The distribution of maxima of approximately Gaussian random fields (Q930661)

Motivated by the problem of testing for the existence of a signal of known parametric structure and unknown location against a noisy background, the authors take up two central themes. First, they develop a method for the derivation of analytic approximations for the tail of the distribution of the maximum of a centered smooth random field. 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. The other theme involves the detailed investigation of a specific case, which is asymptotically Gaussian but where direct application of results for Gaussian fields does not seem adequate. This field arises in the context of testing for the presence of a signal of a given parametric structure within a noisy image. A score statistic is constructed for each candidate signal and an overall test statistic for the presence of some signal is obtained by maximizing the score over the collection of all candidate locations. The authors consider second order approximations for the tail of the distribution of the test statistic under the null hypothesis of the absence of a signal -- the significance level of the test. Similar methods can be applied to obtain an approximation for the power. It is also shown how the method may be adapted to other models, including (under different technical conditions) the frequently discussed case of smooth Gaussian fields, and how it can be used to obtain higher order approximations.