Slepian Models and Regression Approximations in Crossing and Extreme Value Theory

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. International Statistical Institute (ISI) is collaborating with JSTOR to digitize, preserve and extend access to International Statistical Review / Revue Internationale de Statistique. SummaryIn crossing theory for stochastic processes the distribution of quantities such as distances between level crossings, maximum height of an excursion between level crossings, amplitude and wavelength, etc., can only be written in the form of infinite-dimensional integrals, which are difficult to evaluate numerically. A Slepian model is an explicit random function representation of the process after a level crossing and it consists of one regression term and one residual process. The regression approximation of a crossing variable is defined as the corresponding variable in the regression term of the Slepian model, and its distribution can be evaluated numerically as a finite-dimensional integral. This paper reviews the use and structure of the Slepian model and the regression method and shows how they can be used to obtain good numerical approximations to various crossing variables. It gives a detailed account of the regression method for Gaussian processes with auxiliary variables chosen in a recursive way. It also presents a package of computer programs for the numerical calculations, and gives numerical examples on excursion lengths as well as wavelength and amplitude distributions. Further examples deal with an engineering 'jump-andbump' problem, and excursions for a X2-process.

Stochastic Doppler shift and encountered wave period distributions in Gaussian waves

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