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Shape restricted nonparametric regression with Bernstein polynomials


  • Wang, J.
  • Ghosh, S.K.


The objective of this article is to develop a computationally efficient estimator of the regression function subject to various shape constraints. In particular, nonparametric estimators of monotone and/or convex (concave) regression functions are obtained by using a nested sequence of Bernstein polynomials. One of the key distinguishing features of the proposed estimator is that a given shape constraint (e.g., monotonicity and/or convexity) is maintained for any finite sample size and satisfied over the entire support of the predictor space. Moreover, it is shown that the Bernstein polynomial based regression estimator can be obtained as a solution of a constrained least squares method and hence the estimator can be computed efficiently using a quadratic programming algorithm. Finally, the asymptotic properties (e.g., strong uniform consistency) of the estimator are established under very mild conditions, and finite sample properties are explored using several simulation studies and real data analysis. The predictive performances are compared with some of the existing methods.

Suggested Citation

  • Wang, J. & Ghosh, S.K., 2012. "Shape restricted nonparametric regression with Bernstein polynomials," Computational Statistics & Data Analysis, Elsevier, vol. 56(9), pages 2729-2741.
  • Handle: RePEc:eee:csdana:v:56:y:2012:i:9:p:2729-2741
    DOI: 10.1016/j.csda.2012.02.018

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    References listed on IDEAS

    1. S. McKay Curtis & Sujit K. Ghosh, 2011. "A variable selection approach to monotonic regression with Bernstein polynomials," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(5), pages 961-976, February.
    2. Bornkamp, Björn & Ickstadt, Katja, 2009. "A Note on B-Splines for Semiparametric Elicitation," The American Statistician, American Statistical Association, vol. 63(4), pages 373-377.
    3. I-Shou Chang & Chao A. Hsiung & Yuh-Jenn Wu & Che-Chi Yang, 2005. "Bayesian Survival Analysis Using Bernstein Polynomials," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(3), pages 447-466.
    4. Ait-Sahalia, Yacine & Duarte, Jefferson, 2003. "Nonparametric option pricing under shape restrictions," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 9-47.
    5. Chak, Pok Man & Madras, Neal & Smith, Barry, 2005. "Semi-nonparametric estimation with Bernstein polynomials," Economics Letters, Elsevier, vol. 89(2), pages 153-156, November.
    6. Gallant, A. Ronald & Golub, Gene H., 1984. "Imposing curvature restrictions on flexible functional forms," Journal of Econometrics, Elsevier, vol. 26(3), pages 295-321, December.
    7. Terrell, Dek, 1996. "Incorporating Monotonicity and Concavity Conditions in Flexible Functional Forms," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 11(2), pages 179-194, March-Apr.
    8. Sonia Petrone, 1999. "Random Bernstein Polynomials," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(3), pages 373-393.
    9. Melanie Birke & Holger Dette, 2007. "Estimating a Convex Function in Nonparametric Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(2), pages 384-404.
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    Cited by:

    1. Manté, Claude, 2015. "Iterated Bernstein operators for distribution function and density estimation: Balancing between the number of iterations and the polynomial degree," Computational Statistics & Data Analysis, Elsevier, vol. 84(C), pages 68-84.
    2. Roldán López de Hierro, Antonio Francisco & Martínez-Moreno, Juan & Aguilar Peña, Concepción & Roldán López de Hierro, Concepción, 2016. "A fuzzy regression approach using Bernstein polynomials for the spreads: Computational aspects and applications to economic models," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 128(C), pages 13-25.
    3. Hu, Qinqin & Zeng, Peng & Lin, Lu, 2015. "The dual and degrees of freedom of linearly constrained generalized lasso," Computational Statistics & Data Analysis, Elsevier, vol. 86(C), pages 13-26.
    4. Xiaohong Chen & Yin Jia Qiu, 2016. "Methods for Nonparametric and Semiparametric Regressions with Endogeneity: a Gentle Guide," Cowles Foundation Discussion Papers 2032, Cowles Foundation for Research in Economics, Yale University.
    5. repec:bla:scjsta:v:44:y:2017:i:4:p:989-1008 is not listed on IDEAS
    6. Claudia Köllmann & Björn Bornkamp & Katja Ickstadt, 2014. "Unimodal regression using Bernstein–Schoenberg splines and penalties," Biometrics, The International Biometric Society, vol. 70(4), pages 783-793, December.


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