Mixtures of beta weibull G family of distributions and application

Abstract

Mixture models have gained popularity in statistical analyses because of their flexibility in cap turing local variations in heterogeneous populations. Model based approaches to classification use mixture models to fit data via maximum likelihood based approaches and provide labels to unlabelled observations. Over the years model based approaches have grown into an important sub-field of classification because they provide the uncertainty of classifying the unlabelled observations as probabilities. Despite many advances in model based approaches to classification, not much work is evidenced in the literature where reliability data is concerned. The Weibull mixtures are often used in modelling reliability data but they are limited to data with monotone failure rates. To this end we introduce a Beta Weibull G (BWG) mixture that provides an appeal ing framework for handling reliability data with non monotone failure rate functions. Parametric estimation is executed by the Expectation Maximization algorithm, which is an extension of max imum likelihood estimation. The Bayesian Information Criterion is used for model selection. Model based clustering and mixture discriminant analysis techniques are used to assign labels to unlabelled observation. These labels are cross validated by the Adjusted Rand Index. Ad ditionally, parsimony is introduced to the BWG mixtures, by adding constraints on some of the parameter estimates. The constrained models give rise to simple models with great explanatory predictive power. To demonstrate the utility of the proposed approaches, different data sets are simulated to mimic reliability data with non monotone failure rates. The findings of this the sis demonstrate that mixtures of the BWG family of distributions fit heterogeneous population with non monotone hazard rates better than mixtures of the Weibull distributions as evidenced by higher values of BIC for BWG mixtures. The BWG mixtures performed better than Weibull mixtures in both model based clustering and mixture discriminant analysis as demonstrated by high values of the ARI