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2019 Estimation and prediction of Gaussian processes using generalized Cauchy covariance model under fixed domain asymptotics
Moreno Bevilacqua, Tarik Faouzi
Electron. J. Statist. 13(2): 3025-3048 (2019). DOI: 10.1214/19-EJS1597

Abstract

We study estimation and prediction of Gaussian processes with covariance model belonging to the generalized Cauchy (GC) family, under fixed domain asymptotics. Gaussian processes with this kind of covariance function provide separate characterization of fractal dimension and long range dependence, an appealing feature in many physical, biological or geological systems. The results of the paper are classified into three parts.

In the first part, we characterize the equivalence of two Gaussian measures with GC covariance functions. Then we provide sufficient conditions for the equivalence of two Gaussian measures with Matérn (MT) and GC covariance functions and two Gaussian measures with Generalized Wendland (GW) and GC covariance functions.

In the second part, we establish strong consistency and asymptotic distribution of the maximum likelihood estimator of the microergodic parameter associated to GC covariance model, under fixed domain asymptotics. The last part focuses on optimal prediction with GC model and specifically, we give conditions for asymptotic efficiency prediction and asymptotically correct estimation of mean square error using a misspecified GC, MT or GW model.

Our findings are illustrated through a simulation study: the first compares the finite sample behavior of the maximum likelihood estimation of the microergodic parameter of the GC model with the given asymptotic distribution. We then compare the finite-sample behavior of the prediction and its associated mean square error when the true model is GC and the prediction is performed using the true model and a misspecified GW model.

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Moreno Bevilacqua. Tarik Faouzi. "Estimation and prediction of Gaussian processes using generalized Cauchy covariance model under fixed domain asymptotics." Electron. J. Statist. 13 (2) 3025 - 3048, 2019. https://doi.org/10.1214/19-EJS1597

Information

Received: 1 April 2018; Published: 2019
First available in Project Euclid: 20 September 2019

zbMATH: 07113710
MathSciNet: MR4010591
Digital Object Identifier: 10.1214/19-EJS1597

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Vol.13 • No. 2 • 2019
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