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

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.

The Moore and Bilikam model and Burr XII sub-model under progressively type-II censoring scheme

“…Mukhopadhyay et al [16] prepared the two-stage procedures based on the maximum likelihood (ML) and uniformly minimum variance unbiased estimators for the bounded risk point estimation of the parameter and hazard rate in the MB family. Chaturvedi et al [9] focused on the reliability and stress-strength functions of the MB family under the record data sets. Chaturvedi et al [8] considered the robust Bayesian analysis of parameter, reliability and hazard functions of the MB family under the ε-contamination class of priors under the type-II censoring scheme with squared error loss (SEL) and general entropy loss functions.…”