Skip to main content
SpringerLink
Log in

Estimation of the distribution of random shifts deformation

  • Published:
Mathematical Methods of Statistics Aims and scope Submit manuscript

Abstract

Consider discrete values of functions shifted by unobserved translation effects, which are independent realizations of a random variable with unknown distribution μ modeling the variability in the response of each individual. Our aim is to construct a nonparametric estimator of the density of these random translation deformations using semiparametric preliminary estimates of the shifts. Based on the results of Dalalyan et al. [7], semiparametric estimators are obtained in our discrete framework and their performance studied. From these estimates we construct a nonparametric estimator of the target density. Both rates of convergence and an algorithm to construct the estimator are provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Bates and M. Lindstrom, “Nonlinear Mixed Effects Models for Repeated Measures Data”, Biometrics 46(3), 673–687 (1990).

    Article  MathSciNet  Google Scholar 

  2. L. Brumback and M. Lindstrom, “Self-Modeling with Flexible, Random Time Transformations”, Biometrics 60(2), 461–470 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  3. I. Castillo, Penalized Profile Likelihood Methods and Second Order Properties in Semiparametrics, PhD Thesis (Université Paris-Sud, 2006).

  4. I. Castillo, “Semi-Parametric Second-Order Efficient Estimation of the Period of a Signal”, Bernoulli 13(4), 910–932 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  5. L. Cavalier, G. K. Golubev, D. Picard, and A. B. Tsybakov, “Oracle Inequalities for Inverse Problems”, Ann. Statist. 30(3), 843–874 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  6. D. Chafai and J.-M. Loubes, “Maximum Likelihood for a Certain Class of Inverse Problems: An Application to Pharmakocinetics”, SPL 76, 1225–1237 (2006).

    MATH  MathSciNet  Google Scholar 

  7. A. S. Dalalyan, G. K. Golubev, and A. B. Tsybakov, “Penalized Maximum Likelihood and Semiparametric Second Order Efficiency”, Ann. Statist. 34(1), 169–201 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  8. A. S. Dalalyan, “Stein Shrinkage and Second-Order Efficiency for Semiparametric Estimation of the Shift”, Math. Methods Statist. 16(1), 42–62 (2007).

    Article  MathSciNet  Google Scholar 

  9. S. Darolles, J.-P. Florens, and E. Renault, “Nonparametric Instrumental Regression“, Econometrica (2009) (in press).

  10. M. Davidian and D. Giltinan, “Nonlinear Models for Repeated Measurement Data: An Overview and Update”, J. Agricultural, Biological, and Environmental Statist. 8, 387–419 (2003).

    Article  Google Scholar 

  11. F. Gamboa, J.-M. Loubes, and E. Maza, “Shifts Estimation with M-Estimators”, Electronic J. Statist., 1, 616–640 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  12. T. Gasser and A. Kneip, “Searching for Structure in Curve Samples”, J. Amer. Statist. Assoc. 90, 1179–1188 (1995).

    Article  MATH  Google Scholar 

  13. D. Gervini and T. Gasser, “Self-Modelling Warping Functions”, J. R. Statist. Soc. Ser. B 66(4), 959–971 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Kneip and T. Gasser, “Statistical Tools to Analyze Data Representing a Sample of Curves”, Ann. Statist. 20(3), 1266–1305 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Lavielle and C. Lévy-Leduc, “Semiparametric Estimation of the Frequency of Unknown Periodic Functions and Its Application to Laser Vibrometry Signals”, IEEE Transactions on Signal Processing 53(7), 2306–2315 (2005).

    Article  MathSciNet  Google Scholar 

  16. B. Lindsay, “The Geometry of Mixture Likelihoods: a General Theory”, Ann. Statist. 11(1), 86–94 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  17. J.-M. Loubes, É. Maza, M. Lavielle, and L. Rodríguez, “Road Trafficking Description and Short Term Travel Time Forecasting, with a Classification Method”, Canad. J. Statist. 34(3), 475–491 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  18. F. Mentré and A. Mallet, “Handling Covariates in Population Pharmacokinetics”, Int. J. Biomed. Comp. 36, 25–33 (1994).

    Article  Google Scholar 

  19. J. Ramsay and X. Li, “Curve Registration”, J.R. Statist. Soc. Ser. B 60(2), 351–363 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Rønn, “Nonparametric Maximum Likelihood Estimation for Shifted Curves”, J. R. Stat. Soc. Ser. B. 69(2), 243–259 (2001).

    Article  Google Scholar 

  21. A. B. Tsybakov, Introduction à l’estimation non-paramétrique (Introduction to Nonparametric Estimation) in Mathématiques & Applications (Paris) (Springer, Paris, 2004), Vol. 41.

    Google Scholar 

  22. A.W. van der Vaart, Asymptotic Statistics (Cambridge University Press, 1998).

  23. M. Vimond, “Asymptotic Efficiency of Shifts Estimators withM-Estimators”, Ann. Statist. (2009) (in press).

  24. K. Wang and T. Gasser, “Synchronizing Sample Curves Nonparametrically”, Ann. Statist. 27(2), 439–460 (1999).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Exact Sciences, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

    I. Castillo

  2. Inst. de Math. de Toulouse, Univ. Toulouse III, Toulouse, France

    J. -M. Loubes

Authors
  1. I. Castillo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. -M. Loubes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. Castillo.

About this article

Cite this article

Castillo, I., Loubes, J.M. Estimation of the distribution of random shifts deformation. Math. Meth. Stat. 18, 21–42 (2009). https://doi.org/10.3103/S1066530709010025

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066530709010025

Key words

2000 Mathematics Subject Classification

Search

Navigation