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Unbiased risk estimation method for covariance estimation

Published online by Cambridge University Press:  25 July 2014

Hélène Lescornel ,
Jean-Michel Loubes  and
Claudie Chabriac
Hélène Lescornel
Affiliation:
Institut de Mathématiques de Toulouse UMR 5219, 31062 Toulouse, Cedex 9, France. [email protected]; [email protected]; [email protected]
Jean-Michel Loubes
Affiliation:
Institut de Mathématiques de Toulouse UMR 5219, 31062 Toulouse, Cedex 9, France. [email protected]; [email protected]; [email protected]
Claudie Chabriac
Affiliation:
Institut de Mathématiques de Toulouse UMR 5219, 31062 Toulouse, Cedex 9, France. [email protected]; [email protected]; [email protected]
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Abstract

We consider a model selection estimator of the covariance of a random process. Using the Unbiased Risk Estimation (U.R.E.) method, we build an estimator of the risk which allows to select an estimator in a collection of models. Then, we present an oracle inequality which ensures that the risk of the selected estimator is close to the risk of the oracle. Simulations show the efficiency of this methodology.

Keywords

Covariance estimationmodel selectionU.R.E. method
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 251 - 264
Copyright
© EDP Sciences, SMAI, 2014

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References

R.J. Adler, An introduction to continuity, extrema, and related topics for general gaussian processes. Lect. Note Ser. Institute of Mathematical Statistics (1990).
Bickel, P.J. and Levina, E., Covariance regularization by thresholding. Ann. Statist. 36 (2008) 2577–2604. Google Scholar
J. Bigot, R. Biscay, J.-M. Loubes and L.M. Alvarez, Group lasso estimation of high-dimensional covariance matrices. J. Machine Learn. Res. (2011).
Bigot, J., Biscay, R., Loubes, J.-M. and Muñiz-Alvarez, L., Nonparametric estimation of covariance functions by model selection. Electron. J. Statis. 4 (2010) 822–855. Google Scholar
J. Bigot, R. Biscay Lirio, J.-M. Loubes and L. Muniz Alvarez, Adaptive estimation of spectral densities via wavelet thresholding and information projection (2010).
Biscay, R., Rodrguez, L.M. and Daz-Frances, E., Cross-validation of covariance structures using the frobenius matrix distance as a discrepancy function. J. Stat. Comput. Simul. 58 (1997) 195–215. Google Scholar
T. Cai and M. Yuan, Nonparametric covariance function estimation for functional and longitudinal data. Technical report (2010).
N.A.C. Cressie, Statistics for spatial data. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Revised reprint of the 1991 edition, A Wiley-Interscience Publication. John Wiley and Sons Inc., New York (1993).
Diggle, P.J. and Verbyla, A.P., Nonparametric estimation of covariance structure in longitudinal data. Biometrics 54 (1998) 401–415. Google ScholarPubMed
Journel, A.G., Kriging in terms of projections. J. Int. Assoc. Math. Geol. 9 (1977) 563–586. Google Scholar
C.R. Rao, Linear statistical inference and its applications. Wiley ser. Probab. Stastis. Wiley, 2nd edn. (1973).
G.A.F. Seber, A matrix handbook for statisticians. Wiley ser. Probab. Stastis. Wiley (2008).
G.R. Shorack and J.A. Wellner, Empirical processes with applications to statistics. Wiley (1986).
Stein, C.M., Estimation of the mean of a multivariate normal distribution. Ann. Statis. 9 (1981) 1135–1151. Google Scholar
M.L. Stein. Interpolation of spatial data. Some theory for Kriging. Springer Ser. Statis. Springer-Verlag, New York (1999).
A.B. Tsybakov, Introduction à l’estimation non-paramétrique. Vol. 41 of Math. Appl. Springer (2004).