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The asymptotic analysis of covariance parameter estimation of Gaussianprocesses has been subject to intensive investigation. However, this asymptoticanalysis is very scarce for non-Gaussian processes. In this paper, we study aclass of non-Gaussian processes obtained by regular non-linear transformationsof Gaussian processes. We provide the increasing-domain asymptotic propertiesof the (Gaussian) maximum likelihood and cross validation estimators of thecovariance parameters of a non-Gaussian process of this class. We show thatthese estimators are consistent and asymptotically normal, although they aredefined as if the process was Gaussian. They do not need to model or estimatethe non-linear transformation. Our results can thus be interpreted as arobustness of (Gaussian) maximum likelihood and cross validation towardsnon-Gaussianity. Our proofs rely on two technical results that are ofindependent interest for the increasing-domain asymptotic literature of spatialprocesses. First, we show that, under mild assumptions, coefficients ofinverses of large covariance matrices decay at an inverse polynomial rate as afunction of the corresponding observation location distances. Second, weprovide a general central limit theorem for quadratic forms obtained fromtransformed Gaussian processes. Finally, our asymptotic results are illustratedby numerical simulations.

Asymptotic properties of the maximum likelihood and cross validation estimators for transformed Gaussian processes