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Consider one realization of a continuous-time Gaussian process $Z$ whichbelongs to the Mat\' ern family with known ``regularity'' index $\nu >0$. Forestimating the autocorrelation-range and the variance of $Z$ from $n$observations on a fine grid, we studied in Girard (2016) the GE-EV method whichsimply 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 autocorrelationmatrix for ${\bf z}$ associated with a candidate range. The present studyconsiders the case where the observation is ${\bf z}$ plus a Gaussian whitenoise whose variance is known. We propose to simply bias-correct EV and toreplace GE by its conditional mean given the observation. We show that theratio of the large-$n$ mean squared error of the resulting CGEM-EV estimate ofthe range-parameter to the one of its maximum likelihood estimate, and theanalog ratio for the variance-parameter, have the same behavior than in theno-noise case: they both converge, when the grid-step tends to $0$, toward aconstant, only function of $\nu$, surprisingly close to $1$ provided $\nu$ isnot too large. We also obtain, for all $\nu$, convergence to 1 of the analogratio for the microergodic-parameter.

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''