Unbiased risk estimation method for covariance estimation

Hélène Lescornel; Jean-Michel Loubes; Claudie Chabriac

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 251-264
  • ISSN: 1292-8100

Abstract

top
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.

How to cite

top

Lescornel, Hélène, Loubes, Jean-Michel, and Chabriac, Claudie. "Unbiased risk estimation method for covariance estimation." ESAIM: Probability and Statistics 18 (2014): 251-264. <http://eudml.org/doc/274370>.

@article{Lescornel2014,
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.},
author = {Lescornel, Hélène, Loubes, Jean-Michel, Chabriac, Claudie},
journal = {ESAIM: Probability and Statistics},
keywords = {covariance estimation; model selection; U.R.E. method},
language = {eng},
pages = {251-264},
publisher = {EDP-Sciences},
title = {Unbiased risk estimation method for covariance estimation},
url = {http://eudml.org/doc/274370},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Lescornel, Hélène
AU - Loubes, Jean-Michel
AU - Chabriac, Claudie
TI - Unbiased risk estimation method for covariance estimation
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 251
EP - 264
AB - 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.
LA - eng
KW - covariance estimation; model selection; U.R.E. method
UR - http://eudml.org/doc/274370
ER -

References

top
  1. [1] R.J. Adler, An introduction to continuity, extrema, and related topics for general gaussian processes. Lect. Note Ser. Institute of Mathematical Statistics (1990). Zbl0747.60039MR1088478
  2. [2] P.J. Bickel and E. Levina, Covariance regularization by thresholding. Ann. Statist.36 (2008) 2577–2604. Zbl1196.62062MR2485008
  3. [3] J. Bigot, R. Biscay, J.-M. Loubes and L.M. Alvarez, Group lasso estimation of high-dimensional covariance matrices. J. Machine Learn. Res. (2011). Zbl1280.68156MR2877598
  4. [4] J. Bigot, R. Biscay, J.-M. Loubes and L. Muñiz-Alvarez, Nonparametric estimation of covariance functions by model selection. Electron. J. Statis.4 (2010) 822–855. Zbl1329.62365MR2684389
  5. [5] J. Bigot, R. Biscay Lirio, J.-M. Loubes and L. Muniz Alvarez, Adaptive estimation of spectral densities via wavelet thresholding and information projection (2010). 
  6. [6] R. Biscay, L.M. Rodrguez and E. Daz-Frances, Cross-validation of covariance structures using the frobenius matrix distance as a discrepancy function. J. Stat. Comput. Simul.58 (1997) 195–215. Zbl0880.62063
  7. [7] T. Cai and M. Yuan, Nonparametric covariance function estimation for functional and longitudinal data. Technical report (2010). 
  8. [8] 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). Zbl0799.62002MR1239641
  9. [9] P.J. Diggle and A.P. Verbyla, Nonparametric estimation of covariance structure in longitudinal data. Biometrics54 (1998) 401–415. Zbl1058.62600
  10. [10] A.G. Journel, Kriging in terms of projections. J. Int. Assoc. Math. Geol.9 (1977) 563–586. MR456314
  11. [11] C.R. Rao, Linear statistical inference and its applications. Wiley ser. Probab. Stastis. Wiley, 2nd edn. (1973). Zbl0256.62002MR346957
  12. [12] G.A.F. Seber, A matrix handbook for statisticians. Wiley ser. Probab. Stastis. Wiley (2008). Zbl1143.15001MR2365265
  13. [13] G.R. Shorack and J.A. Wellner, Empirical processes with applications to statistics. Wiley (1986). Zbl1170.62365MR838963
  14. [14] C.M. Stein, Estimation of the mean of a multivariate normal distribution. Ann. Statis.9 (1981) 1135–1151. Zbl0476.62035MR630098
  15. [15] M.L. Stein. Interpolation of spatial data. Some theory for Kriging. Springer Ser. Statis. Springer-Verlag, New York (1999). Zbl0924.62100MR1697409
  16. [16] A.B. Tsybakov, Introduction à l’estimation non-paramétrique. Vol. 41 of Math. Appl. Springer (2004). Zbl1029.62034MR2013911

NotesEmbed ?

top

You must be logged in to post comments.