Tilted Euler Characteristic Densities for Central Limit Random Fields, with Application to "Bubbles"

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 ${\Bbb E}_{\varphi}(A_{u})$. Under isotropy, one can use the expansion ${\Bbb E}_{\varphi}(A_{u})=\sum_{k}\scr{V}_{k}\rho _{k}(u)$, where $\scr{V}_{k}$ 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.