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Asymptotic equivalence for nonparametric regression with non-regular errors
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  • Published: 03 November 2011

Asymptotic equivalence for nonparametric regression with non-regular errors

  • Alexander Meister1 &
  • Markus Reiß2 

Probability Theory and Related Fields volume 155, pages 201–229 (2013)Cite this article

Abstract

Asymptotic equivalence in Le Cam’s sense for nonparametric regression experiments is extended to the case of non-regular error densities, which have jump discontinuities at their endpoints. We prove asymptotic equivalence of such regression models and the observation of two independent Poisson point processes which contain the target curve as the support boundary of its intensity function. The intensity of the point processes is of order of the sample size n and involves the jump sizes as well as the design density. The statistical model significantly differs from regression problems with Gaussian or regular errors, which are known to be asymptotically equivalent to Gaussian white noise models.

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References

  1. Brown L.D., Low M.: Asymptotic equivalence of nonparametric regression and white noise. Ann. Stat. 24, 2384–2398 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brown L., Cai T., Low M., Zhang C.-H.: Asymptotic equivalence theory for nonparametric regression with random design. Ann. Stat. 30, 688–707 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brown L., Cai T., Zhou H.H.: Nonparametric regression in exponential families. Ann. Stat. 38, 2005–2046 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brown M.: Discrimination of Poisson processes. Ann. Math. Stat. 42, 773–776 (1971)

    Article  MATH  Google Scholar 

  5. Carter A.: A continuous Gaussian approximation to a nonparametric regression in two dimensions. Bernoulli 12, 143–156 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Carter A.: Asymptotic approximation of nonparametric regression experiments with unknown variances. Ann. Stat. 35, 1644–1673 (2007)

    Article  MATH  Google Scholar 

  7. Carter, A.: Asymptotically sufficient statistics in nonparametric regression experiments with correlated noise. J. Prob. Stat. 2009, 19 (2009) (ID 275308)

  8. Chernozhukov V., Hong H.: Likelihood estimation and inference in a class of nonregular econometric models. Econometrica 72, 1445–1480 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. DeVore R.A., Lorentz G.G.: Constructive Approximation, Grundlehren Series vol. 303. Springer, Berlin (1993)

    Google Scholar 

  10. Gijbels I., Mammen E., Park B., Simar L.: On estimation of monotone and concave frontier functions. J. Am. Stat. Assoc. 94, 220–228 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grama I., Nussbaum M.: Asymptotic equivalence for nonparametric generalized linear models. Prob. Theor. Rel. Fields 111, 167–214 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grama I., Nussbaum M.: Asymptotic equivalence for nonparametric regression. Math. Methods Stat. 11(1), 1–36 (2002)

    MathSciNet  MATH  Google Scholar 

  13. Hall P., van Keilegom I.: Nonparametric “regression” when errors are positioned at end-points. Bernoulli 15, 614–633 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ibragimov, I.A., Hasminskii, R.Z.: Statistical Estimation. Asymptotic Theory. Springer, New York (1981) (translated from the Russian by S. Kotz)

  15. Janssen A., Marohn D.M.: On statistical information of extreme order statistics, local extreme value alternatives and Poisson point processes. J. Multivar. Anal. 48, 1–30 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Karr A.F.: Point Processes and Their Statistical Inference. 2nd edn. Marcel Dekker, New York (1991)

    MATH  Google Scholar 

  17. Knight K.: Limiting distributions of linear programming estimators. Extremes 4, 87–103 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Korostelev A.P., Tsybakov A.B.: Minimax Theory of Image Reconstruction. In: Lecture Notes in Statistics, vol. 82. Springer, New York (1993)

    Book  Google Scholar 

  19. Kutoyants Y.A.: Statistical Inference for Spatial Poisson Processes. In: Lecture Notes in Statistics, vol. 134. Springer, New York (1998)

    Book  Google Scholar 

  20. Le Cam L.M.: Sufficiency and approximate sufficiency. Ann. Math. Stat. 35, 1419–1455 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  21. Le Cam L.M., Yang G.L.: Asymptotics in Statistics, Some Basic Concepts. 2nd edn. Springer, New York (2000)

    MATH  Google Scholar 

  22. Liese F.: Eine informationstheoretische Bedingung für die Äquivalenz unbegrenzt teilbarer Punktprozesse. Math. Nachr. 70, 183–196 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liese F., Vajda I.: Convex Statistical Distances. Teubner, Leipzig (1987)

    MATH  Google Scholar 

  24. Müller U.U., Wefelmeyer W.: Estimation in nonparametric regression with nonregular errors. Comm. Stat. Theor. Methods 39, 1619–1629 (2010)

    Article  MATH  Google Scholar 

  25. Nussbaum M.: Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Stat. 24, 2399–2430 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Reiß M.: Asymptotic equivalence for nonparametric regression with multivariate and random design. Ann. Stat. 36, 1957–1982 (2008)

    Article  MATH  Google Scholar 

  27. Reiß R.-D.: A Course on Point Processes. Springer, New York (1993)

    Book  MATH  Google Scholar 

  28. Tsybakov A.B.: Introduction to Nonparametric Estimation. Springer Series in Statistics, Berlin (2009)

    Book  MATH  Google Scholar 

  29. van de Geer S.A.: Empirical Processes in M-Estimation. Reprint, Cambridge University Press, New York (2006)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Rostock, Ulmenstraße 69, 18051, Rostock, Germany

    Alexander Meister

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Markus Reiß

Authors
  1. Alexander Meister
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  2. Markus Reiß
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Corresponding author

Correspondence to Alexander Meister.

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Meister, A., Reiß, M. Asymptotic equivalence for nonparametric regression with non-regular errors. Probab. Theory Relat. Fields 155, 201–229 (2013). https://doi.org/10.1007/s00440-011-0396-x

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  • Received: 26 January 2011

  • Revised: 11 October 2011

  • Published: 03 November 2011

  • Issue Date: February 2013

  • DOI: https://doi.org/10.1007/s00440-011-0396-x

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Keywords

  • Extreme value statistics
  • Frontier estimation
  • Le Cam distance
  • Le Cam equivalence
  • Poisson point processes

Mathematics Subject Classification (2010)

  • 62B15
  • 62G08
  • 62M30
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