Skip to main content
Log in

Nonparametric curve estimation with bernstein estimates

  • Published:
Metrika Aims and scope Submit manuscript

Abstract

In this paper we propose a Bernstein type estimate of the regression functionm(x)=E[Y|X=x]. Various local and global asymptotic properties of this estimate are studied.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

5 References

  • Bauer H (1991) Wahrscheinlichkeitstheorie. Walter de Gruyter Berlin New York

    MATH  Google Scholar 

  • Bennett G (1962) Probability inequalities for the sum of independent random variables. J Amer Statist Assoc 57:33–45

    Article  MATH  Google Scholar 

  • Chu CK, Marron (1991) Choosing a kernel regression estimator. Statist Sciences 6:404–436

    MATH  MathSciNet  Google Scholar 

  • Davis PJ (1975) Interpolation and approximation. Dover Publications Inc New York

    MATH  Google Scholar 

  • Devroye LP (1978) The uniform convergence of the Nadaraya-Watson regression function estimate. Canad J Statist 6:179–191

    Article  MATH  MathSciNet  Google Scholar 

  • Devroye LP, Wagner TJ (1980a) On theL 1 convergence of kernel estimators of regression functions with applications in discrimination. Z Wahrscheinlichkeitstheorie verw Gebiete 25:15–25

    Article  MathSciNet  Google Scholar 

  • Devroye LP, Wagner TJ (1980b) Distribution-free consistency results in nonparametric discrimination and regression function estimation. Annals of Statistics 8:231–239

    MATH  MathSciNet  Google Scholar 

  • Dvoretzky A, Kiefer J, Wolfowitz J (1956) Asymptotic minimax character of the sample distribution and of the classical multinomial estimator. Ann Math Statist 27:642–669

    MathSciNet  MATH  Google Scholar 

  • Eubank RA (1988) Spline smoothing and nonparametric regression. Marcel Dekker New York

    MATH  Google Scholar 

  • Gasser T, Müller HG (1979) Kernel estimation of regression functions. In: Smoothing techniques for curve estimation. Lecture Notes in Math 757:23–68. Springer New York

    Chapter  Google Scholar 

  • Gawronski W, Stadtmüller U (1980) On density estimation by means of Poisson’s distribution. Scand J Statist 7:90–94

    MathSciNet  MATH  Google Scholar 

  • Gawronski W, Stadtmüller U (1981) Smoothing histograms by lattice and continuous distributions. Metrika 28:155–164

    Article  MATH  MathSciNet  Google Scholar 

  • Gawronski W, Stadtmüller U (1984) Linear combinations of iterated generalized Bernstein functions with an application to density estimation. Acta Sci Math 47:205–221

    MATH  Google Scholar 

  • Gawronski W (1985) Strong laws for density estimators of Bernstein type. Per Math Hungar 16:23–43

    Article  MATH  MathSciNet  Google Scholar 

  • Greblicki WD (1974) Asymptotically optimal probabilistic algorithms for pattern recognition and identification. Monografie No. 3 Prace Naukowe Instytutu Cybernetyki Technicznej Politechniki Wroclawskiej No. 18, Poland

  • Härdle W (1990) Applied nonparametric regression. Cambridge University Press New York

    MATH  Google Scholar 

  • Hastie T, Loader C (1993) Local regression: Automatic kernel carpentry. Statist Sciences 8:120–143

    Google Scholar 

  • Hoeffding W (1963) Probability inequalities for the sum of bounded random variables. J Amer Statist Assoc 58:13–30

    Article  MATH  MathSciNet  Google Scholar 

  • Kiefer J, Wolfowitz J (1958) On the deviations on the empiric distribution function of vector chance variables. Trans Amer Math Soc 87:173–186

    Article  MATH  MathSciNet  Google Scholar 

  • Konakov VD (1977) On a global measure of deviation for an estimate of the regression line. Theory Prob Appl 22:858–868

    Article  MATH  MathSciNet  Google Scholar 

  • Lorentz GG (1986) Bernstein polynomials. Chelsea Publishing Company New York NY

    MATH  Google Scholar 

  • Michels P (1992) Nichtparametrische Analyse and Prognose von Zeitreihen. Arbeiten zur angewandten Statistik 36. Physica-Verlag Heidelberg

    MATH  Google Scholar 

  • Müller HG (1988) Nonparametric regression analysis of longitudinal data. Lecture Notes in Statistices. Springer Verlag New York

    MATH  Google Scholar 

  • Müller HG (1991) Smooth optimum kernel estimators near endpoints. Biometrika 78:521–530

    MathSciNet  Google Scholar 

  • Nadaraya EA (1964) On estimating regression. Theory Prob Appl 9:141–142

    Article  Google Scholar 

  • Nadaraya EA (1965) On nonparametric estimates of density functions and regression curves. Theory Prob Appl 10:186–190

    Article  MATH  Google Scholar 

  • Nadaraya EA (1970) Remarks on some nonparametric estimates of density functions and regression curves. Theory Prob Appl 15:134–137

    Article  MATH  Google Scholar 

  • Nadaraya EA (1974) The limit distribution of the quadratic deviation of nonparametric estimates of the regression function. Soobshch Akad Nauk Gruz SSR 74,1:33–36

    MathSciNet  Google Scholar 

  • Nadaraya EA (1988) Nonparametric estimation of probability densities and regression curves. Mathematics and its applications (Soviet Series), Kluwer Academic Publishers Dordrecht/Boston/London

    Google Scholar 

  • Noda K (1976) Estimation of a regression function by the Parzen kernel-type density estimators. Ann Inst Statist Math 28:221–234

    Article  MATH  MathSciNet  Google Scholar 

  • Rice J (1984) Boundary modifications for kernel regression. Comm Statist Theory Methods 13:893–900

    Article  MATH  MathSciNet  Google Scholar 

  • Rosenblatt M (1969) Conditional probability density and regression estimators. In: Krishaiah PR (ed) Multivariate Analysis II 25–31, Academic Press New York

    Google Scholar 

  • Schuster EF (1972) Joint distribution of the estimated regression function at a finite number of distinct points. Ann Math Statist 43:84–88

    MathSciNet  MATH  Google Scholar 

  • Stadtmüller U (1983) Asymptotic distributions of smoothed histograms. Metrika 30:145–158

    Article  MATH  MathSciNet  Google Scholar 

  • Stadtmüller U (1986) Asymptotic properties of nonparametric curve estimates. Per Math Hung 17:83–108

    Article  MATH  Google Scholar 

  • Stone CJ (1977) Consistent nonparametric regression. Ann Statist 5:595–645

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Tenbusch A (1994b) The equivalence of weak, strong, and complete convergence inL 1[0, 1]d for Bernstein polynomial density estimates. Submitted to Statistics

  • Watson GS (1964) Smooth regression analysis. Sankhŷ Series A 26:359–372

    MATH  Google Scholar 

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

    Article  Google Scholar 

  • Wheeden L, Zygmund A (1977) Measure and integral: An introduction to real analysis. Marcel Dekker Inc New York and Basel

    MATH  Google Scholar 

  • Wahba G (1990) Spline models for observational data. SIAM Philadelphia

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tenbusch, A. Nonparametric curve estimation with bernstein estimates. Metrika 45, 1–30 (1997). https://doi.org/10.1007/BF02717090

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02717090

Key Words

Navigation