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October 2008 Tilted Euler characteristic densities for Central Limit random fields, with application to “bubbles”
N. Chamandy, K. J. Worsley, J. Taylor, F. Gosselin
Ann. Statist. 36(5): 2471-2507 (October 2008). DOI: 10.1214/07-AOS549

Abstract

Local increases in the mean of a random field are detected (conservatively) by thresholding a field of test statistics at a level u chosen to control the tail probability or p-value of its maximum. This p-value is approximated by the expected Euler characteristic (EC) of the excursion set of the test statistic field above u, denoted $\mathbb{E}\varphi(A_{u})$. Under isotropy, one can use the expansion $\mathbb{E}\varphi(A_{u})=\sum_{k}\mathcal{V}_{k}\rho_{k}(u)$, where $\mathcal{V}_{k}$ is an intrinsic volume of the parameter space and ρk is an EC density of the field. EC densities are available for a number of processes, mainly those constructed from (multivariate) Gaussian fields via smooth functions. Using saddlepoint methods, we derive an expansion for ρk(u) for fields which are only approximately Gaussian, but for which higher-order cumulants are available. We focus on linear combinations of n independent non-Gaussian fields, whence a Central Limit theorem is in force. The threshold u is allowed to grow with the sample size n, in which case our expression has a smaller relative asymptotic error than the Gaussian EC density. Several illustrative examples including an application to “bubbles” data accompany the theory.

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N. Chamandy. K. J. Worsley. J. Taylor. F. Gosselin. "Tilted Euler characteristic densities for Central Limit random fields, with application to “bubbles”." Ann. Statist. 36 (5) 2471 - 2507, October 2008. https://doi.org/10.1214/07-AOS549

Information

Published: October 2008
First available in Project Euclid: 13 October 2008

zbMATH: 1226.60075
MathSciNet: MR2458195
Digital Object Identifier: 10.1214/07-AOS549

Subjects:
Primary: 60G60, 62E20, 62M40
Secondary: 53A99, 58E05, 60B12, 60F05

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 5 • October 2008
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