Siegmund, David O.; Worsley, Keith J. Testing for a signal with unknown location and scale in a stationary Gaussian random field. (English) Zbl 0898.62119 Ann. Stat. 23, No. 2, 608-639 (1995). 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. Cited in 3 ReviewsCited in 25 Documents 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 Keywords:Euler characteristic; integral geometry; image analysis; volume of tubes; adaptive filter; random noise; stationary Gaussian random field; Weyl’s tube formula Citations:Zbl 0707.62125; Zbl 0021.35503; Zbl 0898.62120 PDFBibTeX XMLCite \textit{D. O. Siegmund} and \textit{K. J. Worsley}, Ann. Stat. 23, No. 2, 608--639 (1995; Zbl 0898.62119) Full Text: DOI