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On Moran's Property of the Poisson Distribution

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  • Panaretos, John

Abstract

Two interesting results encountered in the literature concerning the Poisson and the negative binomial distributions are due to MORAN (1952) and PATIL & SESHADRI (1964), respectively. MORAN's result provided a fundamental property of the Poisson distribution. Roughly speaking, he has shown that if Y, Z are independent, non-negative, integer-valued random variables with X=Y | Z then, under some mild restrictions, the conditional distribution of Y | X is binomial if and only if Y, Z are Poisson random variables. Motivated by MORAN's result PATIL & SESHADRI obtained a general characterization. A special case of this characterization suggests that, with conditions similar to those imposed by MORAN, Y | X is negative hypergeometric if and only if Y, Z are negative binomials. In this paper we examine the results of MORAN and PATIL & SESHADRI in the case where the conditional distribution of Y | X is truncated at an arbitrary point k-1 (k=1, 2, …). In fact we attempt to answer the question as to whether MORAN's property of the Poisson distribution, and subsequently PATIL & SESHADRI's property of the negative binomial distribution, can be extended, in one form or another, to the case where Y | X is binomial truncated at k-1 and negative hypergeometric truncated at k-1 respectively

Suggested Citation

  • Panaretos, John, 1983. "On Moran's Property of the Poisson Distribution," MPRA Paper 6231, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:6231
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    File URL: https://mpra.ub.uni-muenchen.de/6231/1/MPRA_paper_6231.pdf
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    More about this item

    Keywords

    Poisson Distribution; Binomial Distribution; Negative Binomial Distribution; Negative Hypergeometric Distribution; Moran's Theorem; Patil & Seshadri's Theorem;

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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