Tilted Euler characteristic densities for Central Limit random fields, with application to `bubbles'

This paper is the long version of:

Chamandy, N., Worsley, K.J., Taylor, J.E. & Gosselin, F. (2008). Tilted Euler characteristic densities for Central Limit random fields, with application to `bubbles'. Annals of Statistics, accepted.

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 E(φ(A_u)). Under isotropy one can use the expansion E(φ(A_u))=∑_kV_kρ_k(u), where V_k is an intrinsic volume of the parameter space and ρ_k(u) 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 ρ_k(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 and an application to `bubbles' data accompany the theory.

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