Some new extensions and generalizations of lifetime distributions with applications

Abstract

In this thesis, families and class of new and generalized distributions called exponentiated half logistic-Kumaraswamy-G (EHL-KUM-G) distribution, gamma odd Burr X-G (GOBX-G) distribution, Topp-Leone odd Burr X-G power series (TLOBX-GPS) distribution, Topp-Leoneexponentiated half logistic odd Burr X-G (TLEHLOBX-G) distribution and Marshall-OlkinTopp-Leone odd Burr X-G (MOTLOBX-G) distribution are developed and studied in detail. The development of these distributions is motivated by challenges faced in modelling lifetime real-data. In real-life, datasets are sometimes heavy-tailed and skewed, and traditional distributions may not provide a good fit. Additionally, some of the datasets may have outliers which traditional distributions may also not e↵ectively handle. However, researchers have proposed alternative distributions as alternatives but some may su↵er the issue of parameter identifiability which may hinder estimation accuracy. Most importantly, real-data often deviate from the assumptions of standard symmetric distributions, hence the need for models that may enhance their flexibility and tail behavior. For that, we developed distributions that will accommodate complex data patterns, and capturing the nuances often encountered in real-world scenarios. These newly developed families were subjected to detailed exploration of their mathematical and statistical properties. This includes studying their quantile functions, density expansions, moments and generating functions, incomplete moments, R´enyi entropy and order statistics. Further, special cases of these family which considered baseline distributions such as Weibull and log-logistic distributions were studied in detail to understand their behavior. Even though other parameter estimation techniques were presented, maximum likelihood estimation (MLE) was employed as the primary method in this study due to its widespread application and strong theoretical foundation. The performance of MLE was further assessed using root mean squared error and absolute bias, which indicated that the technique is consistent and e↵ective. The special cases within these families were compared against standard distributions as well as various other existing distributions. This evaluation aimed to gauge their adaptability and practicality across a range of real-world datasets with diverse structures. This comparison was carried out using graphical representations and assessments based on several goodness-of-fit statistics, including 2 log-likelihood, Akaike information criterion, bayesian information criterion, consistent Akaike information criterion, Cram´er-Von Mises, Andersen-Darling, Kolmogorov-Smirnov (and its p-value) and the sum of squares. Our comparison results consistently indicated that the special cases derived from these new families consistently excelled in handling real-life datasets. The implications of this finding are profound, suggesting that these newly developed families have the potential to greatly enhance the accuracy and flexibility of statistical analyses in realworld scenarios.