# SURE shrinkage of gaussian paths and signal identification

Nicolas Privault; Anthony Réveillac

ESAIM: Probability and Statistics (2011)

- Volume: 15, page 180-196
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topPrivault, Nicolas, and Réveillac, Anthony. "SURE shrinkage of gaussian paths and signal identification." ESAIM: Probability and Statistics 15 (2011): 180-196. <http://eudml.org/doc/277133>.

@article{Privault2011,

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.},

author = {Privault, Nicolas, Réveillac, Anthony},

journal = {ESAIM: Probability and Statistics},

keywords = {estimation; sure shrinkage; thresholding; denoising; gaussian processes; Malliavin calculus; SURE shrinkage; Gaussian processes; shrinkage of Gaussian paths; continuous-time Gaussian noise; Stein unbiased risk; input signal; occupation times; signal identification},

language = {eng},

pages = {180-196},

publisher = {EDP-Sciences},

title = {SURE shrinkage of gaussian paths and signal identification},

url = {http://eudml.org/doc/277133},

volume = {15},

year = {2011},

}

TY - JOUR

AU - Privault, Nicolas

AU - Réveillac, Anthony

TI - SURE shrinkage of gaussian paths and signal identification

JO - ESAIM: Probability and Statistics

PY - 2011

PB - EDP-Sciences

VL - 15

SP - 180

EP - 196

AB - 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.

LA - eng

KW - estimation; sure shrinkage; thresholding; denoising; gaussian processes; Malliavin calculus; SURE shrinkage; Gaussian processes; shrinkage of Gaussian paths; continuous-time Gaussian noise; Stein unbiased risk; input signal; occupation times; signal identification

UR - http://eudml.org/doc/277133

ER -

## References

top- [1] J.M. Azaïs and M. Wschebor, Level Sets and Extrema of Random Processes and Fields. Wiley, Hoboken (2009). Zbl1168.60002MR2478201
- [2] E.N. Belitser and B.Y. Levit, On minimax filtering over ellipsoids. Math. Methods Statist.4 (1995) 259–273. Zbl0836.62070MR1355248
- [3] S.M. Berman, Local times and sample function properties of stationary Gaussian processes. Trans. Amer. Math. Soc.137 (1969) 277–299. Zbl0184.40801MR239652
- [4] J. Cuzick, Boundary crossing probabilities for stationary Gaussian processes and Brownian motion. Trans. Amer. Math. Soc.263 (1981) 469–492. Zbl0463.60034MR594420
- [5] D.L. Donoho and I.M. Johnstone, Ideal spatial adaptation by wavelet shrinkage. Biometrika81 (1994) 425–455. Zbl0815.62019MR1311089
- [6] D.L. Donoho and I.M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc.90 (1995) 1200–1224. Zbl0869.62024MR1379464
- [7] D. Geman and J. Horowitz, Occupation densities. Ann. Probab.8 (1980) 1–67. Zbl0499.60081MR556414
- [8] G.K. Golubev, Minimax filtration of functions in L2. Probl. Inf. Transm.18 (1982) 272–278. Zbl0533.62086MR711908
- [9] D. Nualart, The Malliavin calculus and related topics. Probability and its Applications. Springer-Verlag, Berlin, second edition (2006). Zbl1099.60003MR2200233
- [10] M. Nussbaum, Minimax risk, Pinsker bound, in Encyclopedia of Statistical Sciences, S. Kotz Ed. Wiley, New York (1999).
- [11] J. Pickands, Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc.145 (1969) 51–73. Zbl0206.18802MR250367
- [12] M.S. Pinsker, Optimal filtration of square-integrable signals in Gaussian noise. Probl. Inf. Transm.16 (1980) 52–68. Zbl0452.94003MR624591
- [13] H.V. Poor, An introduction to signal detection and estimation. Springer Texts in Electrical Engineering. Springer-Verlag, New York, second edition (1994). Zbl0811.94001MR1270019
- [14] N. Privault and A. Réveillac, Superefficient drift estimation on the Wiener space. C. R. Acad. Sci. Paris Sér. I Math.343 (2006) 607–612. Zbl1101.62067MR2269873
- [15] N. Privault and A. Réveillac, Stein estimation for the drift of Gaussian processes using the Malliavin calculus. Ann. Stat.35 (2008) 2531–2550. Zbl1274.62256MR2458197
- [16] N. Privault and A. Réveillac, Stein estimation of Poisson process intensities. Stat. Inference Stoch. Process.12 (2009) 37–53. Zbl1205.62123MR2486115
- [17] C. Qualls and H. Watanabe, Asymptotic properties of Gaussian processes. Ann. Math. Statist.43 (1972) 580–596. Zbl0247.60031MR307318
- [18] D. Revuz and M. Yor, Continuous martingales and Brownian motion, Vol. 293 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, third edition (1999). Zbl0917.60006MR1725357
- [19] C. Stein, Estimation of the mean of a multivariate normal distribution. Ann. Stat.9 (1981) 1135–1151. Zbl0476.62035MR630098
- [20] M. Weber, The supremum of Gaussian processes with a constant variance. Prob. Th. Rel. Fields81 (1989) 585–591. Zbl0659.60060MR995813

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.