Inference for the Geometric Extreme Exponential Distribution under Progressive Type II Censoring

Reza Pakyari

doi:10.5402/2012/292384

Abstract

Geometric extreme exponential (GE-exponential) is one of the nonnegative right-skewed distribution that is suitable for analyzing lifetime data. It is well known that the maximum likelihood estimators (MLEs) of the parameters lead to likelihood equations that have to be solved numerically. In this paper, we provide explicit estimators through an approximation of the likelihood equations based on progressively Type-II-censored samples. The approximate estimators are then used as starting values to find the MLEs numerically. The bias and variances of the MLEs are calculated for a wide range of sample sizes and different progressive censoring schemes through a Monte Carlo simulation study. Moreover, formulas for the observed Fisher information are given which could be used to construct asymptotic confidence intervals. The coverage probabilities of the confidence intervals and the percentage points of pivotal quantities associated with the MLEs are also calculated. A real dataset has been studied for illustrative purposes.

Highlights

Progressive censoring is one of the important sampling techniques that was first introduced by Herd 1 and its importance in life testing reliability experiments was discussed by Cohen 2
Suppose n units are placed on test
At the time of the first failure, R1 units are randomly removed from the n − 1 surviving units

Summary

Introduction

Progressive censoring is one of the important sampling techniques that was first introduced by Herd 1 and its importance in life testing reliability experiments was discussed by Cohen 2. The progressive Type-II censoring is as follows. We observe m complete failures and R1 R2 · · · Rm items are progressively censored from the n units under test, and so n m R1 R2 · · · Rm. The vector R R1, . For an excellent discussion on progressive Type-II censoring technique see the monograph of Balakrishnan and Aggarwala 3 and the recent discussion paper by Balakrishnan 4. There are numerous articles in the literature dealing with inferential procedures based on the progressively Type-II censoring data for a wide variety of lifetime distributions. We study the inference for the parameters of the GE-exponential distribution based on progressively Type-II censored data.

Maximum Likelihood Estimation

Approximate Maximum Likelihood Estimators

Fisher Information

Simulation Study

Asymptotic Confidence Intervals

Real Data Analysis