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DISTRIBUTIONAL PROPERTIES OF EXCEEDANCE STATISTICS

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JACEK WESOLOWSKI^{1} AND MOHAMMAD AHSANULLAH^{2}

^{1} *Mathematical Institute, Warsaw University of Technology,*

Plac Politechniki 1, 00-661 Warsaw, Poland

^{2} *Department of Management Sciences, Rider University,*

Lawrenceville, NJ 08648-3099, U.S.A.
(Received October 21, 1996; revised May 26, 1997)

**Abstract.**
Behaviour of a sequence of independent
identically distributed random variables with respect to a
random threshold is investigated. Three statistics connected
with exceeding the threshold are introduced, their exact and
asymptotic distributions are derived. Also distribution-free
properties, leading to some common and some new discrete
distributions, are considered. Identification of
equidistribution of observations and the threshold are
discussed. In this context relations between the exponential
and gamma distributions are studied and a new derivation of
the celebrated Laplace expansion for the standard normal
distribution function is given.

*Key words and phrases*:
Exceedance statistics,
random threshold, approximation, discrete distribution,
distribution-free properties, characterization, standard
normal distribution function.

**Source**
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