  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.
