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A New Family of Discrete Distributions with Mathematical Properties, Characterizations, Bayesian and Non-Bayesian Estimation Methods

Mohamed Aboraya (), Haitham M. Yousof (), G.G. Hamedani () and Mohamed Ibrahim ()
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Mohamed Aboraya: Department of Applied Statistics and Insurance, Faculty of Commerce, Damietta University, Damietta 34519, Egypt
Haitham M. Yousof: Department of Statistics, Mathematics and Insurance, Benha University, Banha 13513, Egypt
G.G. Hamedani: Department of Mathematical and Statistical Sciences, Marquette University, Milwaukee, WI 53201-1881, USA
Mohamed Ibrahim: Department of Applied Statistics and Insurance, Faculty of Commerce, Damietta University, Damietta 34519, Egypt

Mathematics, 2020, vol. 8, issue 10, 1-25

Abstract: In this work, we propose and study a new family of discrete distributions. Many useful mathematical properties, such as ordinary moments, moment generating function, cumulant generating function, probability generating function, central moment, and dispersion index are derived. Some special discrete versions are presented. A certain special case is discussed graphically and numerically. The hazard rate function of the new class can be “decreasing”, “upside down”, “increasing”, and “decreasing-constant-increasing (U-shape)”. Some useful characterization results based on the conditional expectation of certain function of the random variable and in terms of the hazard function are derived and presented. Bayesian and non-Bayesian methods of estimation are considered. The Bayesian estimation procedure under the squared error loss function is discussed. Markov chain Monte Carlo simulation studies for comparing non-Bayesian and Bayesian estimations are performed using the Gibbs sampler and Metropolis–Hastings algorithm. Four applications to real data sets are employed for comparing the Bayesian and non-Bayesian methods. The importance and flexibility of the new discrete class is illustrated by means of four real data applications.

Keywords: discretization; Metropolis-Hastings; Markov chain Monte Carlo; maximum likelihood; Cramér–von Mises; squared error loss function; Cramér–von Mises; Bayesian estimation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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