Bayesian Estimation for the Two Log-Logistic Models Under Joint Type II Censoring Schemes

Ranjita Pandey and Pulkit Srivastava*

Department of Statistics, University of Delhi, Delhi, India
E-mail: ranjitapandey111@gmail.com; pulkit.stats@gmail.com
*Corresponding Author

Received 30 January 2022; Accepted 14 March 2022; Publication 16 April 2022

## Abstract

The present paper, discusses classical and Bayesian estimation of unknown combined parameters of two different log-logistic models with common shape parameters and different scale parameters under a new type of censoring scheme known as joint type II censoring scheme. Maximum likelihood estimators are derived. Bayes estimates of parameters are proposed under different loss functions. Classical asymptotic confidence intervals along with the Bayesian credible intervals and Highest Posterior Density region are also constructed. Markov Chain Monte Carlo approximation method is used for simulating the theoretic results. Comparative assessment of the classical and the Bayes results are illustrated through a real archived dataset.

Keywords: Log-logistic model, Bayes estimation, Joint type II censoring scheme, Bayesian credible interval, Markov Chain Monte Carlo.

## 1 Introduction

Identical items produced on different production lines can be classified as having a common shape parameter. However, each production line/time can differ in scale parameter. Thus, within heterogeneous larger groups, the sub population time lines can be regarded as homogeneous with similar shape parameter while allowing variation in scale, which may be caused by shift in production level or by changing production time trend. Inferences of common shape parameter event have been studied by Nelson (2003, 2009), Panza and Vargas (2016), Tripathy and Nagamani (2017) and Chehade et al. (2020).

In any life testing experiment, when the experimenter could not record complete lifetime for all the test items due to time, cost or other limitations, then censored samples are obtained. One of the main motivations for using censoring is reduction of the total experimental time and its associated cost. In the conventional type II censoring scheme, a single sample of pre-defined size from the life-test is obtained. Statistical analysis usually involves various types of one-sample censored data. However, under certain situations the experimenter aims to have simultaneous assessment of different samples. In many situations, two different samples arising from the same model are required to be tested which are combined and subsequently ordered prior to analysis, after getting a desired number of failures. This mechanism termed as joint Type-II censoring scheme (JCS) promulgated by Balakrishnan and Rasouli (2008) enables comparison of sample lifetimes of products coming from different sources within the same facility. Suppose that similar products are being produced on two distinct production belts under the same facility. Under JCS, two independent samples selected from each such production line are simultaneously placed on the life-test in order to save time and capital resources such that the life-test terminates when a certain number of failures (say, r) occur. Balakrishnan and Rasouli (2008) developed likelihood inference for the parameters of two exponential populations, Abdel-Aty (2017) studied two exponential distributions, Al-Matrafi & Abd-Elmougod (2017) gave statistical inferences for two Rayleigh distributions, Ashour & Abo-Kasem (2014) worked with two Weibull distributions. Bayesian inferences for some other distributions under JCS have been undertaken by Ashour and Abo-Kasem (2014), Shafay et al. (2014) and Balakrishnan and Su (2015) among others.

JCS is described as follows: Let $X1,X,2…,Xm$ be m iid random variables with the probability density function (pdf) $f1⁢(x)$ and cumulative density function (cdf) $F1⁢(x)$ and $Y1,Y,2…,Yn$ be n iid random variables with pdf $f2⁢(x)$ and cdf $F2⁢(x)$. Let r be a pre-fixed integer denoting that the experiment will be stopped after recording the first r failures. Further, let $W1≤W2≤⋯≤WN$ be the collective ordered set of $N(=m+n)$ random variables ${X1,X,2…,Xm,…Y1,Y,2…,Yn}$. Then the observable dataset under joint type II censoring scheme will be $(W,δ)$ where $W=(W1,W,2…,Wr)$ and $δ=(δ1,δ,2…,δr)$ such that $δ$ represents an indicator variable with the following demarcation,

 $δi={1 i⁢f⁢Wi∈X0 i⁢f⁢Wi∈Y$

This paper considers Log-logistic distribution (LLD) as a life time model, under JCS, for survival and reliability studies. Maximum likelihood estimates (MLE) of unknown parameters are obtained in Section 2. Construction of Asymptotic Confidence Interval (ACI) based on the asymptotic normality of the MLEs is also undertaken. In Section 3, we consider parameter estimation under Bayesian setup along with the construction of credible intervals. Bayes parametric estimates are derived using the following specifications: Squared error loss function (SELF), general entropy loss function (GELF), linear exponential loss function (LINEX) and non-linear exponential loss function (NLINEX) assuming non-informative priors for scale parameters and gamma prior for the common shape parameter. A Markov chain Monte Carlo (MCMC) simulation has been conducted under Section 4. In Section 5, a real data set has been examined to illustrate the proposed theoretical estimation methods. ACI and BCI are compared for efficiency with the bootstrap confidence intervals.

## 2 Classical Estimation

Let r be a prefixed integer. Let the lifetime distribution have respective pdfs $f1⁢(x;Θ1)$ and $f2⁢(x;Θ2)$ with the corresponding cdfs $F1⁢(x;Θ1)$ and $F2⁢(x;Θ2)$, where $Θ1$ and $Θ2$ represent a vector of parameters. Let $Pr=∑i=1rδi$ be the number of observed X-failures in W and $Qr=∑i=1r(1-δi)=r-Pr$ be the number of observed Y-failures in W. Then the likelihood of $(W,δ)$ is given by

 $L⁢(Θ1,Θ2,W,δ)$ $=m!⁢n!(m-pr)!⁢(n-qr)!⁢∏i=1r$ $×[{f1⁢(wi)}δi⁢{f2⁢(wi)}1-δi]$ $×{1-F1⁢(wr)}(m-pr)⁢{1-F2⁢(wr)}(n-qr)$ (1)

LLD can be considered as a combination of the Gompertz and Gamma distribution with a restriction of unit mean and variance. It is also known as Fisk distribution in economics (Fisk, 1961). Owing to its non-monotonic and decreasing hazard rate function, LLD has been widely used in several life time analyses (Shoukari et al., 1988; Collett, 2003; Ashkar and Mandi, 2006). Its various statistical characteristics have been studied by Reath et al. (2018), Vroon (1987), Singh and Guo (1995), Ahsanullah, and Alzaatreh (2018) and many more. Bayesian Estimation of parameters of LLD was considered by Guure (2015) and Al-Shomrani et al. (2016), Sewailen and Baklizi (2019) under different censoring scheme. The pdf and cdf of LLD is defined as

 $f⁢(x;α,β)=(βα)⁢(xα)β-1{1+(xα)β}2 for x,α,β>0$ (2) $F⁢(x)=11+(xα)-β$ (3)

When two similar kinds of items or products are put on test, we expect that the underlying distributions have some common properties. To mirror such situation, in this paper, we assume that the units or items from two different sample groups follow LLD with the common shape parameter but possibly different scale parameters. Therefore, assuming two different models as $LL⁢(α1,β)$ and $L⁢L⁢(α2,β)$ the likelihood function can be written as

 $L⁢(α1,α2,β,w,δ)$ $∝∏i=1r[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]$ $×{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)$ (4)

Corresponding log-likelihood function can be written as

 $ln⁡L$ $=c⁢o⁢n⁢s⁢t+∑i=1rδi⁢log⁡((βα1)⁢(wiα1)β-1{1+(wiαi)β}2)$ $+∑i=1r(1-δi)⁢log⁡((βα2)⁢(wiα2)β-1{1+(wiα2)β}2)$ $+(m-pr)⁢log⁡(11+(wrα1)β)+(n-qr)⁢log⁡(11+(wrα2)β)$ (5)

There are several classical estimators available in the literature. In this paper, we obtain estimates of unknown parameters using principle of maximum likelihood estimation. Let $λ$ be the unknown parameters, then any function $λ^⁢(x)$ of sample values which maximizes the likelihood function $L⁢(x1,x2,…,xn)$, will be the mle of $λ$. Since logarithm is non-decreasing monotonic function, it will be convenient to work on log-likelihood function. The value of parameter that maximizes the log-likelihood will be obtained using maxima and minima. Thus,

 $α^1$, the mle of $α1$, is the value for which $∂⁡ln⁡L∂⁡α1=0 and ∂2⁡ln⁡L∂⁡α12|α1=α^1<0.$ $α^2$, the mle of $α2$, is the value for which $∂⁡ln⁡L∂⁡α2=0 and ∂2⁡ln⁡L∂⁡α22|α2=α^2<0.$ $β^$, the mle of $β$, is the value for which $∂⁡ln⁡L∂⁡β=0 and ∂2⁡ln⁡L∂⁡β2|β=β^<0.$

Equations of the first partial derivatives of log-likelihood function with respect to individual parameters are analogous to the system of non-linear equations and therefore cannot be solved explicitly as these equations do not have solutions in closed form. Therefore, numerical approximation method of Newton-Raphson is used to evaluate the MLEs.

The asymptotic variance-covariance matrix is needed to construct the confidence intervals. This matrix is obtained by taking inverse of the Fisher’s information matrix $I⁢(λ)$ (Aldrich, 1997). Let $λ^=(α^1,α^2,β^)$ denote the mle of $λ=(α1,α2,β)$. The asymptotic normality result is stated as follows to obtain the confidence interval:

 $n⁢(λ^-λ)⟶N⁢(0,I-1⁢(λ))$

In other words, under certain regularity conditions, $λ^=(α^1,α^2,β^)$ is approximately normal with mean $(α1,α2,β)$ and covariance matrix $I-1⁢(α1,α2,β)$ where

 $I⁢(λ)=-1n⁢E⁢[∂2⁡ln⁡L∂⁡α12 ∂2⁡ln⁡L∂⁡α1⁢∂⁡α2 ∂2⁡ln⁡L∂⁡α1⁢∂⁡β∂2⁡ln⁡L∂⁡α2⁢∂⁡α1 ∂2⁡ln⁡L∂⁡α22 ∂2⁡ln⁡L∂⁡α2⁢∂⁡β∂2⁡ln⁡L∂⁡β⁢∂⁡α1 ∂2⁡ln⁡L∂⁡β2⁢∂⁡α2 ∂2⁡ln⁡L∂⁡β22]$

Such that the matrix entries are defined as

 $∂2⁡ln⁡L∂⁡α12$ $=∑i=1rδi⁢∂2∂⁡α12⁢log⁡((βα1)⁢(wiα1)β-1{1+(wiαi)β}2)$ $+∑i=1r(1-δi)⁢∂2∂⁡α12⁢log⁡((βα2)⁢(wiα2)β-1{1+(wiα2)β}2)$ $+(m-pr)⁢∂2∂⁡α12⁢log⁡(11+(wrα1)β)$ $+(n-qr)⁢∂2∂⁡α12⁢log⁡(11+(wrα2)β)$ (6) $∂2⁡ln⁡L∂⁡α22$ $=∑i=1rδi⁢∂2∂⁡α22⁢log⁡((βα1)⁢(wiα1)β-1{1+(wiαi)β}2)$ $+∑i=1r(1-δi)⁢∂2∂⁡α22⁢log⁡((βα2)⁢(wiα2)β-1{1+(wiα2)β}2)$ $+(m-pr)⁢∂2∂⁡α22⁢log⁡(11+(wrα1)β)$ $+(n-qr)⁢∂2∂⁡α22⁢log⁡(11+(wrα2)β)$ (7) $∂2⁡ln⁡L∂⁡β2$ $=∑i=1rδi⁢∂2∂⁡β2⁢log⁡((βα1)⁢(wiα1)β-1{1+(wiαi)β}2)$ $+∑i=1r(1-δi)⁢∂2∂⁡β2⁢log⁡((βα2)⁢(wiα2)β-1{1+(wiα2)β}2)$ $+(m-pr)⁢∂2∂⁡β2⁢log⁡(11+(wrα1)β)$ $+(n-qr)⁢∂2∂⁡β2⁢log⁡(11+(wrα2)β)$ (8) $∂2⁡ln⁡L∂⁡α1⁢∂⁡β$ $=∑i=1rδi⁢∂2∂⁡α1⁢∂⁡β⁢log⁡((βα1)⁢(wiα1)β-1{1+(wiαi)β}2)$ $+∑i=1r(1-δi)⁢∂2∂⁡α1⁢∂⁡β⁢log⁡((βα2)⁢(wiα2)β-1{1+(wiα2)β}2)$ $+(m-pr)⁢∂2∂⁡α1⁢∂⁡β2⁢log⁡(11+(wrα1)β)$ $+(n-qr)⁢∂2∂⁡α1⁢∂⁡β⁢log⁡(11+(wrα2)β)$ (9) $∂2⁡ln⁡L∂⁡α1⁢∂⁡α2$ $=∑i=1rδi⁢∂2∂⁡α1⁢∂⁡α2⁢log⁡((βα1)⁢(wiα1)β-1{1+(wiαi)β}2)$ $+∑i=1r(1-δi)⁢∂2∂⁡α1⁢∂⁡α2⁢log⁡((βα2)⁢(wiα2)β-1{1+(wiα2)β}2)$ $+(m-pr)⁢∂2∂⁡α1⁢∂⁡α2⁢log⁡(11+(wrα1)β)$ $+(n-qr)⁢∂2∂⁡α1⁢∂⁡α2⁢log⁡(11+(wrα2)β)$ (10) $∂2⁡ln⁡L∂⁡β⁢∂⁡α2$ $=∑i=1rδi⁢∂2∂⁡β⁢∂⁡α2⁢log⁡((βα1)⁢(wiα1)β-1{1+(wiαi)β}2)$ $+∑i=1r(1-δi)⁢∂2∂⁡β⁢∂⁡α2⁢log⁡((βα2)⁢(wiα2)β-1{1+(wiα2)β}2)$ $+(m-pr)⁢∂2∂⁡β⁢∂⁡α2⁢log⁡(11+(wrα1)β)$ $+(n-qr)⁢∂2∂⁡β⁢∂⁡α2⁢log⁡(11+(wrα2)β)$ (11)

The exact mathematical expression for $I⁢(λ)$ is difficult to obtain in a closed form as its elements are intractable in nature. Since $λ$ is unknown, using uniqueness property of mle, we estimate $I-1⁢(λ)$ by $I-1⁢(λ^)$ which provides ACIs for the unknown parameters $α1,α2,β$ as

 $(α^1-zξ2⁢var⁢(α^1),α^1+zξ2⁢var⁢(α^1)),$ $(α^2-zξ2⁢var⁢(α^2),α^2+zξ2⁢var⁢(α^2)) and$ $(β^-zξ2⁢var⁢(β^),β^+zξ2⁢var⁢(β^))$

where $var⁢(α^1)$, $var⁢(α^2)$ and $var⁢(β^)$ are the estimated variances of $α^1,α^2,β^$ respectively given by the main diagonal elements of $I-1⁢(λ^)$ and $zξ2$ represents the right tail probability for standard normal distribution.

## 3 Bayesian Estimation

Any apriori information about parameters can be modelled using a prior distribution. In Bayesian paradigm, choosing such prior distribution is subjective which totally depends on past experience or personal beliefs of the experimenter. Several priors are suggested by different authors. Informative prior should be used in case of availability of any prior information about the concerned parameters (Berger, 1985). A situation where no or little information is available, a better choice is to use non-informative invariant prior as proposed by Jeffreys (1967). For scale parameter, we have used Jeffreys’ weak prior as Jeffreys’ prior is widely used due to its invariance property under one-to-one transformations of parameters. The shape parameter controls the shape of the distribution. Gamma (a,b) distribution is a flexible distribution which can assume variety of shapes. It has, therefore, been used as a prior, in the present paper to represent the shape parameter. Assuming the independence of the scale and shape parameters, the joint prior distribution of $α1,α2,β$ can be written as

 $Π⁢(α1,α2,β)=abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β);α1,α2,β,a,b>0$ (12)

where a, b are hyper parameters. Joint posterior distribution of $α1,α2,β$ is

 $Π⁢(α1,α2,β|x)$ $∝L⁢(x|α1,α2,β)⁢Π⁢(α1,α2,β)$ $∝∏i=1r[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]$ $×{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)$ $×abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)$ (13)

Marginal posterior distribution of $α1$

 $Π⁢(α1|x,α2,β)$ $∝∬∏i=1r[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]$ $×{11+(wrα1)β}(m-pr){11+(wrα2)β}(n-qr)$ $×abα1⁢α2⁢Γ⁢bβb-1exp(-aβ)dα2dβ$ (14)

Marginal posterior distribution of $α2$

 $Π⁢(α2|x,α1,β)$ $∝∬∏i=1r[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]$ $×{11+(wrα1)β}(m-pr){11+(wrα2)β}(n-qr)$ $×abα1⁢α2⁢Γ⁢bβb-1exp(-aβ)dα1dβ$ (15)

Marginal posterior distribution of $β$

 $Π⁢(β|x,α1,α2)$ $∝∬∏i=1r[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]$ $×{11+(wrα1)β}(m-pr){11+(wrα2)β}(n-qr)$ $×abα1⁢α2⁢Γ⁢bβb-1exp(-aβ)dα1dα2$ (16)

Next we derive expressions of Bayes estimates under symmetric and asymmetric loss functions. SELF is taken as symmetric loss function. GELF (Calabria and Pulcini, 1996), LINEX (Varian, 1975) and NLINEX (Islam et. al 2004) are taken as asymmetric loss functions.

Bayes estimate of unknown parameters under SELF It is a symmetric loss function. Underestimation and overestimation both are given equal weights under SELF. For an unknown parameter$λ$, SELF is defined as $L⁢(λ^,λ)=(λ^-λ)2$ where $λ^$ is the estimate of $λ$. Bayes estimate under SELF is

 $λ^SELF=Eλ⁢(λ)$ (17)

• For unknown scale parameters

 $α~1⁢B⁢S$ $∝∭α1⁢∏i=1r$ $×[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]$ $×{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)$ $⁢abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β$ (18) $α~2⁢B⁢S$ $∝∭α2⁢∏i=1r$ $×[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]$ $×{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)$ $×abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β$ (19)

• For unknown shape parameter

 $β~B⁢S$ $∝∭β⁢∏i=1r[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]$ $×{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)$ $×abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β$ (20)

Bayes estimate of unknown parameters under GELF GELF is defined as

 $L⁢(λ^,λ)∝(λ^λ)q-q⁢ln⁡(λ^λ)-1$

The Bayes estimator under GELF is

 $λ^GELF=[Eλ⁢(λ-q)]-1q$ (21)

provided $Eλ⁢(λ-q)$ exists and is finite. $q>0$ represents overestimation and $q<0$ represents underestimation.

For unknown scale parameters

 $(α~1⁢B⁢G)-q$ $∝∭α1-q⁢∏i=1r$ $×[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]$ $×{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)$ $×abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β$ (22) $(α~2⁢B⁢G)-q$ $∝∭α2-q⁢∏i=1r$ $×[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]$ $×{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)$ $×abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β$ (23)

• For unknown shape parameter

 $(β~B⁢G)-q$ $∝∭β-q⁢∏i=1r$ $×[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]$ $×{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)$ $×abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β$ (24)

Bayes estimate of unknown parameter under LINEX LINEX loss function is defined as

 $L⁢(λ^,λ)∝exp⁡(c⁢(λ^-λ))-c⁢(λ^-λ)-1$

The constant c determines the shape of the loss function. The Bayes estimator under the LINEX loss function is

 $λ^LINEX=-1c⁢ln⁡Eλ⁢[exp⁡(-c⁢λ)]$ (25)

provided $Eλ⁢[exp⁡(-c⁢λ)]$ exists and is finite. $c>0$ represents overestimation and $c<0$ represents underestimation.

For unknown scale parameters

 $α~1⁢B⁢L⁢L$ $∝1c⁢ln⁢∭e-c⁢α1⁢∏i=1r$ $×[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]$ $×{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)$ $⁢abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β$ (26) $α~2⁢B⁢L⁢L$ $∝1c⁢ln⁢∭e-c⁢α2⁢∏i=1r$ $×[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]$ $×{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)$ $×abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β$ (27)

• For unknown shape parameter

 $β~B⁢L⁢L$ $∝1c⁢ln⁢∭e-c⁢β⁢∏i=1r$ $×[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]$ $×{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)$ $×abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β$ (28)

Bayes estimate of unknown parameter under NLINEX NLINEX is defined as

 $L⁢(Δ)=k⁢[{exp⁡(c⁢Δ)-c⁢Δ-1}+c⁢Δ2];Δ=λ^-λ,k>0,c>0$ (29)

The Bayes estimates under NLINEX will be

 $λ^NLINEX=-1(c+2)⁢[ln⁡Eλ⁢{exp⁡(-c⁢λ)}-2⁢Eλ⁢(λ)]$ (30)

• For unknown scale parameters

 $α~1⁢B⁢N⁢L$ $∝1(c+2)$ $×[ln⁢∭e-c⁢α1⁢∏i=1r[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β+2⁢∭α1⁢∏i=1r[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β]$ (31)
 $α~2⁢B⁢N⁢L$ $∝1(c+2)$ $×[ln⁢∭e-c⁢α2⁢∏i=1r[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β+2⁢∭α2⁢∏i=1r[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β]$ (32)

• for unknown shape parameter

 $β~2⁢B⁢N⁢L$ $∝1(q+2)$ $⁢[ln⁢∭e-q⁢β⁢∏i=1r[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β+2⁢∭β⁢∏i=1r[{(βα1)⁢(wiα1)β-1{1+(wiαi)β}2}δi⁢{(β2α2)⁢(wiα2)β-1{1+(wiα2)β}2}1-δi]{11+(wrα1)β}(m-pr)⁢{11+(wrα2)β}(n-qr)abα1⁢α2⁢Γ⁢b⁢βb-1⁢exp⁡(-a⁢β)⁢d⁢α1⁢d⁢α2⁢d⁢β]$ (33)

## 4 Markov Chain Monte Carlo Approximation

When the expressions for posterior distributions and Bayes estimates of unknown parameters cannot be solved analytically, a usual procedure is to use MCMC techniques to approximate the complicated integrals. MCMC consists of two methods, namely the Gibbs sampler which is used for simulating from the full conditional posterior distributions and the Metropolis-Hastings algorithms (Metropolis et al., 1953; Hastings, 1970) generates samples from an arbitrary distribution.

The following iterative algorithm is proposed to simulate Bayes estimators:

Step 1: Start with an initial value $ω0=(α10,α20,β0)$ and set $i=0$.

Step 2: Generate a candidate point $α1*,α2*,β*$ from the respective proposal distributions $α1*∼N⁢(α^1,I-1⁢(ω^))$ , $α2*∼N⁢(α^2,I-1⁢(ω^))$, $β*∼N(β^$, $I-1(ω^))$ and a point u from $U⁢(0,1)$.

Then

 $α1(i+1)$ $={α1*⁢w⁢i⁢t⁢h⁢p⁢r⁢o⁢b⁢a⁢b⁢i⁢l⁢i⁢t⁢y⁢κ1⁢(α1*,α1(i)) f⁢o⁢r⁢κ1≤uα1(i)⁢w⁢i⁢t⁢h⁢p⁢r⁢o⁢b⁢a⁢b⁢i⁢l⁢i⁢t⁢y⁢ 1-κ1⁢(α1*,α1(i)) f⁢o⁢r⁢κ1>u$ $α2(i+1)$ $={α2*⁢w⁢i⁢t⁢h⁢p⁢r⁢o⁢b⁢a⁢b⁢i⁢l⁢i⁢t⁢y⁢κ2⁢(α2*,α2(i)) f⁢o⁢r⁢κ2≤uα2(i)⁢w⁢i⁢t⁢h⁢p⁢r⁢o⁢b⁢a⁢b⁢i⁢l⁢i⁢t⁢y⁢ 1-κ2⁢(α2*,α2(i)) f⁢o⁢r⁢κ2>u$ $β(i+1)$ $={β*⁢w⁢i⁢t⁢h⁢p⁢r⁢o⁢b⁢a⁢b⁢i⁢l⁢i⁢t⁢y⁢κ3⁢(β*,β(i)) f⁢o⁢r⁢κ3≤uβ(i)⁢w⁢i⁢t⁢h⁢p⁢r⁢o⁢b⁢a⁢b⁢i⁢l⁢i⁢t⁢y⁢ 1-κ3⁢(β*,β(i)) f⁢o⁢r⁢κ3>u$

Step 3: Set $i=i+1$

Step 4: Repeat steps 2–3. $N=10000$ times, in order to generate the sample observations $ω(1),ω(2),…⁢ω(N)=(α1(1),α2(1),β(1))$, $(α1(2),α2(2),β(2)),…$ $(α1(N),α2(N),β(N)).$

The MCMC algorithm has rapid convergence when the starting value is chosen such that it is in the close neighbourhood of the true value. This is achieved by picking the initial values based on some previous study, experience or some pre-defined criterion. Since initial values are chosen arbitrarily, therefore to revoke its effect, initial M simulated variates are discarded. An approximate posterior sample which is used for further Bayesian analysis is then taken as the remaining residual set corresponding to the position i such that $i=M+1,…⁢N$, for sufficiently large N. Approximate Bayes estimates of the unknown parameters under SELF are given by

 $α~1⁢BSMC$ $=1N-M⁢∑i=M+1Nα1⁢i$ $α~2⁢BSMC$ $=1N-M⁢∑i=M+1Nα2⁢i$ $β~BSMC$ $=1N-M⁢∑i=M+1Nβi$ (34)

Also, the approximate Bayes estimates of the unknown parameters under GELF are given by

 $α~1⁢BGMC=(1N-M⁢∑i=M+1Nα1⁢i-q)-1q$ $α~2⁢BGMC=(1N-M⁢∑i=M+1Nα2⁢i-q)-1q$ $β~BGMC=(1N-M⁢∑i=M+1Nβi-q)-1q$ (35)

The approximate Bayes estimates of the unknown parameters under LINEX are given by

 $α~1⁢BLMC=-1c⁢(1N-M⁢∑i=M+1Ne-c⁢α1⁢i)$ $α~2⁢BLMC=-1c⁢(1N-M⁢∑i=M+1Ne-c⁢α2⁢i)$ $β~BLMC=-1c⁢(1N-M⁢∑i=M+1Ne-c⁢βi)$ (36)

The approximate Bayes estimates of the unknown parameters under NLINEX are given by

 $α~1⁢B⁢N⁢M⁢C$ $=-1(c+2)$ $×[(1N-M⁢∑i=M+1Ne-c⁢α1⁢i)+2⁢(1N-M⁢∑i=M+1Nα1⁢i)]$ $α~2⁢B⁢N⁢M⁢C$ $=-1(c+2)$ $×[(1N-M⁢∑i=M+1Ne-c⁢α2⁢i)+2⁢(1N-M⁢∑i=M+1Nα2⁢i)]$ $β~B⁢N⁢M⁢C$ $=-1(c+2)$ $×[(1N-M⁢∑i=M+1Ne-c⁢βi)+2⁢(1N-M⁢∑i=M+1Nβi)]$ (37)

## 5 Simulation Study

In this section, estimation under JCS for simulated data from LLD is undertaken by taking initial values of unknown parameters as $α1=0.8,β=1.1,α2=0.5$. Assuming a weak prior with mean 0.5 for the shape parameter, we subsequently fix hyper parameters as $a=1,b=2$ OpenBUGS software is used for generating 10,000 posterior samples using MCMC iteration such that the first 2,000 samples are dropped from computation towards burn-in. We have taken 30 iteration of this procedure to get the mean square errors of the estimates along with 10,000 such replications. This results in 30x 10,000 sample frames on which computations for Table No. 15 are based. Approximate Bayes estimates under MCMC are then evaluated using (4)–(4).

Table 1 MLEs and Bayes estimates of $α1$ for different choices of $m,n$ and $r$

 Bayes Estimates MLE SELF GELF LINEX NLINEX (m,n) r $α^1$ $α~1⁢BSMC$ $α~1⁢BGMC1$ $α~1⁢BGMC2$ $α~1⁢BLMC1$ $α~1⁢BLMC2$ $α~1⁢BNMC$ (30,35) 55 Est. 0.5984 0.5847 0.5748 0.5885 0.5805 0.5894 0.5826 MSE 0.0223 0.0464 0.0508 0.0448 0.0482 0.0444 0.0473 60 Est. 0.6898 0.5410 0.5373 0.5425 0.5395 0.5427 0.5403 MSE 0.0525 0.0671 0.0690 0.0664 0.0679 0.0663 0.0675 65 Est. 0.8077 0.9651 0.9237 0.9791 0.9391 0.9937 0.9521 MSE 0.1227 0.0276 0.0156 0.0324 0.0196 0.0380 0.0234 (35,30) 55 Est. 0.6019 0.6039 0.5917 0.6085 0.5986 0.6097 0.6013 MSE 0.0209 0.0385 0.0434 0.0367 0.0406 0.0363 0.0395 60 Est. 0.6886 0.6120 0.5987 0.6170 0.6062 0.6184 0.6091 MSE 0.0498 0.0354 0.0406 0.0336 0.0376 0.0331 0.0365 65 Est. 0.815 0.6432 0.6238 0.6504 0.6343 0.6532 0.6387 MSE 0.1216 0.0248 0.0311 0.0226 0.0276 0.0218 0.0262 (35,35) 60 Est. 0.6073 0.5898 0.5786 0.5941 0.5850 0.5953 0.5874 MSE 0.0227 0.0442 0.0491 0.0424 0.0463 0.0420 0.0453 65 Est. 0.6963 0.6284 0.6107 0.6352 0.6204 0.6376 0.6244 MSE 0.0544 0.0295 0.0359 0.0273 0.0324 0.0265 0.0309 70 Est. 0.8075 0.6969 0.6678 0.7078 0.6826 0.7135 0.6897 MSE 0.1149 0.0108 0.0176 0.0088 0.0139 0.0078 0.0123

Table 2 MLEs and Bayes estimates of $α2$ for different choices of $m,n$ and $r$

 Bayes Estimates MLE SELF GELF LINEX NLINEX (m,n) r $α^2$ $α~2⁢BSMC$ $α~2⁢BGMC1$ $α~2⁢BGMC2$ $α~2⁢BLMC1$ $α~2⁢BLMC2$ $α~2⁢BNMC$ (30,35) 55 Est. 0.4151 0.4361 0.4332 0.4372 0.4352 0.4371 0.4357 MSE 0.3470 0.0041 0.0045 0.0040 0.0042 0.0040 0.0042 60 Est. 0.4561 0.4507 0.4452 0.4528 0.4489 0.4526 0.4498 MSE 0.3023 0.0025 0.0030 0.0023 0.0026 0.0023 0.0026 65 Est. 0.5104 0.4617 0.4545 0.4644 0.4592 0.4643 0.4604 MSE 0.2484 0.0015 0.0021 0.0013 0.0017 0.0013 0.0016 (35,30) 55 Est. 0.4180 0.5204 0.5032 0.5269 0.5139 0.5276 0.5172 MSE 0.3446 0.0005 0.0001 0.0009 0.0003 0.0009 0.0004 60 Est. 0.4598 0.4451 0.4407 0.4468 0.4437 0.4467 0.4444 MSE 0.2993 0.0030 0.0035 0.0028 0.0032 0.0029 0.0031 65 Est. 0.5122 0.5845 0.5563 0.5950 0.5728 0.5978 0.5787 MSE 0.2481 0.0073 0.0033 0.0092 0.0054 0.0097 0.0063 (35,35) 60 Est. 0.4202 0.4576 0.4513 0.4600 0.4555 0.4599 0.4565 MSE 0.3415 0.0019 0.0024 0.0017 0.0020 0.0017 0.0019 65 Est. 0.4554 0.6113 0.5834 0.6213 0.5996 0.6243 0.6054 MSE 0.3030 0.0126 0.0071 0.0149 0.0101 0.0157 0.0113 70 Est. 0.5032 0.4879 0.4754 0.4927 0.4834 0.4929 0.4856 MSE 0.2557 0.0002 0.0006 0.0001 0.0003 0.0001 0.0003

Table 3 MLEs and Bayes estimates of $β$ for different choices of $m,n$ and $r$

 Bayes Estimates MLE SELF GELF LINEX NLINEX (m,n) r $β^$ $β~BSMC$ $β~BGMC1$ $β~BGMC2$ $β~BLMC1$ $β~BLMC2$ $β~BNMC$ (30,35) 55 Est. 2.0146 2.1833 2.1441 2.1962 2.1283 2.2412 2.1558 MSE 0.3449 1.1741 1.0909 1.2023 1.0581 1.3034 1.1153 60 Est. 1.8344 1.7671 1.7349 1.7778 1.7303 1.8063 1.7487 MSE 0.1756 0.4457 0.4039 0.4601 0.3979 0.4995 0.4215 65 Est. 1.5397 1.7634 1.7363 1.7724 1.7324 1.7957 1.7479 MSE 0.0384 0.4405 0.4053 0.4525 0.4003 0.4844 0.4202 (35,30) 55 Est. 2.0254 1.8381 1.8041 1.8494 1.7978 1.8808 1.8180 MSE 0.3690 0.5453 0.4963 0.5621 0.4874 0.6102 0.5160 60 Est. 1.8365 1.8831 1.8508 1.8939 1.8437 1.9253 1.8634 MSE 0.1825 0.6139 0.5643 0.6309 0.5537 0.6818 0.5834 65 Est. 1.5427 1.5479 1.5219 1.5565 1.5217 1.5751 1.5348 MSE 0.0385 0.2009 0.1784 0.2087 0.1782 0.2261 0.1894 (35,35) 60 Est. 1.9796 1.8622 1.8302 1.8726 1.8236 1.9017 1.8429 MSE 0.3039 0.5812 0.5335 0.5973 0.5240 0.6431 0.5522 65 Est. 1.8159 1.6104 1.5846 1.6189 1.5833 1.6383 1.5969 MSE 0.1558 0.2609 0.2351 0.2696 0.2339 0.2902 0.2472 70 Est. 1.5349 1.4918 1.4693 1.4994 1.4698 1.5150 1.4808 MSE 0.0390 0.1538 0.1366 0.1598 0.1370 0.1726 0.1453

Table 4 LL, UL and AL of ACI and BCI of all parameters

 (m,n) r ACI for $α1$ ACI for $α2$ ACI for $β$ BCI for $α1$ BCI for $α2$ BCI for $β$ (30,35) 55 LL 0.3941 0.2870 1.5650 0.5031 0.4010 1.7290 UL 0.8026 0.5433 2.4642 0.7518 0.5194 2.6750 AL 0.4085 0.2564 0.8992 0.2487 0.1184 0.9460 60 LL 0.4430 0.3067 1.4428 0.5012 0.4016 1.3960 UL 0.9365 0.6055 2.226 0.6492 0.5625 2.1650 AL 0.4935 0.2988 0.7833 0.1480 0.1609 0.7690 65 LL 0.4817 0.3197 1.2212 0.6765 0.4020 1.4260 UL 1.1338 0.7012 1.8582 1.3280 0.5917 2.1230 AL 0.6522 0.3815 0.6371 0.6515 0.1897 0.6970 (35,30) 55 LL 0.4122 0.2797 1.5734 0.5048 0.4063 1.4490 UL 0.7915 0.5564 2.4774 0.7858 0.7154 2.2670 AL 0.3793 0.2767 0.9040 0.2810 0.3091 0.8180 60 LL 0.4615 0.2972 1.4441 0.5053 0.4014 1.4990 UL 0.9157 0.6223 2.2289 0.8007 0.5490 2.3030 AL 0.4542 0.3252 0.7847 0.2954 0.1476 0.8040 65 LL 0.5108 0.3056 1.2236 0.5081 0.4163 1.2390 UL 1.1191 0.7188 1.8619 0.8705 0.8443 1.8820 AL 0.6084 0.4132 0.6383 0.3624 0.4280 0.6430 (35,35) 60 LL 0.4134 0.2895 1.5567 0.5034 0.4021 1.4720 UL 0.8012 0.5509 2.4024 0.7662 0.5751 2.2620 AL 0.3879 0.2613 0.8457 0.2628 0.1730 0.7900 65 LL 0.4651 0.3055 1.4435 0.5053 0.4289 1.2950 UL 0.9276 0.6052 2.1884 0.8561 0.8698 1.9530 AL 0.4625 0.2997 0.7449 0.3508 0.4409 0.6580 70 LL 0.5053 0.3145 1.2288 0.5154 0.4034 1.2150 UL 1.1097 0.6920 1.8409 0.9803 0.6576 1.8090 AL 0.6044 0.3774 0.6121 0.4649 0.2542 0.5940

Table 5 LL, UL and AL of 89% HPD and 95% HPD interval of all parameters

 89% HPD 89% HPD 89% HPD 95% HPD 95% HPD 95% HPD (m,n) r for $α1$ for $α2$ for $β$ for $α1$ for $α2$ for $β$ (30,35) 55 LL 0.5000 0.4000 1.7890 0.5000 0.4000 1.7110 UL 0.6735 0.4773 2.5540 0.7159 0.5007 2.6510 AL 0.1735 0.0773 0.7650 0.2159 0.1007 0.9400 60 LL 0.5000 0.4000 1.4560 0.5000 0.4000 1.3990 UL 0.5910 0.5074 2.0800 0.6237 0.5385 2.1670 AL 0.0910 0.1074 0.6240 0.1237 0.1385 0.7680 65 LL 0.7087 0.4001 1.4790 0.6506 0.4000 1.4230 UL 1.2300 0.5264 2.0410 1.2920 0.5623 2.1180 AL 0.5213 0.1263 0.5620 0.6414 0.1623 0.6950 (35,30) 55 LL 0.5000 0.4001 1.4990 0.5000 0.4001 1.4250 UL 0.7014 0.6272 2.1580 0.7508 0.6757 2.2350 AL 0.2014 0.2271 0.6590 0.2508 0.2756 0.8100 60 LL 0.5001 0.4001 1.5410 0.5000 0.4000 1.4830 UL 0.7130 0.4954 2.1920 0.7601 0.5245 2.2820 AL 0.2129 0.0953 0.6510 0.2601 0.1245 0.7990 65 LL 0.5000 0.4031 1.2860 0.5000 0.4002 1.2250 UL 0.7693 0.7264 1.8150 0.8231 0.7917 1.8650 AL 0.2693 0.3233 0.5290 0.3231 0.3915 0.6400 (35,35) 60 LL 0.5000 0.4000 1.5330 0.5000 0.4000 1.4710 UL 0.6818 0.5194 2.1750 0.7275 0.5488 2.2600 AL 0.1818 0.1194 0.6420 0.2275 0.1488 0.7890 65 LL 0.5000 0.4311 1.3260 0.5000 0.4050 1.2770 UL 0.7533 0.7731 1.8590 0.8105 0.8170 1.9320 AL 0.2533 0.3420 0.5330 0.3105 0.4120 0.6550 70 LL 0.5003 0.4000 1.2520 0.5003 0.4000 1.1980 UL 0.8553 0.5764 1.7330 0.9258 0.6224 1.7870 AL 0.3550 0.1764 0.4810 0.4255 0.2224 0.5890

Estimated values and mean squared errors (MSEs) of MLE and Bayes estimates under different loss functions for unknown scale parameter $α1$ are tabulated in Table 1. Similarly, estimated values and MSEs of MLEs and Bayes estimates for unknown parameters $α2,β$ are given in Tables 23 respectively. Table 4 represents lower limit (LL), upper limit (UL) and average length (AL) of ACI and BCI of the unknown parameters. Similarly, Table 5 gives LL, UL and AL 89% HPD and 95% HPD confidence intervals of parameters. The following results are observed:

I. For the unknown scale parameter $α1$, MLEs and Bayes estimates under different loss functions are not comparable as they do not show any unidirectional trend. For some combinations, MLEs have lower MSEs while for other combinations, Bayes estimates show lower MSEs.

II. For the unknown scale parameter $α2$, Bayes procedure gives better estimates than MLEs as they have lower MSEs. Among Bayes estimates, GELF and LINEX give estimates that are closer to true values under the overestimation case.

III. For the common unknown shape parameter $β$ , MLEs give estimates with higher precision than Bayes estimates. Among different Bayes estimates, LINEX under underestimation gives more precise estimates though GELF under underestimation compete quite well with them.

IV. For all the three unknown parameters, 89% HPD intervals have shortest length with the following order observed in their lengths:

 $ACIAL>BCIAL>95%⁢HPDAL>89%⁢HPDAL$

## 6 Real Data Study

In this section, a real data has been taken to illustrate the application of proposed method. Data has been taken from Lawless (2003) (pg-445). The data in Table 6 give the survival times for two groups of laboratory mice. A conventional lab environment and a germ-free environment were set up to keep both separately. Mice of both groups were exposed to radiation of fixed dose (Hoel, 1972). The cause of death for each mouse was thymic lymphoma (C1) as was confirmed after their autopsy.

Table 6 Survival times and causes of death for laboratory mice

 Control group 159, 189, 191, 198, 200, 207, 220, 235, 245, 250, 256, 261, 265, 266, 280, 343, 350, 383, 403, 414, 428, 432 Germ-Free group 158, 192, 193, 194, 195, 202, 212, 215, 229, 230, 237, 240, 244, 247, 259, 300, 301, 321, 337, 415, 434, 444, 485, 496, 529, 537, 624, 707, 800

We first check whether LL distribution fits for the given data set. Results are given in Table 7. Comparative goodness of fit for the selected data set based on negative log likelihood and four information criteria is presented as Log-logistic$>$gamma$>$Weibull$>$exponential. It indicates that LLD gives the best representation in terms of fit to the given data set. Boxplot and TTT plots are also shown in Figure 1 which clearly indicates that the data is right skewed and hence is suitable for LLD.

Table 7 Fitting of data to different distributions

 Sr no. Reliability Model $-$LogL AIC BIC AICC HQC 1. Exponential 344.6566 691.3133 693.2451 691.3949 692.0515 2. Gamma 316.6391 637.2783 641.1419 637.5283 638.7547 3. Weibull 321.5562 647.1125 650.9761 647.3625 648.5889 4. Log logistic 315.0155 634.0310 637.8946 634.2810 635.5074

Figure 1 Boxplot and TTT plot of dataset.

Table 8 JCS real data

 r Joint Type II Censored Data 25 158, 159, 189, 191, 192, 193, 194, 195, 198, 200, 202, 207, 212, 215, 220, 229, 230, 235, 237, 240, 244, 245, 247, 250, 256 35 158, 159, 189, 191, 192, 193, 194, 195, 198, 200, 202, 207, 212, 215, 220, 229, 230, 235, 237, 240, 244, 245, 247, 250, 256, 259, 261, 265, 266, 280, 300, 301, 321, 337, 343

Here, $m=22,n=29$. We take $r=25,35$. JCS sample extracted from Table 6 is given in Table 8. MLE and Bayes estimates of parameters are tabulated in Table 9. AL of ACI, Boot-t, Boot-p, BCI, 89% HPD and 95% HPD confidence intervals of all unknown parameters are presented in Table 10. Among classical intervals, ACI gives best interval in terms of shorter length as compared to BOOT-t and BOOT-p for both unknown scale parameters. However, for the unknown shape parameter, BOOT-t gives shortest length interval. Among Bayesian intervals 89%HPD intervals have shortest length than 95%HPD and BCI for all the three unknown parameters. AL of classical (Bayesian) intervals is seen to increase (decrease) with increasing value of r for both the scale parameters while for the unknown shape parameter, AL decreases consistently with increase in r. Figure 2 shows MCMC trace plot of parameters for both values of r.

Table 9 MLE and Bayes estimates of parameters

 Bayes Estimates r MLE SELF GELF LINEX NLINEX 25 $α1$ 212.3655 210.8308 209.4131 211.2988 139.3936 282.9075 175.1122 $α2$ 213.4356 210.5700 209.5355 210.9213 168.7928 283.0064 189.6814 $β$ 13.2850 7.6307 7.2492 7.7577 6.0704 9.7957 6.8505 35 $α1$ 231.8739 231.1812 229.7675 231.6547 187.1820 304.7064 209.1816 $α2$ 230.6516 229.0967 227.8632 229.5105 185.8919 281.6296 207.4943 $β$ 9.1368 6.4827 6.2682 6.5526 5.6675 7.5724 6.0751

Table 10 AL of different confidence intervals

 r ACI BOOT-t BOOT-p BCI HPD89 HPD95 $α1$ 25 34.390 42.500 35.200 250.393 219.200 236.475 35 43.451 50.100 47.000 110.900 77.400 99.500 $α2$ 25 29.091 33.600 29.600 233.927 197.700 228.791 35 39.962 44.300 41.300 102.500 68.400 97.000 $β$ 25 8.493 6.940 8.340 10.360 9.052 9.780 35 5.004 4.329 5.067 4.965 3.835 4.779

Figure 2 MCMC trace plots of parameters for $r=25,35$.

## 7 Conclusion

In this paper, classical and Bayesian estimation of parameters under JCS for two contemporary samples is considered when lifetimes follow two distinct log-logistic models with a common shape parameter but different scale parameters. Point and interval Bayes estimates are obtained under a symmetric and four asymmetric loss functions and compared for efficiency relative to the respective classical estimates. As the derived estimators are not in closed form, MCMC iterative technique is used to compute approximate estimates. A real dataset has also been discussed for illustration of the methodology developed in the paper. Simulation study shows that the Bayes estimates perform better than the MLEs in terms of minimum MSE and confidence length. However, MLEs compete closely with the Bayes estimates. Among the interval estimators, HPD intervals are found to be more precise than others in terms of shortest average length.

## Acknowledgments

Authors are grateful to the Editor and referees for careful reviewing which greatly strengthened the paper. First author gratefully acknowledges IoE grant from University of Delhi.

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## Biographies

Ranjita Pandey is an esteemed faculty at Department of Statistics, University of Delhi, Delhi. She has more than 20 years of experience of teaching and research. Her research areas include theoretical and applied Bayesian Inference, lifetime distributions, time series models, demography, imputation methods and ecological modelling. She has extensive administrative experience. She has delivered many invited lectures and has served as reviewer for several journals.

Pulkit Srivastava is currently pursuing his Ph.D. at the Department of Statistics, University of Delhi, Delhi. His research areas include Bayesian Inference, Stochastic processes etc.