In a recent paper, Shanbhag (1977) uses an elementary approach from renewal theory to give an extension of a characterization of the Poisson law by Rao-Rubin (1964). In the present paper, a variant of Shanbhag's result is introduced. Using Shanbhag's and this result, several characterizations of truncated and untruncated distributions are obtained.

Let X and Y be two non-negative, integer-valued random variables, such that X > Y, and let Z=X-Y. 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 Rao-Rubin condition. In this thesis an extension of the Rao-Rubin characterization which has been suggested by the work of Shanbhag (J.A.P. 1977) has been used to characterize many well-known 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.

This useful reference/text provides a comprehensive study of the various bivariate discretedistributions that have appeared in the literature- written in an accessible manner thatassumes no more than a first course in mathematical statistics.Supplying individualized treatment of topics while simultaneously exploiting the interrelationshipsof the material, Bivariate Discrete Distributions details the latest techniques ofcomputer simulation for the distributions considered ... contains a general introduction tothe structural properties of discrete distributions, including generating functions, momentrelationships, and the basic ideas of generalizing . . . develops distributions using samplingschemes . .. explores the role of compounding ... covers Waring and "short" distributionsfor use in accident theory ... discusses problems of statistical inference, emphasizing techniquespertinent to the discrete case ... and much more!Containing over 1000 helpful equations, Bivariate Discrete Distributions is

The solutions to these functional equations provide a unified and elegant approach to characterizations of the exponential, geometric, Pareto, Weibull, stable, Poisson and other distributions under a variety of stochastic properties of the random variable. The ICFE also plays an important role in renewal processes, potential theory and other applications of stochastic processes.

Proceedings of the NATO Advanced Study Institute, Trieste, Italy, July 10-August 1, 1980