On Parameters Estimation of Lomax Distribution under General Progressive Censoring

Abstract

We consider the estimation problem of the probability 𝑆=𝑃(𝑌<𝑋) for Lomax distribution based on general progressive censored data. The maximum likelihood estimator and Bayes estimators are obtained using the symmetric and asymmetric balanced loss functions. The Markov chain Monte Carlo (MCMC) methods are used to accomplish some complex calculations. Comparisons are made between Bayesian and maximum likelihood estimators via Monte Carlo simulation study.

1. Introduction

The Lomax distribution, also called “Pareto type II” distribution is a particular case of the generalized Pareto distribution (GPD). The Lomax distribution has been used in the literature in a number of ways. For example, it has been extensively used for reliability modelling and life testing; see, for example, Balkema and de Haan [1]. It also has been used as an alternative to the exponential distribution when the data are heavy tailed; see Bryson [2]. Ahsanullah [3] studied the record values of Lomax distribution. Balakrishnan and Ahsanullah [4] introduced some recurrence relations between the moments of record values from Lomax distribution. The order statistics from nonidentical right-truncated Lomax random variables have been studied by Childs et al. [5]. Also, the Lomax model has been studied, from a Bayesian point of view, by many authors; see, for example, Arnold et al. [6] and El-Din et al. [7]. Howlader and Hossain [8] presented Bayesian estimation of the survival function of the Lomax distribution. Ghitany et al. [9] considered Marshall-Olkin approach and extended Lomax distribution. Cramer and Schmiedt [10] considered progressively type-II censored competing risks data from Lomax distribution. The Lomax distribution has applications in economics, actuarial modelling, queuing problems and biological sciences; for details, we refer to Johnson et al. [11].

A positive random variable 𝑋 is said to have the Lomax distribution, abbreviated as 𝑋∼𝐿(𝛼,𝛽), if it has the probability density function (pdf) 𝑓𝑥;𝛼,𝛽=𝛼𝛽1+𝛽𝑥^−(𝛼+1),𝑥>0,𝛼,𝛽>0. (1) Here, 𝛼 and 𝛽 are the shape and the scale parameters, respectively. The survival function (sf) associated with (1) is 𝐹𝑥;𝛼,𝛽=1+𝛽𝑥^−𝛼,𝑥>0. (2) Further probabilistic properties of this distribution are given, for example, in Arnold [12].

This paper is concerned with the problem of estimating 𝑆=𝑃(𝑌<𝑋) for Lomax based on general progressive censored data. The reliability of a component during a given period of time is defined as the probability that its strength 𝑋 exceeds the stress 𝑌, and symbolically we write 𝑆=𝑃(𝑌<𝑋). We assume 𝑋 and 𝑌 to be independent, and each follows a Lomax distribution. A good overview on estimating 𝑆 can be found in the monograph of Kotz et al. [13]. Later, the problem of estimating 𝑆 attracted the attention of many authors; for example, see Baklizi [14], Raqab et al. [15], Kundu and Raqab [16], and Panahi and Asadi [17], and references cited therein.

The rest of the paper is organized as follows. In Section 2, we give a brief overview of the general progressive censoring. The maximum likelihood estimators (MLEs) are obtained in Section 3. In Section 4, we obtain Bayes estimators using the symmetric and asymmetric balanced loss functions. In Section 5, the MCMC methods are used to accomplish some complex calculations, and, therefore, comparisons are made between Bayesian and maximum likelihood estimators via Monte Carlo simulation study.

2. General Progressive Censoring

We refer to the paper of Soliman et al. [18, page 452], for introducing the general progressive censoring as follows. Consider a general type-II progressive censoring scheme, proposed by Balakrishnan and Sandhu [19]. This scheme of censoring can be explained as follows: at time 𝑋₀≡0, 𝑛 randomly selected components were placed on a life test. The failure times of the first 𝑟 components to fail, 𝑋₁,…,𝑋_𝑟, were not observed. At the time of the (𝑟+1)th failure, 𝑋_{𝑟+1∶𝑛}, 𝑅_𝑟+1 number of surviving components are removed from the test randomly and so on; at the time of the (𝑟+𝑖)th observed failure, 𝑋_{𝑟+𝑖∶𝑛}, 𝑅_𝑟+𝑖 number of surviving components are removed from the test randomly; and finally, at the time of the 𝑚th failure, the remaining 𝑅_𝑚=𝑛−𝑚−𝑅_𝑟+1−𝑅_𝑟+2−⋯−𝑅_𝑚−1 are removed from the test. Suppose that 𝑋_{𝑟+1∶𝑚∶𝑛}≤𝑋_{𝑟+2∶𝑚∶𝑛}≤⋯≤𝑋_{𝑚∶𝑚∶𝑛} are the lifetimes of the completely observed components to fail and that 𝑅_𝑟+1,𝑅_𝑟+2,…,𝑅_𝑚 are the number of components removed from the test at these failure times, respectively. The 𝑅_𝑖's, 𝑚, and 𝑟 are prespecified integers such that 0≤𝑟<𝑚≤𝑛,  0≤𝑅_𝑖≤𝑛−𝑖 for 𝑖=𝑟+1,…,𝑚−1, and 𝑅_𝑚=𝑛−𝑚−∑^𝑚−1_𝑖=𝑟+1𝑅_𝑖. The resulting (𝑚−𝑟) ordered values 𝑋_{𝑟+1∶𝑚∶𝑛},𝑋_{𝑟+2∶𝑚∶𝑛},…,𝑋_{𝑚∶𝑚∶𝑛} are appropriately referred to as general progressively type-II censored order statistics.

Also referring to Soliman et al. [18], it should be noted that (i) if 𝑅_𝑖=0, for 𝑖=𝑟+1,…,𝑚−1, and 𝑅_𝑚=𝑛−𝑚, the general progressively type-II censoring scheme is reduced to the case of type-II doubly censored sample. (ii) If 𝑟=0, this scheme is reduced to the progressive type-II right censoring. (iii) If 𝑟=0 and 𝑅_𝑖=0, for 𝑖=𝑟+1,…,𝑚−1 so that 𝑅_𝑚=𝑛−𝑚, the general progressively type-II censoring scheme is reduced to conventional type-II one-stage right censoring, where just the first 𝑚 usual order statistics are observed. (iv) If 𝑟=0 and 𝑅_𝑖=0, for 𝑖=𝑟+1,…,𝑚, so that 𝑚=𝑛, the general progressively type-II censoring scheme is reduced to the case of no censoring (complete sample case), where all 𝑛 usual order statistics are observed. In this scheme 𝑅₁,𝑅₂,…,𝑅_𝑚 are all prefixed. Saraoglu et al. [20] discussed two examples showing the motivation behind the developments of the stress-strength models under censored samples. For more details, see Balakrishnan and Aggarwala [21].

Suppose that 𝑛 randomly selected components from the L(𝛼,𝛽) are put in test. Further, let 𝑋_{𝑟+1∶𝑚∶𝑛},𝑋_{𝑟+2∶𝑚∶𝑛},…,𝑋_{𝑚∶𝑚∶𝑛} denote a general progressively type-II censored sample from that population, with (𝑅_𝑟+1,𝑅_𝑟+2,…,𝑅_𝑚) being the progressive censoring scheme. For simplicity of notation, we will use 𝑥_𝑖 instead of 𝑥_{𝑖∶𝑚∶𝑛}, and then 𝐱=(𝑥_𝑟+1,…,𝑥_𝑚) is the observed general progressive censored sample. The likelihood function for the parameters 𝛼 and 𝛽 is then ℒ𝛼,𝛽∣𝐱=𝑐^∗1−𝐹𝑥_𝑟+1^𝑟𝑚_𝑖=𝑟+1𝑓𝑥_𝑖𝐹𝑥_𝑖^𝑅𝑖, (3) where 𝑐^∗=𝑛𝑟(𝑛−𝑟)^𝑚_𝑗=𝑟+2𝑛−^𝑗−1_𝑖=𝑟+1𝑅_𝑖−𝑗+1, (4) and the functions 𝑓(𝑥) and 𝐹(𝑥) are given, respectively, by (1) and (2). Substituting (1) and (2) into (3), the likelihood function is ℒ𝛼,𝛽∣𝐱=𝑐^∗1−1+𝛽𝑥_𝑟+1^−𝛼𝑟𝛼𝛽^𝑚−𝑟×^𝑚_𝑖=𝑟+11+𝛽𝑥_𝑖^{−𝛼(1+𝑅𝑖)−1}. (5) Using the binomial expansion, 𝑟 is a positive integer, one can rewrite the likelihood function as follows: ℒ𝛼,𝛽∣𝐱=𝑐^∗(𝛼𝛽)^𝑚−𝑟𝑒^𝑟−𝑢_𝑠=0𝐶_𝑟,𝑠𝑒^{−𝛼𝑇𝑠}, (6) where 𝑢=𝑢𝐱;𝛽=^𝑚_𝑖=𝑟+1ln1+𝛽𝑥_𝑖,𝑇_𝑠=𝑇_𝑠𝐱;𝛽=𝑠ln1+𝛽𝑥_𝑟+1−^𝑚_𝑖=𝑟+11+𝑅_𝑖ln1+𝛽𝑥_𝑖,𝐶_𝑟,𝑠=𝑟𝑠(−1)^𝑠. (7) We focus our attention on the estimation of the probability 𝑆=𝑃(𝑌<𝑋), where 𝑋 and 𝑌 are two independent random variables each is 𝐿(𝛼_𝑗,𝛽_𝑗), 𝑗=1,2 distributed, and the data obtained from both distributions are general progressively type-II censored. Here, 𝑋 and 𝑌 are typically modeled as independent. The probability 𝑃(𝑌<𝑋) has been widely studied under different approaches and distributional assumptions on 𝑋 and 𝑌. The case where 𝑋 and 𝑌 are dependent has been considered by Nandi and Aich [22], Barbiero [23], and Rubio and Steel [24]. We investigate properties of 𝑆 when the common scale parameter 𝛽₁=𝛽₂=𝛽 is known. Then, it can be shown that 𝑆=𝑃(𝑌<𝑋)=_0<𝑦<𝑥𝑓𝛼𝑥,𝑦𝑑𝑥𝑑𝑦=¹𝛼₁+𝛼₂. (8) Here, 𝑓(𝑥,𝑦) is the joint pdf of 𝑋 and 𝑌, 𝑓(𝑥,𝑦)=𝑓₁(𝑥)𝑓₂(𝑦) by the independence. The general case, when 𝛽₁≠𝛽₂, can be studied in a similar manner. We obtain that 𝑆=𝑃(𝑌<𝑋)=𝛼₁𝛽₁^∞₀(1+𝛽₁𝑥)^−(𝛼1+1)(1+𝛽₂𝑥)^−𝛼2𝑑𝑥. (9)

3. Maximum Likelihood Estimation of 𝑆

Suppose that 𝑋₁,𝑋₂,…,𝑋_𝑛1 and 𝑌₁,𝑌₂,…,𝑌_𝑛2 are two independent random samples of size 𝑛₁ and 𝑛₂ from 𝐿(𝛼₁,𝛽₁) and 𝐿(𝛼₂,𝛽₂) distributions, respectively. The log-likelihood function, with ignoring constants, is given by lnℒ(Θ∣𝐱,𝐲)=²_𝑗=1𝑚_𝑗−𝑟_𝑗ln𝛼_𝑗+ln𝛽_𝑗+𝑟₁ln1−(1+𝛽₁𝑥_𝑟1+1)^−𝛼1−^𝑚1_𝑖=𝑟1+1𝛼₁1+𝑅_1𝑖+1ln1+𝛽₁𝑥_𝑖+𝑟₂ln1−(1+𝛽₂𝑦_𝑟2+1)^−𝛼2−^𝑚2_𝑖=𝑟2+1𝛼₂1+𝑅_2𝑖+1ln1+𝛽₂𝑦_𝑖, (10) where Θ=(𝛼₁,𝛼₂,𝛽₁,𝛽₂). The MLEs of (𝛼₁,𝛽₁), say (𝛼_1ML,𝛽_1ML), are obtained as the solution of the system of equations 𝜕lnℒ𝜕𝛼₁=𝑚₁−𝑟₁𝛼₁+𝑟₁ln1+𝛽₁𝑥_𝑟1+11+𝛽₁𝑥_𝑟1+1^𝛼1−−1_{𝑚1𝑖=𝑟1+1}1+𝑅_1𝑖ln1+𝛽₁𝑥_𝑖=0,𝜕lnℒ𝜕𝛽₁=𝑚₁−𝑟₁𝛽₁+𝛼₁𝑟₁𝑥_𝑟1+1(1+𝛽₁𝑥_𝑟1+1)⁻¹(1+𝛽₁𝑥_𝑟1+1)^𝛼1−−1_{𝑚1𝑖=𝑟1+1}𝛼₁1+𝑅_1𝑖𝑥+1^𝑖1+𝛽₁𝑥_𝑖=0. (11) In a similar way, we can obtain the MLEs of (𝛼₂,𝛽₂): say (𝛼_2ML,𝛽_2ML).

The corresponding “ML plug-in estimation” of 𝑆, when 𝛽₁=𝛽₂=𝛽, is obtained by replacing 𝛼₁ and 𝛼₂ by its MLEs, 𝛼_1ML, 𝛼_2ML, and substituting them into relation (8) which yields 𝑆_ML=𝛼_1ML𝛼_1ML+𝛼_2ML. (12) For the general case, 𝛽₁≠𝛽₂, the corresponding “ML plug-in estimation” of 𝑆 is obtained by replacing 𝛼₁, 𝛽₁, 𝛼₂, and 𝛽₂ by its MLEs, 𝛼_1ML, 𝛽_1ML, 𝛼_2ML, and 𝛽_2ML and substituting them into relation (9) which results in obtaining 𝑆_ML as 𝑆_ML=𝛼_1ML𝛽_1ML×^∞₀(1+𝛽_1ML𝑥)⁻⁽𝛼_1ML+1)(1+𝛽_2ML𝑥)⁻𝛼_2ML𝑑𝑥. (13)

4. Bayes Estimation

In this section, Bayesian estimation for the probability 𝑆 in the stress-strength model involving Lomax distribution is obtained. The estimation is based on balanced loss function (BLF) which is introduced by Zellner [25]. We will use an extended class of BLF introduced by Jozani et al. [26]. It is of the following form: 𝐿^𝑞_{𝜌,𝜔,𝛿0}(𝜁(𝜃),𝛿)=𝜔𝑞(𝜃)𝜌𝛿₀,𝛿+(1−𝜔)𝑞(𝜃)𝜌(𝜁(𝜃),𝛿), (14) where 𝑞(⋅) is a suitable positive weight function and 𝜌(𝜁(𝜃),𝛿) is an arbitrary loss function when estimating 𝜁(𝜃) by 𝛿. The parameter 𝛿₀ is a chosen priori estimate of 𝜁(𝜃), obtained for instance from the criterion of maximum likelihood, least squares, or unbiasedness among others. An intuitive interpretation of the BLFs is given by Ahmadi et al. [27] who argue that they give a general Bayesian connection between the case of 𝜔>0, and 𝜔=0 where 0≤𝜔<1. By choosing 𝜌(𝜁(𝜃),𝛿)=(𝛿−𝜁(𝜃))² and 𝑞(𝜃)=1, the BLF is reduced to the balanced squared error loss (BSEL) function, used by Ahmadi et al. [27], in the form 𝐿_𝜔,𝛿0(𝜁(𝜃),𝛿)=𝜔(𝛿−𝛿₀)²+(1−𝜔)𝜌(𝛿−𝜁(𝜃))². (15) The corresponding Bayes estimate of the function 𝜁(𝜃) is given by 𝛿_{𝜔,𝜁,𝛿0}(𝐱)=𝜔𝛿₀+(1−𝜔)𝐸(𝜁(𝜃)∣𝐱). (16) Also, by choosing 𝜌(𝜁(𝜃),𝛿)=𝑒^{𝑎(𝛿−𝜁(𝜃))}−𝑎(𝛿−𝜁(𝜃))−1 and 𝑞(𝜃)=1, we get the balanced LINEX, abbreviated as BLINEX, loss function written as 𝐿_{𝑎,𝜔,𝛿0}(𝜁(𝜃),𝛿)=𝜔𝑒^{𝑎(𝛿−𝛿0)}−𝑎𝛿−𝛿₀+(𝑒−11−𝜔)^{𝑎(𝛿−𝜁(𝜃))}−𝑎(𝛿−𝜁(𝜃))−1. (17) In this case, the Bayes estimate of 𝜁(𝜃) takes the form 𝛿_{𝑎,𝜔,𝜁,𝛿0}(𝐱)1=−𝑎ln𝜔𝑒^−𝑎𝛿0+(1−𝜔)𝐸𝑒^{−𝑎𝜁(𝜃)}∣𝐱, (18) where 𝑎≠0 is the shape parameter of BLINEX loss function.

4.1. Bayes Estimation When 𝛽₁=𝛽₂=𝛽

Assuming that 𝛽₁=𝛽₂=𝛽 (𝛽 is known) and 𝛼₁, 𝛼₂ are random variable each having gamma prior with some parameters, we can write 𝜋𝛼_𝑗=𝛿^𝜈𝑗_𝑗Γ𝜈_𝑗𝛼^𝜈𝑗_𝑗⁻¹𝑒^{−𝛿𝑗𝛼𝑗},𝛼_𝑗>0,𝜈_𝑗,𝛿_𝑗>0,𝑗=1,2. (19) Since 𝛼_𝑗, 𝑗=1,2 are independent, by combining the likelihood function with the priors pdf, the joint posterior density function of 𝛼₁ and 𝛼₂ is given by 𝜋^∗𝛼₁,𝛼₂∣𝐱,𝐲=𝐽²_𝑗=1𝛼^𝐴∗𝑗_𝑗⁻¹𝑒^{−𝛿𝑗𝛼𝑗−𝑢∗𝑗𝑟𝑗}_𝑠=0𝐶_{𝑟𝑗,𝑠}𝑒^{−𝛼𝑗𝑇∗𝑗𝑠}, (20) where 𝐴^∗_𝑗=𝑚_𝑗−𝑟_𝑗+𝜈_𝑗, 𝑇^∗_1𝑠=𝑇_𝑠(𝐱;𝛽), 𝑇^∗_2𝑠=𝑇_𝑠(𝐲;𝛽), 𝑢^∗₁=𝑢(𝐱;𝛽), 𝑢^∗₂=𝑢(𝐲;𝛽), and 𝐽⁻¹=²_𝑗=1^∞₀𝛼^𝐴∗𝑗_𝑗⁻¹𝑒^{−𝛿𝑗𝛼𝑗−𝑢∗𝑗𝑟𝑗}_𝑠=0𝐶_{𝑟𝑗,𝑠}𝑒^{−𝛼𝑗𝑇∗𝑗𝑠}𝑑𝛼_𝑗. (21) Under the BSEL function and using (12) and (20), the proposed “Bayesian estimators” of 𝑆 are actually Bayesian “plug-in” estimators as they are obtained by replacing (𝛼₁,𝛽₁,𝛼₂,𝛽₂) with its Bayesian estimator (𝛼₁,𝛽₁,𝛼₂,𝛽₂)𝑆_BS=𝜔𝑆_ML+(1−𝜔)^∞₀^∞₀𝑆𝜋^∗𝛼₁,𝛼₂∣𝐱,𝐲𝑑𝛼₁𝑑𝛼₂. (22) With the same argument, we can obtain Bayes estimator under the BLINEX loss function. It is obtained as follows: 𝑆_BL1=−𝑎ln𝜔𝑒^−𝑎𝑆_ML+()×1−𝜔_∞0∞0𝑒^−𝑎𝑆𝜋^∗𝛼₁,𝛼₂∣𝐱,𝐲𝑑𝛼₁𝑑𝛼₂, (23) where 𝑆_ML is the ML “plug-in” estimate of 𝑆 as given by (12).

4.2. Bayes Estimation when 𝛽₁≠𝛽₂

We assume that 𝛼_𝑗 and 𝛽_𝑗, 𝑗=1,2, are random variables each having gamma prior with some parameters; that is, 𝜋𝛼_𝑗∣𝛽_𝑗=𝑏^𝑎𝑗_𝑗Γ𝑎_𝑗𝛼^𝑎𝑗_𝑗⁻¹𝑒^{−𝑏𝑗𝛼𝑗},𝛼_𝑗>0,𝑎_𝑗,𝑏_𝑗𝜋𝛽>0,𝑗=1,2,_𝑗=𝑑_𝑗^𝑐𝑗Γ𝑐_𝑗𝛽^𝑐𝑗−1𝑒^{−𝑑𝑗𝛽𝑗},𝛽_𝑗>0,𝑐_𝑗,𝑑_𝑗>0,𝑗=1,2. (24) Since 𝛼_𝑗 and 𝛽_𝑗 are independent, then the joint density function of (𝛼_𝑗,𝛽_𝑗) is given by 𝜋𝛼_𝑗,𝛽_𝑗=𝑏^𝑎𝑗_𝑗𝑑^𝑐𝑗_𝑗Γ𝑎_𝑗Γ𝑐_𝑗𝛼^𝑎𝑗_𝑗⁻¹𝛽^𝑐𝑗_𝑗⁻¹𝑒^{−𝑏𝑗𝛼𝑗−𝑑𝑗𝛽𝑗},𝑗=1,2. (25) Combining the likelihood function with the priors pdf yields the posterior density function of all parameters Θ=(𝛼₁,𝛼₂,𝛽₁,𝛽₂) as follows: 𝜋^∗(Θ∣𝐱,𝐲)=𝐾²_𝑗=1𝛼^𝐴𝑗_𝑗⁻¹𝛽^𝐵𝑗_𝑗⁻¹𝑒^{−𝑢𝑗−𝑑𝑗𝛽𝑗}×^𝑟𝑗_𝑠=0𝐶_{𝑟𝑗,𝑠}𝑒^{−𝛼𝑗(𝑇𝑗𝑠+𝑏𝑗)}, (26) where 𝐴_𝑗=𝑚_𝑗−𝑟_𝑗+𝑎_𝑗, 𝐵_𝑗=𝑚_𝑗−𝑟_𝑗+𝑐_𝑗, 𝑇_1𝑠=𝑇_𝑠(𝐱;𝛽₁), 𝑇_2𝑠=𝑇_𝑠(𝐲;𝛽₂), 𝑢₁=𝑢(𝐱;𝛽₁), 𝑢₂=𝑢(𝐲;𝛽₂), and 𝐾⁻¹=²_𝑗=1^∞₀^∞₀𝛼^𝐴𝑗_𝑗⁻¹𝛽^𝐵𝑗_𝑗⁻¹𝑒^{−𝑢𝑗−𝑑𝑗𝛽𝑗}×^𝑟𝑗_𝑠=0𝐶_{𝑟𝑗,𝑠}𝑒^{−𝛼𝑗(𝑇𝑗𝑠+𝑏𝑗)}𝑑𝛼_𝑗𝑑𝛽_𝑗. (27) Under the BSEL function, and by using (13) and (26), the Bayesian “plug-in” estimate of 𝑆 is given by 𝑆_BS=𝜔𝑆_ML+(1−𝜔)_Θ𝑆𝜋^∗(Θ∣𝐱,𝐲)𝑑Θ. (28) Also, based on the BLINEX loss function, the Bayes estimate of 𝑆 is obtained by using (13) and is written as 𝑆_BL1=−𝑎ln𝜔𝑒^−𝑎𝑆_ML+(1−𝜔)_Θ𝑒^−𝑎𝑆𝜋^∗(Θ∣𝐱,𝐲)𝑑Θ, (29) where 𝑆_ML is the ML “plug-in” estimate of 𝑆 as given by (13).

It may be noted, from (22), (23), (28), and (29), that the Bayes estimates of 𝑆 contain integrals that cannot be obtained in simple closed form, and numerical techniques must be used for computations. We, therefore, propose to consider MCMC methods.

5. MCMC Algorithm for Bayesian Estimation

The MCMC algorithm is conducted to compare the Bayes estimates of 𝑆. We consider the Metropolis-Hastings algorithm to generate samples from the conditional posterior distributions, and then we compute the Bayes estimates. For more details about the MCMC methods, see, for example, Robert and Casella [28], Upadhyaya and Gupta [29], and Jaheen and Al Harbi [30]. The Metropolis-Hastings algorithm generates samples from an arbitrary proposal distribution.

5.1. The Case When 𝛽₁=𝛽₂=𝛽

The conditional posteriors distributions of the parameters 𝛼_𝑗, 𝑗=1,2, can be computed and written, respectively, as 𝜋^∗𝛼₁𝛼₂,𝐱,𝐲∝𝐽𝛼^𝐴∗1₁⁻¹𝑒^{−𝛿1𝛼1𝑟1}_𝑠=0𝐶_𝑟1,𝑠𝑒^{−𝛼1𝑇∗1𝑠},𝜋^∗𝛼₂∣𝛼₁,𝐱,𝐲∝𝐽𝛼^𝐴∗2₂⁻¹𝑒^{−𝛿2𝛼2𝑟2}_𝑠=0𝐶_𝑟2,𝑠𝑒^{−𝛼2𝑇∗2𝑠}. (30)

5.2. The Case When 𝛽₁≠𝛽₂

The conditional posteriors distributions of the parameters 𝛼_𝑗 and 𝛽_𝑗, 𝑗=1,2, can be computed and written, respectively, by 𝜋^∗𝛼₁Θ−𝛼₁,𝐱,𝐲∝𝛼^𝐴1₁^𝑟−11_𝑠=0𝐶_𝑟1,𝑠𝑒^{−𝛼1(𝑇1𝑠+𝑏1)},𝜋^∗𝛽₁Θ−𝛽₁,𝐱,𝐲∝𝛽^𝐵1₁⁻¹𝑒^{−𝑢1−𝑑1𝛽1𝑟1}_𝑠=0𝐶_𝑟1,𝑠𝑒^{−𝛼1(𝑇1𝑠+𝑏1)},𝜋^∗𝛼₂Θ−𝛼₂,𝐱,𝐲∝𝛼^𝐴2₂^𝑟−12_𝑠=0𝐶_𝑟2,𝑠𝑒^{−𝛼2(𝑇2𝑠+𝑏2)},𝜋^∗𝛽₂∣Θ−𝛽₂,𝐱,𝐲∝𝛽^𝐵2₂⁻¹𝑒^{−𝑢2−𝑑2𝛽2𝑟2}_𝑠=0𝐶_𝑟2,𝑠𝑒^{−𝛼2(𝑇2𝑠+𝑏2)}. (31) The following MCMC procedure is proposed to compute Bayes estimators for 𝑆=𝑃(𝑌<𝑋).Step 1. Start with initial guess of 𝛼_𝑗 and 𝛽_𝑗; say 𝛼_𝑗0 and 𝛽_𝑗0,𝑗=1,2. Step 2. Set 𝑖=1. Step 3. Generate 𝛼_1𝑖 from 𝜋^∗(𝛼₁∣Θ−𝛼₁,𝐱,𝐲) and 𝛽_1𝑖 from 𝜋^∗(𝛽₁∣Θ−𝛽₁,𝐱,𝐲). Step 4. Generate 𝛼_2𝑖 from 𝜋^∗(𝛼₂∣Θ−𝛼₂,𝐱,𝐲) and 𝛽_2𝑖 from 𝜋^∗(𝛽₂∣Θ−𝛽₂,𝐱,𝐲). Step 5. Compute 𝑆(𝛼_𝑖,𝛽_𝑖)=𝛼_1𝑖𝛽_1𝑖∫^∞₀(1+𝛽_1𝑖𝑥)^{−(𝛼1𝑖+1)}(1+𝛽_2𝑖𝑥)^−𝛼2𝑖𝑑𝑥. Step 6. Repeat Steps 2–5, 𝑁 times. Now, the approximate means of 𝑆(𝛼_𝑖,𝛽_𝑖) with respect to the posterior distribution are given, respectively, by 𝐸()=1𝑆∣𝐱,𝐲𝑁−𝑀^𝑁_𝑖=𝑀+1𝑆𝛼_𝑖,𝛽_𝑖,𝐸()=1exp−𝑎𝑆∣𝐱,𝐲𝑁−𝑀^𝑁_𝑖=𝑀+1exp−𝑎𝑆𝛼_𝑖,𝛽_𝑖, (32) where 𝑀 is the burn-in period. Therefore, the Bayes estimators of 𝑆=𝑆(𝛼_𝑖,𝛽_𝑖) based on BSEL and BLINEX loss functions are given, respectively, by 𝑆_BS=𝜔𝑆_ML+()𝐸(),𝑆1−𝜔𝑆𝐱,𝐲_BL1=−𝑎ln𝜔𝑒^−𝑎𝑆_ML+(1−𝜔)𝐸exp(−𝑎𝑆)∣𝐱,𝐲. (33)

6. Simulation Study

In order to find the Bayes and likelihood estimates of the parameter 𝑆, a Monte Carlo study is performed following the algorithms as follows.(1)For particular values of 𝛼_𝑗 and 𝛽_𝑗, 𝑗=1,2, Lomax observations of various sizes are generated for different general progressive censored schemes.(2)The ML estimates of 𝛼_𝑗 and 𝛽_𝑗,𝑗=1,2, are computed from the ML equations. The ML estimate of 𝑆 is computed from (13) after replacing 𝛼_𝑗 and 𝛽_𝑗, 𝑗=1,2, by their ML estimates.(3)For 𝑁=20000, 𝑀=2000, the Bayes estimates of 𝑆 are computed from (22), (23), (28), and (29) for BSEL and BLINEX loss functions based on MCMC algorithm.(4)The squared deviations (𝑆^∗−𝑆)² are calculated for different sample sizes and different schemes, where 𝑆^∗ is ML or Bayes estimates of 𝑆.(5)The above steps are repeated 1000 times, and the estimated risk (ER) is computed by averaging the squared deviations over the 1000 repetitions.

The computational results are displayed in Tables 1, 2, and 3. Table 1 shows different censoring schemes used in the simulation study. In the case of 𝛽₁=𝛽₂=𝛽, we take 𝜈₁=4, 𝛿₁=3, 𝜈₂=3, 𝛿₂=5, 𝜔=0.5, 𝛼₁=1.1339, 𝛽=3, 𝛼₂=0.7216, and the true value of 𝑆=0.4516. For this case, Table 2 presents simulation results and the MLE, Bayes estimate, and the corresponding mean squared error is reported within bracket. In the general case, we take 𝑎₁=4, 𝑏₁=3, 𝑎₂=3, 𝑏₂=2, 𝜔=0.5, 𝛼₁=1.5788, 𝛽₁=1.6899, 𝛼₂=1.5690, 𝛽₂=2.2002, and the true value of 𝑆=0.6111. The obtained simulation results for this case are shown in Table 3.

Table 1 

Censoring schemes used in the simulation study.

Table 2 

The simulation results and estimates of 𝑆 when 𝛽₁=𝛽₂=𝛽.

Table 3 

The simulation results and estimates of 𝑆 when 𝛽₁≠𝛽₂.

7. Conclusions

In this paper, the estimation of the stress-strength parameter, 𝑆, for two Lomax distributions under general progressive type-II censoring has been considered. The maximum likelihood and Bayes estimators of the stress-strength parameter have been derived. The MCMC method is used for computing Bayes estimates. It is observed that Bayes estimators outperform the ML estimators in small samples, while the estimators are almost equally efficient in large samples. It may be noted, from Tables 2 and 3, that the Bayes estimates have the smallest mean squared errors as compared with their corresponding maximum likelihood estimates. Based on the obtained results in this study and because of the need to deal with small samples in life testing, we recommend to use Bayes estimators in place of ML estimators.

Acknowledgments

The authors are grateful to the editor for his valuable comments and suggestions which improved the presentation of the paper. This Project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 268/130/1432. The authors, therefore, acknowledge with thanks DSR support for Scientific Research.