Adaptive estimation of covariance functions via wavelet thresholding and information projection

Jérémie Bigot (1) , Rolando Biscay Lirio (2) , Jean-Michel Loubes (1) , Lilian Muniz Alvarez (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Instituto de Cibernetica, Matematica y Fisica
3 Facultad de Matematica y Computacion

In this paper, we study the problem of nonparametric adaptive estimation of the covariance function of a stationary Gaussian process. For this purpose, we consider a wavelet-based method which combines the ideas of wavelet approximation and estimation by information projection in order to warrants the positive semidefiniteness property of the solution. The spectral density of the process is estimated by projecting the wavelet thresholding expansion of the periodogram onto a family of exponential functions. This ensures that the spectral density estimator is a strictly positive function. Then, by Bochner theorem, we obtain a semidefinite positive estimator of the covariance function. The theoretical behavior of the estimator is established in terms of rate of convergence of the Kullback-Leibler discrepancy over Besov classes. We also show the excellent practical performance of the estimator in some numerical experiments.

Domaines: Théorie [stat.TH]; Statistiques [math.ST]

Jérémie Bigot, Rolando Biscay Lirio, Jean-Michel Loubes, Lilian Muniz Alvarez. Adaptive estimation of covariance functions via wavelet thresholding and information projection. 2009. ⟨hal-00440424v2⟩