Maximum Likelihood Estimation for Generalized Pareto Distribution under Progressive Censoring with Binomial Removals

The paper deals with the estimation problem for the generalized Pareto distribution based on progressive type-II cen-soring with random removals. The number of components removed at each failure time is assumed to follow a binomial distribution. Maximum likelihood estimators and the asymptotic variance-covariance matrix of the estimates are obtained. Finally, a numerical example is given to illustrate the obtained results.


Introduction
The generalized Pareto distribution is also known as the Lomax distribution with two parameters, or the Pareto of the second type.It can be considered as a mixture distribution.Suppose that a random variable X has an exponential distribution with some parameter  .Further, suppose that  itself has a gamma distribution, and then the resulting unconditional distribution of X is called the Lomax distribution.This distribution has been extensively used for reliability modeling and life testing, for details see e.g.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].It has applications in economics, actuarial modeling, queuing problems and biological sciences, for more details we refer to Johnson et al. [3].
A random variable X is said to have the generalized Pareto distribution with two parameters, abbreviated as , if it has the probability density function (pdf) Here  and  are the shape and scale parameters, respectively.The survival function (sf) associated to (1) is given by The hazard function is All standard (further probabilistic properties of this distribution are given, for example, in Arnold [4]).
In life testing, the experimenter does not always observe the failure times of all components placed on the test.In such cases, the censored sampling arises.Furthermore, there are some cases in which components are lost or removed from the test before failure.This would lead to progressive censoring.Progressive censoring schemes are very useful in clinical trials and life-testing experiments.Balakrishnan and Aggarwala [5] provided a wealth of information on inferences under progressive censoring sampling.The progressively type-II censored life test is described as follows.The experimenter puts n components on test at time zero.The first failure is observed at 1 X and then 1 of surviving components is randomly selected and removed.When the ith failure component is observed at i R X , i of surviving components are randomly selected and removed, The experiment terminates when the mth failure component is observed at m X and m all removed.In this censoring scheme 1 2 m are all prefixed.However, in some practical experiments, these numbers cannot be pre-fixed and they occur at random.Yuen and Tse [6], Tse and Yuen [7] and Tse et al. [8] considered the estimation problem for Weibull distribution under type-II progressive censoring with binomial removals.Shuo [9] studied the estimation problem for two-parameter Pareto distribution based on progressive censoring with uniform removals.Wu [10] provided estimation for the two-parameter Pareto distribution under progressive censoring with uniform removals.Wu et al. [11] studied the Burr type XII distribution based on progressively censored samples with random removals.This paper is concerned with the estimation problem of the unknown parameters for the generalized Pareto distribution based on progressive type-II censoring with random removals.The number of components removed at each failure time is assumed to follow a binomial distribution.In Section 2, we derive the maximum likelyhood estimators.The asymptotic variance-covariance matrix of the estimates is obtained in Section 3. In Section 3, numerical examples are given to illustrate the obtained results.

Maximum Likelihood Estimation
be the ordered failure times of mcomponents, where is pre-specified before the test.At the ith failure, i components are removed from the test.For progressive censoring with pre-specified number of number of removals , the likelihood function can be defined as follows: (see Cohen [12]) where Using relations (1), ( 2) and ( 3), the log-likelihood function is given by Looking at relation (4), it is noted that the relation is derived conditional on .Each r can be of any i integer value between 0 and We assume that i is a random number and is assumed to follow a binomial distribution with parameter .This means the probability that each component leaves will remain the same, say , and the probability of component leaving after the ith failure occurs is and . Furthermore, we assume that i is independent of X for all i.The joint likelihood funtion of   , P R p is the joint probability distribution of where and is given by

Point Estimation
The maximum likelihood estimators of  and  are found by maximizing , since P R p does not involve the parameters  and  .Thus, the maximum likelihood estimates (MLEs),   ,  of    ,   can be found by solving the following equations:    the estimators of  : From ( 8) and ( 9), we obtain and  can be obtained as the solution of the non-linear equation Therefore,  can be obtained as solution of the nonlinear equation of the form   Ŝince  is a fixed point solution of the non-linear Equation ( 12), therefore, it can be obtained using an iterative scheme as follows where  is the jth iterate of  .The iteration proce- dure should be stopped when     is sufficiently small.Once we obtain  then  can be obtained from (10).The MLEs of reliability and hazard function are Similarly, the MLE of p can be found by maximizing P R p .Independently, the MLE of parameter p is the solution of Hence, the MLE of parameter p is

Interval Estimation
We obtain approximate confidence intervals of the parameters  and  based on the asymptotic dis- tribution of the maximum likelihood estimator of the parameters.Hence, we employ the asymptotic normal approximation to obtain confidence intervals for the unknown parameters.We now obtain the asymptotic Fisher's information matrix.The observed information matrix of

 
, , p The variance-covariance matrix may be approximated as The asymptotic distribution of the maximum likelyhood can be written as follows see e.g.Miller [13].
Since V involves the parameters  ,  and p, we replace the parameters by the corresponding MLEs in order to obtain an estimate of V, which is denoted by .By using (14), approximate 100 confidence intervals for Z  is the upper where 100 - th percentile of the standard normal distribution.

Numerical Study
In our study, we firstly generate the numbers of progresssive censoring with binomial removals, i , and then we get the progressive censoring with binomial removals samples from GP distribution by the Monte-Carlo method.The steps are: Step 1 Generate a group value 3) Step 3 For given values of the progressive censoring scheme , set Then,   , , , is the required from progressive censoring with binomial removals sample of size m from which led to a considerable improvement of the paper.the GP distribution.
Table 1 shows the numerical results and the MLE of the parameters  ,  and p when the actual values of the parameters are 1

Acknowledgements
The author is grateful to the referee for his comments

1 2 m 5 )
are progressive censoring with binomial removals samples of size m from  0,1 .U Step 5 Finally, for given values of parameters  and