This paper presents a kernel estimation of the distribution of the scale parameter of the inverse Gaussian distribution under type II censoring together with the distribution of the remaining time. Estimation is carried out via the Gibbs sampling algorithm combined with a missing data approach. Estimates and confidence intervals for the parameters of interest are also presented.Gibbs sampler, Bayesian inference, Fiducial inference,

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Finite mixture models have been used in many fields of statistical analysis such as pattern recognition, clustering and survival analysis, and have been extensively applied in different scientific areas such as marketing, economics, medicine, genetics and social sciences. Introducing mixtures of new generalized lifetime distributions that exhibit important hazard shapes is a major field of research aiming at fitting and analyzing a wider variety of data sets. The main objective of this article is to present a full mathematical study of the properties of the new finite mixture of the three-parameter Weibull extension model, considered as a generalization of the standard Weibull distribution. The new proposed mixture model exhibits a bathtub-shaped hazard rate among other important shapes in reliability applications. We analytically prove the identifiability of the new mixture and investigate its mathematical properties and hazard rate function. Maximum likelihood estimation of the model parameters is considered. The Kolmogrov-Smirnov test statistic is used to fit two famous data sets from mechanical engineering to the proposed model, the Aarset data and the Meeker and Escobar datasets. Results show that the two-component version of the proposed mixture is a superior fit compared to various lifetime distributions, either one-component or two-component lifetime distributions. The new proposed mixture is a significant statistical tool to study lifetime data sets in numerous fields of study.

This paper focuses on the Bayesian prediction of kth ordered future observations modelled by a two-component mixture of general class of distributions. Samples under consideration are subject to random censoring. A closed form of Bayesian predictive density is obtained under a two-sample scheme. Applications to Weibull and Burr XII components are presented and comparisons with previous results are made. A numerical example is presented for special cases of the exponential and Lomax components to obtain interval prediction of first and last order statistics.