Negative Binomial Distributions For Fixed And Random Parameters

Abstract

The first objective of this project is to construct Negative Binomial Distributions when the two parameters p and r are fixed using various methods based on: Binomial expansion; Poisson – Gamma mixture; Convolution of iid Geometric random variables; Compound Poisson distribution with the iid random variables being Logarithmic series distributions; Katz recursive relation in probability; Experiments where the random variable is the number of failures before the rth success and the total number of trials required to achieve the rth success. Properties considered are the mean, variance, factorial moments, Kurtosis, Skewness and Probability Generating Function. The second objective is to consider p as a random variable within the range 0 and 1. The distributions used are: i. The classical Beta (Beta I) distribution and its special cases (Uniform, Power, Arcsine and Truncated beta distribution). ii. Beyond Beta distributions: Kumaraswamy, Gamma, Minus Log, Ogive and two – sided Power distributions. iii. Confluent and Gauss Hypergeometric distributions. The third objective is to consider r as a discrete random variable. The Logarithmic series and Binomial distributions have been considered. As a continuous random variable, an Exponential distribution is considered for r. The Negative Binomial mixtures obtained have been expressed in at least one of the following forms. a. Explicit form b. Recursive form c. Method of moments form. Comparing explicit forms and the method of moments, some identities have been derived. For further work, other discrete and continuous mixing distributions should be considered. Compound power series distributions with the iid random variables being Geometric or shifted Geometric distributions are Negative Binomial mixtures which need to be studied. Properties, estimations and applications of Negative Binomial mixtures are areas for further research.