Hostname: page-component-7bb8b95d7b-s9k8s Total loading time: 0 Render date: 2024-09-18T16:42:13.049Z Has data issue: false hasContentIssue false
  • English
  • Français

SURE shrinkage of Gaussian paths and signal identification*

Published online by Cambridge University Press:  05 January 2012

Nicolas Privault  and
Anthony Réveillac
Nicolas Privault
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371 Singapore. nprivault@ntu.edu.sg
Anthony Réveillac
Affiliation:
Université Paris Dauphine, CEREMADE UMR CNRS 7534, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France. anthony.reveillac@ceremade.dauphine.fr
Get access

Abstract

Using integration by parts on Gaussian space we construct a Stein Unbiased Risk Estimator (SURE) for the drift of Gaussian processes, based on their local and occupation times. By almost-sure minimization of the SURE risk of shrinkage estimators we derive an estimation and de-noising procedure for an input signal perturbed by a continuous-time Gaussian noise.

Keywords

EstimationSURE shrinkagethresholdingdenoisingGaussian processesMalliavin calculus
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. 180 - 196
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J.M. Azaïs and M. Wschebor, Level Sets and Extrema of Random Processes and Fields. Wiley, Hoboken (2009).
Belitser, E.N. and Levit, B.Y., On minimax filtering over ellipsoids. Math. Methods Statist. 4 (1995) 259–273.
Berman, S.M., Local times and sample function properties of stationary Gaussian processes. Trans. Amer. Math. Soc. 137 (1969) 277–299. CrossRef
Cuzick, J., Boundary crossing probabilities for stationary Gaussian processes and Brownian motion. Trans. Amer. Math. Soc. 263 (1981) 469–492. CrossRef
Donoho, D.L. and Johnstone, I.M., Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 (1994) 425–455. CrossRef
Donoho, D.L. and Johnstone, I.M., Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 (1995) 1200–1224. CrossRef
Geman, D. and Horowitz, J., Occupation densities. Ann. Probab. 8 (1980) 1–67. CrossRef
Golubev, G.K., Minimax filtration of functions in L 2. Probl. Inf. Transm. 18 (1982) 272–278.
D. Nualart, The Malliavin calculus and related topics. Probability and its Applications. Springer-Verlag, Berlin, second edition (2006).
M. Nussbaum, Minimax risk, Pinsker bound, in Encyclopedia of Statistical Sciences, S. Kotz Ed. Wiley, New York (1999).
Pickands, J., Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145 (1969) 51–73. CrossRef
Pinsker, M.S., Optimal filtration of square-integrable signals in Gaussian noise. Probl. Inf. Transm. 16 (1980) 52–68.
H.V. Poor, An introduction to signal detection and estimation. Springer Texts in Electrical Engineering. Springer-Verlag, New York, second edition (1994).
Privault, N. and Réveillac, A., Superefficient drift estimation on the Wiener space. C. R. Acad. Sci. Paris Sér. I Math. 343 (2006) 607–612. CrossRef
Privault, N. and Réveillac, A., Stein estimation for the drift of Gaussian processes using the Malliavin calculus. Ann. Stat. 35 (2008) 2531–2550. CrossRef
Privault, N. and Réveillac, A., Stein estimation of Poisson process intensities. Stat. Inference Stoch. Process. 12 (2009) 37–53. CrossRef
Qualls, C. and Watanabe, H., Asymptotic properties of Gaussian processes. Ann. Math. Statist. 43 (1972) 580–596. CrossRef
D. Revuz and M. Yor, Continuous martingales and Brownian motion, Vol. 293 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, third edition (1999).
Stein, C., Estimation of the mean of a multivariate normal distribution. Ann. Stat. 9 (1981) 1135–1151. CrossRef
Weber, M., The supremum of Gaussian processes with a constant variance. Prob. Th. Rel. Fields 81 (1989) 585–591. CrossRef