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On the empty cells of Poisson histograms

Part of: Decision theory Geometric probability and stochastic geometry Parametric inference

Published online by Cambridge University Press:  14 July 2016

Wilfrid S. Kendall
Wilfrid S. Kendall*
Affiliation:
University of Warwick
*
∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. E-mail: [email protected]
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Abstract

This paper considers the histogram of unit cell size built up from m independent observations on a Poisson (μ) distribution. The following question is addressed: what is the limiting probability of the event that there are no unoccupied cells lying to the left of occupied cells of the histogram? It is shown that the probability of there being no such isolated empty cells (or isolated finite groups of empty cells) tends to unity as the number m of observations tends to infinity, but that the corresponding almost sure convergence fails. Moreover this probability does not tend to unity when the Poisson distribution is replaced by the negative binomial distribution arising when μ is randomized by a gamma distribution. The relevance to empirical Bayes statistical methods is discussed.

Keywords

BOREL–CANTELLI LEMMASGEOMETRIC DISTRIBUTIONGAMMA DISTRIBUTIONEMPIRICAL BAYESNEGATIVE BINOMIAL DISTRIBUTIONPOISSON DISTRIBUTION

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 62C12: Empirical decision procedures; empirical Bayes procedures 62F12: Asymptotic properties of estimators
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 3 , September 1993 , pp. 561 - 574
Copyright
Copyright © Applied Probability Trust 1993 

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References

Courant, R. and Hilbert, D. (1953) Methods of Mathematical Physics Volume I. Wiley, New York.Google Scholar
Holst, L. (1986) On birthdays, collectors, occupancy and other classical urn problems. Internat. Statist. Rev. 54, 15–27.Google Scholar
Kinoshita, K. and Resnick, S. (1992) Convergence of scaled random samples in d. Ann. Prob. 19, 1640–1663.Google Scholar
Maritz, J. S. and Lwin, T. (1989) Empirical Bayes Methods, 2nd edn. Chapman and Hall, New York.Google Scholar
Robbins, H. E. (1955) An empirical Bayes approach to statistics. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 1, 157–163.Google Scholar