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In Bayesian statistics, one frequently encounters priors and posteriors that are product of two probability density functions. In this paper, we discuss three such priors/posteriors, provide motivation and derive expressions for their moments, median and mode. Forty seven motivating examples are discussed. We expect that this paper could serve as a useful reference for practitioners of Bayesian statistics. It could also encourage further research in this area.

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Authors: Saralees Nadarajah and Samuel Kotz


The number of citations of journal papers is an important measure of the impact of research. Thus, the modeling of citation behavior needs attention. Burrell, Egghe, Rousseau and others pioneered this type of modeling. Several models have been proposed for the citation distribution. In this note, we derive the most comprehensive collection of formulas for the citation distribution, covering some 17 flexible families. The corresponding estimation procedures are also derived by the method of moments. We feel that this work could serve as a useful reference for the modeling of citation behavior.

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Consider a two-dimensional discrete random variable (X, Y) with possible values 1, 2, . . . , I for X and 1, 2, . . . , J for Y. For specifying the distribution of (X, Y), suppose both conditional distributions of X given Y and Y given X are specified. In this paper, we address the problem of determining whether a given set of constraints involving marginal and conditional probabilities and expectations of functions are compatible or most nearly compatible. To this end, we incorporate all those information with the Kullback-Leibler (K-L) divergence and power divergence statistics to obtain the most nearly compatible probability distribution when the two conditionals are not compatible, under the discrete set up. Finally, a comparative study is carried out between the K-L divergence and power divergence statistics for some illustrative examples.

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