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Asymptotic near-efficiency of the “Gibbs-energy and empirical-variance” estimating functions for fitting Matérn models. I: Densely sampled processes. (English) Zbl 1419.62216

Summary: Consider one realization of a continuous-time Gaussian process \(Z\) which belongs to the Matérn 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 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 \(\mathbf{z}^T R^{- 1} \mathbf{z} / n\) where \(\mathbf{z}\) is the vector of observations and \(R\) is the autocorrelation matrix for \(\mathbf{z}\) 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 \(\nu\), surprisingly close to \(1\) provided \(\nu\) is not too large. This latter condition on \(\nu\) 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.

MSC:

62M09 Non-Markovian processes: estimation
60G15 Gaussian processes
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12 Asymptotic properties of parametric estimators
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