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Connections of the Poisson weight function to overdispersion and underdispersion. (English) Zbl 1133.62007

Summary: We establish several connections of the Poisson weight function to overdispersion and underdispersion. Specifically, we establish that the logconvexity (logconcavity) of the mean weight function is a necessary and sufficient condition for overdispersion (underdispersion) when the Poisson weight function does not depend on the original Poisson parameter. We also discuss some properties of the weighted Poisson distributions (WPD). We then introduce a notion of pointwise duality between two WPDs and discuss some associated properties. Next, we present some illustrative examples and provide a discussion on various Poisson weight functions used in practice. Finally, some concluding remarks are made.

MSC:

62E10 Characterization and structure theory of statistical distributions
62E15 Exact distribution theory in statistics
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