# A New Discrete Distribution: Properties, Characterizations, Modeling Real Count Data, Bayesian and Non-Bayesian Estimations

## DOI:

https://doi.org/10.6092/issn.1973-2201/11635## Keywords:

Discretization, Characterizations, Discrete Burr-Hatke Distribution, Bayesian estimation, Metropolis Hastings, Markov chain Monte Carlo, Maximum Likelihood, Cramer-von-Mises.## Abstract

In this work, a new discrete distribution which includes the discrete Burr-Hatke distribution is defined and studied. Relevant statistical properties are derived. The probability mass function of the new distribution can be "right skewed" with different shapes, bimodal and "uniformed". Also, the corresponding hazard rate function can be "monotonically decreasing", "upside down", "monotonically increasing", "upside down increasing", and "upside down-constant-increasing". A numerical analysis for the mean, variance, skewness, kurtosis and the index of dispersion is presented. The new distribution could be useful in the modeling of "under-dispersed" or "overdispersed" count data. Certain characterizations of the new distribution are presented. These characterizations are based on the conditional expectation of a certain function of the random variable and in terms of the hazard rate function. Bayesian and non-Bayesian estimation methods are considered. Numerical simulations for comparing Bayesian and non-Bayesian estimation methods are performed. The new model is applied for modeling carious teeth data and counts of cysts of kidneys data.

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*Statistica*,

*81*(2), 135–162. https://doi.org/10.6092/issn.1973-2201/11635

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