• Bernoulli
  • Volume 13, Number 4 (2007), 910-932.

Semi-parametric second-order efficient estimation of the period of a signal

I. Castillo

Full-text: Open access


This paper is concerned with the estimation of the period of an unknown periodic function in Gaussian white noise. A class of estimators of the period is constructed by means of a penalized maximum likelihood method. A second-order asymptotic expansion of the risk of these estimators is obtained. Moreover, the minimax problem for the second-order term is studied and an estimator of the preceding class is shown to be second order efficient.

Article information

Bernoulli, Volume 13, Number 4 (2007), 910-932.

First available in Project Euclid: 9 November 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

exact minimax asymptotics penalized maximum likelihood second-order efficiency semi-parametric estimation unknown period


Castillo, I. Semi-parametric second-order efficient estimation of the period of a signal. Bernoulli 13 (2007), no. 4, 910--932. doi:10.3150/07-BEJ5077.

Export citation


  • Bickel, P.J., Klaassen, C.A.J., Ritov, Y. and Wellner, J.A. (1998)., Efficient and Adaptive Estimation for Semiparametric Models. New York: Springer.
  • Castillo, I. (2006). Penalized profile likelihood methods and second order properties in semiparametrics. Ph.D. thesis, Université Paris-Sud. Available at,
  • Castillo, I., Lévy-Leduc, C. and Matias, C. (2006). Exact adaptive estimation of the shape of a periodic function with unknown period corrupted by white noise., Math. Methods Statist. 15 146--175.
  • Chen, Z.-G., Wu, K.H. and Dahlhaus, R. (2000). Hidden frequency estimation with data tapers., J. Time Ser. Anal. 21 113--142.
  • Dalalyan, A. (2007). Stein shrinkage and second-order efficiency for semiparametric estimation of the shift., Math. Methods Statist. To appear.
  • Dalalyan, A., Golubev, G. and Tsybakov, A. (2006). Penalized maximum likelihood and semiparametric second order efficiency., Ann. Statist. 34 169--201.
  • Gassiat, E. and Lévy-Leduc, C. (2006). Efficient semi-parametric estimation of the periods in a superposition of periodic functions with unknown shape., J. Time Ser. Anal. 27 877--910.
  • Gill, R.D. and Levit, B.Y. (1995). Applications of the Van Trees inequality: A Bayesian Cramér--Rao bound., Bernoulli 1 59--79.
  • Golubev, G.K. (1988). Estimating the period of a signal of unknown shape corrupted by white noise., Problems Inform. Transmission 24 288--299.
  • Golubev, G.K. and Härdle, W. (2000). Second order minimax estimation in partial linear models., Math. Methods Statist. 9 160--175.
  • Golubev, G.K. and Härdle, W. (2002). On adaptive smoothing in partial linear models., Math. Methods Statist. 11 98--117.
  • Ibragimov, I.A. and Has'minskii, R.Z. (1981)., Statistical Estimation: Asymptotic Theory. New York: Springer.
  • Kimeldorf, G.S. and Wahba, G. (1970). A correspondence between Bayesian estimation on stochastic processes and smoothing by splines., Ann. Math. Statist. 41 495--502.
  • Lavielle, M. and Lévy-Leduc, C. (2005). Semiparametric estimation of the frequency of unknown periodic functions and its application to laser vibrometry., IEEE Transactions on Signal Processing 53 2306--2315.
  • Pinsker, M.S. (1980). Optimal filtration of square-integrable signals in Gaussian noise., Problems Inform. Transmission 16 120--133.
  • Prenat, M. (2001). Vibration modes and laser vibrometry performance with noise., Proceedings of the Physics in Signal and Image Processing (PSIP2001) Conference, Marseille, France.
  • Tsybakov, A.B. (2004)., Introduction à l'estimation non-paramétrique. (Introduction to Nonparametric Estimation.) Paris: Springer.
  • van der Vaart, A.W. (1998)., Asymptotic Statistics. Cambridge Univ. Press.
  • Yoshida, K. (1978)., Functional Analysis. Berlin: Springer.