The Annals of Statistics

Stein estimation for the drift of Gaussian processes using the Malliavin calculus

Nicolas Privault and Anthony Réveillac

Full-text: Open access


We consider the nonparametric functional estimation of the drift of a Gaussian process via minimax and Bayes estimators. In this context, we construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and superharmonic functionals on Gaussian space. Our results are illustrated by numerical simulations and extend the construction of James–Stein type estimators for Gaussian processes by Berger and Wolpert [J. Multivariate Anal. 13 (1983) 401–424].

Article information

Ann. Statist., Volume 36, Number 5 (2008), 2531-2550.

First available in Project Euclid: 13 October 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 60H07: Stochastic calculus of variations and the Malliavin calculus 31B05: Harmonic, subharmonic, superharmonic functions

Nonparametric drift estimation Stein estimation Gaussian space Malliavin calculus harmonic analysis


Privault, Nicolas; Réveillac, Anthony. Stein estimation for the drift of Gaussian processes using the Malliavin calculus. Ann. Statist. 36 (2008), no. 5, 2531--2550. doi:10.1214/07-AOS540.

Export citation


  • [1] Alòs, E., Mazet, O. and Nualart, D. (2001). Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 766–801.
  • [2] Berger, J. and Wolpert, R. (1983). Estimating the mean function of a Gaussian process and the Stein effect. J. Multivariate Anal. 13 401–424.
  • [3] Fourdrinier, D., Strawderman, W. E. and Wells, M. T. (1998). On the construction of Bayes minimax estimators. Ann. Statist. 26 660–671.
  • [4] Ibragimov, I. A. and Rozanov, Y. A. (1978). Gaussian Random Processes. Springer, New York.
  • [5] James, W. and Stein, C. (1961). Estimation with quadratic loss. Proc. 4th Berkeley Sympos. Math. Statist. Probab. I 361–379. Univ. California Press, Berkeley.
  • [6] Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of Random Processes, II. Springer, Berlin.
  • [7] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
  • [8] Privault, N. and Réveillac, A. (2008). Stochastic analysis on Gaussian space applied to drift estimation. Preprint. arXiv:0805.2002v1.
  • [9] Prakasa Rao, B. L. S. (1999). Statistical Inference for Diffusion Type Processes. Edward Arnold, London.
  • [10] Stein, C. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 1135–1151.
  • [11] Wolpert, R. and Berger, J. (1982). Incorporating prior information in minimax estimation of the mean of a Gaussian process. In Statistical Decision Theory and Related Topics III 2 (West Lafayette, Ind., 1981) 451–464. Academic Press, New York.