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A Poisson approximation with applications to the number of maxima in a discrete sample

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  • Olofsson, Peter

Abstract

We present a Poisson approximation with applications to extreme value theory. Let X1, X2,... be i.i.d. and let Mn(1)[greater-or-equal, slanted]Mn(2)[greater-or-equal, slanted]...[greater-or-equal, slanted]Mn(j) be the j largest order statistics. Then the asymptotic behavior of the vector (Mn(1),...,Mn(j)) is the same as that of (MN(1),...,MN(j)) where N is a random variable which is independent of X1, X2,... and has a Poisson distribution with mean n. The distribution of (MN(1),...,MN(j)) is easy to obtain since the points X1, X2,...,XN form a Poisson process on the real line. The mean measure is n dF where F is the distribution function of the Xi. We apply this to the problem of multiple maxima in discrete samples, in particular from the geometric distribution where it is known that the number of maxima has no limiting distribution.

Suggested Citation

  • Olofsson, Peter, 1999. "A Poisson approximation with applications to the number of maxima in a discrete sample," Statistics & Probability Letters, Elsevier, vol. 44(1), pages 23-27, August.
    • Handle: RePEc:eee:stapro:v:44:y:1999:i:1:p:23-27
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    1. Baryshnikov, Yuliy & Eisenberg, Bennett & Stengle, Gilbert, 1995. "A necessary and sufficient condition for the existence of the limiting probability of a tie for first place," Statistics & Probability Letters, Elsevier, vol. 23(3), pages 203-209, May.
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    Cited by:

    1. Eisenberg, Bennett, 2008. "On the expectation of the maximum of IID geometric random variables," Statistics & Probability Letters, Elsevier, vol. 78(2), pages 135-143, February.

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