulation is carried out to compare the
effectiveness of the proposed methods. Further, three different optimality
criteria have been taken into account to obtain the most effective censoring
plans. Finally, a real-life data set has been analyzed to illustrate the
operability and applicability of the proposed methods.

Keywords: Adaptive Type II progressive hybrid censoring; Dependent competing
risk; Gamma-Dirichlet Prior; HPD credible interval; MCMC convergence;
Optimality.

MATHEMATICS SUBJECT CLASSIFICATIONS 2010: 62F10; 62F15; 62E15.

##  1 Introduction

In life-testing experiments, it is very common that due to complexity of the
internal structure and external environment, failure of any product occurs due
to various competing causes. In the literature, such types of models are
dubbed as the competing risk models. Competing risks data frequently appear in
many fields, such as engineering, biology, social science and medical science.
In reliability analysis, to get around the issue of model identifiability,
many researchers assume that the causes of failure are independent. See, for
example Panwar et al. (2015), Ashour and Nassar (2017), Koley and Kundu
(2017), Wang (2018), Wang and Li (2019), Mahto et al. (2021), Dutta and Kayal
(2022b), Dutta and Kayal (2022a) and Dutta et al. (2023). However, in general,
it is not easy to verify the assumption that the causes of failure are
independent to each other. Further, due to complex system design and operating
conditions in real-life experiments, the causes of failure affect each other
and become inter-dependent. Thus, in practical situations, the assumption of
dependent causes of failure is more reasonable than the assumption of
independent causes of failure. When analysing data to address dependent
competing risks models, it has been shown that the Marshall-Olkin bivariate
distributions are frequently used. For details, one may refer to Kundu and
Gupta (2013) in this matter. In recent years, statistical inference of
dependent competing risks model has gained much attention from many
researchers. Feizjavadian and Hashemi (2015) discussed the classical
estimation of the parameters of Marshall-Olkin bivariate Weibull (MOBW)
distribution in the presence of a dependent competing risks model under
progressive hybrid censored sample. Liang et al. (2019) considered inference
of the Marshall-Olkin bivariate exponential (MOBE) distribution for dependent
competing risks model under generalized progressive hybrid censoring scheme.
Du and Gui (2021) considered the inference for dependence competing risk model
by using MOBE distribution under adaptive Type-II progressive hybrid
censoring.

Censoring has become a commonly used phenomenon in survival analysis and
reliability studies. Compared to complete data, if one gets the exact failure
time of only a few experimental units in a lifetime experiment, such data are
called censored. Nowadays, the quality and lifetime of any industrial product
have become higher because of modern science and technology. Thus, it has
become more challenging to increase product qualities with reliability testing
in a short period with low expenditure. In this context, censoring is sensible
and important to study the failure data in a reliability experiment. A variety
of censoring schemes have already been put forth by several researchers in
order to improve the efficiency of an experiment. In survival analysis, Type-I
and Type-II censoring schemes are most commonly used. The mixture of these two
censoring schemes is known as hybrid censoring schemes. These conventional
censoring schemes do not allow removal of experimental units at intermediate
points except at the termination point. To overcome such disadvantage,
progressive Type-II censoring scheme is introduced, which allows an
experimenter to take out units from an experiment at different phases. To
improve the efficiency of the experiments, Kundu and Joarder (2006) introduced
progressive hybrid censoring scheme (PHCS), which is a combination of the
conventional hybrid and progressive censoring schemes. The drawback of PHCS is
that the effective sample size may become zero, which is as similar as Type-I
censoring scheme. To get beyond such obstacle, Ng et al. (2009) introduced
adaptive Type-II progressive hybrid censoring scheme (AT-II PHCS), where the
experimental time may exceed the predetermined time $T$ and the effective
sample size $m$ is prefixed. In AT-II PHCS, a progressive scheme
$(R_{1},R_{2},\cdots,R_{m})$ is provided. Consider that the time of $i$-th
failure of units is represented as $X_{i:m:n}$. In AT-II PHCS, if
$X_{m:m:n}<T$ then the experiment is terminated at $X_{m:m:n}$ with the
censoring scheme $(R_{1},R_{2},\cdots,R_{m})$. Otherwise, if $X_{m:m:n}>T,$
then there exists $d$ such that $X_{d:m:n}<T<X_{d+1:m:n}$ and the experiment
terminates at $X_{m:m:n}$ with no removal by setting $R_{i}=0$, where
$i=d+1,\cdots,m-1$ and $R_{m}=n-m-\sum_{i=1}^{d}R_{i}$. This adaptation
ensures that the experiment will end once the predetermined number of failures
has been obtained. In recent years, AT-II PHCS has received extensive
consideration in the statistical literature. Hemmati and Khorram (2011)
discussed the competing risks model based on exponential distributions under
AT-II PHCS. Ismail (2014) considered the classical estimation of parameters of
Weibull distribution under step-stress partially accelerated life test model
based on AT-II PHCS. Nassar and Abo-Kasem (2017) discussed frequentist and
Bayesian estimation for the parameters of inverse Weibull (IW) distribution
based on AT-II PHCS. Panahi and Moradi (2020) discussed the estimation of
parameters of inverted exponentiated Rayleigh distribution under AT-II PHCS.

We recall that the AT-II PHCS can be applied in quality inspections of
manufacturing plants that require enough failure information in a brief time
period. For instance, we cannot observe an adequate number of failures from
ten thousand or more electronic product keystrokes in between the specific
time and the test price increases there. AT-II PHCS is relevant to apply in
such cases. Sometimes the failure risks may be correlated and affects each
other to make an individual failure early. Hence the competing risks are
usually dependent. For example, Lin et al. (1999) investigated colon cancer
data in which cancer recurrence or death were the causes of failure.
Obviously, these two causes are dependent and correlated with each other. Note
that the MOBE distribution may not be applicable when bivariate data shows a
non-constant hazard rate function or a unimodal marginal probability density
function. The MOBW distribution is more appropriate to fit such models.
Furthermore, when dependent competing risks exist, a MOBW distribution has a
correlation control parameter that may be applied. the duration of paired
organ (eyes, lungs and kidneys) failure can be analyzed by using MOBW model in
survival analysis. Also in reliability analysis, this model fits to the
duration of paired components (aircraft engines) failure. Motivated by such
reasons, we have considered a dependent competing risks model using MOBW
distributions to discuss statistical inferences based on AT-II PHCS.

In this study, our main goal is to analyze and investigate the statistical
inference of the parameters of MOBW distribution using both frequentist and
Bayesian approaches under AT-II PHC dependent competing risks model. In the
main results, maximum likelihood estimates (MLEs) are observed to be
inaccessible in closed form. In order to obtain MLEs, a suitable numerical
method has been applied. Using the asymptotic normality properties of the
MLEs, approximate confidence intervals (ACIs) have been derived. Bayes
estimators have been obtained under two different types of loss functions.
Since the Bayes estimates cannot be derived in closed form, Markov Chain Monte
Carlo (MCMC) method has been implemented. Highest posterior density (HPD)
credible intervals are also constructed using MCMC samples. MCMC convergence
is also discussed. Further, a Monte Carlo simulation study has been conducted
to assess the effectiveness of the proposed various estimates based on the
absolute bias (AB), mean squared error (MSE), average width (AW), and coverage
probability (CP). In addition, we have proposed optimal life-testing plans
based on three different optimality criteria.

The remaining portion of the paper is structured as follows. In Section 2, the
model description has been provided. The maximum likelihood estimates of the
unknown parameters with their corresponding approximate confidence intervals
have been derived in Section 3. In Section 4, the MCMC approach for Bayesian
analysis has been demonstrated, and the associated HPD credible intervals have
been constructed. MCMC convergence is also tested there. A Monte Carlo
simulation study has been carried out in Section 5. In Section 6, three
different optimality criteria have been discussed to choose an optimal
progressive censoring plan. In Section 7, a real data set has been analyzed to
verify the effectiveness of the proposed methods in practical situations.
Finally, some concluding remarks have been added in Section 8.

##  2 Model and data description

###  2.1 MOBW distribution

In this paper, MOBW distribution has been addressed for statistical inference.
First, we describe this model very briefly. Marshall and Olkin (1967) proposed
a multivariate lifetime model known as the Marshall-Olkin model, that can be
explained as follows. For $i=0,~{}1,~{}2$, assume that $V_{i}$ $\sim$
$Weibull$ $(\alpha,\lambda_{i})$ be independent to each other. The cumulative
distribution function (CDF), probability density function (PDF), and survival
function of $V_{i}$ are given by

$\displaystyle
F_{WE}(v;\alpha,\lambda_{i})=1-e^{-\lambda_{i}v^{\alpha}},~{}~{}f_{WE}(v;\alpha,\lambda_{i})=\alpha\lambda_{i}v^{\alpha-1}e^{-\lambda_{i}v^{\alpha}},~{}~{}\mbox{and}~{}~{}S_{WE}(v;\alpha,\lambda_{i})=e^{-\lambda_{i}v^{\alpha}},$

where $v>0,~{}\alpha,~{}\lambda_{i}>0,$ $i=0,~{}1,~{}2.$ Define
$Y_{1}=min\\{V_{0},V_{1}\\}$ and $Y_{2}=min\\{V_{0},V_{2}\\}$, then
$(Y_{1},Y_{2})$ has MOBW distribution with shape parameter $\alpha$ and scale
parameters $\lambda_{0}$, $\lambda_{1}$ and $\lambda_{2}$. Henceforth, we
denote $(Y_{1},Y_{2})\sim MOBW(\lambda_{0},\lambda_{1},\lambda_{2},\alpha)$.
Here, $Y_{1}$ and $Y_{2}$ are independent if $\lambda_{0}=0$; otherwise, they
are dependent. Hence $\lambda_{0}$ is the parameter describing correlation
between these two variables $Y_{1}$ and $Y_{2}$. Let us assume that
$\lambda_{012}=\lambda_{0}+\lambda_{1}+\lambda_{2}$ and
$\lambda_{ab}=\lambda_{a}+\lambda_{b}$, for $a,~{}b=0,~{}1,~{}2$ with $a\neq
b$.

######  Theorem 2.1.

If we consider, $(Y_{1},Y_{2})\sim
MOBW(\lambda_{0},\lambda_{1},\lambda_{2},\alpha)$, then the joint survival
function of $(Y_{1},Y_{2})$ is given by

$\displaystyle
S_{Y_{1},Y_{2}}(y_{1},y_{2})=\begin{cases}S_{WE}(y_{1};\alpha,\lambda_{1})~{}S_{WE}(y_{2};\alpha,\lambda_{02}),~{}~{}~{}\mbox{if}~{}~{}y_{1}<y_{2}\\\
S_{WE}(y_{1};\alpha,\lambda_{01})~{}S_{WE}(y_{2};\alpha,\lambda_{2}),~{}~{}~{}\mbox{if}~{}~{}y_{2}<y_{1}\\\
S_{WE}(y;\alpha,\lambda_{012}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{if}~{}~{}y=y_{1}=y_{2}.\end{cases}$
(2.1)

###### Proof.

See Appendix $A.1$. ∎

######  Corollary 2.1.

If $(Y_{1},Y_{2})\sim MOBW(\lambda_{0},\lambda_{1},\lambda_{2},\alpha)$, then
the joint PDF of $(Y_{1},Y_{2})$ is given by

$\displaystyle
f_{Y_{1},Y_{2}}(y_{1},y_{2})=\begin{cases}f_{WE}(y_{1};\alpha,\lambda_{1})~{}f_{WE}(y_{2};\alpha,\lambda_{02}),~{}~{}~{}\mbox{if}~{}~{}y_{1}<y_{2}\\\
f_{WE}(y_{1};\alpha,\lambda_{01})~{}f_{WE}(y_{2};\alpha,\lambda_{2}),~{}~{}~{}\mbox{if}~{}~{}y_{2}<y_{1}\\\
\frac{\lambda_{0}}{\lambda_{012}}f_{WE}(y;\alpha,\lambda_{012}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{if}~{}~{}y=y_{1}=y_{2}.\end{cases}$
(2.2)

###### Proof.

See Appendix $A.2$. ∎

Figure 1: Surface plots for joint PDF of MOBW.

MOBW is one of the important Marshall-Olkin type distributions due to having
singular and absolute continuous parts. The joint PDF
$f_{Y_{1},Y_{2}}(y_{1},y_{2})$ given by $(\ref{2.2})$ yields that densities
with two dimensional Lebesgue measure are represented by first two expressions
and a density with single dimensional Lebesgue measure is represented by third
one. The unimodality property of the bivariate density function of MOBW
distribution can be easily observed from the surface plots given in Fig. $1$.
In addition, for $\alpha=1$, MOBW distribution becomes Marshall-Olkin
bivariate exonential distribution. Further, MOBW distribution has Weibull
models as marginals which have monotone hazard rate functions and decreasing
or unimodal PDFs. Due to having such statistical properties and simpler
mathematical structure, analysis of MOBW distribution under AT-II PHCS has
been taken into account here may be considered as a proper model of
discussion.

###  2.2 Data description

Suppose that $n$ identical individuals are placed on a life test with lifetime
$Y_{i}$ and assume that two causes of failures are known with
$Y_{i}=min\\{Y_{1i},Y_{2i}\\}$, where $Y_{ki}$ is the latent failure time of
$i$-th individual because of $k$-th ($k=1,2$) cause of failure. Then competing
risk data under AT-II PHCS can be expressed as

Case I: $(y_{1:m:n},\delta_{1},R_{1}),\cdots,(y_{m:m:n},\delta_{m},R_{m})$,
where $y_{m:m:n}<T$

Case II: $(y_{1:m:n},\delta_{1},R_{1}),\cdots,(y_{d:m:n},\delta_{d},R_{d})$,
where $y_{d:m:n}<T<y_{d+1:m:n}$,

where $y_{i:m:n}$ is the AT-II PHC sample of $Y_{i}$ and the associated
failure indicator $\delta_{i}$ can be expressed as

$\displaystyle\delta_{i}=\begin{cases}0,~{}~{}\mbox{if}~{}y_{1i}=y_{2i}\\\
1,~{}~{}\mbox{if}~{}y_{1i}<y_{2i}\\\ 2,~{}~{}\mbox{if}~{}y_{1i}>y_{2i}\\\
3,~{}~{}\mbox{unknown cause of failure}\end{cases}$ (2.3)

and $d$ is a number for which $y_{d:m:n}<T<y_{d+1:m:n}$.

######  Remark 2.1.

Nowadays the cause of failures can be identified in most cases due to the
improvement of quality in research. However, due to complex external and
internal interferences, there are some practical situations where each and
every failure mode can not be discovered. Therefore, it is more reasonable to
consider the unknown cause of failures in the study of the competing risks
model.

##  3 Classical estimation

###  3.1 Maximum likelihood estimation

Suppose that $Y_{i:m:n}$ is the $i$-th failure time of experimental units from
a sample of $n$ units which are produced from two Weibull populations that are
dependent on each other and have competing risks. For convenience, $y_{i:m:n}$
has been expressed as $y_{i}$ and the joint PDF is given by

$\displaystyle L(data)=C\prod_{i=0}^{m}$
$\displaystyle\big{[}f_{Y_{1},Y_{2}}(y_{i},y_{i})\big{]}^{\delta_{i0}}\bigg{[}-\frac{\partial
S_{Y_{1},Y_{2}}(y_{1},y_{2})}{\partial
y_{1}}\bigg{]}_{(y_{i},y_{i})}^{\delta_{i1}}\bigg{[}-\frac{\partial
S_{Y_{1},Y_{2}}(y_{1},y_{2})}{\partial
y_{2}}\bigg{]}_{(y_{i},y_{i})}^{\delta_{i2}}$ $\displaystyle\times$
$\displaystyle
f(y_{i};\alpha,\lambda_{012})^{\delta_{i3}}\prod_{i=1}^{J}S_{Y_{1},Y_{2}}(y_{1},y_{2})^{R_{i}}S_{Y_{1},Y_{2}}(y_{m},y_{m})^{R^{*}}$
(3.1)

where $C=\prod_{i=1}^{m}\prod_{j=1}^{m}(R_{j}+1)$,
$\delta_{ij}=I(\delta_{i}=j)(j=0,1,2,3)$ such that
$\sum_{j=0}^{3}\delta_{ij}=1$,

$\displaystyle J=\begin{cases}m,~{}~{}\mbox{if}~{}~{}y_{m}<T\\\
d,~{}~{}~{}\mbox{if}~{}~{}y_{d}<T<y_{d+1}\end{cases}~{}~{}\mbox{and}~{}~{}R^{*}=\begin{cases}0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{if}~{}~{}y_{m}<T\\\
n-m-\sum_{i=1}^{d}R_{i},~{}~{}\mbox{if}~{}~{}y_{d}<T<y_{d+1}.\end{cases}$

Suppose that $(Y_{i1},Y_{i2})\sim
MOBW(\lambda_{0},\lambda_{1},\lambda_{2},\alpha)$, $i=1,\cdots,n$ are
independent and identically distributed. Using equations $(\ref{2.2})$ and
$(\ref{3.1})$, the likelihood function is expressed as

$\displaystyle L(\lambda_{0},\lambda_{1},\lambda_{2},\alpha)=$
$\displaystyle~{}C\alpha^{m}\prod_{i=1}^{m}\lambda_{0}^{\delta_{i0}}\lambda_{1}^{\delta_{i1}}\lambda_{2}^{\delta_{i2}}\lambda_{012}^{\delta_{i3}}~{}y_{i}^{\alpha-1}e^{-\lambda_{012}\big{(}\sum_{i=1}^{m}y_{i}^{\alpha}+\sum_{i=1}^{J}R_{i}y_{i}^{\alpha}+R^{*}y_{m}^{\alpha}\big{)}}$
$\displaystyle=$
$\displaystyle~{}C\alpha^{m}\lambda_{0}^{m_{0}}\lambda_{1}^{m_{1}}\lambda_{2}^{m_{2}}\lambda_{012}^{m_{3}}\prod_{i=1}^{m}y_{i}^{\alpha-1}e^{-\lambda_{012}\big{(}\sum_{i=1}^{m}y_{i}^{\alpha}+\sum_{i=1}^{J}R_{i}y_{i}^{\alpha}+R^{*}y_{m}^{\alpha}\big{)}},$
(3.2)

where $m_{j}=\sum_{j=0}^{3}\delta_{ij}$ represents the number of failures due
to four different cases given in $(\ref{2.3})$ such that
$\sum_{j=0}^{3}m_{j}=m$. Then the log likelihood function can be expressed as

$\displaystyle\log L=$
$\displaystyle~{}m\log\alpha+m_{0}\log\lambda_{0}+m_{1}\lambda_{1}+m_{2}\log\lambda_{2}+m_{3}\log\lambda_{012}$
$\displaystyle~{}+(\alpha-1)\sum_{i=1}^{m}\log
y_{i}-\lambda_{012}~{}A(\alpha),$ (3.3)

where
$A(\alpha)=\sum_{i=1}^{m}y_{i}^{\alpha}+\sum_{i=1}^{J}R_{i}y_{i}^{\alpha}+R^{*}y_{m}^{\alpha}$.
Now, taking derivative of $(\ref{3.3})$ with respect to $\alpha$,
$\lambda_{0}$, $\lambda_{1}$ and $\lambda_{2}$ and equating to zero, we get

$\displaystyle\begin{cases}\frac{\partial\log
L}{\partial\lambda_{0}}=&~{}\frac{m_{0}}{\lambda_{0}}+\frac{m_{3}}{\lambda_{012}}-A(\alpha)=0,\\\
\frac{\partial\log
L}{\partial\lambda_{1}}=&~{}\frac{m_{1}}{\lambda_{1}}+\frac{m_{3}}{\lambda_{012}}-A(\alpha)=0,\\\
\frac{\partial\log
L}{\partial\lambda_{2}}=&~{}\frac{m_{2}}{\lambda_{2}}+\frac{m_{3}}{\lambda_{012}}-A(\alpha)=0,\\\
\frac{\partial\log L}{\partial\alpha}=&~{}\frac{m}{\alpha}+\sum_{i=1}^{m}\log
y_{i}-\lambda_{012}~{}A^{\prime}(\alpha)=0,\end{cases}$ (3.4)

where $A^{\prime}(\alpha)=\sum_{i=1}^{m}y_{i}^{\alpha}\log
y_{i}+\sum_{i=1}^{J}R_{i}y_{i}^{\alpha}\log y_{i}+R^{*}y_{m}^{\alpha}\log
y_{m}$. The MLEs of $\lambda_{j}$, $j=0,~{}1,~{}2$ for known shape parameter
$\alpha$, are provided by the subsequent theorem.

######  Theorem 3.1.

Suppose $m_{j}>0~{}(j=0,1,2)$. Then the MLE of $\lambda_{j}$ given $\alpha$ is
obtained as

$\displaystyle\widehat{\lambda}_{j}=\frac{m_{j}}{m_{012}}\frac{m}{A(\alpha)},~{}~{}~{}j=0,1,2$
(3.5)

where $m_{012}=m_{0}+m_{1}+m_{2}$.

###### Proof.

See Appendix $A.3$. ∎

Using $(\ref{3.4})$ and $(\ref{3.5})$, we have

$\displaystyle\frac{m}{\alpha}+\sum_{i=1}^{m}\log
y_{i}-(\widehat{\lambda}_{0}+\widehat{\lambda}_{1}+\widehat{\lambda}_{2})~{}A^{\prime}(\alpha)=0.$
(3.6)

Since $(\ref{3.11})$ can not be solved explicitly, an iteration method as
discussed in Kundu (2007) will be employed to obtain the MLE of $\alpha$.
Using $(\ref{3.11})$, the MLE $\widehat{\alpha}$ can be obtained as follows

$\displaystyle\alpha=h(\alpha),$ (3.7)

where

$\displaystyle
h(\alpha)=\frac{m}{(\widehat{\lambda}_{0}+\widehat{\lambda}_{1}+\widehat{\lambda}_{2})~{}A^{\prime}(\alpha)-\sum_{i=1}^{m}\log
y_{i}}.$

Assume an initial value for $\alpha$ as $\alpha^{(0)}$. Substitute
$\alpha^{(0)}$ in $(\ref{3.12})$ to obtain $\alpha^{(1)}=h(\alpha^{(0)})$.
This procedure will be continued until we get an $\alpha^{(k)}$ such that
$|\alpha^{(k)}-\alpha^{(k-1)}|<\epsilon$, where $k$ is a positive integer and
$\epsilon$ is very small positive real number. This $\alpha^{(k)}$ will be
considered as the MLE of $\alpha$. Then replacing $\widehat{\alpha}$ in
$(\ref{3.5})$ we get

$\displaystyle\widehat{\lambda}_{j}=\frac{m_{j}}{m_{012}}\frac{m}{A(\widehat{\alpha})},~{}~{}~{}j=0,~{}1,~{}2.$

###  3.2 Approximate confidence interval

In this subsection, the $100(1-\gamma)\%$ approximate confidence intervals
(ACIs) of unknown parameters $\alpha$, $\lambda_{0}$, $\lambda_{1}$ and
$\lambda_{2}$ are constructed based on large sample approximation. Under some
mild regularity conditions, it can be shown that

$\displaystyle\begin{pmatrix}\widehat{\lambda}_{0}\\\ \widehat{\lambda}_{1}\\\
\widehat{\lambda}_{2}\\\ \widehat{\alpha}\\\
\end{pmatrix}-\begin{pmatrix}{\lambda}_{0}\\\ {\lambda}_{1}\\\
{\lambda}_{2}\\\ {\alpha}\\\ \end{pmatrix}\rightarrow
N\big{(}0,I^{-1}(\widehat{\lambda}_{0},\widehat{\lambda}_{1},\widehat{\lambda}_{2},\widehat{\alpha})\big{)},$
(3.8)

where the observed Fisher information matrix
$I(\widehat{\lambda}_{0},\widehat{\lambda}_{1},\widehat{\lambda}_{2},\widehat{\alpha})=\bigg{[}-\frac{\partial^{2}\log
L}{\partial\Theta_{i}\partial\Theta_{j}}\bigg{]}_{\Theta=\widehat{\Theta}},$
$i,~{}j=0,~{}1,~{}2,~{}3$ and
$\Theta=(\Theta_{1},\Theta_{2},\Theta_{3},\Theta_{4})=(\lambda_{0},\lambda_{1},\lambda_{2},\alpha)$.
All these elements $\frac{\partial^{2}\log
L}{\partial\Theta_{i}\partial\Theta_{j}}$ can be expressed as
$\frac{\partial^{2}\log
L}{\partial\Theta_{i}^{2}}=-\frac{m_{i}}{\lambda_{i}^{2}}-\frac{m_{3}}{\lambda_{012}^{2}}$,
$\frac{\partial^{2}\log
L}{\partial\Theta_{i}\partial\Theta_{j}}=-\frac{m_{3}}{\lambda_{012}^{2}}$,
$\frac{\partial^{2}\log
L}{\partial\Theta_{i}\partial\Theta_{3}}=\frac{\partial^{2}\log
L}{\partial\Theta_{3}\partial\Theta_{i}}=-A^{\prime}(\alpha)$, for $i,j=0,1,2$
and $i\neq j$;
$\frac{\partial^{2}\log
L}{\partial\Theta_{3}^{2}}=-\frac{m}{\alpha^{2}}-\lambda_{012}A^{\prime\prime}(\alpha)$,
where $A^{\prime\prime}(\alpha)=\sum_{i=1}^{m}y_{i}^{\alpha}(\log
y_{i})^{2}+\sum_{i=1}^{J}R_{i}y_{i}^{\alpha}(\log
y_{i})^{2}+R^{*}y_{m}^{\alpha}(\log y_{m})^{2}$. For $0<\gamma<1$, the ACIs of
the parameters with significance level $\gamma$ are constructed as

$\displaystyle\widehat{\alpha}\underline{+}z_{\gamma/2}\sqrt{Var(\widehat{\alpha})}~{}~{}\mbox{and}~{}~{}\widehat{\lambda}_{j}\underline{+}z_{\gamma/2}\sqrt{Var(\widehat{\lambda}_{j})},~{}\mbox{for}~{}j=0,1,2$
(3.9)

where $z_{\gamma/2}$ is the upper $\gamma/2$-th quantile of the standard
normal distribution.

##  4 Bayesian estimation

In the study of reliability analysis, sometimes it has been observed that the
classical estimation through the MLE approach may fail when the data don’t
provide enough sampling details. In order to overcome this problem, prior
information can be used in conjunction with Bayesian analysis. This section
deals with Bayesian approach to estimate unknown parameters and the
accompanying HPD credible intervals.

###  4.1 Prior information

In order to obtain Bayes estimates, prior knowledge is crucial, particularly
when the data from sampling-based information about the unknown parameters is
insufficient. As similar as Kundu and Gupta (2013), let us consider that
$\lambda_{012}\sim$ Gamma $(a,b)$ with PDF as follows

$\displaystyle\pi_{0}(\lambda_{012}|a,b)=\frac{b^{a}}{\Gamma(a)}\lambda_{012}^{a-1}e^{-b\lambda_{012}},~{}~{}a>0,b>0,\lambda_{012}>0.$
(4.1)

According to Cai et al. (2017), let us consider that for given
$\lambda_{012}$,
$(\frac{\lambda_{1}}{\lambda_{012}},\frac{\lambda_{2}}{\lambda_{012}})$
follows a Dirichlet prior, say $\pi_{1}(\cdot|d_{0},d_{1},d_{2})$ and that is
given by

$\displaystyle\pi_{1}(\frac{\lambda_{1}}{\lambda_{012}},\frac{\lambda_{2}}{\lambda_{012}}|\lambda_{012},d_{0},d_{1},d_{2})=\frac{\Gamma(d_{012})}{\Gamma(d_{0})\Gamma(d_{1})\Gamma(d_{2})}\prod_{i=0}^{2}\bigg{(}\frac{\lambda_{i}}{\lambda_{012}}\bigg{)}^{d_{i}},~{}~{}\lambda_{0}>0,\lambda_{1}>0,\lambda_{2}>0,$
(4.2)

where $d_{0}$, $d_{1}$ and $d_{2}>0$ are the Dirichlet distribution’s
parameters with $d_{012}=d_{0}+d_{1}+d_{2}$ and known as the gamma-Dirichlet
distribution, which is an extension to higher dimension of a gamma-beta
distribution. Through the Jacobian transformation, the joint density function
of $\lambda_{0}$, $\lambda_{1}$ and $\lambda_{2}$ can be expressed as

$\displaystyle\pi_{1}(\lambda_{0},\lambda_{1},\lambda_{2})=\frac{\Gamma(d_{012})}{\Gamma(a)}\prod_{i=0}^{2}\frac{b^{d_{i}}}{\Gamma(d_{i})}\lambda_{i}^{d_{i}-1}e^{-b\lambda_{i}}(\lambda_{012}b)^{a-d_{012}}.$
(4.3)

This type of prior distribution can be explained different situations
depending on the proper hyperparameter values whether the parameters are
independent or not. Hence it becomes highly flexible. When $a=d_{012}$, then
the parameters $\lambda_{0}$, $\lambda_{1}$ and $\lambda_{2}$ in $(\ref{4.3})$
are independent. Since the shape parameter $\alpha$ is not known, then the
conjugate prior of the joint prior of
$(\lambda_{0},\lambda_{1},\lambda_{2},\alpha)$ never existed and there is no
way to presume any particular variation of the prior on $\alpha$, say
$\pi_{2}(\alpha)$. Only the density function $\pi_{2}(\alpha)$ with a non-
negative support $(0,\infty)$ may possibly be log-concave. Moreover, the joint
prior of $(\lambda_{0},\lambda_{1},\lambda_{2})$ is not affected by
$\pi_{2}(\alpha)$. Based on these assumptions, the joint prior of the model
parameters, say $\pi(\lambda_{0},\lambda_{1},\lambda_{2},\alpha)$ is obtained
as

$\displaystyle\pi(\lambda_{0},\lambda_{1},\lambda_{2},\alpha)=\pi_{1}(\lambda_{0},\lambda_{1},\lambda_{2})\cdot\pi_{2}(\alpha),$
(4.4)

where it should be noted that, for the purpose of data analysis the choices of
the hyperparameters is important for any specific form of $\pi_{2}(\alpha)$.

###  4.2 Posterior analysis

After combining $(\ref{3.2})$ and $(\ref{4.4})$, it is possible to express the
joint posterior density function of $\lambda_{0}$, $\lambda_{1}$,
$\lambda_{2}$ and $\alpha$ as

$\displaystyle\pi(\Theta|data)$
$\displaystyle=\frac{L(\Theta;data)\pi(\Theta)}{\int_{0}^{\infty}\cdots\int_{0}^{\infty}L(\Theta;data)\pi(\Theta)~{}d\Theta}$
$\displaystyle\propto~{}\alpha^{m}\bigg{(}\prod_{j=0}^{2}\lambda_{j}^{m_{j}+d_{j}-1}\bigg{)}\lambda_{012}^{m_{3}+a-d_{012}}\bigg{(}\prod_{i=1}^{m}y_{i}^{\alpha-1}\bigg{)}e^{-\lambda_{012}(b+A(\alpha))}\cdot\pi_{2}(\alpha).$
(4.5)

In order to obtain Bayes estimates, the posterior expected value of a loss
function has to be minimized. In such case, the choice of different loss
function reflects different estimation error effects. Thus, we choose two
different loss functions and these are squared error loss function (SELF) and
LINEX loss function (LLF). Here, LLF is asymmetric in nature, whereas SELF is
symmetric. The loss functions SELF and LLF are defined as

$\displaystyle L_{SE}(W,\widehat{W})=\big{(}\widehat{W}-W\big{)}^{2},$ (4.6)

and

$\displaystyle
L_{LI}(W,\widehat{W})=e^{p\big{(}\widehat{W}-W\big{)}}-p\big{(}\widehat{W}-W\big{)}-1,~{}~{}p\neq
0,$ (4.7)

respectively, where $\widehat{W}$ is the estimate of a parametric function
$W$. Under these SELF and LLF, the Bayes estimates of $W(\theta)$ can be
obtained as

$\displaystyle\widehat{W}_{SE}(\Theta)=\frac{\int_{0}^{\infty}\cdots\int_{0}^{\infty}W(\Theta)\pi(\Theta|data)d\Theta}{\int_{0}^{\infty}\cdots\int_{0}^{\infty}\pi(\Theta|data)d\Theta},$
(4.8)

and

$\displaystyle\widehat{W}_{LI}(\Theta)=-\bigg{(}\frac{1}{p}\bigg{)}\log\bigg{(}\frac{\int_{0}^{\infty}\cdots\int_{0}^{\infty}e^{-pW(\Theta)}\pi(\Theta|data)d\Theta}{\int_{0}^{\infty}\cdots\int_{0}^{\infty}\pi(\Theta|data)d\Theta}\bigg{)},$
(4.9)

respectively. These above equations $(\ref{4.8})$ and $(\ref{4.9})$ can not be
solved explicitly. Thus, the Gibbs sampling technique can be used to generate
MCMC samples which can be used to obtain the Bayes estimates and the
corresponding HPD credible intervals. The following results are useful to
generate the MCMC samples with the help of Gibbs sampling technique.

######  Theorem 4.1.

If $m>1$ and $y_{1},\cdots,y_{m}$ are AT-II PHC competing risks samples, then
the marginal posterior density function of $\alpha$ is

$\displaystyle\pi_{1}^{*}(\alpha|data)\propto~{}\pi_{2}(\alpha)\alpha^{m}\bigg{(}\prod_{i=1}^{m}y_{i}^{\alpha-1}\bigg{)}\bigg{(}b+A(\alpha)\bigg{)}^{-(a+m)},$
(4.10)

and the joint posterior density of $(\lambda_{0},\lambda_{1},\lambda_{2})$ for
the provided $\alpha$ and data, is

$\displaystyle\pi_{2}^{*}(\lambda_{0},\lambda_{1},\lambda_{2}|\alpha,data)\propto~{}\lambda_{012}^{m_{3}+a-d_{012}}\bigg{(}\prod_{j=0}^{2}Gamma\big{(}\lambda_{j};m_{j}+d_{j},b+A(\alpha)\big{)}\bigg{)},$
(4.11)

which follows gamma-Dirichlet distribution, denoted by
$GD(a+m,b+A(\alpha),m_{0}+d_{0},m_{1}+d_{1},m_{2}+d_{2})$.

###### Proof.

See Appendix $A.4$. ∎

Since it is assumed that $\pi_{2}(\alpha)$ is log-concave, then it is
considered as

$\displaystyle\pi_{2}(\alpha)\propto\alpha^{a_{1}-1}e^{-b_{1}\alpha},$ (4.12)

where $a_{1}>0,b_{1}>0$. Now, combining $(\ref{4.10})$ and $(\ref{4.12})$, the
marginal posterior density of $\alpha$ is given by

$\displaystyle\pi_{1}^{*}(\alpha|data)\propto\alpha^{m+a_{1}-1}e^{-\alpha\big{(}b_{1}-\sum_{i=1}^{m}\log
y_{i}\big{)}}\big{(}b+A(\alpha)\big{)}^{-(a+m)}.$ (4.13)

######  Theorem 4.2.

The marginal posterior density $\pi_{1}^{*}(\alpha|data)$ expressed by
$(\ref{4.13})$ is log-concave.

###### Proof.

See Appendix $A.5$. ∎

MCMC samples of the unknown parameters can be generated to compute Bayes
estimates and associated HPD credible intervals by using Theorem 4.1 and 4.2.
To generate samples the following algorithm is proposed :

Step 1: Generate $\alpha$ by using the method proposed by Devroye (1984) from
$(\ref{4.13})$.
Step 2: For each given $\alpha$, generate the samples for
$(\lambda_{0},\lambda_{1},\lambda_{2})$ from
$GD(a+m,b+A(\alpha),m_{0}+d_{0},m_{1}+d_{1},m_{2}+d_{2})$.
Step 3: Repeat Steps 1-2, upto $M$ times to get the samples
$\\{(\alpha^{1},\lambda_{0}^{1},\lambda_{1}^{1},\lambda_{2}^{1}),\cdots,(\alpha^{M},\lambda_{0}^{M},\lambda_{1}^{M},\lambda_{2}^{M})\\}$.

Using the MCMC samples, Bayes estimates of $W(\Theta)$ under SELF and LLF are
expressed as

$\displaystyle\widehat{W}_{SE}(\Theta)=\frac{1}{M}\sum_{i=1}^{M}W(\Theta^{i}),$

and

$\displaystyle\widehat{W}_{LI}(\Theta)=-\frac{1}{p}\log\bigg{(}\frac{1}{M}e^{-pW(\Theta^{i})}\bigg{)},$

respectively, where
$\Theta^{i}=(\alpha^{i},\lambda_{0}^{i},\lambda_{1}^{i},\lambda_{2}^{i})$.
Replacing $W(\Theta)$ by the parameters, Bayes estimates of the parameters can
be obtained.

###  4.3 Highest posterior density credible interval

In order to obtain HPD credible intervals with $\gamma$ level of significance
for the model parameters, the following steps are employed using the MCMC
samples generated in Section $4.2$.

Step 1: Generated MCMC samples of the parameters have been arranged in
ascending order as $(\alpha^{(1)},\cdots,\alpha^{(M)})$,
$(\lambda_{0}^{(1)},\cdots,\lambda_{0}^{(M)})$,
$(\lambda_{1}^{(1)},\cdots,\lambda_{1}^{(M)})$.
Step 2: HPD credible intervals of $\Theta$ with $\gamma$ significance level
are given by

$\displaystyle(\Theta^{(j)},\Theta^{(j+(1-\frac{\gamma}{2})M)}),$

where $j$ can be obtained from the equation
$\Theta^{(j+(1-\frac{\gamma}{2})M)})-\Theta^{(j)}=min_{i}~{}(\Theta^{(i+(1-\frac{\gamma}{2})M)})-\Theta^{(i)})$,
where $1\leq i\leq M-(1-\frac{\gamma}{2})M$.

###  4.4 MCMC convergence criterion

In this subsection, a convergence test suggested by Gelman and Rubin (1992)
has been used to discuss the convergence of MCMC samples. The following steps
have been used to check whether the MCMC samples are convergent or not.

Step I: $M$ number of samples have been generated for the parameters
$\Theta=(\theta_{1},\theta_{2},\theta_{3},\theta_{4})=(\alpha,\lambda_{0},\lambda_{1},\lambda_{2})$
as $(\theta_{1}^{1},\cdots,\theta_{1}^{M})$,
$(\theta_{2}^{1},\cdots,\theta_{2}^{M})$,
$(\theta_{3}^{1},\cdots,\theta_{3}^{M})$ and
$(\theta_{4}^{1},\cdots,\theta_{4}^{M})$. Then consider these samples as $M$
number of chains of the parameters as
$\\{(\theta_{1}^{1},\theta_{2}^{1},\theta_{3}^{1},\theta_{4}^{1}),\cdots,(\theta_{1}^{M},\theta_{2}^{M},\theta_{3}^{M},\theta_{4}^{M})\\}$
Step II: Calculate the mean for each chain of parameters as

$\displaystyle\widehat{\theta}_{j}=\frac{1}{4}\sum_{i=1}^{4}\theta_{i}^{j},$

where $i\in\\{1,2,3,4\\}$ and $j\in\\{1,\cdots,M\\}$.
Step III: For each chain of parameters, the intra-chain variance can be
computed as

$\displaystyle\sigma_{j}^{2}=\frac{1}{3}\sum_{i=1}^{4}\big{(}\theta_{i}^{j}-\widehat{\theta}_{j}\big{)}^{2}.$

Step IV: Compute mean as
$\widehat{\theta}=\frac{1}{M}\sum_{j=1}^{M}\widehat{\theta}_{j}$ and variance
of all chains as $W=\frac{1}{M}\sum_{j=1}^{M}\sigma_{j}^{2}$.
Step V: Calculate the variance of the average mean over all chains as

$\displaystyle
B=\frac{4}{M-1}\sum_{j=1}^{M}\big{(}\widehat{\theta}_{j}-\widehat{\theta}\big{)}^{2}.$

Step VI: Calculate $V=\frac{3}{4}W+\frac{M+1}{4M}B$ and $G=\sqrt{V/W}$.

Finally, the MCMC sample converges if $G\approx 1$, otherwise not.

##  5 Simulation study

In this section, some results based on Monte Carlo simulations have been
provided to evaluate the performance of the proposed methods given in the
preceding sections. Based on the following considerations, the performance of
the estimates has been compared.

  * •

Absolute Bias (AB):
$\frac{1}{N}\sum_{i=1}^{N}|\Theta_{i}-\widehat{\Theta}_{i}|$, where
$\Theta_{i}$ represents parameters, whereas $\widehat{\Theta}_{i}$ represents
their estimates and $N$ is the number of iterations. The lower value of AB
suggests that the experimental data and prediction model are more accurately
correlated.

  * •

Mean squared error (MSE):
$\frac{1}{N}\sum_{i=1}^{N}\big{(}\Theta_{i}-\widehat{\Theta}_{i}\big{)}^{2}$.
The greater performance of the estimations is indicated by the smaller value
of MSE.

  * •

Average width (AW): Average width of $100(1-\gamma)\%$ interval estimates has
been assessed. Better performance of the interval estimates is correlated with
narrower width.

  * •

Coverage probability (CP): The probability of containing the true values of
the parameters in between the interval estimates.

To generate AT-II PHCS with competing risks, the following algorithm has been
constructed.

—————————————————————————————————————————— Algorithm
—————————————————————————————————————————— 1\. According to inverse
distribution function method min $\\{Y_{1},Y_{2}\\}\sim
MOBW(\lambda_{0},\lambda_{1},\lambda_{2},\alpha)$, then the progressive Type-
II censored samples following MOBW distribution have been generated.
2\. According to the given time threshold $T$, the value of $J$ has been
obtained such that $Y_{j:m:n}<T<Y_{j+1:m:n}$.
3\. Then failure risks $\delta_{i}=0,1,2,3$, where $i=1,\cdots,m$ have been
assigned to the obtained censored data.
—————————————————————————————————————————— In that case when only partial
failure reasons have been observed during an experiment, then it has been
considered that each observation is independent of each other with the
probability of the unknown reason for each failure as $P(\delta_{i}=3)=q$. The
probability of failure risks can be obtained as

$\displaystyle
P(\delta_{i}=0)=\frac{\lambda_{0}}{g(\lambda)},~{}~{}P(\delta_{i}=1)=\frac{\lambda_{2}\lambda_{01}}{g(\lambda)},~{}\mbox{and}~{}P(\delta_{i}=2)=\frac{\lambda_{1}\lambda_{02}}{g(\lambda)},$

where
$g(\lambda)=\lambda_{0}+\lambda_{1}\lambda_{02}+\lambda_{2}\lambda_{01}$.

The Monte Carlo simulation study has been performed with the initial guess of
the parameters as $\lambda_{0}=0.5,~{}\lambda_{1}=1,~{}\lambda_{2}=1.5,$ and
$\alpha=1$. Set $(n,m)=(50,30),~{}(50,40),~{}(80,40),~{}(80,60)$ with
different values of $T=(0.5,1)$ and $q=(0.1,0.3)$. Three following censoring
schemes are considered here.

Scheme I: $R_{1}=\cdots=R_{m-1}=0,R_{m}=(n-m)$.
Scheme II: $R_{1}=(n-m),R_{2}=\cdots=R_{m}=0$.
Scheme III: $R_{2m-n+1}=\cdots=R_{m}=1$, and $R_{i}=0$, others.

In order to obtain Bayes estimates two different loss functions such as SELF
and LLF have been considered. Gamma distribution is thought to be the prior
distribution of $\alpha$. Here, all the hyper-parameters are considered as
$0.001$. To obtain Bayes estimates, an MCMC algorithm is simulated with 10,000
repetitions. Also the value of the statistic $G$ has been computed to check
the convergence of the MCMC sample. Further, $95\%$ ACIs and HPD credible
intervals for the unknown model parameters are also constructed. This
simulation study has been evaluated using $R~{}4.0.4$ software via three
recommended packages. These packages are ‘nleqslv’ introduced by Hasselman and
Hasselman (2018) to compute classical estimates using Newton-Raphson method,
‘coda’ introduced by Plummer et al. (2015) to obtain Bayes estimates using
MCMC samples, and ‘matlib’ introduced by Friendly et al. (2021). The
simulation results of point and interval estimates are tabulated in Tables
1-4. From these tables, some observations can be summarized as follows:

  * •

With the increase of $(n,m)$ and $T$ when $q$ is fixed, the AB and MSE of
point estimates decrease. That is, the estimated results are more accurate
when the effective sample size becomes larger.

  * •

According to ABs and MSEs, Bayes estimates outperform than MLEs. Further, the
performance of Bayes estimates under LLF is superior to that under SELF.

  * •

With the increase of $(n,m)$ and $T$ when $q$ is fixed, the AW of the
intervals decrease, which yields that the estimation has more accuracy. But no
trends of CP have been observed throughout these simulation results.

  * •

HPD intervals perform better than ACIs based on AWs and CPs.

  * •

With an increase of $q$, the point and interval estimation results based on
AB, MSE, and AW become less accurate since the increasing value of $q$
indicates less available information.

  * •

The values of $G$\- statistic are almost $1$, which yields that the MCMC
samples are convergent.

From the above results it has been concluded that the performance of point and
interval estimates are quite satisfactory. Bayes estimates has been computed
based on non-informative prior,and the results of the point estimates based on
Bayesian estimation are more effective. For interval estimation one can prefer
to HPD intervals over ACIs to get more accurate results.

Table 1: Simulated ABs and MSEs (in parentheses) of the MLE, Bayes estimates
under SELF and LLF for different values of $T$ with $q=0.1$.

| MLE | SELF | LLF
---|---|---|---
$(n,m)$ | $T$ | Scheme | $\lambda_{0}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\alpha$ | $\lambda_{0}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\alpha$ | $\lambda_{0}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\alpha$
(50,30) | 0.5 | I | 0.2284 | 0.6347 | 0.4445 | 0.1482 | 0.1090 | 0.5578 | 0.2442 | 0.1171 | 0.1014 | 0.4556 | 0.2149 | 0.1106
|  |  | (0.0605) | (0.8188) | (0.4862) | (0.0382) | (0.0179) | (0.4086) | (0.1119) | (0.0203) | (0.0178) | (0.2752) | (0.1062) | (0.0196)
|  | II | 0.2319 | 0.4854 | 0.3017 | 0.1160 | 0.1119 | 0.4250 | 0.2740 | 0.1102 | 0.1044 | 0.3553 | 0.2686 | 0.1077
|  |  | (0.0589) | (0.3670) | (0.1438) | (0.0230) | (0.0186) | (0.2862) | (0.1260) | (0.0203) | (0.0185) | (0.2031) | (0.1092) | (0.0192)
|  | III | 0.2345 | 0.5686 | 0.4120 | 0.1360 | 0.1118 | 0.4593 | 0.3269 | 0.1138 | 0.1043 | 0.3778 | 0.3077 | 0.1116
|  |  | (0.0625) | (0.6372) | (0.3660) | (0.0317) | (0.0187) | (0.3545) | (0.1818) | (0.0212) | (0.0181) | (0.2383) | (0.1453) | (0.0202)
| 1 | I | 0.2259 | 0.6277 | 0.4259 | 0.1476 | 0.1243 | 0.3910 | 0.2385 | 0.1250 | 0.1176 | 0.3483 | 0.3054 | 0.1222
|  |  | (0.0547) | (0.6246) | (0.3936) | (0.0358) | (0.0165) | (0.3668) | (0.0951) | (0.0192) | (0.0158) | (0.2407) | (0.0868) | (0.0187)
|  | II | 0.2294 | 0.4757 | 0.2779 | 0.1077 | 0.1107 | 0.4058 | 0.2659 | 0.1087 | 0.1032 | 0.3464 | 0.2618 | 0.1041
|  |  | (0.0551) | (0.3523) | (0.1385) | (0.0222) | (0.0185) | (0.2793) | (0.1261) | (0.0204) | (0.0194) | (0.1969) | (0.1104) | (0.0190)
|  | III | 0.2302 | 0.5446 | 0.3889 | 0.1335 | 0.1114 | 0.4426 | 0.3095 | 0.1115 | 0.1019 | 0.3676 | 0.2885 | 0.1108
|  |  | (0.0602) | (0.6102) | (0.3595) | (0.0313) | (0.0186) | (0.3457) | (0.1693) | (0.0209) | (0.0179) | (0.2262) | (0.1314) | (0.0200)
(50,40) | 0.5 | I | 0.2098 | 0.5228 | 0.2813 | 0.1373 | 0.0999 | 0.3203 | 0.2123 | 0.1160 | 0.1010 | 0.2756 | 0.2043 | 0.0986
|  |  | (0.0487) | (0.4148) | (0.1361) | (0.0232) | (0.0160) | (0.2243) | (0.1052) | (0.0198) | (0.0154) | (0.2181) | (0.1032) | (0.0193)
|  | II | 0.2089 | 0.4689 | 0.2443 | 0.1046 | 0.1092 | 0.4160 | 0.2384 | 0.1013 | 0.1017 | 0.3373 | 0.2256 | 0.0991
|  |  | (0.0475) | (0.3316) | (0.1036) | (0.0183) | (0.0175) | (0.2822) | (0.0944) | (0.0189) | (0.0184) | (0.2011) | (0.0935) | (0.0181)
|  | III | 0.2245 | 0.4630 | 0.3309 | 0.1188 | 0.1053 | 0.4261 | 0.3060 | 0.1073 | 0.1018 | 0.3671 | 0.3010 | 0.1061
|  |  | (0.0561) | (0.3658) | (0.2720) | (0.0269) | (0.0184) | (0.2925) | (0.1416) | (0.0198) | (0.0180) | (0.2158) | (0.1394) | (0.0191)
| 1 | I | 0.2054 | 0.5138 | 0.2783 | 0.1176 | 0.0926 | 0.3040 | 0.1820 | 0.0936 | 0.1053 | 0.3723 | 0.1638 | 0.0915
|  |  | (0.0497) | (0.4268) | (0.1454) | (0.0235) | (0.0158) | (0.2213) | (0.0866) | (0.0176) | (0.0154) | (0.1645) | (0.0848) | (0.0171)
|  | II | 0.2042 | 0.4659 | 0.2412 | 0.1038 | 0.1089 | 0.3931 | 0.2137 | 0.1012 | 0.1004 | 0.3245 | 0.2107 | 0.0979
|  |  | (0.0449) | (0.3275) | (0.0983) | (0.0181) | (0.0174) | (0.2765) | (0.0912) | (0.0180) | (0.0172) | (0.1932) | (0.0903) | (0.0176)
|  | III | 0.2106 | 0.4237 | 0.2803 | 0.1153 | 0.1025 | 0.4182 | 0.2863 | 0.1040 | 0.1012 | 0.3489 | 0.2653 | 0.1055
|  |  | (0.0491) | (0.3458) | (0.2358) | (0.0245) | (0.0175) | (0.2670) | (0.1144) | (0.0194) | (0.0173) | (0.2064) | (0.1121) | (0.0183)
(80,40) | 0.5 | I | 0.2044 | 0.5165 | 0.2620 | 0.1273 | 0.0961 | 0.2884 | 0.1829 | 0.1072 | 0.0915 | 0.2617 | 0.1613 | 0.0954
|  |  | (0.0478) | (0.4081) | (0.1283) | (0.0227) | (0.0152) | (0.2101) | (0.1044) | (0.0182) | (0.0151) | (0.2049) | (0.1017) | (0.0178)
|  | II | 0.2059 | 0.4517 | 0.2394 | 0.0975 | 0.1074 | 0.4052 | 0.2186 | 0.0955 | 0.1012 | 0.3264 | 0.2064 | 0.0936
|  |  | (0.0457) | (0.3225) | (0.1027) | (0.0178) | (0.0170) | (0.2726) | (0.0933) | (0.0176) | (0.0180) | (0.1952) | (0.0916) | (0.0173)
|  | III | 0.2147 | 0.4345 | 0.3217 | 0.1091 | 0.1022 | 0.4172 | 0.2941 | 0.0976 | 0.0948 | 0.3461 | 0.2891 | 0.0959
|  |  | (0.0525) | (0.3449) | (0.2014) | (0.0227) | (0.0181) | (0.2769) | (0.1385) | (0.0183) | (0.0176) | (0.2076) | (0.1288) | (0.0178)
| 1 | I | 0.2014 | 0.4957 | 0.2534 | 0.1148 | 0.0915 | 0.2573 | 0.1678 | 0.0917 | 0.0902 | 0.2371 | 0.1591 | 0.0912
|  |  | (0.0459) | (0.3984) | (0.1225) | (0.0216) | (0.0156) | (0.2052) | (0.0868) | (0.0172) | (0.0151) | (0.1595) | (0.0824) | (0.0169)
|  | II | 0.2016 | 0.4442 | 0.2318 | 0.0960 | 0.1030 | 0.3876 | 0.2116 | 0.0940 | 0.0986 | 0.3180 | 0.2050 | 0.0934
|  |  | (0.0436) | (0.3109) | (0.0929) | (0.0175) | (0.0167) | (0.2671) | (0.0898) | (0.0164) | (0.0168) | (0.1906) | (0.0891) | (0.0165)
|  | III | 0.2082 | 0.4065 | 0.2781 | 0.1080 | 0.1031 | 0.3869 | 0.2697 | 0.0961 | 0.0915 | 0.3304 | 0.2532 | 0.0957
|  |  | (0.0486) | (0.3280) | (0.1916) | (0.0198) | (0.0175) | (0.2567) | (0.1083) | (0.0176) | (0.0168) | (0.1987) | (0.1112) | (0.0173)
(80,60) | 0.5 | I | 0.1991 | 0.4874 | 0.2477 | 0.1144 | 0.0950 | 0.2646 | 0.1670 | 0.1032 | 0.0898 | 0.2560 | 0.1500 | 0.0937
|  |  | (0.0432) | (0.3299) | (0.1038) | (0.0218) | (0.0137) | (0.2032) | (0.0974) | (0.0161) | (0.0142) | (0.1932) | (0.0928) | (0.0157)
|  | II | 0.2015 | 0.4391 | 0.2182 | 0.0926 | 0.1029 | 0.3367 | 0.1897 | 0.0924 | 0.0995 | 0.3158 | 0.1828 | 0.0911
|  |  | (0.0430) | (0.2653) | (0.0975) | (0.0172) | (0.0167) | (0.2439) | (0.0889) | (0.0160) | (0.0179) | (0.1921) | (0.0846) | (0.0156)
|  | III | 0.2072 | 0.4079 | 0.3035 | 0.0967 | 0.0980 | 0.3374 | 0.2854 | 0.0907 | 0.0910 | 0.3240 | 0.2754 | 0.0898
|  |  | (0.0514) | (0.2819) | (0.1907) | (0.0196) | (0.0175) | (0.2033) | (0.1174) | (0.0173) | (0.0171) | (0.1952) | (0.1116) | (0.0167)
| 1 | I | 0.1979 | 0.4622 | 0.2389 | 0.1035 | 0.0861 | 0.2412 | 0.1356 | 0.0902 | 0.0867 | 0.2292 | 0.1470 | 0.0877
|  |  | (0.0427) | (0.3064) | (0.0947) | (0.0195) | (0.0136) | (0.1941) | (0.0869) | (0.0153) | (0.0133) | (0.1859) | (0.0838) | (0.0148)
|  | II | 0.1940 | 0.4270 | 0.2053 | 0.0960 | 0.1008 | 0.3301 | 0.1771 | 0.0916 | 0.0935 | 0.3044 | 0.1695 | 0.0862
|  |  | (0.0424) | (0.2623) | (0.0861) | (0.0162) | (0.0164) | (0.2433) | (0.0670) | (0.0146) | (0.0160) | (0.1825) | (0.0628) | (0.0141)
|  | III | 0.1985 | 0.3863 | 0.2397 | 0.0918 | 0.0897 | 0.2761 | 0.2287 | 0.0802 | 0.0823 | 0.2647 | 0.2144 | 0.0785
|  |  | (0.0428) | (0.2704) | (0.1861) | (0.0189) | (0.0171) | (0.2020) | (0.1028) | (0.0154) | (0.0165) | (0.1911) | (0.0987) | (0.0151)

Table 2: Simulated AWs and CPs (in parentheses) of ACIs and HPD credible
intervals along with corresponding $G$ values for different values of $T$ when
$q=0.1$.

| ACI | HPD |
---|---|---|---
$(n,m)$ | $T$ | Scheme | $\lambda_{0}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\alpha$ | $\lambda_{0}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\alpha$ | $G$
(50,30) | 0.5 | I | 0.8257 | 2.5102 | 2.5598 | 0.6901 | 0.2916 | 1.5883 | 1.5579 | 0.4513 | 0.9864
|  |  | (0.8574) | (0.9399) | (0.9507) | (0.9516) | (0.9030) | (0.9541) | (0.9735) | (0.9752) |
|  | II | 0.7068 | 1.7502 | 1.7670 | 0.5555 | 0.2918 | 1.3613 | 1.3650 | 0.4364 | 0.9892
|  |  | (0.8646) | (0.9397) | (0.9409) | (0.9527) | (0.8923) | (0.9483) | (0.9672) | (0.9621) |
|  | III | 0.7657 | 2.1742 | 2.2108 | 0.6089 | 0.2938 | 1.4627 | 1.4585 | 0.4311 | 0.9925
|  |  | (0.8154) | (0.9393) | (0.9412) | (0.9375) | (0.8925) | (0.9531) | (0.9632) | (0.9562) |
| 1 | I | 0.8224 | 2.5012 | 2.5497 | 0.6889 | 0.2841 | 1.2439 | 1.2842 | 0.4373 | 0.9951
|  |  | (0.8603) | (0.9402) | (0.9514) | (0.9535) | (0.9089) | (0.9685) | (0.9776) | (0.9794) |
|  | II | 0.7009 | 1.7460 | 1.7306 | 0.5459 | 0.2892 | 1.3608 | 1.3584 | 0.4324 | 0.9963
|  |  | (0.8436) | (0.9378) | (0.9436) | (0.9524) | (0.8913) | (0.9259) | (0.9646) | (0.9628) |
|  | III | 0.7413 | 2.0253 | 2.1649 | 0.5980 | 0.2905 | 1.4471 | 1.4389 | 0.4276 | 0.9927
|  |  | (0.8587) | (0.9425) | (0.9378) | (0.9497) | (0.8912) | (0.9583) | (0.9592) | (0.9582) |
(50,40) | 0.5 | I | 0.6621 | 1.7138 | 1.6679 | 0.5569 | 0.2780 | 1.3737 | 1.2928 | 0.4365 | 1.0102
|  |  | (0.8795) | (0.9349) | (0.9438) | (0.9533) | (0.9117) | (0.9614) | (0.9625) | (0.9848) |
|  | II | 0.6313 | 1.4962 | 1.4572 | 0.4942 | 0.2869 | 1.2400 | 1.2026 | 0.4056 | 0.9975
|  |  | (0.8789) | (0.9380) | (0.9401) | (0.9504) | (0.8927) | (0.9577) | (0.9606) | (0.9655) |
|  | III | 0.6236 | 1.5862 | 1.5432 | 0.5229 | 0.2935 | 1.2601 | 1.2235 | 0.4090 | 0.9938
|  |  | (0.8609) | (0.9391) | (0.9322) | (0.9142) | (0.9088) | (0.9504) | (0.9629) | (0.9583) |
| 1 | I | 0.6430 | 1.6976 | 1.6523 | 0.5451 | 0.2726 | 1.2782 | 1.2770 | 0.4056 | 1.0423
|  |  | (0.8694) | (0.9345) | (0.9625) | (0.9495) | (0.8907) | (0.9473) | (0.9884) | (0.9798) |
|  | II | 0.6301 | 1.4898 | 1.4351 | 0.4923 | 0.2843 | 1.2347 | 1.2008 | 0.4042 | 0.9942
|  |  | (0.8783) | (0.9385) | (0.9488) | (0.9495) | (0.8944) | (0.9443) | (0.9644) | (0.9651) |
|  | III | 0.6201 | 1.5631 | 1.5290 | 0.5126 | 0.2869 | 1.2395 | 1.2101 | 0.4017 | 1.0028
|  |  | (0.8735) | (0.9409) | (0.9426) | (0.9514) | (0.8962) | (0.9672) | (0.9591) | (0.9583) |
(80,40) | 0.5 | I | 0.6543 | 1.6950 | 1.6208 | 0.5319 | 0.2672 | 1.2802 | 1.2762 | 0.3559 | 0.9880
|  |  | (0.8621) | (0.9438) | (0.9480) | (0.9532) | (0.9084) | (0.9542) | (0.9623) | (0.9695) |
|  | II | 0.6287 | 1.4871 | 1.4485 | 0.4711 | 0.2864 | 1.2333 | 1.2019 | 0.3897 | 0.9973
|  |  | (0.8615) | (0.9169) | (0.9453) | (0.9538) | (0.8975) | (0.9532) | (0.9609) | (0.9712) |
|  | III | 0.6272 | 1.5765 | 1.5460 | 0.4960 | 0.2907 | 1.2534 | 1.2105 | 0.3780 | 0.9905
|  |  | (0.8735) | (0.9372) | (0.9072) | (0.9332) | (0.8925) | (0.9676) | (0.9351) | (0.9623) |
| 1 | I | 0.6306 | 1.6841 | 1.6118 | 0.5298 | 0.2346 | 1.1070 | 1.1245 | 0.3260 | 0.9961
|  |  | (0.8705) | (0.9480) | (0.9431) | (0.9530) | (0.9059) | (0.9897) | (0.9776) | (0.9830) |
|  | II | 0.6247 | 1.4847 | 1.4402 | 0.4695 | 0.2781 | 1.2197 | 1.1974 | 0.3829 | 1.0214
|  |  | (0.8721) | (0.9317) | (0.9457) | (0.9497) | (0.8929) | (0.9584) | (0.9627) | (0.9686) |
|  | III | 0.6167 | 1.5581 | 1.5155 | 0.4844 | 0.2853 | 1.2370 | 1.2072 | 0.3704 | 1.0325
|  |  | (0.8731) | (0.9373) | (0.9567) | (0.9499) | (0.9016) | (0.9609) | (0.9772) | (0.9616) |
(80,60) | 0.5 | I | 0.5458 | 1.3901 | 1.3323 | 0.4551 | 0.2673 | 1.1507 | 1.1418 | 0.3583 | 1.0403
|  |  | (0.8054) | (0.9465) | (0.9379) | (0.9524) | (0.9023) | (0.9745) | (0.9698) | (0.9889) |
|  | II | 0.5145 | 1.1850 | 1.1413 | 0.3953 | 0.2808 | 1.0497 | 1.0054 | 0.3424 | 0.9945
|  |  | (0.8040) | (0.8675) | (0.9430) | (0.9508) | (0.8854) | (0.9299) | (0.9605) | (0.9733) |
|  | III | 0.5056 | 1.2538 | 1.2033 | 0.4179 | 0.2880 | 1.0839 | 1.0395 | 0.3476 | 1.0210
|  |  | (0.8567) | (0.9421) | (0.9067) | (0.9057) | (0.8691) | (0.9654) | (0.9599) | (0.9629) |
| 1 | I | 0.5381 | 1.3862 | 1.3280 | 0.4517 | 0.2579 | 1.1462 | 1.1191 | 0.3524 | 0.9978
|  |  | (0.8154) | (0.9416) | (0.9442) | (0.9510) | (0.8948) | (0.9536) | (0.9794) | (0.9806) |
|  | II | 0.5064 | 1.1789 | 1.1369 | 0.3910 | 0.2781 | 1.0464 | 1.0008 | 0.3407 | 0.9953
|  |  | (0.8652) | (0.9049) | (0.9512) | (0.9497) | (0.8835) | (0.9559) | (0.9731) | (0.9775) |
|  | III | 0.5025 | 1.2368 | 1.2018 | 0.4159 | 0.2811 | 1.0569 | 1.0136 | 0.3373 | 1.0089
|  |  | (0.8126) | (0.9317) | (0.9411) | (0.9456) | (0.8921) | (0.9556) | (0.9653) | (0.9663) |

Table 3: Simulated ABs and MSEs (in parentheses) of the MLE, Bayes estimates
under SELF and LLF for different values of $T$ with $q=0.3$.

| MLE | SELF | LLF
---|---|---|---
$(n,m)$ | $T$ | Scheme | $\lambda_{0}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\alpha$ | $\lambda_{0}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\alpha$ | $\lambda_{0}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\alpha$
(50,30) | 0.5 | I | 0.2309 | 0.6494 | 0.4589 | 0.1495 | 0.1387 | 0.5705 | 0.2984 | 0.1282 | 0.1332 | 0.5167 | 0.2551 | 0.1214
|  |  | (0.0649) | (0.8394) | (0.5048) | (0.0391) | (0.0290) | (0.4389) | (0.1841) | (0.0235) | (0.0280) | (0.3760) | (0.1701) | (0.0216)
|  | II | 0.2408 | 0.5043 | 0.3132 | 0.1259 | 0.1246 | 0.4494 | 0.2867 | 0.1221 | 0.1218 | 0.4084 | 0.2789 | 0.1113
|  |  | (0.0617) | (0.3891) | (0.1559) | (0.0252) | (0.0237) | (0.3047) | (0.1295) | (0.0214) | (0.0239) | (0.2394) | (0.1158) | (0.0204)
|  | III | 0.2459 | 0.5719 | 0.4255 | 0.1452 | 0.1313 | 0.5080 | 0.3850 | 0.1239 | 0.1131 | 0.4723 | 0.3172 | 0.1166
|  |  | (0.0650) | (0.6559) | (0.3820) | (0.0332) | (0.0198) | (0.4242) | (0.3088) | (0.0242) | (0.0193) | (0.4072) | (0.1679) | (0.0219)
| 1 | I | 0.2320 | 0.6365 | 0.4382 | 0.1491 | 0.1171 | 0.4780 | 0.3657 | 0.1207 | 0.1094 | 0.4363 | 0.3221 | 0.1175
|  |  | (0.0580) | (0.7425) | (0.4486) | (0.0382) | (0.0234) | (0.3637) | (0.1888) | (0.0276) | (0.0212) | (0.2470) | (0.1673) | (0.0259)
|  | II | 0.2378 | 0.4858 | 0.2808 | 0.1177 | 0.1186 | 0.4251 | 0.2925 | 0.1127 | 0.1101 | 0.4052 | 0.2684 | 0.1086
|  |  | (0.0581) | (0.3763) | (0.1442) | (0.0237) | (0.0189) | (0.3027) | (0.1308) | (0.0219) | (0.0184) | (0.2890) | (0.1201) | (0.0206)
|  | III | 0.2320 | 0.5629 | 0.3943 | 0.1361 | 0.1222 | 0.4855 | 0.3114 | 0.1165 | 0.1161 | 0.3805 | 0.3019 | 0.1129
|  |  | (0.0630) | (0.6195) | (0.3608) | (0.0328) | (0.0252) | (0.3558) | (0.1762) | (0.0215) | (0.0238) | (0.3041) | (0.1496) | (0.0206)
(50,40) | 0.5 | I | 0.2162 | 0.5477 | 0.2881 | 0.1383 | 0.1164 | 0.3440 | 0.2290 | 0.1218 | 0.1191 | 0.2952 | 0.2109 | 0.1083
|  |  | (0.0518) | (0.4387) | (0.1474) | (0.0237) | (0.0202) | (0.2511) | (0.1128) | (0.0212) | (0.0192) | (0.2298) | (0.1089) | (0.0205)
|  | II | 0.2120 | 0.4790 | 0.2543 | 0.1092 | 0.1229 | 0.4335 | 0.2483 | 0.1047 | 0.1213 | 0.3633 | 0.2366 | 0.1011
|  |  | (0.0489) | (0.3482) | (0.1059) | (0.0211) | (0.0215) | (0.3033) | (0.1235) | (0.0202) | (0.0207) | (0.2566) | (0.1154) | (0.0197)
|  | III | 0.2286 | 0.4868 | 0.3513 | 0.1289 | 0.1230 | 0.4488 | 0.3404 | 0.1163 | 0.1205 | 0.3739 | 0.3349 | 0.1125
|  |  | (0.0582) | (0.3826) | (0.2748) | (0.0268) | (0.0216) | (0.3160) | (0.1615) | (0.0206) | (0.0208) | (0.2393) | (0.1368) | (0.0197)
| 1 | I | 0.2194 | 0.5265 | 0.2854 | 0.1221 | 0.1207 | 0.3227 | 0.2011 | 0.1161 | 0.1169 | 0.2959 | 0.1873 | 0.1126
|  |  | (0.0535) | (0.4489) | (0.1494) | (0.0241) | (0.0209) | (0.2641) | (0.1290) | (0.0203) | (0.0196) | (0.2422) | (0.1210) | (0.0193)
|  | II | 0.2200 | 0.4807 | 0.2526 | 0.1142 | 0.1294 | 0.4207 | 0.2332 | 0.1009 | 0.1238 | 0.3931 | 0.2343 | 0.0985
|  |  | (0.0526) | (0.3401) | (0.1061) | (0.0199) | (0.0274) | (0.2809) | (0.0995) | (0.0186) | (0.0262) | (0.2369) | (0.0969) | (0.0180)
|  | III | 0.2162 | 0.4977 | 0.2881 | 0.1183 | 0.1164 | 0.4040 | 0.2790 | 0.1062 | 0.1121 | 0.3725 | 0.2709 | 0.1083
|  |  | (0.0518) | (0.3847) | (0.2474) | (0.0257) | (0.0202) | (0.3151) | (0.1428) | (0.0182) | (0.0195) | (0.2487) | (0.1389) | (0.0188)
(80,40) | 0.5 | I | 0.2140 | 0.5848 | 0.3923 | 0.1261 | 0.1150 | 0.3033 | 0.1959 | 0.1021 | 0.1091 | 0.2922 | 0.1807 | 0.0995
|  |  | (0.0532) | (0.4333) | (0.1416) | (0.0267) | (0.0252) | (0.2494) | (0.1238) | (0.0215) | (0.0219) | (0.2244) | (0.1159) | (0.0188)
|  | II | 0.2115 | 0.4623 | 0.2428 | 0.0986 | 0.1121 | 0.4137 | 0.2384 | 0.0961 | 0.1054 | 0.3615 | 0.2106 | 0.0949
|  |  | (0.0486) | (0.3395) | (0.1081) | (0.0187) | (0.0212) | (0.2951) | (0.1037) | (0.0179) | (0.0205) | (0.2347) | (0.1008) | (0.0175)
|  | III | 0.2214 | 0.4943 | 0.3426 | 0.1125 | 0.1185 | 0.4279 | 0.3082 | 0.1017 | 0.1121 | 0.4026 | 0.2923 | 0.0984
|  |  | (0.0554) | (0.3691) | (0.2106) | (0.0235) | (0.0208) | (0.2937) | (0.1418) | (0.0211) | (0.0189) | (0.2168) | (0.1357) | (0.0191)
| 1 | I | 0.2143 | 0.5918 | 0.2996 | 0.1262 | 0.1082 | 0.2830 | 0.1783 | 0.1013 | 0.1027 | 0.2461 | 0.1694 | 0.0955
|  |  | (0.0534) | (0.4302) | (0.1371) | (0.0231) | (0.0196) | (0.2218) | (0.1059) | (0.0189) | (0.0184) | (0.1837) | (0.0974) | (0.0182)
|  | II | 0.2223 | 0.4507 | 0.2419 | 0.1218 | 0.1137 | 0.4098 | 0.2249 | 0.1048 | 0.1089 | 0.3552 | 0.2113 | 0.0985
|  |  | (0.0495) | (0.3293) | (0.1051) | (0.0191) | (0.0183) | (0.2752) | (0.0938) | (0.0173) | (0.0176) | (0.2250) | (0.0902) | (0.0169)
|  | III | 0.2191 | 0.4751 | 0.3055 | 0.1145 | 0.1108 | 0.4115 | 0.2811 | 0.1049 | 0.1047 | 0.3863 | 0.2653 | 0.0993
|  |  | (0.0517) | (0.3560) | (0.2187) | (0.0206) | (0.0215) | (0.2893) | (0.1210) | (0.0184) | (0.0205) | (0.2439) | (0.1158) | (0.0179)
(80,60) | 0.5 | I | 0.2212 | 0.6078 | 0.4124 | 0.1343 | 0.1309 | 0.3315 | 0.2060 | 0.1113 | 0.1238 | 0.3176 | 0.1962 | 0.1078
|  |  | (0.0542) | (0.4427) | (0.1634) | (0.0286) | (0.0261) | (0.2819) | (0.1325) | (0.0229) | (0.0240) | (0.2430) | (0.1270) | (0.0208)
|  | II | 0.2069 | 0.4424 | 0.2577 | 0.1281 | 0.1161 | 0.3696 | 0.1930 | 0.1091 | 0.1090 | 0.3237 | 0.1875 | 0.0989
|  |  | (0.0452) | (0.2943) | (0.1036) | (0.0194) | (0.0191) | (0.2634) | (0.0928) | (0.0178) | (0.0185) | (0.2218) | (0.0899) | (0.0171)
|  | III | 0.2193 | 0.4270 | 0.3197 | 0.0981 | 0.1039 | 0.3645 | 0.2890 | 0.0945 | 0.0984 | 0.3317 | 0.2805 | 0.0921
|  |  | (0.0525) | (0.3055) | (0.2108) | (0.0215) | (0.0195) | (0.2431) | (0.1325) | (0.0196) | (0.0182) | (0.2039) | (0.1250) | (0.0189)
| 1 | I | 0.2007 | 0.4699 | 0.2414 | 0.1095 | 0.1016 | 0.2736 | 0.1406 | 0.0982 | 0.0956 | 0.2385 | 0.1524 | 0.0917
|  |  | (0.0459) | (0.3183) | (0.1094) | (0.0207) | (0.0169) | (0.2108) | (0.0985) | (0.0179) | (0.0154) | (0.2017) | (0.0934) | (0.0168)
|  | II | 0.2067 | 0.4385 | 0.2276 | 0.0985 | 0.1043 | 0.3583 | 0.1923 | 0.0933 | 0.0986 | 0.3212 | 0.1794 | 0.0910
|  |  | (0.0450) | (0.2883) | (0.1025) | (0.0184) | (0.0199) | (0.2551) | (0.0821) | (0.0169) | (0.0186) | (0.2105) | (0.0783) | (0.0159)
|  | III | 0.2009 | 0.4282 | 0.2652 | 0.0964 | 0.1031 | 0.3106 | 0.2360 | 0.0912 | 0.0966 | 0.2883 | 0.2254 | 0.0869
|  |  | (0.0440) | (0.3014) | (0.1954) | (0.0207) | (0.0186) | (0.2118) | (0.1095) | (0.0186) | (0.0178) | (0.1994) | (0.1023) | (0.0175)

Table 4: Simulated AWs and CPs (in parentheses) of of ACIs and HPD credible
intervals along with corresponding $G$ values for different values of $T$ when
$q=0.3$.

| ACI | HPD |
---|---|---|---
$(n,m)$ | $T$ | Scheme | $\lambda_{0}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\alpha$ | $\lambda_{0}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\alpha$ | $G$
(50,30) | 0.5 | I | 0.9562 | 2.5978 | 2.6565 | 0.6979 | 0.3434 | 1.6724 | 1.6901 | 0.4808 | 0.9893
|  |  | (0.8527) | (0.9489) | (0.9374) | (0.9498) | (0.8986) | (0.9693) | (0.9529) | (0.9802) |
|  | II | 0.7909 | 1.8249 | 1.8625 | 0.5857 | 0.3186 | 1.4035 | 1.4543 | 0.5166 | 0.9931
|  |  | (0.8585) | (0.9495) | (0.9487) | (0.9530) | (0.8743) | (1.0000) | (0.9795) | (0.9885) |
|  | III | 0.8859 | 2.2575 | 2.3049 | 0.6290 | 0.3071 | 1.8277 | 1.8676 | 0.4478 | 1.0047
|  |  | (0.8244) | (0.9494) | (0.9495) | (0.9394) | (0.8974) | (0.9763) | (0.9690) | (0.9715) |
| 1 | I | 0.9466 | 2.5678 | 2.6272 | 0.6900 | 0.3137 | 1.6128 | 1.6698 | 0.4589 | 1.0225
|  |  | (0.8578) | (0.9499) | (0.9578) | (0.9565) | (0.8858) | (0.9728) | (0.9805) | (0.9872) |
|  | II | 0.7891 | 1.7606 | 1.7934 | 0.5559 | 0.3062 | 1.3845 | 1.4140 | 0.5026 | 1.0239
|  |  | (0.8798) | (0.9397) | (0.9472) | (0.9487) | (0.9066) | (0.9691) | (0.9851) | (0.9899) |
|  | III | 0.8638 | 2.1073 | 2.1547 | 0.6166 | 0.2903 | 1.2284 | 1.2525 | 0.4259 | 0.9959
|  |  | (0.8654) | (0.9460) | (0.9403) | (0.9480) | (0.8973) | (0.9691) | (0.9707) | (0.9694) |
(50,40) | 0.5 | I | 0.7263 | 1.7879 | 1.7905 | 0.5775 | 0.3151 | 1.3893 | 1.4403 | 0.4453 | 1.0187
|  |  | (0.8199) | (0.9496) | (0.9417) | (0.9506) | (0.8937) | (0.9708) | (0.9695) | (0.9786) |
|  | II | 0.6813 | 1.5456 | 1.5306 | 0.4989 | 0.3141 | 1.3024 | 1.2532 | 0.4202 | 1.0306
|  |  | (0.8368) | (0.9390) | (0.9473) | (0.9514) | (0.8732) | (0.9501) | (0.9796) | (1.0000) |
|  | III | 0.6835 | 1.6496 | 1.6481 | 0.5277 | 0.3158 | 1.3038 | 1.3123 | 0.4214 | 0.9915
|  |  | (0.8204) | (0.9375) | (0.9409) | (0.9501) | (0.8776) | (0.9610) | (0.9649) | (0.9764) |
| 1 | I | 0.7286 | 1.7961 | 1.8036 | 0.5870 | 0.3123 | 1.4514 | 1.4383 | 0.4506 | 0.9930
|  |  | (0.8140) | (0.9392) | (0.9477) | (0.9517) | (0.8749) | (0.9591) | (0.9603) | (0.9699) |
|  | II | 0.6956 | 1.5847 | 1.5856 | 0.4983 | 0.3358 | 1.2633 | 1.2520 | 0.4181 | 1.0355
|  |  | (0.8348) | (0.9417) | (0.9476) | (0.9492) | (0.8723) | (0.9705) | (0.9721) | (0.9799) |
|  | III | 0.7263 | 1.7879 | 1.7905 | 0.5575 | 0.3151 | 1.3893 | 1.4403 | 0.4153 | 0.9961
|  |  | (0.8199) | (0.9496) | (0.9417) | (0.9506) | (0.8937) | (0.9608) | (0.9695) | (0.9786) |
(80,40) | 0.5 | I | 0.8104 | 2.3216 | 2.3345 | 0.6007 | 0.3266 | 1.3214 | 1.2846 | 0.4021 | 0.9893
|  |  | (0.8294) | (0.9459) | (0.9472) | (0.9567) | (0.8538) | (0.9697) | (0.9687) | (0.9798) |
|  | II | 0.6828 | 1.5502 | 1.5361 | 0.4812 | 0.3141 | 1.2960 | 1.2824 | 0.4046 | 0.9947
|  |  | (0.8492) | (0.9486) | (0.9517) | (0.9512) | (0.8745) | (0.9699) | (0.9668) | (0.9805) |
|  | III | 0.7256 | 1.8515 | 1.8544 | 0.5150 | 0.3123 | 1.7063 | 1.6556 | 0.4317 | 0.9981
|  |  | (0.8199) | (0.9494) | (0.9327) | (0.9461) | (0.8874) | (0.9583) | (0.9647) | (0.9762) |
| 1 | I | 0.7140 | 2.3356 | 2.3471 | 0.6010 | 0.2862 | 1.1717 | 1.2151 | 0.4178 | 1.0414
|  |  | (0.8197) | (0.9509) | (0.9567) | (0.9535) | (0.8843) | (0.9714) | (0.9695) | (0.9772) |
|  | II | 0.6910 | 1.5728 | 1.5745 | 0.4722 | 0.3010 | 1.2718 | 1.2769 | 0.4190 | 1.0228
|  |  | (0.8278) | (0.9391) | (0.9499) | (0.9518) | (0.8825) | (0.9644) | (0.9598) | (0.9750) |
|  | III | 0.6713 | 1.6960 | 1.7040 | 0.5028 | 0.3247 | 1.3028 | 1.2559 | 0.3881 | 0.9937
|  |  | (0.8171) | (0.9447) | (0.9541) | (0.9496) | (0.8717) | (0.9670) | (0.9705) | (0.9645) |
(80,60) | 0.5 | I | 0.6053 | 1.4517 | 1.4229 | 0.4651 | 0.2821 | 1.2037 | 1.2129 | 0.3659 | 0.9915
|  |  | (0.8374) | (0.9473) | (0.9593) | (0.9541) | (0.8601) | (0.9683) | (0.9789) | (0.9894) |
|  | II | 0.5551 | 1.2368 | 1.2126 | 0.4165 | 0.3023 | 1.2418 | 1.1750 | 0.3711 | 0.9837
|  |  | (0.8476) | (0.9315) | (0.9528) | (0.9508) | (0.8783) | (0.9604) | (0.9760) | (0.9718) |
|  | III | 0.5598 | 1.3175 | 1.2921 | 0.4284 | 0.2831 | 1.1426 | 1.0765 | 0.3685 | 0.9945
|  |  | (0.8196) | (0.9456) | (0.9523) | (0.9518) | (0.8995) | (0.9672) | (0.9863) | (0.9905) |
| 1 | I | 0.6070 | 1.4553 | 1.4261 | 0.4549 | 0.2823 | 1.1215 | 1.0667 | 0.3543 | 1.0124
|  |  | (0.8289) | (0.9453) | (0.9513) | (0.9503) | (0.9055) | (0.9641) | (0.9787) | (0.9706) |
|  | II | 0.5730 | 1.2584 | 1.2374 | 0.4060 | 0.2937 | 1.1039 | 1.0783 | 0.3519 | 1.0309
|  |  | (0.8357) | (0.9414) | (0.9539) | (0.9508) | (0.9032) | (0.9624) | (0.9759) | (0.9730) |
|  | III | 0.6032 | 1.4234 | 1.3960 | 0.4371 | 0.3034 | 1.1356 | 1.0987 | 0.3423 | 0.9959
|  |  | (0.8481) | (0.9477) | (0.9523) | (0.9561) | (0.8926) | (0.9678) | (0.9889) | (1.0000) |

##  6 Optimal censoring scheme

In reliability analysis, to obtain sufficient information about unknown
parameters, a practitioner seeks out the optimal censoring plan within a class
of all feasible chosen schemes. In our study, in order to get enough details
about model parameters, choosing ideal time $T$ with a progressive censoring
scheme $(R_{1},\cdots,R_{D})$, where $D=m$ for case-I and $D=J$ for case-II,
for prefixed values of $n,m$, may be of interest to an experimenter. However,
comparing two (or more) different censoring plans has gained a lot of
attention in past few years by several authors. For instance one can see Ng et
al. (2004), Kundu (2008) and Singh et al. (2015). Here, three commonly used
criteria have been considered to compare between the chosen class of
progressive censoring schemes, see Table 5.

Table 5: Different optimality criterion.

Criterion |  |  |  |  | Goal |
---|---|---|---|---|---|---
A-optimality |  |  |  |  |  minimum trace $(I^{-1}(\widehat{\Theta}))$ |
D-optimality |  |  |  |  |  minimum det $(I^{-1}(\widehat{\Theta}))$ |
F-optimality |  |  |  |  |  maximum trace $(I(\widehat{\Theta}))$ |

$A$-optimality and $D$-optimality have been considered as criterion I and II,
respectively. These two criteria are widely used in the statistical literature
and intend to minimize the trace and determinant of the variance-covariance
(V-C) matrix based on the observed Fisher information matrix associated with
$\widehat{\Theta}$, respectively. The trace of the V-C matrix equals to the
sum of the diagonal elements of $I^{-1}(\widehat{\Theta})$. According to
$F$-optimality, which has been considered as criterion III, the trace of the
observed Fisher information matrix has been minimized. The related optimal
progressive censoring plans have been taken into consideration based on the
minimum values of the $A$\- and $D$\- optimality criteria, and maximum value
of the $F$-optimality criterion (see Table $9$).

##  7 Real data analysis

To illustrate the usefulness of the proposed methods, a real life soccer game
data set from Meintanis (2007) has been considered in this section. This data
set represents that at least one goal was scored by home team, and at least
one goal was scored directly by a kick (i.e., penalty, foul, or any other
kick) by any team. Let $Y_{1}$ be the time of the first goal scored by any
team, and $Y_{2}$ be the time of the first goal scored by the home team. Since
the soccer game is 90 minutes long, the data is divided by 90. The complete
failure data with the observed failure risks $\delta_{i}$ is tabulated in
Table 6.

Table 6: Complete failure data with corresponding risks of the soccer game.

Original soccer game data ($Y_{1}$,$Y_{2}$)
---
(0.0222, 0.0222) (0.4556, 0.0333) (0.4667, 0.0333) (0.6000, 0.0778) (0.0889,
0.0889) (0.2778, 0.1000)
(0.6111, 0.1222) (0.4889, 0.1444) (0.2444, 0.1556) (0.2778, 0.1556) (0.7111,
0.1667) (0.1778, 0.1778)
(0.1778, 0.8333) (0.7000, 0.2000) (0.2000, 0.2000) (0.2111, 0.2111) (0.2889,
0.2222) (0.2667, 0.2667)
(0.2889, 0.5333) (0.3000, 0.5222) (0.3111, 0.3111) (0.5667, 0.3111) (0.4889,
0.3333) (0.3778, 0.3778)
(0.4000, 0.5778) (0.4333, 0.4333) (0.5889, 0.4333) (0.4444, 0.4444) (0.4667,
0.4667) (0.9111, 0.5333)
(0.5444, 0.5444) (0.5444, 0.5444) (0.7333, 0.6889) (0.8444, 0.7111) (0.7333,
0.9444) (0.7667, 0.7889)
(0.8000, 0.8000)
Competing risks data ($Y_{i}$,$\delta_{i}$)
(0.0222,0) (0.0333,2) (0.0333,2) (0.0778,2) (0.0889,0) (0.1000,2)
(0.1222,2) (0.1444,2) (0.1556,2) (0.1556,2) (0.1667,2) (0.1778,0)
(0.1778,1) (0.2000,2) (0.2000,0) (0.2111,0) (0.2222,2) (0.2667,0)
(0.2889,1) (0.3000,1) (0.3111,0) (0.3111,2) (0.3333,2) (0.3778,0)
(0.4000,1) (0.4333,0) (0.4333,2) (0.4444,0) (0.4667,0) (0.5333,2)
(0.5444,0) (0.5444,0) (0.6889,2) (0.7111,2) (0.7333,1) (0.7667,1)
(0.8000,0)

Table 7:  Three censoring schemes for AT-II PHC data with competing risks are
used as an illustration.

Data I: $n=37$, $m=28$, $T=0.4$ and $R=(0*27,9)$
---
(0.0222,0) (0.0333,2) (0.0333,2) (0.0778,2) (0.0889,0) (0.1000,2)
(0.1222,2) (0.1444,2) (0.1556,2) (0.1556,2) (0.1667,2) (0.1778,0)
(0.1778,1) (0.2000,2) (0.2000,0) (0.2111,0) (0.2222,2) (0.2667,0)
(0.2889,1) (0.3000,1) (0.3111,0) (0.3111,2) (0.3333,2) (0.3778,0)
(0.4000,1) (0.4333,0) (0.4333,2) (0.4444,0)
Data II: $n=37$, $m=28$, $T=0.4$ and $R=(9,0*27)$
(0.0222,0) (0.0333,2) (0.0333,2) (0.0778,0) (0.1000,2) (0.1222,2)
(0.1444,2) (0.1556,2) (0.1667,2) (0.1778,1) (0.2000,2) (0.2000,0)
(0.2111,0) (0.2222,2) (0.2667,0) (0.2889,1) (0.3000,1) (0.3111,0)
(0.3111,2) (0.3333,2) (0.4333,0) (0.4333,2) (0.4444,0) (0.4667,0)
(0.5333,2) (0.5444,0) (0.6889,2) (0.7111,2)
Data III: $n=37$, $m=28$, $T=0.4$ and $R=(1*9,0*19)$
(0.0333,2) (0.0333,2) (0.0778,0) (0.1000,2) (0.1222,2) (0.1444,2)
(0.1556,2) (0.1667,2) (0.1778,1) (0.2000,2) (0.2000,0) (0.2111,0)
(0.2222,2) (0.2667,0) (0.2889,1) (0.3000,1) (0.3111,0) (0.3111,2)
(0.3333,2) (0.4333,0) (0.4333,2) (0.4444,0) (0.4667,0) (0.5333,2)
(0.5444,0) (0.6889,2) (0.7111,2) (0.7333,1)

Table 8: Point and interval estimates of the unknown parameters based on
generated samples from the real life data set.

Data | $\Theta$ | MLE | ACI | Bayes | HPD
---|---|---|---|---|---
I | $\lambda_{0}$ | 1.2739 | (0.4204,2.5274)[2.1070] | 1.5833 | (1.2779,1.8847)[0.6068]
| $\lambda_{1}$ | 0.5159 | (0.0317,1.1037)[1.0720] | 0.6482 | (0.5628,0.7054)[0.1426]
| $\lambda_{2}$ | 2.0098 | (0.7108,3.3089)[2.5981] | 2.2199 | (1.7334,2.7056)[0.9722]
| $\alpha$ | 1.3846 | (0.9609,1.8082)[0.8473] | 1.5704 | (1.3746,1.8544)[0.4798]
II | $\lambda_{0}$ | 1.5688 | (0.4711,2.6665)[2.1954] | 1.9522 | (1.5872,2.3524)[0.7652]
| $\lambda_{1}$ | 0.6275 | (0.0203,1.2753)[1.2550] | 0.5812 | (0.5253,0.6364)[0.1111]
| $\lambda_{2}$ | 2.5101 | (1.0347,3.9856)[2.9509] | 2.3674 | (1.8621,2.8271)[0.9650]
| $\alpha$ | 1.4685 | (1.0578,1.8793)[0.8215] | 1.5056 | (1.3367,1.6168)[0.2801]
III | $\lambda_{0}$ | 1.5100 | (0.4596,2.5605)[2.1009] | 1.5457 | (1.4224,1.6723)[0.2499]
| $\lambda_{1}$ | 0.6040 | (0.0179,1.2260)[1.2081] | 0.4944 | (0.2626,0.6696)[0.4070]
| $\lambda_{2}$ | 2.4161 | (1.0076,3.8246)[2.8170] | 2.5246 | (2.1011,2.9496)[0.8485]
| $\alpha$ | 1.6001 | (1.1654,2.0346)[0.8692] | 1.6663 | (1.4724,1.8089)[0.3365]

Prior to conducting data analysis, we examine how well this set of data can be
fitted with the MOBW distribution using the Kolmogorov-Smirnov (K-S)
statistic. Since there is no such goodness-of-fit test for bivariate
distributions as univariate distributions, here we have investigated whether
the marginals $Y_{1}$, $Y_{2}$ and $min(Y_{1},Y_{2})$ fit Weibull distribution
or not. The K-S distances and the corresponding p-values (given in bracket) of
$Y_{1}$, $Y_{2}$ and $min(Y_{1},Y_{2})$ are $0.0834$ $(0.9602)$, $0.1055$
$(0.8051)$ and $0.0689$ $(0.9947)$, respectively. Further, the empirical CDF
(ECDF), probability-probability (P-P) and quantile- quantile (Q-Q) plots along
with associated fitted Weibull distribution are provided in Figures $2-4$.
From these observations, we conclude that this data set can be considered to
model Weibull distribution for the marginals $Y_{1}$, $Y_{2}$ and their
minimum. This yields that MOBW distribution can be used for analyzing this
bivariate data set given in Table 6.

Figure 2: ECDF plots along with fitted Weibull models for real data set.

Figure 3: P-P plots along with fitted Weibull models for real data set.

Figure 4: Q-Q plots along with fitted Weibull models for real data set.

Based on these data set, three different censored data sets have been
generated under different censoring schemes for min $(Y_{1},Y_{2})$ with
$q=0$, see Table 7. The point and interval estimates are calculated and
tabulated in Table 8, where the interval widths are reported in square
brackets. Since there is no prior information, the Bayes estimates have been
computed under non-informative prior with hyperparameters $0.0001$. Based on
three above mentioned data sets, MCMC samples for model parameters are plotted
in Figures $5-7$. It has been noticed from Table 8 that the MLEs and Bayes
estimates are extremely comparable. HPD intervals performs better than ACIs in
terms of width, which yields that the results are consistent with the
previously simulated results. From Table 9, it has been observed that the
censoring scheme $R=(0*29,7)$ is optimal corresponding to the other proposed
censored schemes according to $A$-, $D$-, and $F$-optimality criteria given in
Table 5. It is apparent that the suggested algorithm is adequate for
processing AT-II PHC dependent competing risks data.

Figure 5: MCMC samples of (a) $\lambda_{0}$, (b) $\lambda_{1}$, (c)
$\lambda_{2}$ and (d) $\alpha$ for Data I.

Figure 6: MCMC samples of (a) $\lambda_{0}$, (b) $\lambda_{1}$, (c)
$\lambda_{2}$ and (d) $\alpha$ for Data II.

Figure 7: MCMC samples of (a) $\lambda_{0}$, (b) $\lambda_{1}$, (c)
$\lambda_{2}$ and (d) $\alpha$ for Data III. Table 9: Optimal censoring scheme
using three AT-II PHCS based on real life data set.

Data | Criterion I | Criterion II | Criterion III
---|---|---|---
I | 0.8587 |  1.7845 $\times 10^{-4}$ | 71.1219
II | 1.0333 |  4.1915 $\times 10^{-4}$ | 58.2624
III | 0.9533 |  3.7313 $\times 10^{-4}$ | 54.2736

##  8 Conclusion

In this article, statistical inference for a MOBW distribution based on
adaptive Type II progressive hybrid censored data under a dependent competing
risk model is developed. Based on both frequentist and Bayesian approaches,
point and interval estimation procedures are proposed. Since MLEs of the
unknown parameters cannot be derived explicitly, Newton’s iterative method has
been implemented for this purpose. The existence of MLEs of the unknown
parameters are also established. Similarly, due to complex form, Bayes
estimates have been obtained by using the MCMC method. Convergence of this
MCMC technique has been tested. To evaluate the effectiveness of the suggested
techniques, a Monte Carlo simulation has been carried out. It has been
discovered that the Bayesian technique produces superior outcomes to the
frequentist approach. In the case of interval estimation, HPD credible
intervals perform better than the ACIs in terms of their average width. A
real-life data set has been analyzed in order to demonstrate the usefulness of
the offered methodologies in practice. An optimal progressive censoring plan
has been suggested by using different optimality criteria.

Acknowledgement: The author S. Dutta, expresses gratitude to the Council of
Scientific and Industrial Research (C.S.I.R. Grant No.
09/983(0038)/2019-EMR-I), India, for providing financial support received for
this research project. Both authors thank the Department of Mathematics,
National Institute of Technology Rourkela, India, for providing research
facilities.

Conflicts of interest: Both authors declare that they have no conflict of
interest.

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## Appendix A.1 Proof of Theorem 2.1

When $(Y_{1},Y_{2})\sim MOBW(\lambda_{0},\lambda_{1},\lambda_{2},\alpha)$,
then the joint survival function of $(Y_{1},Y_{2})$ can be expressed as

$\displaystyle S_{Y_{1},Y_{2}}(y_{1},y_{2})=$
$\displaystyle~{}P(Y_{1}>y_{1},Y_{2}>y_{2})$ $\displaystyle=$
$\displaystyle~{}P(V_{1}>y_{1},V_{2}>y_{2},V_{0}>max\\{y_{1},y_{2}\\})$
$\displaystyle=$
$\displaystyle~{}S_{WE}(y_{1};\alpha,\lambda_{1})S_{WE}(y_{2};\alpha,\lambda_{2})S_{WE}(max\\{y_{1},y_{2}\\};\alpha,\lambda_{0}),$

where $max\\{y_{1},y_{2}\\}\in(0,\infty)\times(0,\infty)$. Furthermore, the
joint survival function can be expressed as

$\displaystyle
S_{Y_{1},Y_{2}}(y_{1},y_{2})=\begin{cases}S_{WE}(y_{1};\alpha,\lambda_{1})~{}S_{WE}(y_{2};\alpha,\lambda_{02}),~{}~{}~{}\mbox{if}~{}~{}y_{1}<y_{2}\\\
S_{WE}(y_{1};\alpha,\lambda_{01})~{}S_{WE}(y_{2};\alpha,\lambda_{2}),~{}~{}~{}\mbox{if}~{}~{}y_{2}<y_{1}\\\
S_{WE}(y;\alpha,\lambda_{012}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{if}~{}~{}y=y_{1}=y_{2}.\end{cases}$

## Appendix A.2 Proof of Corollary 2.1

Using the joint survival function from Theorem 2.1, the joint PDF can be
derived from the expression
$-\frac{\partial^{2}S_{(Y_{1},Y_{2})}(y_{1},y_{2})}{\partial y_{1}\partial
y_{2}}$ when $y_{1}<y_{2}$ and $y_{2}<y_{1}$, respectively.
We know that,

$\displaystyle\int_{0}^{\infty}\int_{0}^{y_{2}}f_{WE}(y_{1};\alpha,\lambda_{1})f_{WE}(y_{2};\alpha,\lambda_{02})~{}dy_{1}dy_{2}=$
$\displaystyle~{}\frac{\lambda_{1}}{\lambda_{012}},$
$\displaystyle\int_{0}^{\infty}\int_{0}^{y_{2}}f_{WE}(y_{1};\alpha,\lambda_{01})f_{WE}(y_{2};\alpha,\lambda_{2})~{}dy_{1}dy_{2}=$
$\displaystyle~{}\frac{\lambda_{2}}{\lambda_{012}},$

and further using the full probability formula, when $y_{1}=y_{2}=y$ the joint
PDF $f_{(Y_{1},Y_{2})}(y_{1},y_{2})$ can be derived as
$\frac{\lambda_{0}}{\lambda_{012}}f_{WE}(y;\alpha,\lambda_{012})$. Hence the
result is provided.

## Appendix A.3 Proof of Theorem 3.1

From $(\ref{3.4})$, we already have

$\displaystyle\frac{\partial\log L}{\partial\lambda_{j}}=$
$\displaystyle~{}\frac{m_{j}}{\lambda_{j}}+\frac{m_{3}}{\lambda_{012}}-A(\alpha),~{}~{}\mbox{for}~{}j=0,1,2.$
(A.1.1)

For a fixed $\alpha$ equating (A.1.1) to zero, one can easily obtain that
$\lambda_{j}=\frac{m_{j}}{m_{012}}\frac{m}{A(\alpha)}$, for $j=0,1,2$.
Furthermore, it can be proved that the associated conditional MLEs maximize
the log-likelihood function $\log
L(\alpha,\lambda_{0},\lambda_{1},\lambda_{2})$.

We know that, $\log v\leq v-1$ and using these for
$v=\frac{\lambda_{j}}{\widehat{\lambda}_{j}}$, where $j=0,1,2$ and
$v=\frac{\lambda_{012}}{\widehat{\lambda}_{012}}$ we have

$\displaystyle m_{j}\log\lambda_{j}\leq
m_{j}\frac{\lambda_{j}}{\widehat{\lambda}_{j}}-m_{j}+m_{j}\log\widehat{\lambda}_{j}=\frac{m_{012}\lambda_{j}A(\alpha)}{m}-m_{j}+m_{j}\log\widehat{\lambda}_{j},$
(A.1.2)

and

$\displaystyle m_{3}\ln\lambda_{012}\leq
m_{3}\frac{\lambda_{012}}{\widehat{\lambda}_{012}}-m_{3}+m_{3}\log\widehat{\lambda}_{012}=\frac{m_{3}\lambda_{012}A(\alpha)}{m}-m_{3}+m_{3}\log\widehat{\lambda}_{012}.$
(A.1.3)

From Equation $(\ref{3.3})$, we can write that

$\displaystyle\log L=$
$\displaystyle~{}m\log\alpha+\sum_{j=0}^{2}m_{j}\log\lambda_{j}+m_{3}\log\lambda_{012}+(\alpha-1)\sum_{i=1}^{m}\log
y_{i}-\lambda_{012}A(\alpha).$ (A.1.4)

Using (A.1.2), (A.1.3) and (A.1.4) we have

$\displaystyle\log L(\lambda_{0},\lambda_{1},\lambda_{2},\alpha)\leq
m\log\alpha+\sum_{j=0}^{2}m_{j}\log\widehat{\lambda}_{j}+m_{3}\log\widehat{\lambda}_{012}+(\alpha-1)\sum_{i=1}^{m}\log
y_{i}-m.$ (A.1.5)

Since $m=\widehat{\lambda}_{012}A(\alpha)$, then we have

$\displaystyle\log L(\lambda_{0},\lambda_{1},\lambda_{2},\alpha)\leq$
$\displaystyle~{}m\log\alpha+\sum_{j=0}^{2}m_{j}\log\widehat{\lambda}_{j}+m_{3}\log\widehat{\lambda}_{012}+(\alpha-1)\sum_{i=1}^{m}\log
y_{i}-\widehat{\lambda}_{012}A(\alpha)$ $\displaystyle=$ $\displaystyle~{}\log
L(\widehat{\lambda}_{0},\widehat{\lambda}_{1},\widehat{\lambda}_{2},\alpha),$

and equality holds iff
$(\lambda_{0},\lambda_{1},\lambda_{2})=(\widehat{\lambda}_{0},\widehat{\lambda}_{1},\widehat{\lambda}_{2})$.

## Appendix A.4 Proof of Theorem 4.1

The marginal posterior density of $\alpha$ from $(\ref{4.5})$ can be obtained
as follows

$\displaystyle\pi_{1}^{*}(\alpha|data)$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\pi(\lambda_{0},\lambda_{1},\lambda_{2},\alpha|data)~{}d\lambda_{0}~{}d\lambda_{1}~{}d\lambda_{2}$
$\displaystyle\propto\pi_{2}(\alpha)\alpha^{m}\bigg{(}\prod_{i=1}^{m}y_{i}^{\alpha-1}\bigg{)}\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\lambda_{012}^{m_{3}+a-d_{012}}\bigg{(}\prod_{j=0}^{2}\lambda_{j}^{m_{j}+d_{j}-1}\bigg{)}$
$\displaystyle~{}~{}\times
e^{-\lambda_{012}\big{(}b+A(\alpha)\big{)}}~{}d\lambda_{0}~{}d\lambda_{1}~{}d\lambda_{2}.$
(A.2.6)

Let us assume that $u_{1}=\lambda_{012}$,
$u_{2}=\frac{\lambda_{1}}{\lambda_{012}}$, and
$u_{3}=\frac{\lambda_{2}}{\lambda_{012}}$. The transformation from
$(\lambda_{0},\lambda_{1},\lambda_{2})$ to $(u_{1},u_{2},u_{3})$ is one-to-one
transformation such that

$\lambda_{1}=u_{1}u_{2},~{}~{}\lambda_{2}=u_{1}u_{3},~{}~{}\mbox{and}~{}\lambda_{0}=u_{1}(1-u_{2}-u_{3}),$

where $0<u_{1}<\infty$, $0<u_{2}+u_{3}<1$. The Jacobian of the above
transformation is

$G=\frac{\partial(\lambda_{0},\lambda_{1},\lambda_{2})}{\partial(u_{1},u_{2},u_{3})}={\begin{bmatrix}1-u_{2}-u_{3}&-u_{1}&-u_{1}\\\
u_{2}&u_{1}&0\\\ u_{3}&0&u_{3}\\\
\end{bmatrix}},~{}~{}\mbox{and}~{}~{}det(G)=u_{1}^{2}.$

Therefore, we have

$\displaystyle\int_{0}^{\infty}$
$\displaystyle\int_{0}^{\infty}\int_{0}^{\infty}\lambda_{012}^{m_{3}+a-d_{012}}\bigg{(}\prod_{j=0}^{2}\lambda_{j}^{m_{j}+d_{j}-1}\bigg{)}e^{-\lambda_{012}\big{(}b+A(\alpha)\big{)}}~{}d\lambda_{0}~{}d\lambda_{1}~{}d\lambda_{2}$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\int_{0<u_{2}+u_{3}<1}u_{1}^{a+m-1}u_{2}^{m_{1}+d_{1}-1}u_{3}^{m_{2}+d_{2}-1}(1-u_{2}-u_{3})^{m_{0}+d_{0}-1}e^{-u_{1}(b+A(\alpha))}~{}du_{1}~{}du_{2}~{}du_{3}$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}u_{1}^{a+m-1}e^{-u_{1}(b+A(\alpha))}du_{1}\times\int_{0<u_{2}+u_{3}<1}u_{2}^{m_{1}+d_{1}-1}u_{3}^{m_{2}+d_{2}-1}(1-u_{2}-u_{3})^{m_{0}+d_{0}-1}~{}du_{2}~{}du_{3}$
$\displaystyle=$
$\displaystyle\Gamma(a+m)\big{(}b+A(\alpha)\big{)}^{-(a+m)}\cdot
B(m_{1}+d_{1},m_{0}+d_{0}+m_{2}+d_{2})\cdot B(m_{2}+d_{2},m_{0}+d_{0}),$

where $B(\cdot,\cdot)$ represents the Beta function. This completes the proof
of the first part. The joint posterior density of
$(\lambda_{0},\lambda_{1},\lambda_{2})$ for given data and $\alpha$, can be
written as follows

$\displaystyle\pi_{2}^{*}(\lambda_{0},\lambda_{1},\lambda_{2}|\alpha,data)\propto\pi(\lambda_{0},\lambda_{1},\lambda_{2},\alpha|data)/\pi(\alpha|data)$

which completes the proof of second part of the theorem.

## Appendix A.5 Proof of Theorem 4.2

The marginal posterior density of $\alpha$ from $(\ref{4.13})$ can be written
as

$\displaystyle\pi_{1}^{*}(\alpha|data)\propto\alpha^{m+a_{1}-1}e^{-\alpha\big{(}b_{1}-\sum_{i=1}^{m}\log
y_{i}\big{)}}\big{(}b+A(\alpha)\big{)}^{-(a+m)}.$ (A.3.7)

Now taking logarithm on both sides of (A.3.7) we get

$\displaystyle\log\pi_{1}^{*}(\alpha|data)=(m+a_{1}-1)\log\alpha-\bigg{(}b_{1}-\sum_{i=1}^{m}\log
y_{i}\bigg{)}\alpha-(a+m)\log\big{(}b+A(\alpha)\big{)}.$ (A.3.8)

After differentiating (A.3.8) partially with respect to $\alpha$ twice we get

$\displaystyle\frac{\partial^{2}\log\pi_{1}^{*}}{\partial\alpha^{2}}=-\frac{m+a_{1}-1}{\alpha^{2}}-(a+m)\frac{(b+A(\alpha))A^{\prime\prime}(\alpha)-\big{(}A^{\prime}(\alpha)\big{)}^{2}}{(b+A(\alpha))^{2}},$

where
$A^{\prime\prime}(\alpha)=\sum_{i=1}^{m}y_{i}^{\alpha}\log^{2}y_{i}+\sum_{i=1}^{J}R_{i}y_{i}^{\alpha}\log^{2}y_{i}+R^{*}y_{m}^{\alpha}\log^{2}y_{m}\geq
0$. According to Cauchy-Schwartz inequality, we know that
$A(\alpha)A^{\prime\prime}(\alpha)-\big{(}A^{\prime}(\alpha)\big{)}^{2}\geq 0$
and this implies that
$\frac{\partial^{2}\log\pi_{1}^{*}}{\partial\alpha^{2}}\leq 0$. Hence the
result is proved.