Statistical Science

Gaussian Integrals and Rice Series in Crossing Distributions—to Compute the Distribution of Maxima and Other Features of Gaussian Processes

Georg Lindgren

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We describe and compare how methods based on the classical Rice’s formula for the expected number, and higher moments, of level crossings by a Gaussian process stand up to contemporary numerical methods to accurately deal with crossing related characteristics of the sample paths.

We illustrate the relative merits in accuracy and computing time of the Rice moment methods and the exact numerical method, developed since the late 1990s, on three groups of distribution problems, the maximum over a finite interval and the waiting time to first crossing, the length of excursions over a level, and the joint period/amplitude of oscillations.

We also treat the notoriously difficult problem of dependence between successive zero crossing distances. The exact solution has been known since at least 2000, but it has remained largely unnoticed outside the ocean science community.

Extensive simulation studies illustrate the accuracy of the numerical methods. As a historical introduction an attempt is made to illustrate the relation between Rice’s original formulation and arguments and the exact numerical methods.

Article information

Statist. Sci., Volume 34, Number 1 (2019), 100-128.

First available in Project Euclid: 12 April 2019

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Computational statistics distribution of maximum Durbin’s formula excursion length distribution first passage independent interval assumption level crossings multivariate normal probabilities period/amplitude distribution Rice’s formula RIND program statistical computation stochastic process successive crossing distance distribution truncated normal moments Wafo toolbox


Lindgren, Georg. Gaussian Integrals and Rice Series in Crossing Distributions—to Compute the Distribution of Maxima and Other Features of Gaussian Processes. Statist. Sci. 34 (2019), no. 1, 100--128. doi:10.1214/18-STS662.

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Supplemental materials

  • Supplementary material. The supplement contains instructions how to obtain the Wafo package, example codes to some of the figures, and additional Matlab m-files.