Translator Disclaimer
2020 Asymptotic properties of the maximum likelihood and cross validation estimators for transformed Gaussian processes
François Bachoc, José Betancourt, Reinhard Furrer, Thierry Klein
Electron. J. Statist. 14(1): 1962-2008 (2020). DOI: 10.1214/20-EJS1712

Abstract

The asymptotic analysis of covariance parameter estimation of Gaussian processes has been subject to intensive investigation. However, this asymptotic analysis is very scarce for non-Gaussian processes. In this paper, we study a class of non-Gaussian processes obtained by regular non-linear transformations of Gaussian processes. We provide the increasing-domain asymptotic properties of the (Gaussian) maximum likelihood and cross validation estimators of the covariance parameters of a non-Gaussian process of this class. We show that these estimators are consistent and asymptotically normal, although they are defined as if the process was Gaussian. They do not need to model or estimate the non-linear transformation. Our results can thus be interpreted as a robustness of (Gaussian) maximum likelihood and cross validation towards non-Gaussianity. Our proofs rely on two technical results that are of independent interest for the increasing-domain asymptotic literature of spatial processes. First, we show that, under mild assumptions, coefficients of inverses of large covariance matrices decay at an inverse polynomial rate as a function of the corresponding observation location distances. Second, we provide a general central limit theorem for quadratic forms obtained from transformed Gaussian processes. Finally, our asymptotic results are illustrated by numerical simulations.

Citation

Download Citation

François Bachoc. José Betancourt. Reinhard Furrer. Thierry Klein. "Asymptotic properties of the maximum likelihood and cross validation estimators for transformed Gaussian processes." Electron. J. Statist. 14 (1) 1962 - 2008, 2020. https://doi.org/10.1214/20-EJS1712

Information

Received: 1 December 2019; Published: 2020
First available in Project Euclid: 28 April 2020

zbMATH: 07200248
MathSciNet: MR4091860
Digital Object Identifier: 10.1214/20-EJS1712

Subjects:
Primary: 62M30
Secondary: 62F12

JOURNAL ARTICLE
47 PAGES


SHARE
Vol.14 • No. 1 • 2020
Back to Top