MLE and the estimated expected test time for the two-parameter Gompertz distribution under progressive censoring with binomial removals

doi:10.1016/j.amc.2006.03.020

Applied Mathematics and Computation

Volume 181, Issue 2, 15 October 2006, Pages 1657-1670

https://doi.org/10.1016/j.amc.2006.03.020 Get rights and content

Abstract

This study considers the estimation of two-parameter Gompertz distribution under Type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial or a uniform distribution. The maximum likelihood estimates of two parameters and their asymptotic distribution are derived. The expected termination point to complete the censoring test is computed and analyzed for different censoring schemes. For binomial removals, the effect of various p on the expected termination point under progressive censoring and the relative expected termination point over the complete sample (REET1) are investigated in this article. For random removals (PCR), the expected termination point is computed and the relative expected termination point over the complete sample (REET2) are also investigated.

Introduction

Life-tests are often the main research topics in many experimental designs. There are many applications of Gompertz distribution in the lifetime data analysis (see [4]). Censoring arises when some lifetimes of products are missing or for implementing some purposes of experimental designs. There are several types of censoring schemes and the Type II censoring scheme is most common one. The Progressively Type II censoring scheme is described as follows. First, the experimenter place n units on test. The first failure is observed and then r₁ of surviving units are randomly selected and removed. When the ith failure unit is observed, r_i of surviving units are randomly selected and removed, i = 2, … , m. This experiment terminates when the mth failure unit is observed and r_m = n − r₁ − ⋯ − r_m−1 − m of surviving units are all removed. When the censoring scheme r₁, … , r_m are all pre-fixed, Cohen [2], Cohen and Norgaard [3] had studied the statistical inference on the parameter of several failure time distributions under Type II Progressive censoring. But in some reliability experiment, the number of patients dropped out the experiment cannot be pre-fixed and they are random. Yuen and Tse [7] and Tse et al. [5] had studied the parameters estimation for weibull distributed lifetimes under Progressive censoring with random removals and Binomial Removals respectively. Wu et al. [6] proposed the MLE and the Estimated Expected Test Time for Pareto Distribution for Progressive Censoring Data. In this paper, we proposed the MLE and the Estimated Expected Test Time for Gompertz Distribution under Progressive Censoring. The MLE formulas are derived in Section 2 and the comparisons of the expected test time under Progressive Censoring with the complete sample are given in Section 3. The final conclusion is made in Section 4.

Section snippets

Model

Let random variable X have an Gompertz distribution with parameter λ and β, where λ is the scale parameter and β is the shape parameter. The probability density function of X is given by $f (x) = λ e^{β x} \exp [- \frac{λ}{β} (e^{β x} - 1)], x > 0, λ > 0, β > 0 .$ The cumulative distribution function F(x) is given by $F (x) = 1 - \exp [- \frac{λ}{β} (e^{β x} - 1)]$ . Let X₁ < X₂ < ⋯ < X_m denote a Progressively type II censored sample. With pre-determined number of removals R = (R₁ = r₁, … , R_m−1 = r_m−1), the conditional likelihood function can be written as [2] $L_{1} (x; θ, β | R = r) = c^{*} \prod_{i = 1}^{m}$

Expected test time

In practical applications, it is often useful to have an idea of the termination time of the whole test. For progressively type II censoring sampling plan with random or binomial removals, the termination point for the experiment is given by the expectation of the mth order statistic X_m.

Conclusions

This paper discusses some results of the Gompertz distributed data under Type II progressive censoring with binomial and random removals. We gain the MLE of the parameters from the likelihood function and derive its asymptotic distribution. We also compute the expected termination point for Type II progressive censoring with binomial and random removals and give some figures for REET₁ and REET₂ respectively. From the numerical results, it is observed that the expected termination point for Type

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MLE and the estimated expected test time for the two-parameter Gompertz distribution under progressive censoring with binomial removals

Abstract

Introduction

Section snippets

Model

Expected test time

Conclusions

Progressive Censoring: Theory, Methods and Applications

Progressively censored samples in the life testing

Technometrics

Progressively censored sampling in the three parameter gamma distribution

Technometrics