COM-type generalizations of hypergeometric, negative hypergeometric and negative binomial distributions
The Poisson, binomial, hypergeometric, negative hypergeometric and negative binomial are the most basic and well-known discrete probability distributions in statistics literature. These distributions have found applications in many areas of scientific investigation. However, they have limitations. The Poisson distribution is equi-dispersed, the binomial and hypergeometric distributions are both under-dispersed while the negative hypergeometric and the negative binomial are both over-dispersed. In order to make the Poisson distribution more flexible, Shmueli et al. introduced a generalization of the Poisson distribution called the COMP distribution. Subsequently, Borges et al. introduced a similar generalization of the binomial distribution, called the COMPB distribution. Both of these models have become very popular and useful. In this thesis, following a similar spirit, we develop generalizations of the remaining three models: hypergeometric, negative hypergeometric and negative binomial. These generalizations are appropriately called the COM-H, COM-NH and COM-NB respectively. The proposed models are obtained by introducing a shape parameter γ which makes the new models more flexible than their original counterparts. The COM-H model can accommodate both over- and under-dispersion. The COM-NH and COM-NB can display lower and higher index of dispersion than their ordinary counterparts. We have compared the shapes of the probability mass functions of the proposed models for different values of γ. Some salient characteristics of these models such as their limiting behavior, moments, probability generating function, log-concavity and log-convexity and monotonicity of the failure rates are investigated. We express them as weighted distributions and study the properties of the weight functions. We have developed estimation of the parameters of these models by the method of maximum likelihood and examined the behavior of the resulting estimators extensively by simulation. For these models, we have formulated likelihood ratio tests for testing H0 : γ =1 to ascertain if the more general model is appropriate for a given data set. We have investigated power of this test by simulation. For each distribution, we present some examples to show their applicability in various areas by fitting the models to some available data from literature. Finally, to show the versatility of the COM-NB model, we also formulate a COM-NB regression model and present an example.