The distribution of maxima of approximately Gaussian random fields

TL;DR

This paper develops an approximation method for the tail distribution of the maximum of smooth Gaussian random fields, aiding in signal detection against noise by estimating significance levels.

Contribution

It introduces a novel approach using likelihood-ratio identities and local field approximations to estimate the distribution tail for maxima of Gaussian fields.

Findings

The approximation accurately estimates tail probabilities in numerical tests.

The method effectively determines significance levels for maximum score tests.

Applications demonstrate the approach's utility in signal detection problems.

Abstract

Motivated by the problem of testing for the existence of a signal of known parametric structure and unknown ``location'' (as explained below) against a noisy background, we obtain for the maximum of a centered, smooth random field an approximation for the tail of the distribution. For the motivating class of problems this gives approximately the significance level of the maximum score test. The method is based on an application of a likelihood-ratio-identity followed by approximations of local fields. Numerical examples illustrate the accuracy of the approximations.