Parameter Estimation of Kumaraswamy Distribution Based on Progressive Type II Censoring Scheme Using Expectation-Maximization Algorithm

Wafula Mike Erick

doi:10.11648/j.ajtas.20160503.21

Abstract

This project considers the parameter estimation problem of test units from Kumaraswamy distribution based on progressive Type-II censoring scheme. The progressive Type-II censoring scheme allows removal of units at intermediate stages of the test other than the terminal point. The Maximum Likelihood Estimates (MLEs) of the parameters are derived using Expectation-Maximization (EM) algorithm. Also the expected Fisher information matrix based on the missing value principle is computed. By using the obtained expected Fisher information matrix of the MLEs, asymptotic 95% confidence intervals for the parameters are constructed. Through simulations, the behaviour of these estimates are studied and compared under different censoring schemes and parameter values. It’s concluded that for an increasing sample; the estimated parameter values become closer to the true values, the variances and widths of the confidence intervals reduce. Also, more efficient estimates are obtained with censoring schemes concerned with removals of units from their right.

Highlights

Censored sampling arises in a life testing experiment whenever the experimenter does not observe the failure times of all units placed on a life test.“According to Horst, a data sample is said to be censored when, either by accident or design the value of the variables under investigation is unobserved for some of the items in the sample.”[1]
It is observed that irrespective of the censoring rate and the position at which the censored units are removed from the sample, for increasing sample size; (i) the estimated value of the parameter becomes closer to the true value, (ii) the variances of the Maximum Likelihood Estimates (MLEs) decrease
This study has addressed the problem of estimation of parameters of the Kumaraswamy distribution based on progressive Type-II censored data

Summary

Introduction

Censored sampling arises in a life testing experiment whenever the experimenter does not observe (either intentionally or un-intentionally) the failure times of all units placed on a life test. Feroze et al [7] estimated the parameters of Kumaraswamy distribution under progressive type II censoring with random removals using maximum likelihood method. Mostafa et al [8] derived parameter estimators of Kumaraswamy distribution based on general progressive type II censoring scheme using maximum likelihood and Bayesian approaches. As far as we know, no one has described the EM algorithm for determining the MLEs of the parameters of the Kumaraswamy distribution based on progressive type-II censoring scheme. The purpose of this study is to estimate the shape and scale parameters of the Kumaraswamy distribution under progressive type-II censoring using the EM algorithm and to compare the results under different censoring schemes. This is because the EM algorithm is relatively robust against the initial values compared to the traditional Newton-Raphson (NR) method. [15, 16] For some of the recently relevant references on EM algorithm and censoring include [17 and 20]

Progressive Type II Censoring

Maximum Likelihood Estimation

EM Algorithm

Asymptotic Variance-Covariance Matrix of the MLEs

Numerical Results and Discussion

Conclusion