A Flexible Discrete Rayleigh-G Family for Engineering and Reliability Modeling: Properties, Characterizations, Bayesian and Non-Bayesian Inference

Abstract

A novel flexible probability tool for modeling extreme and zero-inflated count data with various hazard rate shapes is introduced in this work. Numerous pertinent statistical and mathematical features are developed and examined. Some important mathematical features are obtained, including, ordinary moments, central moment, dispersion index, L-moments, cumulant generating function and moment generating function. A specific example is investigated numerically and visually examined. The new class of hazard rate function offers a broad range of flexibility, including "monotonically decreasing," "upside down," "monotonically increasing," "constant," "decreasing-constant," and "decreasing-constant-increasing (U-hazard rate function)". Furthermore, the new mass function accommodates many useful forms in the field of modeling, including the "right skewed with one peak", "right skewed with two peaks (right skewed and bimodal)", "symmetric mass function" "left skewed with one peak". The conditional expectation of a certain function of the random variable as well as the hazard function are used to provide relevant characterization results.  For the estimation process, evaluating and comparing inferential effectiveness, Bayesian and non-Bayesian estimation approaches are taken into consideration. We propose and explain the Bayesian estimation method for the squared error loss function. For comparing non-Bayesian versus Bayesian estimates, Markov chain Monte Carlo simulation experiments are carried out using the Metropolis Hastings algorithm and the Gibbs sampler. The Bayesian and non-Bayesian approaches are compared using four real-life applications of count data sets. By using four additional real count data applications, the significance and adaptability of the new discrete class are demonstrated.