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**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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\textit{N. Chamandy} et al., Ann. Stat. 36, No. 5, 2471--2507 (2008; Zbl 1226.60075)

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