A New Weibull Model with Inliers at Zero and One Based on Type-II Censored Samples

Abstract

Inliers (instantaneous or early failures) are natural occurrences of a life test, where some of the items fail immediately or within a short time of the life test due to mechanical failure, inferior quality or faulty construction of items and components. The inconsistency of such life data is modeled using a nonstandard mixture of distributions; with degeneracy occurring at zero and one, and a probability distribution for positive observations. In this paper, the estimation of parameters based on type-II censored sample from a Weibull distribution with discrete mass at zero and one is studied. The maximum likelihood estimators (MLE) are developed for estimating the unknown parameters. The Fisher information matrix, as well as the asymptotic variance–covariance matrix of the MLEs are derived. Uniformly minimum variance unbiased estimate (UMVUE) of model parameters as well as UMVUE of density function, reliability function and some parametric function is obtained along with UMVUE of the different estimators. The model is implemented on a real data of tumor size in invasive ductal breast carcinoma of female patients.