Electronic Journal of Statistics

Sharp template estimation in a shifted curves model

Jérémie Bigot, Sébastien Gadat, and Clément Marteau

Full-text: Open access


This paper considers the problem of adaptive estimation of a template in a randomly shifted curve model. Using the Fourier transform of the data, we show that this problem can be transformed into a linear inverse problem with a random operator. Our aim is to approach the estimator that has the smallest risk on the true template over a finite set of linear estimators defined in the Fourier domain. Based on the principle of unbiased empirical risk minimization, we derive a nonasymptotic oracle inequality in the case where the law of the random shifts is known. This inequality can then be used to obtain adaptive results on Sobolev spaces as the number of observed curves tend to infinity. Some numerical experiments are given to illustrate the performances of our approach.

Article information

Electron. J. Statist., Volume 4 (2010), 994-1021.

First available in Project Euclid: 7 October 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 42C40: Wavelets and other special systems 41A29: Approximation with constraints

Template estimation curve alignment inverse problem oracle inequality adaptive estimation random operator


Bigot, Jérémie; Gadat, Sébastien; Marteau, Clément. Sharp template estimation in a shifted curves model. Electron. J. Statist. 4 (2010), 994--1021. doi:10.1214/10-EJS576. https://projecteuclid.org/euclid.ejs/1286455791

Export citation


  • [BG10] J. Bigot and S. Gadat. A deconvolution approach to estimation of a common shape in a shifted curves model., Annals of statistics, to be published, 2010.
  • [BGV09] J. Bigot, F. Gamboa, and M. Vimond. Estimation of translation, rotation and scaling between noisy images using the fourier mellin transform., SIAM Journal on Imaging Sciences, 2:614–645, 2009.
  • [Big06] J. Bigot. Landmark-based registration of curves via the continuous wavelet transform., Journal of Computational and Graphical Statistics, 15(3):542–564, 2006.
  • [BL96] Lawrence D. Brown and Mark G. Low. Asymptotic equivalence of nonparametric regression and white noise., Ann. Statist., 24(6) :2384–2398, 1996.
  • [CGPT02] L. Cavalier, G.K. Golubev, D. Picard, and A.B. Tsybakov. Oracle inequalities for inverse problems., Ann. Statist., 30(3):843–874, 2002. Dedicated to the memory of Lucien Le Cam.
  • [CH05] L. Cavalier and N.W. Hengartner. Adaptive estimation for inverse problems with noisy operators., Inverse Problems, 21(4) :1345–1361, 2005.
  • [CL09] I. Castillo and J.-M. Loubes. Estimation of the distribution of random shifts deformation., Math. Methods Statist., 18(1):21–42, 2009.
  • [CR07] L. Cavalier and M. Raimondo. Wavelet deconvolution with noisy eigenvalues., IEEE Trans. on Signal Processing, 55 :2414–2424, 2007.
  • [CT02] L. Cavalier and A.B. Tsybakov. Sharp adaptation for inverse problems with random noise., Probability Theory and Related Fields, (123):323–354, 2002.
  • [Dal07] A. Dalalyan. Penalized maximum likelihood and semiparametric second-order efficiency., Math. Methods of Statist., 16(1):43–63, 2007.
  • [DGT06] A. Dalalyan, G.K. Golubev, and A.B. Tsybakov. Penalized maximum likelihood and semiparametric second-order efficiency., Ann. Statist., 34(1):169–201, 2006.
  • [DJKP95] D.L. Donoho, I.M. Johnstone, G. Kerkyacharian, and D. Picard. Wavelet shrinkage: Asymptopia?, J. Roy. Statist. Soc. Ser. B, 57:301–369, 1995.
  • [Don95] D.L. Donoho. Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition., Appl. Comput. Harmon. Anal., 2(2):101–126, 1995.
  • [EK01] S. Efromovich and V. Koltchinskii. On inverse problems with unknown operators., IEEE Transactions on Information Theory, 47(7) :2876–2894, 2001.
  • [Fan91] J. Fan. On the optimal rates of convergence for nonparametric deconvolution problems., Ann. Statist., 19 :1257–1272, 1991.
  • [GK92] T. Gasser and A. Kneip. Statistical tools to analyze data representing a sample of curves., Ann. Statist., 20(3) :1266–1305, 1992.
  • [GK95] T. Gasser and A. Kneip. Searching for structure in curve samples., JASA, 90(432) :1179–1188, 1995.
  • [GLM07] F. Gamboa, J-M. Loubes, and E. Maza. Semi-parametric estimation of shifts., Electron. J. Stat., 1:616–640, 2007.
  • [HR08] M. Hoffmann and M. Reiß. Nonlinear estimation for linear inverse problems with error in the operator., Annals of Statistics, 36:310–336, 2008.
  • [IRT08] U. Isserles, Y. Ritov, and T. Trigano. Semiparametric density estimation of shifts between curves., preprint, 2008.
  • [KG88] A. Kneip and T. Gasser. Convergence and consistency results for self-modelling regression., Ann. Statist., 16:82–112, 1988.
  • [LM04] X. Liu and H.G. Müller. Functional convex averaging and synchronization for time-warped random curves., JASA, 99(467):687–699, 2004.
  • [Mar06] C. Marteau. Regularization of inverse problems with unknown operator., Mathematical Methods of Statistics, 15:415–443, 2006.
  • [Mar09] C. Marteau. On the stability of the risk hull method for projection estimator., Journal of Statistical Planning and Inference, 139 :1821–1835, 2009.
  • [RL01] J.O. Ramsay and X. Li. Curve registration., Journal of the Royal Statistical Society, Series B, 63:243–259, 2001.
  • [Ron98] Birgitte B. Ronn. Nonparametric maximum likelihood estimation for shifted curves., JRSS B, 60:351–363, 1998.
  • [RS02] J.O. Ramsay and B.W. Silverman., Functional Data Analysis. Lecture Notes in Statistics, New York: Springer-Verlag, 2002.
  • [RS05] J.O. Ramsay and B.W. Silverman., Applied Functional Data Analysis. Lecture Notes in Statistics, New York: Springer-Verlag, 2005.
  • [Vim10] M. Vimond. Efficient estimation for a subclass of shape invariant models., Annals of statistics, 38(3) :1885–1912, 2010.
  • [WG97] K. Wang and T. Gasser. Alignment of curves by dynamic time warping., Ann. Statist., 25(3) :1251–1276, 1997.