Asymptotic near-efficiency of the "Gibbs-energy and empirical-variance" estimating functions for fitting Matern models to a dense (noisy) series

Details Asymptotic near-efficiency of the "Gibbs-energy and empirical-variance" estimating functions for fitting Matern models to a dense (noisy) series Asymptotic near-efficiency of the "Gibbs-energy and empirical-variance" estimating functions for fitting Matern models to a dense (noisy) series

2009-09-05

arxiv.org/abs/0909.1046v2

Mathematics

Statistics Theory

Scientific paper

Let us call as "Gaussian Gibbs energy" the quadratic form appearing in the maximum likelihood (ML) criterion when fitting a zero-mean multidimensional Gaussian distribution to one realization. We consider a continuous-time Gaussian process $Z$ which belongs to the Mat\'ern family with known "regularity" index $\nu \geq 1/2$. For estimating the range and the variance of $Z$ from observations on a "dense" regular grid, corrupted by a Gaussian white noise of known variance, we propose two simple estimating functions based on the conditional Gibbs energy mean (CGEM) and the empirical variance (EV). We show that the ratio of the large sample mean squared error of the resulting CGEM-EV estimate of the range-parameter to the one of its ML 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 may benefit from very easy numerical implementations, are briefly discussed.

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