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Asymptotic near-efficiency of the “Gibbs-energy and empirical-variance” estimating functions for fitting Matérn models — I: Densely sampled processes

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  • Girard, Didier A.

Abstract

Consider one realization of a continuous-time Gaussian process Z which belongs to the Matérn family with known regularity index ν>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 zTR−1z/n where z is the vector of observations and R is the autocorrelation matrix for 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 ν, surprisingly close to 1 provided ν is not too large. This latter condition on ν 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.

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  • Girard, Didier A., 2016. "Asymptotic near-efficiency of the “Gibbs-energy and empirical-variance” estimating functions for fitting Matérn models — I: Densely sampled processes," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 191-197.
  • Handle: RePEc:eee:stapro:v:110:y:2016:i:c:p:191-197
    DOI: 10.1016/j.spl.2015.12.021
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    References listed on IDEAS

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    1. C. G. Kaufman & B. A. Shaby, 2013. "The role of the range parameter for estimation and prediction in geostatistics," Biometrika, Biometrika Trust, vol. 100(2), pages 473-484.
    2. Peter Guttorp & Tilmann Gneiting, 2006. "Studies in the history of probability and statistics XLIX On the Matern correlation family," Biometrika, Biometrika Trust, vol. 93(4), pages 989-995, December.
    3. Wu, Wei-Ying & Lim, Chae Young & Xiao, Yimin, 2013. "Tail estimation of the spectral density for a stationary Gaussian random field," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 74-91.
    4. Zhang, Hao, 2004. "Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 250-261, January.
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    1. Girard, Didier A., 2020. "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 & Probability Letters, Elsevier, vol. 162(C).

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