Asymptotic near-efficiency of the “Gibbs-energy and empirical-variance” estimating functions for fitting Matérn models - I: Densely sampled processes

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 propose two simple estimating functions based on the “candidate Gibbs energy” (GE) and the empirical variance (EV). Here a candidate GE designates 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. We show that the ratio of the large-n mean squared error of the resulting GE-EV estimate of the range-parameter to the one of its maximum likelihood estimate, and the analog ratio for the variance-parameter, 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. This latter condition on ν has not to be imposed to obtain the convergence to 1 of the analog ratio for the microergodic-parameter. Possible extensions of this approach, which could be rather easily implemented, are briefly discussed.