On Parameters Estimation of Lomax Distribution under General Progressive Censoring

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.TheMarkov chainMonte Carlo (MCMC)methods are used to accomplish some complex calculations. Comparisons are made between Bayesian and maximum likelihood estimators via Monte Carlo simulation study.


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

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  0 ≡ 0,  randomly selected components were placed on a life test.The failure times of the first  components to fail,  1 , . . .,   , were not observed.At the time of the ( + 1)th failure,  +1: ,  +1 number of surviving components are removed from the test randomly and so on; at the time of the ( + )th observed failure,  +: ,  + number of surviving components are removed from the test randomly; and finally, at the time of the th failure, the remaining   =  −  −  +1 −  +2 − ⋅ ⋅ ⋅ −  −1 are removed from the test.Suppose that  +1:: ≤  +2:: ≤ ⋅ ⋅ ⋅ ≤  :: are the lifetimes of the completely observed components to fail and that  +1 ,  +2 , . . .,   are the number of components removed from the test at these failure times, respectively.The   's, , and  are prespecified integers such that 0 ≤  <  ≤ , 0 ≤   ≤  −  for  =  + 1, . . .,  − 1, and The resulting ( − ) ordered values  +1:: ,  +2:: , . . .,  :: 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   = 0, for  = +1, . . ., −1, and   = −, the general progressively type-II censoring scheme is reduced to the case of type-II doubly censored sample.(ii) If  = 0, this scheme is reduced to the progressive type-II right censoring.(iii) If  = 0 and   = 0, for  =  + 1, . . .,  − 1 so that   =  − , the general progressively type-II censoring scheme is reduced to conventional type-II one-stage right censoring, where just the first  usual order statistics are observed.(iv) If  = 0 and   = 0, for  =  + 1, . . ., , so that  = , the general progressively type-II censoring scheme is reduced to the case of no censoring (complete sample case), where all  usual order statistics are observed.In this scheme  1 ,  2 , . . .,   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  randomly selected components from the L(, ) are put in test.Further, let  +1:: ,  +2:: , . . .,  :: denote a general progressively type-II censored sample from that population, with ( +1 ,  +2 , . . .,   ) being the progressive censoring scheme.For simplicity of notation, we will use   instead of  :: , and then x = ( +1 , . . .,   ) is the observed general progressive censored sample.The likelihood function for the parameters  and  is then where and the functions () and () are given, respectively, by (1) and (2).Substituting (1) and ( 2) into (3), the likelihood function is Using the binomial expansion,  is a positive integer, one can rewrite the likelihood function as follows: where We focus our attention on the estimation of the probability  = ( < ), where  and  are two independent random variables each is (  ,   ),  = 1,2 distributed, and the data obtained from both distributions are general progressively type-II censored.Here,  and  are typically modeled as independent.The probability ( < ) has been widely studied under different approaches and distributional assumptions on  and .The case where  and  are dependent has been considered by Nandi and Aich [22], Barbiero [23], and Rubio and Steel [24].We investigate properties of  when the common scale parameter  1 =  2 =  is known.Then, it can be shown that Here, (, ) is the joint pdf of  and , (, ) =  1 () 2 () by the independence.The general case, when  1 ̸ =  2 , can be studied in a similar manner.We obtain that
The corresponding "ML plug-in estimation" of , when  1 =  2 = , is obtained by replacing  1 and  2 by its MLEs, α1ML , α2ML , and substituting them into relation (8) which yields For the general case,  1 ̸ =  2 , the corresponding "ML plug-in estimation" of  is obtained by replacing  1 ,  1 ,  2 , and  2 by its MLEs, α1ML , β1ML , α2ML , and β2ML and substituting them into relation (9) which results in obtaining ŜML as

Bayes Estimation
In this section, Bayesian estimation for the probability  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: where (⋅) is a suitable positive weight function and ((), ) is an arbitrary loss function when estimating () by .The parameter  0 is a chosen priori estimate of (), 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  > 0, and  = 0 where 0 ≤  < 1.By choosing ((), ) = ( − ()) 2 and () = 1, the BLF is reduced to the balanced squared error loss (BSEL) function, used by Ahmadi et al. [27], in the form The corresponding Bayes estimate of the function () is given by In this case, the Bayes estimate of () takes the form where  ̸ = 0 is the shape parameter of BLINEX loss function.

Bayes Estimation when 𝛽
We assume that   and   ,  = 1, 2, are random variables each having gamma prior with some parameters; that is, Since   and   are independent, then the joint density function of (  ,   ) is given by Combining the likelihood function with the priors pdf yields the posterior density function of all parameters Θ = ( 1 ,  2 ,  1 ,  2 ) as follows: where   =   −   +   ,   =   −   +   ,  1 =   (x;  1 ),  2 =   (y;  2 ),  1 = (x;  1 ),  2 = (y;  2 ), and Under the BSEL function, and by using ( 13) and ( 26), the Bayesian "plug-in" estimate of  is given by Also, based on the BLINEX loss function, the Bayes estimate of  is obtained by using ( 13) and is written as where ŜML is the ML "plug-in" estimate of  as given by ( 13).
It may be noted, from ( 22), ( 23), (28), and (29), that the Bayes estimates of  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.

MCMC Algorithm for Bayesian Estimation
The MCMC algorithm is conducted to compare the Bayes estimates of .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.

The Case
When  1 =  2 = .The conditional posteriors distributions of the parameters   ,  = 1, 2, can be computed and written, respectively, as

The Case When
The conditional posteriors distributions of the parameters   and   ,  = 1, 2, can be computed and written, respectively, by The following MCMC procedure is proposed to compute Bayes estimators for  = ( < ).
Step 1. Start with initial guess of   and   ; say   0 and (33)

Simulation Study
In order to find the Bayes and likelihood estimates of the parameter , a Monte Carlo study is performed following the algorithms as follows.
(1) For particular values of   and   ,  = 1, 2, Lomax observations of various sizes are generated for different general progressive censored schemes.(2) The ML estimates of   and   ,  = 1, 2, are computed from the ML equations.The ML estimate of  is computed from (13) after replacing   and   ,  = 1, 2, by their ML estimates.
(4) The squared deviations ( * − ) 2 are calculated for different sample sizes and different schemes, where  * is ML or Bayes estimates of .
(5) The above steps are repeated 1000 times, and the estimated risk (ER) is computed by averaging the squared deviations over the 1000 repetitions.

Conclusions
In this paper, the estimation of the stress-strength parameter, , 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.

Table 1 :
Censoring schemes used in the simulation study.

Table 2 :
The simulation results and estimates of  when  1 =  2 = .

Table 3 :
The simulation results and estimates of  when  1 ̸ =  2 .