Study of the New Finite Mixture of Weibull Extension Model: Identifiability, Properties and Estimation

Noura S Mohamed,Moshira A Ismail,Sanaa A Ismail

doi:10.13189/ms.2022.100104

Abstract

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.

Highlights

Survival analysis has been studied with the classical statistical distributions such as Gamma, exponential and Weibull distributions among others
The descriptive statistics of the datasets (see, Table (2)) reveal heterogeneity and that it can be better represented by a mixture model with a hazard rate that exhibits bathtub shape
It is noticed that the two-component mixture of Chen model, which is a reduced model of MWEM does not fit the data which shows the importance of including the scale parameters β1 and β2

Summary

Introduction

Survival analysis has been studied with the classical statistical distributions such as Gamma, exponential and Weibull distributions among others Generalizations of such lifetime distributions have been introduced in the literature to fit a wider variety of data sets and to present multiple shapes of useful hazard functions. The Weibull distribution is perhaps the most widely used lifetime distribution model because of its flexibility and simple expressions for the density, survival, and hazard functions. It cannot capture the behavior of lifetime data that exhibit non-monotone hazard shapes (such as bathtub, inverse bathtub (unimodal)...etc.), often encountered in reliability engineering studies. Reference [3] studied a new generalization of the flexible Weibull distribution with three parameters referred to as the exponentiated flexible

Objectives

Results

Conclusion