Abstract
Siegmund and Worsley (1995) considered the problem of testing for a signal with unknown location and scale in a Gaussian random field defined on R^N. The test statistic was the maximum of a Gaussian random field in an N+1 dimensional `scale space', N dimensions for location and 1 dimension for the scale of a smoothing kernel. Two methods were used to derive an approximate null distribution for N=2 and N=3: one based on the method of volumes of tubes, the other based on the expected Euler characteristic of the excursion set. The purpose of this paper is two-fold: to show how the latter method can be extended to higher dimensions, and to apply this more general result to chi^2 fields. The result of Siegmund and Worsley (1995) then follows as a special case. We apply these results to the problem of searching for activation in brain images obtained by functional magnetic resonance imaging (fMRI).