Minimum Variance Unbiased Estimation in the Gompertz Distribution under Progressive Type II Censored Data with Binomial Removals

This paper deals with the problem of uniformly minimum variance unbiased estimation for the parameter of the Gompertz distribution based on progressively Type II censored data with binomial removals. We have obtained the uniformly minimum variance unbiased estimator (UMVUE) for powers of the shape parameter and its functions. The UMVUE of the variance of these estimators is also given. The UMVUE of (i) pdf, (ii) cdf, (iii) reliability function, and (iv) hazard function of the Gompertz distribution is derived. Further, an exact (1 − α)100% confidence interval for the pth quantile is obtained. The UMVUE of pdf is utilized to obtain the UMVUE of P(X < Y). An illustrative numerical example is presented.


Introduction
A Type II censored sample is one for which only  smallest observations in a sample of  items are observed.A generalization of Type II censoring is a progressive Type II censoring.Under this scheme, units of the same kind are placed on test at time zero, and  failures are observed.When the first failure is observed, a number  1 of surviving units are randomly withdrawn from the test; at the second failure time,  2 surviving units are selected randomly and taken out of the experiment, and so on.At the time of th failure, the remaining   =  −  1 −  2 − ⋅ ⋅ ⋅ −  −1 −  units are removed.Balakrishnan et al. [1] indicated that such scheme can arise in clinical trials where the drop out of patients may be caused by migration or by lack of interest.In such situations, the progressive censoring scheme with random removals is required.Many authors have discussed inference under progressive Type II censored samples using different life distributions including El-Din and Shafay [2], Kim et al. [3] Kim and Han [4], Ali Mousa and Jaheen [5], and Pérez-González and Fernández [6].For a detailed discussion of progressive censoring, we refer to Balakrishnan and Aggarwala [7] and Balakrishnan [8].Note that if  1 =  2 = ⋅ ⋅ ⋅ =  −1 = 0, this scheme reduces to the Type II censoring scheme.Also note that if  1 =  2 = ⋅ ⋅ ⋅ =   = 0, so that  = , the progressively Type II censoring scheme reduces to the case of no censoring, that is, the case of a complete sample.In this paper, we use progressively Type II censoring scheme with binomial removals where the number of units removed at each failure time follows a binomial distribution.
The Gompertz distribution was first introduced by Gompertz [9] to describe human mortality and establish actuarial tables.Since then, many investigators have used the Gompertz distribution or some related forms of it in a variety of studies.There are many forms of the Gompertz distribution in the literature.
The Gompertz distribution is applied in actuarial science, reliability and life testing studies, and epidemiological and biomedical studies.Several such situations have been discussed by Ananda et al. [10], Walker and Adham [11], Jaheen [12], and many others.For a review of literature on estimating parameters of the Gompertz distribution, one may refer to Gordon [13], Chen [14], Wu et al. [15], Garg et al. [16], Ismail [17], Al-Khedhairi and El-Gohary [18], and many others.
Inference for The Gompertz distribution based on progressively Type II censored data is discussed by many authors.Wu et al. [19] obtained the maximum likelihood estimators of the two-parameter Gompertz distribution under progressive Type II censoring with binomial removals.They had also given the expected test time to complete the censoring test.Wu et al. [20] discussed the problem of interval estimation for the two-parameter Gompertz distribution under progressive Type II censored data.Many authors have studied the problem of estimation of  = ( < ) for various distributions.This model involves two independent random variables  and .The term  = ( < ) is the reliability of a system of strength  is subjected to a stress .The system fails if the applied stress exceeds its strength.An extensive review of this topic is given in Kotz.et al. [21].Sarac ¸oglu and kaya [22] obtained the maximum likelihood estimate of stress strength reliability for the Gompertz distribution.Sarac ¸oglu et al. [23] have obtained maximum likelihood estimate and UMVUE of stress strength reliability for the Gompertz distribution when  and are independent but not identically random variables belonging to the Gompertz distribution when complete sample is available.
In this paper, we discuss the problem of UMVUE for shape parameter of the Gompertz distribution based on progressively Type II censored data with binomial removals.In Section 2, the conditional likelihood function is given.In Section 3, the UMVUE of parameter of  and its functions are derived.Also, the UMVUE of the (i) pdf, (ii) cdf, (iii) reliability function (iv) hazard function are obtained.In Section 4. the UMVUE of ( < ) is obtained by using the UMVUE of p.d.f.In Section 5 an exact (1−)100% confidence interval for th quantile is obtained.An illustrative numerical example is presented.

The Model
Let the failure time distribution be the Gompertz with probability density function, where  and  are the parameters.We assume that  is known.The cumulative distribution function is given by The survival function is given by The density given in (1) can be written as where Let ( 1 ,  1 ), ( 2 ,  2 ), . . ., (  ,   ) denote a progressively Type II censored sample, where   =  :: , for  = 1, 2, . . ., and  1 <  2 < ⋅⋅⋅ <   .The conditional likelihood function can be written as, see Cohen [24], where Substituting ( 1) and ( 3) in ( 6) we get Suppose that an individual unit being removed from the life test is independent of others but with the same probability .
Then the number of units removed at each failure time follows a binomial distribution, and, following Wu et al. [19], the joint probability mass function of  1 ,  2 , . . . −1 is given by that is, .
The unconditional likelihood function is Using ( 7) and ( 9) in (10) we can write the full likelihood function as . (11)

Unbiased Estimation
Let then   have exponential distribution with mean 1/.We can show that  1 <  2 < ⋅ ⋅ ⋅ <   is a progressive Type II censored sample from an exponential distribution with mean 1/.Let us consider the following transformation: In order to derive the distribution of   ,  = 1, 2, . . .,  consider the inverse transformation  1 =  1 / and   = ∑  =2 (  /( −  −1 −  + 1)),  = 2, 3, . . ..The variables  1 ,  2 , . . .,   defined in ( 13) are all independent and identically distributed with exponential distribution with mean 1/, see Thomas and Wilson [25].The joint density of  1 ,  2 , . . .,   , is It can be seen that Using ( 12) in ( 15), we have Let Since ( 14) is a member of exponential family of distributions, is a complete sufficient statistic for .The distribution of  is gamma with parameters  and , which is again a member of exponential family of distributions.The pdf of  is given by where Jani and Dave [26] have studied the problem of minimum variance unbiased estimation in a class of exponential family of distributions.They have shown that if  1 ,  2 , . . .  , be a random sample from density of the type given in (4) and the p.d.f. of its complete sufficient statistics can be written as the one given in (18), then the UMVUE of [ℎ()]  is given by and the UMVUE of [()]  is Following the results derived in Jani and Dave [26], we get the UMVUE of some important parametric functions as given below.
(i) Using (20), the UMVUE of exp[−] is (ii) Using ( 22), the UMVUE of the variance of  , , is given by, (iii) Using ( 21), the UMVUE of (1/)  , is given by (iv) Using ( 24), the UMVUE of the variance of  , is given by (v) The UMVUE of density () given in (1), for fixed , is given by, where (vi) The UMVUE of variance of  , ,  > 2 is given by where  is given by ( 27).
(vii) Considering  as fixed, the UMVUE of reliability function () = ( > ),  ≥ 0 is obtained as follows.Since () = [ℎ()] [  −1]/ , where ℎ() is given in (5) and using (22) with  = (1/)[  − 1], the UMVUE R() of () is given by (viii) Using ( 29), the UMVUE of the variance of R() is given by Special Cases (a) Substituting  = 1 in ( 22), we get the UMVUE of exp[−] as, (c) Substituting  = 1 in (24), we get the UMVUE of (1/) as (d) Substituting  = −1 in (24), we get the UMVUE of  as (e) The hazard function for the Gompertz distribution is ℎ() =   .Using (35), the UMVUE of hazard function, for fixed , can be given as Shanubhogue and Jain [27] have studied the problem of minimum variance unbiased estimation in exponential distribution under progressive Type II censored data with binomial removals.They have given the UMVUE for parameter  and various functions of .Since the joint density ( = ) given in ( 9) is independent of , one gets the same estimators of , and its various functions as given in Shanubhogue and Jain [27].

Illustrative Example
In this section, we illustrate the use of the estimation methods given in this paper.
The following are the numbers of tumor-free days of 30 rats fed with unsaturated diet, see King et al. [28] These data are presented by Lee [29] and studied by Chen [14] and Wu et al. [20].Chen [14] and Wu et al. [20] assumed a Gompertz distribution for tumor-free times.Wu et al. [20] obtained the MLE of  as ĉ = 0.0505 for progressive Type II censored data.We generate a progressive Type II censored data with binomial removals from these data, assuming  = 0.0505.The progressive censored sample size is  = 18.

Table 1 :
Progressive Type II censored sample with binomial removals generated from the tumor-free time data.

Table 2 :
The UMVU estimates of different parametric functions of  based on data given in Table1.