Inference and Application of Modified Lehmann Type-II Fr\'echet Distribution using Progressively Censored Data

Abstract

In this article, we introduce a new three-parameter Fr\'echet distribution via the modified Lehmann Type-II class of distributions and investigate its properties, inferential methods, and real-world application. Fundamental distributional properties such as the quantile function, moments, moment generating function, entropy, and order statistics have been discussed. Inferential results have been obtained within the classical and Bayesian frameworks, utilizing a progressive censoring scheme. In the classical estimation framework, maximum likelihood estimation and maximum product spacing estimation are considered to obtain the estimates using the Newton-Raphson method. In addition, approximate confidence intervals have been derived using maximum likelihood estimation estimates. Meanwhile, we have used informative and non-informative prior via likelihood and product spacing functions to find the Bayes estimates in the Bayesian framework. Since the posterior distributions cannot be expressed in closed form, we employ a combination of Gibbs sampling and the Metropolis-Hastings algorithm to obtain the Bayes estimates.Furthermore, credible intervals are constructed using the Bayes estimates under informative and non-informative Prior. A comprehensive simulation study is carried out to assess the performance of the proposed estimation techniques. To demonstrate the practical utility of the proposed model, a real-life dataset is analyzed, showcasing the effectiveness of the proposed methodologies.