  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.
