2019/11/25 by François Bachoc, José Betancourt, Bachoc, François +5
Environmental Science · Mathematics · #Advanced Statistical Methods and Models #FOS: Mathematics #Soil Geostatistics and Mapping #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1911.11199
openalex publication_date 2019/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The asymptotic analysis of covariance parameter estimation of Gaussian\nprocesses has been subject to intensive investigation. However, this asymptotic\nanalysis is very scarce for non-Gaussian processes. In this paper, we study a\nclass of non-Gaussian processes obtained by regular non-linear transformations\nof Gaussian processes. We provide the increasing-domain asymptotic properties\nof the (Gaussian) maximum likelihood and cross validation estimators of the\ncovariance parameters of a non-Gaussian process of this class. We show that\nthese estimators are consistent and asymptotically normal, although they are\ndefined as if the process was Gaussian. They do not need to model or estimate\nthe non-linear transformation. Our results can thus be interpreted as a\nrobustness of (Gaussian) maximum likelihood and cross validation towards\nnon-Gaussianity. Our proofs rely on two technical results that are of\nindependent interest for the increasing-domain asymptotic literature of spatial\nprocesses. First, we show that, under mild assumptions, coefficients of\ninverses of large covariance matrices decay at an inverse polynomial rate as a\nfunction of the corresponding observation location distances. Second, we\nprovide a general central limit theorem for quadratic forms obtained from\ntransformed Gaussian processes. Finally, our asymptotic results are illustrated\nby numerical simulations.\n