Mixture Distributions and Parameter Estimation for the Reliability Models

Abstract

The finite mixture model of continuous distributions is a robust statistical framework for modeling the complex real-life problems characterized by multiple subpopulations with distinct distributional forms. This thesis explores the use of mixed distributions in data sets and covers the parameters estimation advancements, reliability indices and practical applications of finite mixture models, elucidating its utility across diverse domains. Our study presents novel contributions to the finite mixture models using real data of survival and continuous distributions. By developing various finite mixture models, statistical tests have been carried out. The investigation done resolves the critical challenges such as model construction, parameter estimation, and choice of best-fit model, enriching the available models using flexible weights, delay distribution, memory concepts, etc. The present thesis is dedicated to establishing the reliability performance indices of finite mixture models for mixed distributions and their applications in COVID-19 vaccination, software systems, and failure of devices. We develop the mixture model to ensure the reliability growth of the real-time software systems using datasets. The study explores many statistical as well as reliability indices including hazard rate function, Mills ratio, mean time to failure, and reliability perspectives within the context of survival data testing. The thesis is divided into ten chapters including the first introductory chapter. Chapters 2-9 explore mixture models composed of mixed distributions and provide investigations on various reliability characteristics and parameter estimation. Some chapters are further divided into sections to address similar investigations with different frameworks. The chapter-wise organization of the thesis is as follows. Chapter 1 introduces the motivation, basic concepts of continuous lifetime distributions, methodological aspects, reliability indices, and outlines of the investigations presented in the thesis. Chapter 2 focuses on the performance modeling and analysis of the mixture model with two-fold distribution incorporating constant and Poisson dynamic weight functions. The parameter estimation of the mixture model is carried out using maximum likelihood estimation and expectation maximization methods. Chapter 3 addresses the performance issues of constant and dynamic weight function-based mixture models. The finite model is developed using the heterogeneous specific continuous lifetime distributions such as Weibull, lognormal and exponential distributions. The parameter estimation and validation of the mixture models have been done using statistical techniques and the failure dataset. Chapter 4 deals with the performance prediction of a shifted mixture model which is composed of Weibull, lognormal and gamma distributions. The parameter values have been estimated using MLE and EM algorithms. Taking experimental datasets, the goodness-of-fit tests have been done to evaluate the best mixture model. Chapter 5 discusses the mixture model of homogeneous and non-homogeneous exponential distributions. The parameter i values are estimated using maximum likelihood estimation (MLE), expectation-maximization (EM) and least square estimation (LSE) methods. The mixture model is validated using goodness of fit tests for the real-time shared memory processors (SMP) lab data sets. Chapter 6 addresses the performance prediction of a heterogeneous mixture model using a soft computing technique. The EM algorithm is employed for the parameter estimation. The ANFIS and goodness of fit tests are employed to validate the mixture model on the experimental datasets. Chapter 7 focuses on the parameter estimation and the construction of the shifted mixture model involving Weibull, lognormal, and Gompertzian distributions. This chapter utilizes EM and meta-heuristic techniques such as PSO and DE for the parameter estimation. Chapter 8 presents a mixture model of fractional distributions. The performance prediction of reliability indices for the mixture model has exponential and Weibull distributions as components. It estimates the parameter values of the proposed mixture models using the EM algorithm. The validation of the mixture model is done for the failure datasets and using AIC, BIC, and AICc tests. Chapter 9 is devoted to the three-fold fractional mixture model composed of exponential, Weibull, and Gompertzian distributions. It estimates the parameter values using Maximum Likelihood Estimation (MLE) and Expectation Maximization (EM) methods. The mixture model is validated through goodness-of-fit tests using failure datasets. Chapter 10 contains the conclusions by highlighting the noble features and future scope of the research works done. The key contributions and significance of the works carried out in the thesis are mentioned. The investigations done in the thesis will provide valuable insights into the applicability of the finite mixture model developed in different frameworks by using the lifetime data. The relevant references are listed alphabetically at the end of the thesis.