We suppose that our observations can be decomposed into a fixed signal plus random noise, where the noise is modelled as a particular stationary Gaussian random field in $N$-dimensional Euclidean space. The signal has the form of a known function centered at an unknown location and multiplied by an unknown amplitude, and we are primarily interested in a test to detect such a signal. There are many examples where the signal scale or width is assumed known, and the test is based on maximising a Gaussian random field over all locations in a subset of $N$-dimensional Euclidean space. The novel feature of this work is that the width 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 location plus one dimension for the width. Two convergent approaches are used to approximate the null distribution: one based on the method of Knowles and Siegmund, which uses a version of Weyl's tube formula for manifolds with boundaries, and the other based on some recent work by Worsley, which uses the Hadwiger characteristic of excursion sets as introduced by Adler. Finally we compare the power of our method with one based on a fixed but perhaps incorrect signal width.
"Testing for a Signal with Unknown Location and Scale in a Stationary Gaussian Random Field." Ann. Statist. 23 (2) 608 - 639, April, 1995. https://doi.org/10.1214/aos/1176324539