Distributional Properties of Exceedance Statistics

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.