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

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 nonlinear transformations of Gaussian processes. We provide the increasingdomain 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.

Bounds in $L^1$ Wasserstein distance on the normal approximation of general M-estimators

“…The next condition is interpreted as a global identifiability of the correlation parameter. This condition is already made in the increasing-domain asymptotic literature on cross validation and is not restrictive on the sequence (x i ) i∈N and the set {k θ } [10,12]. Condition 15.…”

The effective sample size for multivariate spatial processes with an application to soil contamination

“…The ML estimate of the ESS is A recent discussion about the asymptotic analysis of covariance parameter estimation for Gaussian processes in the misspecified case and for transformed Gaussian processes can be found in Bachoc (2018) and Bachoc et al (2020), respectively.…”

Asymptotic analysis of ML-covariance parameter estimators based on covariance approximations

“…The measurability condition further ensures that random matrices based upon first, second, and third order partial derivatives (with respect to θ) of c θ are well defined. Note that (1) and (2) of Assumption 3.1 are different to the conditions assumed in [3] (compare also to Condition 3.2 imposed in [4], or Condition 4 stated in [6]). In [3] it is assumed that the given covariance function K θ is not only bounded on R d , but also it decays sufficiently fast in the Euclidean norm on R d .…”

“…Similar remarks can be made with regard to the conditions imposed on the partial derivatives of c θ with respect to θ (see Lemma A.5). Finally, (3) of Assumption 3.1 is also imposed in [3] (compare also to [7] and [6]). It guarantees that the minimal eigenvalues of Σ θ x (n) are bounded from below, uniformly in n ∈ N + , x (n) ∈ (S X ) n , and θ ∈ Θ (see Lemma A.3 and also Lemma 4.1 for the probabilistic statement).…”

“…The expression pseudo is used because for fixed ω ∈ Ω, for the functions θ → l n (θ)(ω) and θ → l n (θ)(ω), the respective matrices Σ θ x (n) and Σ θ x (n) are not necessarily positive definite and thus we rely on the notion of the pseudoinverse of an n × n matrix. Note that in the literature about ML estimators for covariance parameters under an increasing-domain asymptotic framework, the notion of modified opposite log-likelihoods can be considered as standard (see for instance [3], [4], and also [6]). There, the notion of pseudo-likelihoods is not of interest, as imposed regularity conditions will ensure that covariance matrices are anyway positive definite, regardless of the chosen parameter values, the sampled observation points and the number of sampled observation points.…”