DISTRIBUTIONAL PROPERTIES OF EXCEEDANCE STATISTICS

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

1Mathematical Institute, Warsaw University of Technology, Plac Politechniki 1, 00-661 Warsaw, Poland 2Department 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.