Asymptotic near-efficiency of the ''Gibbs-energy (GE) and empirical-variance'' estimating functions for fitting Mat{é}rn models -- II: Accounting for measurement errors via ''conditional GE mean''

Abstract

Consider one realization of a continuous-time Gaussian process $Z$ which belongs to the Mat\' ern family with known ``regularity'' index $\nu >0$. For estimating the autocorrelation-range and the variance of $Z$ from $n$ observations on a fine grid, we studied in Girard (2016) the GE-EV method which simply retains the empirical variance (EV) and equates it to a candidate ``Gibbs energy (GE)'' i.e.~the quadratic form ${\mathbf{z}}^T{R}^{-1}\mathbf{z}/n$ where $\mathbf{z}$ is the vector of observations and $R$ is the autocorrelation matrix for $\mathbf{z}$ associated with a candidate range. The present study considers the case where the observation is $\mathbf{z}$ plus a Gaussian white noise whose variance is known. We propose to simply bias-correct EV and to replace GE by its conditional mean given the observation. We show that the ratio of the large-$n$ mean squared error of the resulting CGEM-EV estimate of the range-parameter to the one of its maximum likelihood estimate, and the analog ratio for the variance-parameter, have the same behavior than in the no-noise case: they both converge, when the grid-step tends to $0$, toward a constant, only function of $\nu$, surprisingly close to $1$ provided $\nu$ is not too large. We also obtain, for all $\nu$, convergence to 1 of the analog ratio for the microergodic-parameter.

arXiv commentThe previous version (version 2) considered both the case with measurement errors (also called ''nugget-effect'' or simply ''noise'') and the no-noise case. The no-noise case is now in Girard (2016) with more detailed proofs and two additional (wrt version 2) results: a consistency result is proved and the restriction $\nu \ge 1/2$ is eliminated. This version 3 is devoted to the case with measurement errors, and also gives the analogs of these two additional results