On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains

DOI10.1214/aoap/1026915624zbMath1016.60042OpenAlexW1979472779MaRDI QIDQ1872403

Akimichi Takemura, Satoshi Kuriki

Publication date: 6 May 2003

Published in: The Annals of Applied Probability (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1214/aoap/1026915624

zbMATH Keywords

Gauss-Bonnet theoremMorse functionKarhunen-Loeve expansionNaiman's inequalitychi-bar-squared distributionkinematic fundamental formula

Mathematics Subject Classification ID

Random fields (60G60) Gaussian processes (60G15) Sample path properties (60G17) Integral geometry (53C65)

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Cites Work

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• Curvature measures and generalized Morse theory
• Hotelling's theorem on the volume of tubes: Some illustrations in simultaneous inference and data analysis
• Conservative confidence bands in curvilinear regression
• On Hotelling's formula for the volume of tubes and Naiman's inequality
• Weights of \(\overline{\chi}^2\) distribution for smooth or piecewise smooth cone alternatives
• Tail probability via the tube formula when the critical radius is zero
• On excursion sets, tube formulas and maxima of random fields.
• Some geometry of the cone of nonnegative definite matrices and weights of associated \(\overline{\chi}^2\) distribution
• Tail probabilities of the maxima of multilinear forms and their applications.
• Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images
• Tail probabilities of the maxima of Gaussian random fields
• Curvature Measures
• On Hotelling's Approach to Testing for a Nonlinear Parameter in Regression
• Towards a Unified Theory of Inequality Constrained Testing in Multivariate Analysis
• Morse Theory. (AM-51)
• Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics
• Tubes and Spheres in n-Spaces, and a Class of Statistical Problems
• On the Volume of Tubes
• Shrinkage estimation towards a closed convex set with a smooth boundary