A novel framework for constructing flexible lifetime distributions with applications in reliability

Abstract

The growing complexity of real-world data demands highly adaptable prob- ability models capable of capturing diverse distributional behaviors. This thesis proposes a novel approach for constructing flexible families of probability distri- butions, termed the QT -transformation and its dual concept, QTd -transformation. By combining the quantile function Q(·) with a continuous function T (·) (or Td ), the approach accommodates a variety of mathematical forms, enabling the generation of diverse distributions with different shapes and tail behaviors. Sev- eral functional forms of T (·), including widely used sigmoid functions (logistic, arctangent, and error sigmoid), as well as known transformations (Yun and sig- moidal), are employed. Besides these established functions, a simple arbitrary function satisfying the necessary conditions of T (·) is used to develop a new family of distributions. The study mainly focuses on modifications of the expo- nential distribution; the structural and reliability properties, as well as parameter estimation of all the submodels using the exponential as the baseline, are exam- ined. The thesis further explores the utility of the proposed model in reliability through stress-strength reliability analysis. For single-component systems, es- timation is carried out using both maximum likelihood estimation (MLE) and Bayesian estimation, accompanied by the construction of asymptotic confidence intervals, Bayesian credible intervals, and highest posterior density intervals. This analysis is then extended to s-out-of-k systems under progressive type II censoring, employing MLE, Bayesian, and uniformly minimum variance unbiased estimation techniques. In all cases, the efficacy and applicability of the proposed methods are validated through comprehensive simulation studies and real data applications. The thesis concludes by highlighting the capability of the proposed methods to generate flexible distributional forms, thereby paving the way for future research opportunities and practical implementations