Title:

Characterization of discrete distributions based on conditionality and damage models : contribution to the theory of discrete univariate and multivariate distributions, with particular emphasis on conditionality characterizations using methods related to the RaoRubin property

Let X and Y be two nonnegative, integervalued random variables, such that X > Y, and let Z=XY. When the conditional distribution of Y/X=n is used for making inferences about the distribution of X or the distribution of Y, this model is called a conditionality model. Rao (Classical and Contagious discrete distributions 1963), introduced a new version of the conditionality model; he called this a damage model. In this model X represents an observation which is produced by some natural process and which may be partially damaged; Y/X=n is the destructive process. Thus Y stands for what we actually observe of X (the remaining part of X). Rao and Rubin (Sankhya 1964) obtained a characterization for the Poisson distribution using damage model theory and a condition which has come to be known as the RaoRubin condition. In this thesis an extension of the RaoRubin characterization which has been suggested by the work of Shanbhag (J.A.P. 1977) has been used to characterize many wellknown discrete distributions as the distribution of X or as the distribution of Y/X=n when the other one of the two is given. This model is extended to provide characterizations for truncated distributions. A new model is suggested enabling us to characterize finite discrete distributions, truncated and untruncated. Bivariate and Multivariate extensions of all the results obtained in the Univariate case are derived. Finally, the damage model is examined in the more general situation where either the distribution of X or the distribution of Y/X=n is a compound distribution. Some interesting characterizations are provided by this situation. Many of the results existing in the literature in this field are found to be special cases of our results.
