If X and Y are random variables such that P (X> Y) = 1 and the conditional distribution of Y given X is binomial, then Moran (1952) showed that Y and (X-Y) are independent if X is Poisson. This document extends Moran's result to a more general type of conditional distribution of Y given X, using only partial independence of Y and X-Y. This provides a generalization of a recent results of Janardhan and Rao (1982) on the characterization of generalized Polya-Eggenberger distribution. A variant of Moran's theorem is proved which generalizes the results of Patil and Seshadri (1964) on the characterization of the distribution of a random variable x based on some conditions on the conditional distribution of Y given X and the independence of Y and X-Y.

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 Set Contains: Continuous Multivariate Distributions, Volume 1, Models and Applications, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Continuous Univariate Distributions, Volume 1, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Continuous Univariate Distributions, Volume 2, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Discrete Multivariate Distributions by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Univariate Discrete Distributions, 3rd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson Discover the latest advances in discrete distributions theory The Third Edition of the critically acclaimed Univariate Discrete Distributions provides a self-contained, systematic treatment of the theory, derivation, and application of probability distributions for count data. Generalized zeta-function and q-series distributions have been added and are covered in detail. New families of distributions, including Lagrangian-type distributions, are integrated into this thoroughly revised and updated text. Additional applications of univariate discrete distributions are explored to demonstrate the flexibility of this powerful method. A thorough survey of recent statistical literature draws attention to many new distributions and results for the classical distributions. Approximately 450 new references along with several new sections are introduced to reflect the current literature and knowledge of discrete distributions. Beginning with mathematical, probability, and statistical fundamentals, the authors provide clear coverage of the key topics in the field, including: Families of discrete distributions Binomial distribution Poisson distribution Negative binomial distribution Hypergeometric distributions Logarithmic and Lagrangian distributions Mixture distributions Stopped-sum distributions Matching, occupancy, runs, and q-series distributions Parametric regression models and miscellanea Emphasis continues to be placed on the increasing relevance of Bayesian inference to discrete distribution, especially with regard to the binomial and Poisson distributions. New derivations of discrete distributions via stochastic processes and random walks are introduced without unnecessarily complex discussions of stochastic processes. Throughout the Third Edition, extensive information has been added to reflect the new role of computer-based applications. With its thorough coverage and balanced presentation of theory and application, this is an excellent and essential reference for statisticians and mathematicians.

Timely, comprehensive, practical--an important working resource for all who use this critical statistical method Discrete Multivariate Distributions is the only comprehensive, single-source reference for this increasingly important statistical subdiscipline. It covers all significant advances that have occurred in the field over the past quarter century in the theory, methodology, computational procedures, and applications of discrete multivariate distributions in a wide range of disciplines. Distributions covered include multinomial, binomial, negative binomial, Poisson, power series, hypergeometric, Polya-Eggenberger, Ewens, orders, and some families of distributions. Each distribution is presented in its own chapter, along with necessary details and descriptions of real-world applications gleaned from the current literature on discrete multivariate distributions. Discrete Multivariate Distributions is the fourth volume of the ongoing revision of Johnson and Kotz's acclaimed Distributions in Statistics--universally acknowledged to be the definitive work on statistical distributions. Originally planned as a revision of Chapter 11 of that classic, this project soon blossomed into a substantial volume as a result of the unprecedented growth that has occurred in the literature on discrete multivariate distributions and their applications over the past quarter century. The only comprehensive, single-volume work on the subject, this valuable reference affords statisticians direct access to all of the latest developments concerning discrete multivariate distributions. Concentrating primarily on areas of interest to theoretical as well as applied statisticians, the authors provide complete coverage of several important discrete multivariate distributions. These include multinomial, binomial, negative binomial, Poisson, power series, hypergeometric, Polya-Eggenberger, Ewens, orders, and some families of distributions. Discrete Multivariate Distributions begins with a general overview of the multivariate method in which the authors lay the basic theoretical groundwork for the discussions that follow. For clarity and consistency, subsequent chapters follow a similar format, beginning with a concise historical account followed by a discussion of properties and characteristics. Coverage then advances to in-depth explorations of inferential issues and applications, liberally supplemented with helpful details and a collection of real-world applications obtained from the authors' extensive searches of current literature worldwide. Discrete Multivariate Distributions is an essential working resource for researchers, professionals, practitioners, and graduate students in statistics, mathematics, computer science, engineering, medicine, and the biological sciences.