## Tilted Euler characteristic densities for central limit random fields, with application to “bubbles”.(English)Zbl 1226.60075

Summary: 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 $$\rho _k$$ 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 $$\rho _k(u)$$ 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 including an application to “bubbles” data accompany the theory.

### MSC:

 60G60 Random fields 62E20 Asymptotic distribution theory in statistics 62M40 Random fields; image analysis 53A99 Classical differential geometry 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60F05 Central limit and other weak theorems
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