Abstract

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 % confidence interval for the th quantile is obtained. The UMVUE of pdf is utilized to obtain the UMVUE of . An illustrative numerical example is presented.

1. 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 When the first failure is observed, a number of surviving units are randomly withdrawn from the test; at the second failure time, surviving units are selected randomly and taken out of the experiment, and so on. At the time of failure, the remaining 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 , this scheme reduces to the Type II censoring scheme. Also note that if , 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]. Saraçoğlu and kaya [22] obtained the maximum likelihood estimate of stress strength reliability for the Gompertz distribution. Saraçoğlu 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 100% confidence interval for th quantile is obtained. An illustrative numerical example is presented.

2. 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 denote a progressively Type II censored sample, where , for , and . The conditional likelihood function can be written as, see Cohen [24], where and for .

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 is given by that is, The unconditional likelihood function is Using (7) and (9) in (10) we can write the full likelihood function as

3. Unbiased Estimation

Let then have exponential distribution with mean . We can show that is a progressive Type II censored sample from an exponential distribution with mean . Let us consider the following transformation: In order to derive the distribution of , consider the inverse transformation and . The variables defined in (13) are all independent and identically distributed with exponential distribution with mean , see Thomas and Wilson [25]. The joint density of , 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 , 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 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 is

(ii) Using (22), the UMVUE of the variance of , is given by,

(iii) Using (21), the UMVUE of , 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 is given by

where is given by (27).

(vii) Considering as fixed, the UMVUE of reliability function , is obtained as follows. Since , where is given in (5) and using (22) with , the UMVUE of is given by

(viii) Using (29), the UMVUE of the variance of is given by

(ix) The UMVUE of cumulative distribution function given in (2) is

Special Cases (a)Substituting in (22), we get the UMVUE of as, (b)Substituting in (22), we get UMVUE of as (c)Substituting in (24), we get the UMVUE of as (d)Substituting in (24), we get the UMVUE of as (e)The hazard function for the Gompertz distribution is . 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].

4. UMVU Estimator of

In the following theorem, we derive the UMVUE of . Let units (out of ) on and units (out of ) on are recorded which follow the Gompertz distributions, given in (1) with parameters and , respectively. Let and be corresponding removals. We denote

Theorem 1. Under progressive Type II censored data, the UMVU estimator of for the density given in (1) is given by where and are given by (37) and (38), respectively.

Proof. We have where Using (26) in (40) and let , we have Now After substituting (43) into (42), we get
Further simplification of (44) and applying the result , we get Further simplification of (45) gives Similarly, we can show that for the case , the UMVUE of is

5. Exact Confidence Interval for th Quantile

Theorem 2. Under progressive Type II censored data an exact confidence interval for quantile, is given by,