The Annals of Statistics

Tilted Euler characteristic densities for Central Limit random fields, with application to “bubbles”

N. Chamandy, K. J. Worsley, J. Taylor, and F. Gosselin

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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.

Article information

Source
Ann. Statist., Volume 36, Number 5 (2008), 2471-2507.

Dates
First available in Project Euclid: 13 October 2008

Permanent link to this document
https://projecteuclid.org/euclid.aos/1223908100

Digital Object Identifier
doi:10.1214/07-AOS549

Mathematical Reviews number (MathSciNet)
MR2458195

Zentralblatt MATH identifier
1226.60075

Subjects
Primary: 60G60: Random fields 62E20: Asymptotic distribution theory 62M40: Random fields; image analysis
Secondary: 53A99: None of the above, but in this section 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.) 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60F05: Central limit and other weak theorems

Keywords
Saddlepoint approximation isotropic field Poisson process threshold excursion set

Citation

Chamandy, N.; Worsley, K. J.; Taylor, J.; Gosselin, F. Tilted Euler characteristic densities for Central Limit random fields, with application to “bubbles”. Ann. Statist. 36 (2008), no. 5, 2471--2507. doi:10.1214/07-AOS549. https://projecteuclid.org/euclid.aos/1223908100


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