# Asymptotic near-efficiency of the "Gibbs-energy (GE) and empirical-variance" estimating functions for fitting Matérn models - II: Accounting for measurement errors via "conditional GE mean''

1 IPS [2020-....] - Inférence Processus Stochastiques [2020-....]
LJK [2020-....] - Laboratoire Jean Kuntzmann [2020-....]
Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-00413693
Contributor : Didier A. Girard <>
Submitted on : Friday, June 7, 2019 - 10:11:30 AM
Last modification on : Wednesday, August 5, 2020 - 3:24:06 AM

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• HAL Id : hal-00413693, version 3
• ARXIV : 0909.1046

### Citation

Didier A. Girard. Asymptotic near-efficiency of the "Gibbs-energy (GE) and empirical-variance" estimating functions for fitting Matérn models - II: Accounting for measurement errors via "conditional GE mean''. Statistics and Probability Letters, Elsevier, 2020, Statistics and Probability Letters, 162, pp.1-11. ⟨hal-00413693v3⟩

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