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\jyear{2023}%

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\begin{document}
%old
%\title[A latent variable approach for parametric\dots]{A latent variable approach for parametric modeling of recall-based time-to-event data}
%new
\title[]{Estimation from Recall-Based Competing Risk Data: A Multilevel Framework}

%%=============================================================%%
%% Prefix	-> \pfx{Dr}
%% GivenName	-> \fnm{Joergen W.}
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%% NatureName	-> \tanm{Poet Laureate} -> Title after name
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%% \author*[1,2]{\pfx{Dr} \fnm{Joergen W.} \spfx{van der} \sur{Ploeg} \sfx{IV} \tanm{Poet Laureate} 
%%                 \dgr{MSc, PhD}}\email{iauthor@gmail.com}
%%=============================================================%%

\author[1]{\fnm{M. S.} \sur{Panwar}}\email{mspanwarstat@bhu.ac.in}
\equalcont{Co-first authors.}

\author[1]{\fnm{Sanjeev K.} \sur{Tomer}}\email{sktomer73@gmail.com}

\author*[2]{\fnm{C. P.} \sur{Yadav}}\email{chandraprakashy29@gmail.com}

\equalcont{Co-first authors.}


\affil[1]{\orgdiv{Department of Statistics}, \orgname{Banaras Hindu University}, \orgaddress{\city{Varanasi}, \postcode{221005}, \country{India}}}

\affil[2]{\orgdiv{Saw Swee Hock School of Public Health}, \orgname{National University of Singapore}, \orgaddress{\postcode{117549}, \city{Singapore}}}


\maketitle

%================================================================%
\section{Calculation for Two Causes Setup} \label{4.a1}
\subsection{Likelihood Construction} \label{4.A1}
Under two causes setup, we can write the likelihood function under observed data vector $d$ as  
{\small \begin{align}
	L(\theta \vert d)&=
	\prod\limits_{i=1}^{n_{r_1}} \Big[ f(t_i,\theta_1)\bar{F}(t_i,\theta_2) \{1-\psi_1(s_i,t_i)\}\Big] \prod\limits_{i=n_{r_1}+1}^{n_{r_2}} \Big[ f(t_i,\theta_2)\bar{F}(t_i,\theta_1)\{1-\psi_2(s_i,t_i)\} \Big] \nonumber \\
	&\quad\prod\limits_{i=n_r+1}^{n_{nr}}\Big[\int_{0}^{s_i} f(u,\theta_1)\bar{F}(u,\theta_2)\psi_1(s_i,u)du+\int_{0}^{s_i} f(u,\theta_2)\bar{F}(u,\theta_1)\psi_2(s_i,u)du\Big] \nonumber \\ &\quad\prod\limits_{i=n_{nr}+1}^{n}\bar{F}(s_i;\theta_1,\theta_2). \label{eq4.3}
\end{align}
}
The symbols $n_{r_1}$ and $n_{r_2}$ denotes the number of individuals who had experienced the event of interest before monitoring time and recalled the exact causes of occurrence of the event as $1$ and $2$ respectively. 

Assume $T\sim\mathcal{W}(\alpha_k,\beta_k)$; $k=1, 2$ for causes $1$ and $2$ and  denote the parameter vector by $\Theta=(\alpha_1,\beta_1,\lambda_1,\alpha_2,\beta_2,\lambda_2)$. Putting all the functional forms of the densities in (\ref{eq4.3}), the likelihood function can be written as below
{\small \begin{eqnarray}\nonumber
	L(\Theta \vert d)&=&
	\alpha_1^{n_{r_1}} \beta_1^{n_{r_1}}  \alpha_2^{n_{r_2}-n_{r_1}} \beta_2^{n_{r_2}-n_{r_1}} \prod\limits_{i=1}^{n_{r_1}} t_i^{\alpha_1-1}\exp\{-\beta_1 t_i^{\alpha_1}-\beta_2 t_i^{\alpha_2}-\lambda_1(s_i-t_i)\} \nonumber \\
	&& \prod\limits_{i=n_{r_1}+1}^{n_{r_2}} t_i^{\alpha_2-1}\exp\{-\beta_1 t_i^{\alpha_1}-\beta_2 t_i^{\alpha_2}-\lambda_2(s_i-t_i)\} \prod\limits_{i=n_{nr}+1}^{n}\exp\{-\beta_1s_i^{\alpha_1}-\beta_2s_i^{\alpha_2}\} \nonumber \\
	&& \prod\limits_{i=n_r+1}^{n_{nr}}\left[ \int_{0}^{s_i} \alpha_1 \beta_1 u^{\alpha_1-1} \exp\{-\beta_1 u^{\alpha_1}-\beta_2 u^{\alpha_2}\} \Big(1-\exp\{-\lambda_1(s_i-u)\}\Big)du \right. \nonumber \\
	&& \left.+\int_{0}^{s_i}  \alpha_2 \beta_2 u^{\alpha_2-1} \exp\{-\beta_1 u^{\alpha_1}-\beta_2 u^{\alpha_2}\} \Big(1-\exp\{-\lambda_2(s_i-u)\}\Big)du \right].  \label{eq4.4}
\end{eqnarray}
}
%-----------------------------------------------------------------------------------------------------------------------%
\subsection{Classical Inference} \label{A2}
The likelihood function (\ref{eq4.4}) is incompatible to deal further. Since for the non-recall observations, partial information is available on time to event and causes of the event is also unknown. We treat it as a missing data problem and apply the E-M algorithm for point estimation. 
\subsubsection{Expectation Maximization Algorithm}\label{subsec4.3.1} 
Since any observation under non-recall is exposed to two cause, hence a latent variable $Z_i$ following Bernoulli distribution with probability of success $P_i$ is introduced. The probability of success $(Z_i=1)$ is defined as
\begin{equation}
	P_i=\frac{I_k(s_i)}{\sum\limits_{
			k}^{} I_k(s_i)}; \quad i=n_r+1,n_r+2,\ldots, n_{nr}, \ k=1,2, \label{eq4.5}
\end{equation} 
where, $I_1(.)$ and $I_2(.)$ is given by
\begin{equation*}
	I_1(s_i)=\int\limits_{0}^{s_i} \alpha_1 \beta_1  u^{\alpha_1-1} \exp\{-\beta_1 u^{\alpha_1}-\beta_2 u^{\alpha_2}\} \Big(1-\exp\{-\lambda_1(s_i-u)\}\Big)du 
\end{equation*}
\text{and}
\begin{equation*}
	I_2(s_i)=\int\limits_{0}^{s_i} \alpha_2  \beta_2 u^{\alpha_2-1} \exp\{-\beta_1 u^{\alpha_1}-\beta_2 u^{\alpha_2}\} \Big(1-\exp\{-\lambda_2(s_i-u)\}\Big)du. 
\end{equation*}
With the help of these probabilities and intoduced latent variable, we can assign exact causes to each observation under non-recall. Incorporating these, the likelihood can be written as  
{\small \begin{align}\nonumber
	L(\Theta \vert  d)&=
	\alpha_1^{n_{r_1}} \beta_1^{n_{r_1}}  \alpha_2^{n_{r_2}-n_{r_1}} \beta_2^{n_{r_2}-n_{r_1}} \prod\limits_{i=1}^{n_{r_1}} t_i^{\alpha_1-1}\exp\{-\beta_1 t_i^{\alpha_1}-\beta_2 t_i^{\alpha_2}-\lambda_1(s_i-t_i)\} \nonumber \\
	&\quad \prod\limits_{i=n_{r_1}+1}^{n_{r_2}} t_i^{\alpha_2-1}\exp\{-\beta_1 t_i^{\alpha_1}-\beta_2 t_i^{\alpha_2}-\lambda_2(s_i-t_i)\} \prod\limits_{i=n_{nr}+1}^{n}\exp\{-\beta_1s_i^{\alpha_1}-\beta_2s_i^{\alpha_2}\} \nonumber \\
	& \quad \prod\limits_{i=n_r+1}^{n_{nr}}\left[ \int_{0}^{s_i} \alpha_1 \beta_1 u^{\alpha_1-1} \exp\{-\beta_1 u^{\alpha_1}-\beta_2 u^{\alpha_2}\} \Big(1-\exp\{-\lambda_1(s_i-u)\}\Big)du \right]^{z_i} \nonumber \\
	&\quad \prod\limits_{i=n_r+1}^{n_{nr}}\left[\int_{0}^{s_i} \alpha_2 \beta_2  u^{\alpha_2-1} \exp\{-\beta_1 u^{\alpha_1}-\beta_2 u^{\alpha_2}\}  \Big(1-\exp\{-\lambda_2(s_i-u)\}\Big)du \right]^{1-z_i}.  \label{eq4.6}
\end{align}
}
Following equivalent quantity approach, in light of given causes, quantity $T^*$ is introduced, which lies in the interval $(0, S)$. Now the likelihood function (\ref{eq4.6}) in the light of conditional density of $T^*$ can be re-written as
{\small \begin{align}\nonumber
	L(\Theta \vert d)&=
	\alpha_1^{n_{r_1}} \beta_1^{n_{r_1}}  \alpha_2^{n_{r_2}-n_{r_1}} \beta_2^{n_{r_2}-n_{r_1}} \prod\limits_{i=1}^{n_{r_1}} t_i^{\alpha_1-1}\exp\{-\beta_1 t_i^{\alpha_1}-\beta_2 t_i^{\alpha_2}-\lambda_1(s_i-t_i)\} \nonumber \\
	&\quad \prod\limits_{i=n_{r_1}+1}^{n_{r_2}} t_i^{\alpha_2-1}\exp\{-\beta_1 t_i^{\alpha_1}-\beta_2 t_i^{\alpha_2}-\lambda_2(s_i-t_i)\} \prod\limits_{i=n_{nr}+1}^{n}\exp\{-\beta_1s_i^{\alpha_1}-\beta_2s_i^{\alpha_2}\} \nonumber \\
	&\quad \prod\limits_{i=n_r+1}^{n_{nr}}\left[ \alpha_1 \beta_1  {t_i^*}^{\alpha_1-1} \exp\{-\beta_1 {t_i^*}^{\alpha_1}-\beta_2 {t_i^*}^{\alpha_2}\} \Big(1-\exp\{-\lambda_1(s_i-t_i^*)\}\Big) \right]^{z_i} \nonumber \\
	&\quad \prod\limits_{i=n_r+1}^{n_{nr}}\left[  \alpha_2 \beta_2 {t_i^*}^{\alpha_2-1} \exp\{-\beta_1 {t_i^*}^{\alpha_1}-\beta_2 {t_i^*}^{\alpha_2}\}  \Big(1-\exp\{-\lambda_2(s_i-t_i^*)\}\Big) \right]^{1-z_i}.  \label{eq4.8}
\end{align}    
}
Also, for making the functional form of non-recall probabilities compatible, we introduce two latent variables $U^*$ and $V^*$ corresponding to causes $1$ and $2$ following exponential distribution with mean $\frac{1}{\lambda_1}$ and $\frac{1}{\lambda_2}$ truncated at $(S-T^*)$. For non-recall observations, let us denote the missing data by $\underline{D}^*=(Z, T^*,U^*,V^*)$. For $i^{th}$ observation belonging to non-recall category, denote the missing data by $d^*=(z_i, t_i^*,u_i^*,v_i^*)$. The complete data can be written as ${(d,d^*)}$. Now under the complete data the likelihood function (\ref{eq4.8}) can be written as 

{\small \begin{align}
		L_c(\Theta\vert d,d^*)&=
		\alpha_1^{n_{r_1}+Z^{*}} \beta_1^{n_{r_1}+Z^{*}} \lambda_1^{Z^{*}} \alpha_2^{n_{r_2}+n_{nr}-n_r-Z^{*}} \beta_2^{n_{r_2}+n_{nr}-n_r-Z^{*}} \lambda_2^{n_{nr}-n_r-Z^{*}}  \nonumber \\
		&\quad \prod\limits_{i=1}^{n_{r_1}} t_i^{\alpha_1-1} \exp\{-\beta_1 t_i^{\alpha_1}-\beta_2 t_i^{\alpha_2}-\lambda_1(s_i-t_i)\} \nonumber \\ 
		&\prod\limits_{i=n_{r_1}+1}^{n_{r_2}} t_i^{\alpha_2-1}\exp\{-\beta_1 t_i^{\alpha_1}-\beta_2 t_i^{\alpha_2}-\lambda_2(s_i-t_i)\}  \nonumber \\ 
		&\quad \prod\limits_{i=n_r+1}^{n_{nr}}\left[{t_i^*}^{\alpha_1-1} \exp\{-\beta_1 {t_i^*}^{\alpha_1}-\beta_2 {t_i^*}^{\alpha_2}-\lambda_1u_i^*\} \right]^{z_i} \prod\limits_{i=n_{nr}+1}^{n}\exp\{-\beta_1s_i^{\alpha_1}-\beta_2s_i^{\alpha_2}\} 
		\nonumber \\
		&\quad
		\prod\limits_{i=n_r+1}^{n_{nr}}\left[ {t_i^*}^{\alpha_2-1} \exp\{-\beta_1 {t_i^*}^{\alpha_1}-\beta_2 {t_i^*}^{\alpha_2}-\lambda_2v_i^*\} \right]^{1-z_i}, \label{eq4.9}
\end{align}}
where, $Z^*=\sum_{i=n_r+1}^{n_{nr}}z_i$. Taking natural logarithm of (\ref{eq4.9}), the log-likelihood function can be written as
{\footnotesize \begin{align}
		l_c(\Theta\vert d,d^*)&=
		(n_{r_1}+Z^{*}) \ln(\alpha_1)+(n_{r_1}+Z^{*}) \ln(\beta_1) +Z^{*}\ln(\lambda_1) +(n_{r_2}+n_{nr}-n_r-Z^{*})\ln(\alpha_2)\nonumber \\
		&+(n_{r_2}+n_{nr}-n_r-Z^{*}) \ln(\beta_2) + (n_{nr}-n_r-Z^{*})\ln(\lambda_2)+(\alpha_1-1)\sum_{i=1}^{n_{r_1}}\ln(t_i)-\beta_1 \sum_{i=1}^{n_{r_1}} t_i^{\alpha_1} \nonumber \\
		&-\beta_2 \sum_{i=1}^{n_{r_1}} t_i^{\alpha_2}-\lambda_1\sum_{i=1}^{n_{r_1}}(s_i-t_i)+(\alpha_2-1)\sum_{i=n_{r_1}+1}^{n_{r_2}}\ln(t_i)-\beta_1 \sum_{i=n_{r_1}+1}^{n_{r_2}} t_i^{\alpha_1}-\beta_2 \sum_{i=n_{r_1}+1}^{n_{r_2}} t_i^{\alpha_2}\nonumber \\
		&-\lambda_2\sum_{i=n_{r_1}+1}^{n_{r_2}}(s_i-t_i)+(\alpha_1-1)\sum_{i=1}^{n_{nr}}z_i\ln(t_i^*)-\beta_1\sum_{i=n_r+1}^{n_{nr}}z_i {t_i^*}^{\alpha_1}-\beta_2\sum_{i=n_r+1}^{n_{nr}}z_i {t_i^*}^{\alpha_2}\nonumber \\
		&-\lambda_1\sum_{i=n_r+1}^{n_{nr}}z_iu_i^*+(\alpha_2-1)\sum_{i=n_r+1}^{n_{nr}}(1-z_i)\ln(t_i^*)-\beta_1\sum_{i=n_r+1}^{n_{nr}}(1-z_i) {t_i^*}^{\alpha_1} \nonumber \\
		&-\beta_2\sum_{i=n_r+1}^{n_{nr}}(1-z_i) {t_i^*}^{\alpha_2}-\lambda_2\sum_{i=n_r+1}^{n_{nr}}(1-z_i)v_i^*-\beta_1\sum_{i=n_{nr}+1}^{n}s_i^{\alpha_1}-\beta_2\sum_{i=n_{nr}+1}^{n}s_i^{\alpha_2}.
		\label{eq4.10}
\end{align}}
We define the quantity $Q$ as 
{\footnotesize \begin{align}
		Q\Big(\Theta\vert\hat{\Theta}^{(m)}\Big)&=
		(n_{r_1}+P) \ln(\alpha_1)+(n_{r_1}+P) \ln(\beta_1) +P\ln(\lambda_1) +(n_{r_2}+n_{nr}-n_r-P)\ln(\alpha_2)\nonumber \\
		&+(n_{r_2}+n_{nr}-n_r-P) \ln(\beta_2) + (n_{nr}-n_r-P)\ln(\lambda_2)+(\alpha_1-1)\sum_{i=1}^{n_{r_1}}\ln(t_i)-\beta_1 \sum_{i=1}^{n_{r_1}} t_i^{\alpha_1} \nonumber \\
		&-\beta_2 \sum_{i=1}^{n_{r_1}} t_i^{\alpha_2}-\lambda_1\sum_{i=1}^{n_{r_1}}(s_i-t_i)+(\alpha_2-1)\sum_{i=n_{r_1}+1}^{n_{r_2}}\ln(t_i)-\beta_1 \sum_{i=n_{r_1}+1}^{n_{r_2}} t_i^{\alpha_1}-\beta_2 \sum_{i=n_{r_1}+1}^{n_{r_2}} t_i^{\alpha_2}\nonumber \\
		&-\lambda_2\sum_{i=n_{r_1}+1}^{n_{r_2}}(s_i-t_i)+(\alpha_1-1)\sum_{i=n_r+1}^{n_{nr}}P_iE\Big[\ln(t_i^*)\mid t_i^*<s_i\Big]-\beta_1\sum_{i=n_r+1}^{n_{nr}}P_i E\Big[{t_i^*}^{\alpha_1}\mid t_i^*<s_i\Big]\nonumber \\
		&-\beta_2\sum_{i=n_r+1}^{n_{nr}}P_i E\Big[{t_i^*}^{\alpha_2}\mid t_i^*<s_i\Big]-\lambda_1\sum_{i=n_r+1}^{n_{nr}}P_iE\Big[u_i^*\mid u_i^*<(s_i-t_i^*)\Big]\nonumber \\
		&+(\alpha_2-1)\sum_{i=n_r+1}^{n_{nr}}(1-P_i)E\Big[\ln(t_i^*)\mid t_i^*<s_i\Big]-\beta_1\sum_{i=n_r+1}^{n_{nr}}(1-P_i) E\Big[{t_i^*}^{\alpha_1}\mid t_i^*<s_i\Big] \nonumber \\
		&-\beta_2\sum_{i=n_r+1}^{n_{nr}}(1-P_i) E\Big[{t_i^*}^{\alpha_2}\mid t_i^*<s_i\Big]-\lambda_2\sum_{i=n_r+1}^{n_{nr}}(1-P_i)E\Big[v_i^*\mid v_i^*<(s_i-t_i^*)\Big]\nonumber \\
		&-\beta_1\sum_{i=n_{nr}+1}^{n}s_i^{\alpha_1}-\beta_2\sum_{i=n_{nr}+1}^{n}s_i^{\alpha_2}.
		\label{eq4.11}
\end{align}}

Since $z_i\sim\mathcal{B}(1,P_i)$, hence $E(z_i)=P_i$ and $E(Z^{*})=E(\sum_{i=n_r+1}^{n_{nr}}z_i)=P$. 
The conditional densities of $U^*$ and $V^*$ can be calculated by using density below
\begin{align}
	h_{Y}\Big(y_i \vert y_i<(s_i-t_i^*),z_i,\lambda_k\Big)= \frac{h(y_i;\lambda_k)}{1-\bar{H}\Big((s_i-t_i^*);\lambda_k\Big)} =\frac{\lambda_k \exp\{-\lambda_k y_i\}}{1-\exp\{-\lambda_k(s_i-t_i^*) \}} \label{eq4.23}
\end{align}
where $h(.)$ and $\bar{H}(.)$ denotes the truncated density and survival functions of $Y$ at point $(S-T^*)$.

The expectation terms used in E-step of the E-M algorithm can be calculated from conditional densities of latent variables given below
{\footnotesize \begin{align*}
		&\xi_1(t_i^*;\alpha_1,\beta_1)=E\Big[{t_i^*}^{\alpha_1} \vert t_i^*<s_i,k=1,\alpha_1,\beta_1\Big]=\frac{\int_{0}^{s_i}u^{2\alpha_1-1}\alpha_1 \beta_1  \exp\{-\beta_1 u^{\alpha_1}\}du}{1-\exp\{-\beta_1 s_i^{\alpha_1}\}} \\
		&\qquad \qquad \quad \ \  = \frac{1-(1+\beta_1 s_i^{\alpha_1})\exp\{-\beta_1 s_i^{\alpha_1}\}}{\beta_1 \Big(1-\exp\{-\beta_1 s_i^{\alpha_1}\}\Big)},\\
		&\xi_2(t_i^*;\alpha_1,\beta_1,\alpha_2)=E\Big[{t_i^*}^{\alpha_2}\vert t_i^*<s_i,k=1,\alpha_1,\beta_1\Big]= \frac{\int_{0}^{s_i}u^{\alpha_1+\alpha_2-1} \alpha_1 \beta_1 \exp\{-\beta_1u^{\alpha_1}\}du}{1-\exp\{-\beta_1 s_i^{\alpha_1}\}}\\
		&\qquad \qquad \qquad \quad \ = \frac{\Gamma\left(\frac{\alpha_1+\alpha_2}{\alpha_1}\right)-\Gamma\left(\frac{\alpha_1+\alpha_2}{\alpha_1},\beta_1 s_i^{\alpha_1}\right)}{(\beta_1)^{\alpha_2/\alpha_1}\Big(1-\exp\{-\beta_1 s_i^{\alpha_1}\}\Big)},
\end{align*}}
{\footnotesize \begin{align*}
		&\xi_3(t_i^*;\alpha_1,\beta_1)=E\Big[{\ln(t_i^*)} \vert t_i^*<s_i, k=1,\alpha_1,\beta_1\Big]=\frac{\int_{0}^{s_i}u^{\alpha_1-1}\ln(u)\alpha_1 \beta_1 \exp\{-\beta_1 u^{\alpha_1}\}du}{1-\exp\{-\beta_1 s_i^{\alpha_1}\}},\\
		&\xi_4(t_i^*;\alpha_1,\beta_1)=E\Big[{t_i^*}^{\alpha_1} \ln(t_i^*) \vert t_i^*<s_i,k=1,\alpha_1,\beta_1\Big]= \frac{\int_{0}^{s_i} u^{2\alpha_1-1}\ln(u) \alpha_1 \beta_1 \exp\{-\beta_1u^{\alpha_1}\}du}{1-\exp\{-\beta_1 s_i^{\alpha_1}\}} \\
		&\qquad \qquad \quad \ \ = \frac{K_2(s_i)-\ln(\beta_1)\Big(1-(1+\beta_1 s_i^{\alpha_1})\exp\{-\beta_1 s_i^{\alpha_1}\}\Big)}{\alpha_1\beta_1\Big(1-\exp\{-\beta_1 s_i^{\alpha_1}\}\Big)},\\
		&\xi_5(t_i^*;\alpha_1,\beta_1,\alpha_2)=E\Big[{t_i^*}^{\alpha_2} \ln(t_i^*) \vert t_i^*<s_i,k=1,\alpha_1,\beta_1\Big]\\
		&\qquad \qquad \qquad \quad  =\frac{\int_{0}^{s_i} u^{\alpha_1+\alpha_2-1}\ln(u) \alpha_1 \beta_1 \exp\{-\beta_1u^{\alpha_1}\}du}{1-\exp\{-\beta_1 s_i^{\alpha_1}\}} \\
		&\qquad \qquad \qquad \quad  = \frac{K_3(s_i)-\ln(\beta_1)\left\{\Gamma\left(\frac{\alpha_1+\alpha_2}{\alpha_1}\right)-\Gamma\left(\frac{\alpha_1+\alpha_2}{\alpha_1},\beta_1 s_i^{\alpha_1}\right)\right\}}{\alpha_1(\beta_1)^{\alpha_2/\alpha_1}\Big(1-\exp\{-\beta_1 s_i^{\alpha_1}\}\Big)},\\
		&\xi_6(u_i^*;\lambda_1)=E\Big[{u_i^*} \vert u_i^*<(s_i-t_i^*),k=1,\lambda_1\Big]=\frac{\int_{0}^{s_i-t_i^*}u\lambda_1 \exp\{-\lambda_1 u\}}{1-\exp\{-\lambda_1(s_i-t_i^*)\}}  \\
		&\qquad \qquad \ =\frac{1}{\lambda_1} \left[\frac{1-\{1+\lambda_1(s_i-t_i^*)\}\exp\{-\lambda_1 (s_i-t_i^*)\}}{1-\exp\{-\lambda_1(s_i-t_i^*)\}} \right],	\\	
		&\xi_7(t_i^*;\alpha_1,\beta_1)=E\Big[{t_i^*}^{\alpha_1}\Big(\ln({t_i^*})\Big)^2 \vert t_i^*<s_i,k=1,\alpha_1,\beta_1\Big]\\
		&\qquad \qquad \quad \ = \frac{\int_{0}^{s_i}u^{2\alpha_1-1} \Big(\ln(u)\Big)^2\alpha_1 \beta_1  \exp\{-\beta_1u^{\alpha_1}\}du}{1-\exp\{-\beta_1s_i^{\alpha_1}\}} \\
		&\qquad \qquad \quad \ =\frac{K_4(s_i)+\Big(\ln(\beta_1)\Big)^2\Big(1-(1+\beta_1 s_i^{\alpha_1})\exp\{-\beta_1 s_i^{\alpha_1}\}\Big)-2\ln(\beta_1)K_2(s_i)}{\alpha_1^2 \beta_1 \Big({1-\exp\{-\beta_1 s_i^{\alpha_1}\}\Big)}}, \\
		&\xi_8(t_i^*;\alpha_1,\beta_1,\alpha_2)=E\Big[{t_i^*}^{\alpha_2}\Big(\ln({t_i^*})\Big)^2 \vert t_i^*<s_i,k=1,\alpha_1,\beta_1\Big] \nonumber \\
		&\qquad \qquad \qquad \quad = \frac{\int_{0}^{s_i}u^{\alpha_1+\alpha_2-1} \Big(\ln(u)\Big)^2 \alpha_1 \beta_1  \exp\{-\beta_1u^{\alpha_1}\}du}{1-\exp\{-\beta_1s_i^{\alpha_1}\}} \\
		&\qquad \qquad \qquad \quad =\frac{K_5(s_i)+\Big(\ln(\beta_1)\Big)^2\left\{\Gamma\left(\frac{\alpha_1+\alpha_2}{\alpha_1}\right)-\Gamma\left(\frac{\alpha_1+\alpha_2}{\alpha_1},\beta_1 s_i^{\alpha_1}\right)\right\}-2\ln(\beta_1)K_3(s_i)}{\alpha_1^2 (\beta_1)^{\alpha_2/\alpha_1}\Big({1-\exp\{-\beta_1 s_i^{\alpha_1}\}\Big)}},\\
		&\xi_9(t_i^*;\alpha_1,\alpha_2,\beta_2)=E\Big[{t_i^*}^{\alpha_1} \vert t_i^*<s_i,k=2,\alpha_2,\beta_2\Big]=\frac{\int_{0}^{s_i}u^{\alpha_1+\alpha_2-1} \alpha_2 \beta_2 \exp\{-\beta_2 u^{\alpha_2}\}du}{1-\exp\{-\beta_2 s_i^{\alpha_2}\}} \\
		&\qquad \qquad \qquad \quad =\frac{\Gamma\left(\frac{\alpha_1+\alpha_2}{\alpha_2}\right)-\Gamma\left(\frac{\alpha_1+\alpha_2}{\alpha_2},\beta_2 s_i^{\alpha_2}\right)}{(\beta_2)^{\alpha_1/\alpha_2}\Big(1-\exp\{-\beta_2 s_i^{\alpha_2}\}\Big)},\\
		&\xi_{10}(t_i^*;\alpha_2,\beta_2)=E\Big[{t_i^*}^{\alpha_2} \vert t_i^*<s_i,k=2,\alpha_2,\beta_2\Big]= \frac{\int_{0}^{s_i}u^{2\alpha_2-1} \alpha_2 \beta_2 \exp\{-\beta_2 u^{\alpha_2}\}du}{1-\exp\{-\beta_2 s_i^{\alpha_2}\}}\\
		&\qquad \qquad \qquad  =\frac{1-(1+\beta_2 s_i^{\alpha_2})\exp\{-\beta_2 s_i^{\alpha_2}\}}{\beta_2 \Big(1-\exp\{-\beta_2 s_i^{\alpha_2}\}\Big)}, 
\end{align*}}

{\footnotesize \begin{align*}
		&\xi_{11}(t_i^*;\alpha_2,\beta_2)=E\Big[{\ln(t_i^*)}\vert t_i^*<s_i,k=2,\alpha_2,\beta_2\Big]=\frac{\int_{0}^{s_i}u^{\alpha_2-1} \ln(u) \alpha_2 \beta_2 \exp\{-\beta_2 u^{\alpha_2}\}du}{1-\exp\{-\beta_2 s_i^{\alpha_2}\}} \\
		&\qquad \qquad \qquad  =\frac{K_6(s_i)-\ln(\beta_2)\Big(1-\exp\{-\beta_2 s_i^{\alpha_2}\}\Big)}{\alpha_2\Big(1-\exp\{-\beta_2 s_i^{\alpha_2}\}\Big)},\\
		&\xi_{12}(t_i^*;\alpha_1,\alpha_2,\beta_2)=E\Big[{t_i^*}^{\alpha_1} \ln(t_i^*)\vert t_i^*<s_i,k=2,\alpha_2,\beta_2\Big]=\frac{\int_{0}^{s_i} u^{\alpha_1+\alpha_2-1}\ln(u)\alpha_2\beta_2  \exp\{-\beta_2 u^{\alpha_2}\}du}{1-\exp\{-\beta_2 s_i^{\alpha_2}\}} \\
		&\qquad \qquad \qquad  \quad \  =\frac{K_7(s_i)-\ln(\beta_2)\left\{\Gamma\left(\frac{\alpha_1+\alpha_2}{\alpha_2}\right)-\Gamma\left(\frac{\alpha_1+\alpha_2}{\alpha_2},\beta_2 s_i^{\alpha_2}\right)\right\}}{\alpha_2(\beta_2)^{\alpha_1/\alpha_2}\Big(1-\exp\{-\beta_2 s_i^{\alpha_2}\}\Big)},\\
		&\xi_{13}(t_i^*;\alpha_2,\beta_2)=E\Big[{t_i^*}^{\alpha_2} \ln(t_i^*)\vert t_i^*<s_i,k=2,\alpha_2,\beta_2\Big]= \frac{\int_{0}^{s_i} u^{2\alpha_2-1}\ln(u) \alpha_2 \beta_2  \exp\{-\beta_2 u^{\alpha_2}\}du}{1-\exp\{-\beta_2 s_i^{\alpha_2}\}} \\
		&\qquad \qquad \qquad = \frac{K_8(s_i)-\ln(\beta_2)\Big({1-(1+\beta_2 s_i^{\alpha_2})\exp\{-\beta_2 s_i^{\alpha_2}\}\Big)}}{\alpha_2\beta_2\Big(1-\exp\{-\beta_2 s_i^{\alpha_2}\}\Big)},	\\
		&\xi_{14}(v_i^*;\lambda_2)=E\Big[{v_i^*}\vert v_i^*<(s_i-t_i^*),k=2,\lambda_2\Big]=\frac{\int_{0}^{s_i-t_i^*}u\lambda_2 \exp\{-\lambda_2 u\}}{1-\exp\{-\lambda_2(s_i-t_i^*)\}}  \\
		&\qquad \quad \quad \ = \frac{1}{\lambda_2} \left[\frac{1-\Big(1+\lambda_2(s_i-t_i^*)\Big)\exp\{-\lambda_2 (s_i-t_i^*)\}}{1-\exp\{-\lambda_2(s_i-t_i^*)\}} \right], \\
		&\xi_{15}(t_i^*;\alpha_1,\alpha_2,\beta_2)=E\Big[{t_i^*}^{\alpha_1}\Big(\ln({t_i^*})\Big)^2\vert t_i^*<s_i,k=2,\alpha_2,\beta_2\Big] \nonumber \\
		&\qquad \qquad \qquad \ \ \ = \frac{\int_{0}^{s_i}u^{\alpha_1+\alpha_2-1} \Big(\ln(u)\Big)^2 \alpha_2 \beta_2 \exp\{-\beta_2 u^{\alpha_2}\}du}{1-\exp\{-\beta_2 s_i^{\alpha_2}\}} \\
		&\qquad \qquad \qquad \ \ \ =\frac{K_{9}(s_i)+\Big(\ln(\beta_2)\Big)^2\left\{\Gamma\left(\frac{\alpha_1+\alpha_2}{\alpha_2}\right)-\Gamma\left(\frac{\alpha_1+\alpha_2}{\alpha_2},\beta_2 s_i^{\alpha_2}\right)\right\}-2\ln(\beta_2)K_7(s_i)}{\alpha_2^2 (\beta_2)^{\alpha_1/\alpha_2}\Big(1-\exp\{-\beta_2 s_i^{\alpha_2}\}\Big)},\\		
		&\xi_{16}(t_i^*;\alpha_2,\beta_2)=E\Big[{t_i^*}^{\alpha_2}\Big(\ln({t_i^*})\Big)^2\vert t_i^*<s_i,k=2,\alpha_2,\beta_2\Big]\\
		&\qquad \qquad \qquad  \ = \frac{\int_{0}^{s_i}u^{2\alpha_2-1} \Big(\ln(u)\Big)^2 \alpha_2 \beta_2  \exp\{-\beta_2 u^{\alpha_2}\}du}{1-\exp\{-\beta_2 s_i^{\alpha_2}\}} \\
		&\qquad \qquad \qquad  \ =\frac{K_{10}(s_i)+\Big(\ln(\beta_2)\Big)^2\Big(1-(1+\beta_2 s_i^{\alpha_2})\exp\{-\beta_2 s_i^{\alpha_2}\}\Big)-2\ln(\beta_2)K_8(s_i)}{\alpha_2^2\beta_2\Big(1-\exp\{-\beta_2 s_i^{\alpha_2}\}\Big)},
\end{align*}}
where, 
{\small \begin{align*}
	K_{1}(y)&=\int_{0}^{\beta_1 y^{\alpha_1}}\ln(y) \exp\{-y\}dy, \quad K_{2}(y)=\int_{0}^{\beta_1 y^{\alpha_1}} y \ln(y) \exp\{-y\}dy, \\ K_{3}(y)&=\int_{0}^{\beta_1 y^{\alpha_1}} y^{\alpha_2/\alpha_1} \ln(y) \exp\{-y\}dz, \quad 
	K_{4}(y)=\int_{0}^{\beta_1 y^{\alpha_1}} y \Big(\ln(y)\Big)^2 \exp\{-y\}dy \\
	K_{5}(z)&=\int_{0}^{\beta_1 y^{\alpha_1}} y^{\alpha_2/\alpha_1} \Big(\ln(y)\Big)^2 \exp\{-y\}dy, \quad
	K_{6}(y)=\int_{0}^{\beta_2 y^{\alpha_2}}\ln(y) \exp\{-y\} dy, \\ K_{7}(y)&=\int_{0}^{\beta_2 y^{\alpha_2}} y^{\alpha_1/\alpha_2} \ln(y) \exp\{-y\}dy, \quad K_{8}(y)=\int_{0}^{\beta_2 y^{\alpha_2}} y \ln(y) \exp\{-y\}dy\\
	K_{9}(y)&=\int_{0}^{\beta_2 y^{\alpha_2}} y^{\alpha_1/\alpha_2} \Big(\ln(y)\Big)^2 \exp\{-y\}dy, \quad
	K_{10}(y)=\int_{0}^{\beta_2 y^{\alpha_2}} y \Big(\ln(y)\Big)^2 \exp\{-y\}dy.
\end{align*}}
After updating the missing data with help of expectation terms calculated in the E-step, the log-likelihood is maximized in the M-step and let at the $m^{th}$ iteration $\hat{\Theta}^{(m)}$ be the vector of unknown parameter's estimate of $\Theta$. Then the updated value of parameters at $(m+1)^{th}$ iteration can be computed by using the following expressions
{\footnotesize \begin{align}
		{\hat{\alpha}_1}^{(m+1)}&= \frac{\Big(n_{r_1}+P^{(m)}\Big)}{A} \label{eq4.28} \\
		{\hat{\alpha}_2}^{(m+1)}&= \frac{\Big(n_{r_2}+n_{nr}-n_r-P^{(m)}\Big)}{B} \label{eq4.29} \\ 
		{\hat{\beta}_1}^{(m+1)}&=\frac{\Big(n_{r_1}+P^{(m)}\Big)}{\left[\splitfrac{\sum_{i=1}^{n_{r_1}} t_i^{{\hat{\alpha}_1}^{(m+1)}}+\sum_{i=n_{r_1}+1}^{n_{r_2}}t_i^{{\hat{\alpha}_1}^{(m+1)}}+\sum_{i=n_r+1}^{n_{nr}}{P_i}^{(m)}\xi_1\Big(t_i^*;{\alpha_1}^{(m)},{\beta_1}^{(m)}\Big)}{+\sum_{i=n_r+1}^{n_{nr}}\Big(1-{P_i}^{(m)}\Big)\xi_9\Big(t_i^*;{\alpha_1}^{(m)},{\alpha_2}^{(m)},{\beta_2}^{(m)}\Big)+\sum_{i=n_{nr}+1}^{n}s_i^{{\hat{\alpha}_1}^{(m+1)}}}\right]} \label{eq4.24} \\
		{\hat{\lambda}_1}^{(m+1)}&= \frac{P^{(m)}}{\sum_{i=1}^{n_{r_1}}(s_i-t_i)+\sum_{i=n_r+1}^{n_{nr}}{P_i}^{(m)}\xi_6\Big(u_i^*;{\lambda_1}^{(m)}\Big)} \label{eq4.25} \\
		{\hat{\beta}_2}^{(m+1)}&=\frac{\Big(n_{r_2}+n_{nr}-n_r-P^{(m)}\Big)}{\left[\splitfrac{\sum_{i=1}^{n_{r_1}}t_i^{{\hat{\alpha}_2}^{(m+1)}}+\sum_{i=n_{r_1}+1}^{n_{r_2}}t_i^{{\hat{\alpha}_2}^{(m+1)}}+\sum_{i=n_r+1}^{n_{nr}}{P_i}^{(m)}\xi_2\Big(t_i^*;{\alpha_1}^{(m)},{\beta_1}^{(m)},{\alpha_2}^{(m)}\Big)}{+\sum_{i=n_r+1}^{n_{nr}}\Big(1-{P_i}^{(m)}\Big)\xi_{10}\Big(t_i^*;{\alpha_2}^{(m)},{\beta_2}^{(m)}\Big)+\sum_{i=n_{nr}+1}^{n}s_i^{{\hat{\alpha}_2}^{(m+1)}}}\right]} \label{eq4.26} \\
		{\hat{\lambda}_2}^{(m+1)}&=  \frac{\Big(n_{nr}-n_r-P^{(m)}\Big)}{\sum_{i=n_{r_1}+1}^{n_{r_2}}(s_i-t_i)+\sum_{i=n_r+1}^{n_{nr}}\Big(1-{P_i}^{(m)}\Big)\xi_{14}\Big(v_i^*;{\lambda_2}^{(m)}\Big)} \label{eq4.27}
\end{align}}
where, 
{\footnotesize \begin{align*}
		A&=-\sum_{i=1}^{n_{r_1}}\ln(t_i)+{\hat{\beta}_1}^{(m)}\sum_{i=1}^{n_{r_1}}t_i^{{\hat{\alpha}_1}^{(m)}}\ln(t_i)+{\hat{\beta}_1}^{(m)}\sum_{i=n_{r_1}+1}^{n_{r_2}}t_i^{{\hat{\alpha}_1}^{(m)}}\ln(t_i)-\sum_{i=n_r+1}^{n_{nr}}{P_i}^{(m)}\xi_3\Big(t_i^*;{\alpha_1}^{(m)},{\beta_1}^{(m)}\Big)\\
		&+{\hat{\beta}_1}^{(m)}\sum_{i=n_r+1}^{n_{nr}}{P_i}^{(m)}\xi_4\Big(t_i^*;{\alpha_1}^{(m)},{\beta_1}^{(m)}\Big)+{\hat{\beta}_1}^{(m)}\sum_{i=n_r+1}^{n_{nr}}\Big(1-{P_i}^{(m)}\Big)\xi_{12}\Big(t_i^*;{\alpha_1}^{(m)},{\alpha_2}^{(m)},{\beta_2}^{(m)}\Big)\\
		&+{\hat{\beta}_1}^{(m)}\sum_{i=n_{nr}+1}^{n}s_i^{{\hat{\alpha}_1}^{(m)}}\ln(s_i); \\
		B&=-\sum_{i=n_{r_1}+1}^{n_{r_2}}\ln(t_i)+{\hat{\beta}_2}^{(m)}\sum_{i=1}^{n_{r_1}}t_i^{{\hat{\alpha}_2}^{(m)}}\ln(t_i)+{\hat{\beta}_2}^{(m)}\sum_{i=n_{r_1}+1}^{n_{r_2}}t_i^{{\hat{\alpha}_2}^{(m)}}\ln(t_i)+{\hat{\beta}_2}^{(m)}\sum_{i=n_{nr}+1}^{n}s_i^{{\hat{\alpha}_2}^{(m)}}\ln(s_i) \\
		&+{\hat{\beta}_2}^{(m)}\sum_{i=n_r+1}^{n_{nr}}{P_i}^{(m)}\xi_5\Big(t_i^*;{{\alpha_1}^{(m)},{\alpha_2}^{(m)},\beta_2}^{(m)}\Big)+{\hat{\beta}_2}^{(m)}\sum_{i=n_r+1}^{n_{nr}}(1-{P_i}^{(m)})\xi_{13}\Big(t_i^*;{\alpha_2}^{(m)},{\beta_2}^{(m)}\Big)  \\
		&-\sum_{i=n_r+1}^{n_{nr}}\Big(1-{P_i}^{(m)}\Big)\xi_{11}\Big	(t_i^*;{\alpha_2}^{(m)},{\beta_2}^{(m)}\Big)
	\end{align*}
}

%-------------------------------------------------------------------------------------------------------------------------%
\subsubsection{Observed Fisher Information Matrix}\label{subsec4.3.2}
The terms used for the construction of observed Fisher information matrix are given below
{\footnotesize \begin{align*}
		&\frac{\partial^2 Q }{\partial \alpha_1^2}= -\frac{(n_{r_1}+P)}{\alpha_1^2}-\beta_1\sum_{i=1}^{n_{r_1}}t_i^{\alpha_1}\Big(\ln(t_i)\Big)^2-\beta_1\sum_{i=n_{r_1}+1}^{n_{r_2}}t_i^{\alpha_1}\Big(\ln(t_i)\Big)^2-\beta_1\sum_{i=n_r+1}^{n_{nr}}P_i\xi_7\Big(t_i^*;\alpha_1,\beta_1\Big)\\
		&\quad \qquad -\beta_1\sum_{i=n_r+1}^{n_{nr}}(1-P_i)\xi_{15}\Big(t_i^*;\alpha_1,\alpha_2,\beta_2\Big)-\beta_1\sum_{i=n_{nr}+1}^{n}s_i^{\alpha_1} \Big(\ln(s_i)\Big)^2,\\
		&\frac{\partial^2 Q }{\partial \beta_1^2}=-\frac{\Big(n_{r_1}+P\Big)}{\beta_1^2}; \quad \frac{\partial^2 Q }{\partial \beta_2^2}=-\frac{(n_{r_2}+n_{nr}-n_r-P)}{\beta_2^2},\\
		&\frac{\partial^2 Q }{\partial \alpha_2^2}= -\frac{\Big(n_{r_2}+n_{nr}-n_r-P\Big)}{\alpha_2^2}-\beta_2\sum_{i=1}^{n_{r_1}}t_i^{\alpha_2}\Big(\ln(t_i)\Big)^2-\beta_2\sum_{i=n_{r_1}+1}^{n_{r_2}}t_i^{\alpha_2}\Big(\ln(t_i)\Big)^2\\
		&\quad \qquad  -\beta_2\sum_{i=n_r+1}^{n_{nr}}P_i\xi_{8}\Big(t_i^*;\alpha_1,\beta_1,\alpha_2\Big)-\beta_2\sum_{i=n_r+1}^{n_{nr}}(1-P_i)\xi_{16}\Big(t_i^*;\alpha_2,\beta_2\Big) -\beta_2\sum_{i=n_{nr}+1}^{n}s_i^{\alpha_2} \Big(\ln(s_i)\Big)^2, \\
		&\frac{\partial^2 Q }{\partial \lambda_1^2}=-\frac{P}{\lambda_1^2}; \quad \frac{\partial^2 Q }{\partial \lambda_2^2}=-\frac{\Big(n_{nr}-n_r-P\Big)}{\lambda_2^2},
\end{align*}}
{\footnotesize \begin{align*}
		&\frac{\partial^2 Q }{\partial \beta_1 \partial \alpha_1}=-\sum_{i=1}^{n_{r_1}}t_i^{\alpha_1}\ln(t_i)-\sum_{i=n_{r_1}+1}^{n_{r_2}}t_i^{\alpha_1}\ln(t_i)-\sum_{i=n_r+1}^{n_{nr}}P_i\xi_4\Big(t_i^*;\alpha_1,\beta_1\Big)-\sum_{i=n_{nr}+1}^{n}s_i^{\alpha_1}\ln(s_i)\\
		&\quad \qquad \quad -\sum_{i=n_r+1}^{n_{nr}}\Big(1-P_i\Big)\xi_{12}\Big(t_i^*;\alpha_1,\alpha_2,\beta_2\Big)=\frac{\partial^2 Q }{\partial \alpha_1 \partial \beta_1},\\
		&\frac{\partial^2 Q }{\partial \beta_2 \partial \alpha_2}=-\sum_{i=1}^{n_{r_1}}t_i^{\alpha_2}\ln(t_i)-\sum_{i=n_{r_1}+1}^{n_{r_2}}t_i^{\alpha_2}\ln(t_i)-\sum_{i=n_r+1}^{n_{nr}}P_i\xi_5\Big(t_i^*;\alpha_1,\beta_1,\alpha_2\Big)-\sum_{i=n_{nr}+1}^{n}s_i^{\alpha_2}\ln(s_i)\\
		&\quad \qquad \quad -\sum_{i=n_r+1}^{n_{nr}}\Big(1-P_i\Big)\xi_{13}\Big(t_i^*;\alpha_2,\beta_2\Big)=\frac{\partial^2 Q }{\partial \alpha_2 \partial \beta_2}.
\end{align*}}
%-------------------------------------------------------------------------------------------------------------------------%
\subsection{Bayesian Inference} \label{A3}
All the inferences in the Bayesian paradigm are drawn from the posterior distribution. The likelihood in (\ref{eq4.4}) indicates that it is difficult to study the properties of posterior analytically. So, we use likelihood given in (\ref{eq4.9}) in this section for further study. 
\subsubsection{Prior Distribution}\label{subsec4.4.1}
Consider $\alpha_k$, $\beta_k$, $\lambda_k; \ (k=1,2)$ as Gamma distributed with parameters $(a_l,b_l); \ l=1,2,\ldots,6$. Assuming the independence of priors the joint prior density is written up to proportionality constants as below
{\footnotesize \begin{equation}
		\pi(\Theta)\propto{\alpha_1}^{a_1-1} {\beta_1}^{a_2-1}
		{\lambda_1}^{a_3-1} {\alpha_2}^{a_4-1} {\beta_2}^{a_5-1}
		{\lambda_2}^{a_6-1} \exp\left\{-(\alpha_1 b_1+\beta_1 b_2+\lambda_1 b_3+\alpha_2 b_4+\beta_2 b_5+\lambda_2 b_6)\right\} \label{eq4.30}
\end{equation}}
\subsubsection{Gibbs Sampling}\label{subsec4.2}
For applying the Gibbs Sampling the posterior distribution is required which can be obtained by merging complete likelihood (\ref{eq4.9}) and the joint prior (\ref{eq4.30}). The posterior distribution up-to proportionality constant is written below
%\newpage
{\tiny \begin{align}
		\Pi(\Theta\vert d,d^*)&\propto L_c(\Theta\vert d,d^*)\pi(\Theta) \nonumber \\
		&\propto  \alpha_1^{(n_{r_1}+Z^{*}+a_1-1)} \beta_1^{(n_{r_1}+Z^{*}+a_2-1)} \lambda_1^{(Z^{*}+a_3-1)}  \alpha_2^{(n_{r_2}+n_{nr}-n_r-Z^{*}+a_4-1)}  \beta_2^{(n_{r_2}+n_{nr}-n_r-Z^{*}+a_5-1)} \nonumber \\
		& \lambda_2^{(n_{nr}-n_r-Z^{*}+a_6-1)} \prod\limits_{i=1}^{n_{r_1}} t_i^{\alpha_1-1}\exp\{-\beta_1 t_i^{\alpha_1}-\beta_2 t_i^{\alpha_2}-\lambda_1(s_i-t_i)\} \prod\limits_{i=n_{nr}+1}^{n}\exp\{-\beta_1s_i^{\alpha_1}-\beta_2s_i^{\alpha_2}\} \nonumber \\ 
		&\prod\limits_{i=n_{r_1}+1}^{n_{r_2}} t_i^{\alpha_2-1}\exp\{-\beta_1 t_i^{\alpha_1}-\beta_2 t_i^{\alpha_2}-\lambda_2(s_i-t_i)\}  \prod\limits_{i=n_r+1}^{n_{nr}}\left[{t_i^*}^{\alpha_1-1} \exp\{-\beta_1 {t_i^*}^{\alpha_1}-\beta_2 {t_i^*}^{\alpha_2}-\lambda_1u_i^*\} \right]^{z_i} 
		\nonumber \\ 
		&\prod\limits_{i=n_r+1}^{n_{nr}}\left[ {t_i^*}^{\alpha_2-1}  \exp\{-\beta_1 {t_i^*}^{\alpha_1}-\beta_2 {t_i^*}^{\alpha_2}-\lambda_2v_i^*\} \right]^{1-z_i} \exp\{-\alpha_1 b_1-\beta_1 b_2-\lambda_1 b_3-\alpha_2 b_4-\beta_2 b_5-\lambda_2 b_6\}
		\label{eq4.31}
\end{align}}
In order to draw the samples from posterior, we write the full conditionals for parameters based on the posterior distribution (\ref{eq4.31}) which are given below
{\tiny \begin{align}
		&\alpha_1\vert \Theta_{(-\alpha_1)} \propto \alpha_1^{(n_{r_1}+Z^{*}+a_1-1)}\left(\prod_{i=1}^{n_{r_1}}t_i^{\alpha_1-1}\right) \left(\prod_{i=n_r+1}^{n_{nr}}{t_i^*}^{z_i(\alpha_1-1)}\right) \nonumber \\
		&\qquad \quad \quad \  \ \ \ \ \exp\left(-\beta_1\sum_{i=1}^{n_{r_1}}t_i^{\alpha_1}-\beta_1\sum_{i=n_{r_1}+1}^{n_{r_2}}t_i^{\alpha_1}-\beta_1\sum_{i=n_r+1}^{n_{nr}}z_i{t_i^*}^{\alpha_1}\right) \nonumber \\
		&\qquad \quad \quad \  \ \ \ \ \exp\left(-\beta_1\sum_{i=n_r+1}^{n_{nr}}(1-z_i){t_i^*}^{\alpha_1}-\beta_1\sum_{i=n_{nr}+1}^{n}s_i^{\alpha_1}-\alpha_1 b_1 \right), \label{eq4.36} \\
		&\alpha_2\vert \Theta_{(-\alpha_2)} \propto \alpha_2^{(n_{r_2}+n_{nr}-n_r-Z^{*}+a_4-1)}\left(\prod_{i=n_{r_1}+1}^{n_{r_2}}t_i^{\alpha_2-1}\right) \left(\prod_{i=n_r+1}^{n_{nr}}{t_i^*}^{(1-z_i)(\alpha_2-1)}\right) \nonumber \\ 
		& \qquad \quad \quad \  \ \ \ exp\left(-\beta_2\sum_{i=1}^{n_{r_1}}t_i^{\alpha_2}-\beta_2\sum_{i=n_{r_1}+1}^{n_{r_2}}t_i^{\alpha_2}\right)\nonumber \\
		&\qquad \quad \quad \ \ \ \ \ \  \exp\left(-\beta_2\sum_{i=n_r+1}^{n_{nr}}z_i{t_i^*}^{\alpha_2}-\beta_2\sum_{i=n_r+1}^{n_{nr}}(1-z_i){t_i^*}^{\alpha_2}-\beta_2\sum_{i=n_{nr}+1}^{n}s_i^{\alpha_2}-\alpha_2b_4 \right),  \label{eq4.37} \\
		&\beta_1\vert \Theta_{(-\beta_1)} \sim \mathcal{G}\left(n_{r_1}+Z^{*}+a_2,\ \sum_{i=1}^{n_{r_1}}t_i^{\alpha_1}+\sum_{i=n_{r_1}+1}^{n_{r_2}}t_i^{\alpha_1}+\sum_{i=n_r+1}^{n_{nr}}z_i {t_i^*}^{\alpha_1}+\sum_{i=n_r+1}^{n_{nr}}(1-z_i) {t_i^*}^{\alpha_1}+\sum_{i=n_{nr}+1}^{n}s_i^{\alpha_1}+b_2\right), \label{eq4.32}\\
		&\lambda_1\vert \Theta_{(-\lambda_1)} \sim \mathcal{G}\left(Z^{*}+a_3,\ \sum_{i=1}^{n_{r_1}}(s_i-t_i)+\sum_{i=n_r+1}^{n_{nr}}z_iu_i^*+b_3 \right),   \label{eq4.33}  \\
		&\beta_2\vert \Theta_{(-\beta_2)} \sim \mathcal{G}\left(\tilde{Z^*},\ \sum_{i=1}^{n_{r_1}}t_i^{\alpha_2}+\sum_{i=n_{r_1}+1}^{n_{r_2}}t_i^{\alpha_2}+\sum_{i=n_r+1}^{n_{nr}}z_i {t_i^*}^{\alpha_2}+\sum_{i=n_r+1}^{n_{nr}}(1-z_i) {t_i^*}^{\alpha_2}+\sum_{i=n_{nr}+1}^{n}s_i^{\alpha_2}+b_5 \right), \label{eq4.34} \\
		&\lambda_2\vert \Theta_{(-\lambda_2)} \sim \mathcal{G}\left(n_{nr}-n_r-Z^{*}+a_6,\ \sum_{i=n_{r_1}+1}^{n_{r_2}}(s_i-t_i)+\sum_{i=n_r+1}^{n_{nr}}(1-z_i)v_i^*+b_6 \right). \label{eq4.35}
\end{align}}
Here, $\tilde{Z^*}=(n_{r_2}+n_{nr}-n_r-Z^{*}+a_5)$. From equations (\ref{eq4.36}) to (\ref{eq4.35}), we can see that the full conditionals of $\alpha_1$ and $\alpha_2$ are not in standard density form. So, samples from these full conditionals are generated by using Metropolis-Hasting (M-H) algorithm taking Normal distribution as proposal density. Further full conditionals of $\beta_1$, $\lambda_1$, $\beta_2$ and $\lambda_2$ are following Gamma distribution with varying shapes and scales and observations from these can be drawn directly for given values of latent variables $z$, $t^*$, $u^*$ and $v^*$ and hyper-parameters.  
%-------------------------------------------------------------------------------------%
\subsubsection{Data Augmentation Algorithm}\label{subsec4.4.3}
The algorithm used for generating samples from full conditionals using nested Gibbs sampling is given below:
\begin{itemize}
	\item \textbf{Step 1}: Set initial values of parameters, say  $\Theta^{(0)}=\Big(\alpha_1^{(0)},\beta_1^{(0)},\lambda_1^{(0)},\alpha_2^{(0)},\beta_2^{(0)},\lambda_2^{(0)}\Big)$ and generate $z_{i}\sim\mathcal{B}(1,P_{i});\ i=n_r+1,n_r+2,\cdots,n_{nr}$.
	\item \textbf{Step 2}: For given values of $z_{i}$ and $\Theta$, generate observations on $t_{i}^*$ using expression
{\small 	\begin{equation}
		t_{i}^*=\left[-\frac{1}{\beta_k}\ln \left\{1-w_i\left(1-\exp\{-\beta_ks_i^{\alpha_k}\}\right)\right\}\right]^{1/\alpha_k};\ i=n_r+1,n_r+2,\ldots,n_{nr}, \ k=1,2,\label{eq4.38}
	\end{equation}}
	where $w \sim \mathcal{U}(0,1)$.
	\item \textbf{Step 3}: Based on generated values of $z_{i}$, $t_{i}^*$ and $\Theta$ from previous steps, observations on $u_{i}^*$ and $v_i^*$ can be generated by using expression
	{\small \begin{equation}
		y_i^*=-\frac{1}{\lambda_k} \ln\Big[1-w_i\Big(1-\exp\{-\lambda_k(s_i-t_{i}^*)\}\Big) \Big];\ i=n_r+1,n_r+2,\ldots,n_{nr}, \ k=1,2. \label{eq4.39}
	\end{equation}}
	\item \textbf{Step 4}: For given values of $z_{i}$, $t_{i}^*$, $u_{i}^*$ and $v_{i}^*$ generate observations on $\alpha_1^{(1)}$ and $\alpha_2^{(1)}$ using M-H algorithm taking normal as proposal density. 
	\item \textbf{Step 5}: In the next step observations on $\beta_1^{(1)}$, $\lambda_1^{(1)}$, $\beta_2^{(1)}$ and $\lambda_2^{(1)}$ can be generated for given $\alpha_1^{(1)}$ and $\alpha_2^{(1)}$  using (\ref{eq4.32}) to (\ref{eq4.35}) respectively. 
\end{itemize}
Now, the current state is $\Theta^{(1)}=\Big(\alpha_1^{(1)},\beta_1^{(1)},\lambda_1^{(1)},\alpha_2^{(1)},\beta_2^{(1)},\lambda_2^{(1)}\Big)$. Steps $1$ to $5$ are replicated $M$ times to obtain a sequence of random variables $\Big(\Theta^{(1)},\Theta^{(2)},\cdots,\Theta^{(M)}\Big)$. After discarding the burn-in from generated chains, the stationarity of the chains are checked by trace plot and Gelman and Rubin's test statistics, we left out with a reduced chain of length $M'$. All the Bayesian inferences are drawn based on these samples.

\section{Posterior Plots for Menopause data}

\begin{figure}[ht]
	\centering
	\includegraphics[width=0.95\textwidth, height=0.39\textheight]{Cause Known_MatPlot.pdf}
	\caption{MCMC trace plots for posterior samples when causes are known for non-recall.}
	\label{Fig1}
\end{figure}

\begin{figure}[ht]
	\centering
	\includegraphics[width=0.95\textwidth, height=0.39\textheight]{Cause Known_Density.pdf}
	\caption{MCMC density plots for posterior samples when causes are known for non-recall.}
	\label{Fig2}
\end{figure}

\begin{figure}[ht]
	\centering
	\includegraphics[width=0.95\textwidth, height=0.39\textheight]{MatPlot_Cause_Time_Unknown.pdf}
	\caption{MCMC trace plots for posterior samples when causes are unknown for non-recall.}
	\label{Fig3}
\end{figure}

\begin{figure}[ht]
	\centering
	\includegraphics[width=0.95\textwidth, height=0.39\textheight]{Density_Cause_Time_Unknown.pdf}
	\caption{MCMC density plots for posterior samples when causes are unknown for non-recall.}
	\label{Fig4}
\end{figure}
 


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