TESTING FOR A SIGNAL WITH UNKNOWN LOCATION AND SCALE IN A STATIONARY GAUSSIAN RANDOM-FIELD

We suppose that our observations can be decomposed into a fixed signal plus random noise, where the noise is modelled as a particular statio nary Gaussian random field in N-dimensional Euclidean space. The signa l has the form of a known function centered at an unknown location and multiplied by an unknown amplitude, and we are primarily interested i n a test to detect such a signal. There are many examples where the si gnal scale or width is assumed known, and the test is based on maximis ing a Gaussian random field over all locations in a subset of N-dimens ional Euclidean space. The novel feature of this work is that the widt h of the signal is also unknown and the test is based on maximising a Gaussian random field in N + 1 dimensions, N dimensions for the locati on plus one dimension for the width. Two convergent approaches are use d to approximate the null distribution: one based on the method of Kno wles and Siegmund, which uses a version of Weyl's tube formula for man ifolds with boundaries, and the other based on some recent work by Wor sley, which uses the Hadwiger characteristic of excursion sets as intr oduced by Adler. Finally we compare the power of our method with one b ased on a fixed but perhaps incorrect signal width.