The Annals of Statistics

On the Nonparametric Estimation of Covariance Functions

Peter Hall, Nicholas I. Fisher, and Branka Hoffmann

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We describe kernel methods for estimating the covariance function of a stationary stochastic process, and show how to ensure that the estimator has the positive semidefiniteness property. From a practical viewpoint, our method is significant because it does not demand a parametric model for covariance. From a technical angle, our results exhibit a striking departure from those in more familiar cases of kernel estimation. For example, in the context of covariance estimation, kernel estimators can have the same convergence rates as maximum likelihood estimators, and can have exceptionally fast convergence rates when employed to estimate variance.

Article information

Ann. Statist., Volume 22, Number 4 (1994), 2115-2134.

First available in Project Euclid: 11 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62G10: Hypothesis testing

Convergence rate correlation covariance positive semidefinite stochastic process kernel variance variogram


Hall, Peter; Fisher, Nicholas I.; Hoffmann, Branka. On the Nonparametric Estimation of Covariance Functions. Ann. Statist. 22 (1994), no. 4, 2115--2134. doi:10.1214/aos/1176325774.

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