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Title: Estimation of expected Euler characteristic curves of nonstationary Gaussian random fields Authors:  Fabian Telschow - University of California San Diego (United States) [presenting]
Armin Schwartzman - University of California, San Diego (United States)
Dan Cheng - Texas Tech University (United States)
Pratyush Pranav - Ecole Normale Superieure de Lyon (France)
Abstract: The expected Euler characteristic (EEC) curve of excursion sets of a Gaussian random field is used to approximate the distribution of its supremum for high thresholds. Viewed as a function of the excursion threshold it is expressed by the Gaussian kinematic formula (GKF) as a linear function of the Lipschitz-Killing curvatures (LKCs) of the field, which solely depend on the domain and covariance function of the field. So far its use for non-stationary Gaussian fields over non-trivial domains, has been limited because in this case the LKCs are difficult to estimate. Consistent estimators of the LKCs are proposed as linear projections of ``pinned" observed Euler characteristic curves and a linear parametric estimator of the EEC curve is obtained, which is more efficient than its nonparametric counterpart for repeated observations. A multiplier bootstrap modification reduces the variance of the estimator, and allows estimation of LKCs and EEC of the limiting field of non-Gaussian fields satisfying a functional CLT. The proposed methods are evaluated using simulations and applications are presented, e.g., 3D fMRI brain activation.