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

TL;DR

This paper extends the GE-EV estimation method for Matérn Gaussian processes to account for measurement errors, demonstrating that the asymptotic efficiency ratios remain close to 1, similar to the noise-free case.

Contribution

It introduces a bias-corrected empirical variance and a conditional mean version of the GE estimator for noisy observations, analyzing their asymptotic efficiency.

Findings

Ratios of mean squared errors converge to 1 as grid step tends to 0.

Efficiency ratios are close to 1 for a range of smoothness parameters.

Results hold uniformly for the microergodic parameter.

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 ${\bf z}^T R^{-1} {\bf z}/n$ where ${\bf z}$ is the vector of observations and $R$ is the autocorrelation matrix for ${\bf z}$ associated with a candidate range. The present study considers the case where the observation is ${\bf z}$ 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.