Asymptotic near-efficiency of the “Gibbs-energy (GE) and empirical-variance” estimating functions for fitting Matérn 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 Matérn family with known “regularity” index ν>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 zTR−1z∕n where z is the vector of observations and R is the autocorrelation matrix for z associated with a candidate range. The present study considers the case where the observation is 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 ν, surprisingly close to 1 provided ν is not too large. We also obtain, for all ν, convergence to 1 of the analog ratio for the microergodic-parameter. Furthermore, we discuss possible non-normality of the noise, and the impact of a “not small enough” grid-step.