×

Testing for a signal with unknown location and scale in a stationary Gaussian random field. (English) Zbl 0898.62119

Summary: 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 M. Knowles and D. Siegmund [Int. Stat. Rev. 57, No. 3, 205-220 (1989; Zbl 0707.62125)] which uses a version of Weyl’s tube formula [H. Weyl, Am. J. Math. 61, 461-472 (1939; Zbl 0021.35503)] for manifolds with boundaries, and the other based on some recent work by K. J. Worsley [ibid., 640-669 (1995; see the following entry, Zbl 0898.62120)] 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.

MSC:

62M40 Random fields; image analysis
52A22 Random convex sets and integral geometry (aspects of convex geometry)
62M09 Non-Markovian processes: estimation
60D05 Geometric probability and stochastic geometry
PDFBibTeX XMLCite
Full Text: DOI