Bernstein estimation for a copula derivative with application to conditional distribution and regression functionals

Abstract

Bernstein estimators attracted considerable attention as smooth nonparametric estimators for distribution functions, densities, copulas and copula densities. The present paper adds a parallel result for the first-order derivative of a copula function. This result then leads to Bernstein estimators for a conditional distribution function and its important functionals such as the regression and quantile functions. Results of independent interest have been derived such as an almost sure oscillation behavior of the empirical copula process and a Bahadur-type almost sure asymptotic representation for the Bernstein estimator of a regression quantile function. Simulations demonstrate the good performance of the proposed estimators.

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References

  1. Babu GJ, Canty AJ, Chaubey YP (2002) Application of Bernstein polynomials for smooth estimation of a distribution and density function. J Stat Plan Inf 105:377–392

    MathSciNet  Article  MATH  Google Scholar 

  2. Babu GJ, Chaubey YP (2006) Smooth estimation of a distribution and density function on a hypercube using Bernstein polynomials for dependent random vectors. Stat Prob Lett 76:959–969

    MathSciNet  Article  MATH  Google Scholar 

  3. Bojanic R, Cheng F (1989) Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation. J Math Anal Appl 141:136–151

    MathSciNet  Article  MATH  Google Scholar 

  4. Bouezmarni T, Rolin J-M (2007) Bernstein estimator for unbounded density function. J Nonparametr Stat 19:145–161

    MathSciNet  Article  MATH  Google Scholar 

  5. Bouezmarni T, El Gouch A, Taamouti T (2013) Bernstein estimator for unbounded copula densities. Stat Risk Model 30:343–360

    MathSciNet  MATH  Google Scholar 

  6. Bouezmarni T, Funke F, Camirand Lemyre F (2014) Regression estimation based on Bernstein density copulas (submitted)

  7. Chiu S-T (1996) A comparative review of bandwidth selection for kernel density estimation. Stat Sin 6:129–145

    MathSciNet  MATH  Google Scholar 

  8. Fan J, Gijbels I (1992) Variable bandwidth and local linear regression smoothers. Ann Stat 20:2008–2036

    MathSciNet  Article  MATH  Google Scholar 

  9. Gaenssler P, Stute W (1987) Seminar on empirical processes. In: DMV seminar 9. Birkhäuser, Basel

  10. Ghosh JK (1971) A new proof of the Bahadur representation of quantiles and an application. Ann Math Stat 42:1957–1961

    MathSciNet  Article  MATH  Google Scholar 

  11. Janssen P, Swanepoel JWH, Veraverbeke N (2012) Large sample behavior of the Bernstein copula estimator. J Stat Plan Inf 142:1189–1197

    MathSciNet  Article  MATH  Google Scholar 

  12. Janssen P, Swanepoel JWH, Veraverbeke N (2014) A note on the asymptotic behavior of the Bernstein estimator of a copula density. J Multivar Anal 124:480–487

    MathSciNet  Article  MATH  Google Scholar 

  13. Lee TCM, Solo V (1999) Bandwidth selection for local linear regression: a simulation study. Comput Stat 14:515–532

    Article  MATH  Google Scholar 

  14. Leblanc A (2009) Chung–Smirnov property for Bernstein estimators of distribution functions. J Nonparametr Stat 21:133–142

    MathSciNet  Article  MATH  Google Scholar 

  15. Leblanc A (2010) A bias-reduced approach to density estimation using Bernstein polynomials. J Nonparametr Stat 22:459–475

    MathSciNet  Article  MATH  Google Scholar 

  16. Leblanc A (2012) On estimating distribution functions using Bernstein polynomials. Ann Inst Stat Math 64:919–943

    MathSciNet  Article  MATH  Google Scholar 

  17. Lehmann EL (1966) Some concepts of dependence. Ann Math Stat 37:1137–1153

    MathSciNet  Article  MATH  Google Scholar 

  18. Mason D, Swanepoel JWH (2011) A general result on the uniform in bandwidth consistency of kernel-type function estimators. Test 20:72–94

    MathSciNet  Article  MATH  Google Scholar 

  19. Nelsen R (2006) An introduction to copulas, 2nd edn. Springer, New York

    Google Scholar 

  20. Noh H, El Ghouch A, Bouezmarni T (2013) Copula-based regression estimation and inference. J Am Stat Assoc 108:676–688

    MathSciNet  Article  MATH  Google Scholar 

  21. Parzen E (1979) Nonparametric statistical data modeling. J Am Stat Assoc 74:105–121

    MathSciNet  Article  MATH  Google Scholar 

  22. Sancetta A, Satchell S (2004) The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econ Theory 20:535–562

    MathSciNet  Article  MATH  Google Scholar 

  23. Serfling RJ (1980) Approximation theorems of mathematical statistics. Wiley, New York

    Google Scholar 

  24. Sklar A (1959) Fonctions de répartition à \(n\) dimensions et leurs marges. Pub Inst Stat Univ Paris 8:229–231

    MathSciNet  MATH  Google Scholar 

  25. Stone CJ (1977) Consistent nonparametric regression. Ann Stat 4:595–814

    MathSciNet  Article  MATH  Google Scholar 

  26. Stute W (1984) The oscillation behavior of empirical processes: the multivariate case. Ann Prob 12:361–379

    MathSciNet  Article  MATH  Google Scholar 

  27. Swanepoel JWH, Allison JS (2013) Some new results on the empirical copula estimator with applications. Stat Prob Lett 83:1731–1739

    MathSciNet  Article  MATH  Google Scholar 

  28. Tenbusch A (1994) Two dimensional Bernstein polynomial density estimators. Metrika 41:233–253

    MathSciNet  Article  MATH  Google Scholar 

  29. Tenbusch A (1997) Nonparametric curve estimation with Bernstein estimates. Metrika 45:1–30

    MathSciNet  Article  MATH  Google Scholar 

  30. Vitale RA (1973) A Bernstein polynomial approach to density estimation. Commun Stat 2:493–506

    Google Scholar 

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Acknowledgments

The authors thank Mr. Charl Pretorius for his important help with the simulation section. They also thank the editor, associate editor and two referees for their valuable remarks and suggestions. The work was supported by the IAP Research Network P7/13 of the Belgian State (Belgian Science Policy). J. Swanepoel thanks the National Research Foundation of South Africa for financial support. N. Veraverbeke is also extraordinary professor at the North-West University, Potchefstroom, South Africa.

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Correspondence to Noël Veraverbeke.

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Janssen, P., Swanepoel, J. & Veraverbeke, N. Bernstein estimation for a copula derivative with application to conditional distribution and regression functionals. TEST 25, 351–374 (2016). https://doi.org/10.1007/s11749-015-0459-x

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Keywords

  • Asymptotic normality
  • Asymptotic representation
  • Bernstein estimation
  • Copula
  • Copula density
  • Oscillation of empirical copula process
  • Quantile function

Mathematics Subject Classification

  • 62G05
  • 62G07
  • 62G20