Validity of the expected Euler characteristic heuristic

By Jonathan E. Taylor, Akimichi Takemura, Robert J. Adler. 2005

110 citations recorded by the source corpus.

We study the accuracy of the expected Euler characteristic approximation to the distribution of the maximum of a smooth, centered, unit variance Gaussian process f. Using a point process representation of the error, valid for arbitrary smooth processes, we show that the error is in general exponentially smaller than any of the terms in the approximation. We also give a lower bound on this exponential rate of decay in terms of the maximal variance of a family of Gaussian processes f x , derived from the original process f.

Subfields: Applied Mathematics, Environmental Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/12a113ab2fec774fecb34b3874659ed5bcf99e73

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In Applied Mathematics

In Environmental Engineering

In Statistics and Probability

Citations in the corpus

Some new random field tools for spatial analysis (2008, background): “ube(A;r))= ¥ å j=0 rj j! Mk j (A); (18) for small enough r. We call the coefﬁcients Mk j (A) the Gaussian Minkowski functionals (GMFs) of A. Clearly Mk 0 (A) = g k(A). Note that, unlike the expansion (17) in Steiner’s theorem, this expansion is an inﬁnite one. To see how this expansion works in the ”
Random Fields and Geometry (2007, background): “It has a number of precursors, the closest being a recent paper [149] with Akimichi Takemura, from which most of this chapter is taken.”
Random fields of multivariate test statistics, with applications to shape analysis (2008, unknown): “The Annals of Statistics 2008, Vol. 36, No. 1, 1–27 DOI: 10.1214/009053607000000406 © Institute of Mathematical Statistics, 2008”
False Discovery Control for Random Fields (2004, methodology): “Second, as shown by Takemura and Kuriki (2003) (see also Taylor, Takemura, and Adler 2003), when search regions are nonconvex, not all terms in the Euler approximation are accurate, whereas the volume term used in the approximation of Piterbarg (1996) is accurate.”
A gaussian kinematic formula (2006, methodology): “A heuristic justification of the above approximation can be found in [2, 6], with a rigorous justification in [17].”
Topological complexity of smooth random functions (2011, background): “In the Gaussian case, this heuristic is now a well established theorem, and the error term is known to be of order exp(−u2(1+ η)/2) for some (often identifiable) η > 0, while both the probability and expectation are of order u M−1 exp(−u2/2) [81].”
Likelihood ratio tests and singularities (2007, background): “Moreover, as pointed out by a referee, the “tube” and “Euler characteristic” methods may be useful for approximating limiting distributions; see [32, 33] and the references therein.”