Preliminary Test Estimators and Confidence Intervals for the Parametric Functions of the Moore and Bilikam Family of Lifetime Distributions Based on Records

Ananya Malhotra,Suk-Bok Kang,Ajit Chaturvedi

doi:10.17713/ajs.v48i4.844

Abstract

We consider two measures of reliability functions namely R(t)=P(X>t) and P=P(X>Y) for the Moore and Bilikam (1978) family of lifetime distributions which covers fourteen distributions as specific cases. For record data from this family of distributions, preliminary test estimators (PTEs) and preliminary test confidence interval (PTCI) based on uniformly minimum variance unbiased estimator (UMVUE), maximum likelihood estimator (MLE), empirical Bayes estimator (EBE) are obtained for the parameter. The bias and mean square error (MSE) (exact and asymptotic) of the proposed estimators are derived to study their relative efficiency and through simulation studies we establish that PTEs perform better than ordinary UMVUE, MLE and EBE. We also obtain the coverage probability (CP) and the expected length of the PTCI of the parameter and establish that the confidence intervals based on MLE are more precise. An application of the ordinary preliminary test estimator is also considered. To the best of the knowledge of the authors, no PTEs have been derived for R(t) and P based on records and thus we define improved PTEs based on MLE and UMVUE of R(t) and P. A comparative study of different methods of estimation done through simulations establishes that PTEs perform better than ordinary UMVUE and MLE.

Highlights

In statistical inference, we often come across problems where some prior information on the parameters is available, which give rise to restricted models
In this paper we construct some preliminary test estimators (PTEs) on the basis of records for the powers of the parameter and reliability functions of Moore and Bilikam (1978) family of lifetime distribution which covers as many as fourteen distributions as its specific cases, in two different situations
We study the efficiency of PTEs of P based on maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) over the usual estimators of P based on MLE and UMVUE

Summary

Introduction

We often come across problems where some prior information on the parameters (often regarded as constraints) is available, which give rise to restricted models. In this paper we construct some PTEs on the basis of records for the powers of the parameter and reliability functions of Moore and Bilikam (1978) family of lifetime distribution which covers as many as fourteen distributions as its specific cases, in two different situations. In this case, using asymptotic normality of the MLE, we obtain superiority conditions for the proposed PTE. Asymptotic bias, MSE and asymptotic relative efficiency of the proposed PTE of the parameter are obtained

Proposed preliminary test estimators

Bias and mean square error

Proposed preliminary test confidence interval

Numerical findings

An example on real data

Discussion on the proposed estimation methods

Estimation of parametric functions when the scale parameter is known

Estimation of parametric functions when both parameters are unknown

Proposed preliminary test estimator

Asymptotic bias and mean square error

Comparison

Conclusion