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A variable selection approach to monotonic regression with Bernstein polynomials

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  • S. McKay Curtis
  • Sujit K. Ghosh

Abstract

One of the standard problems in statistics consists of determining the relationship between a response variable and a single predictor variable through a regression function. Background scientific knowledge is often available that suggests that the regression function should have a certain shape (e.g. monotonically increasing or concave) but not necessarily a specific parametric form. Bernstein polynomials have been used to impose certain shape restrictions on regression functions. The Bernstein polynomials are known to provide a smooth estimate over equidistant knots. Bernstein polynomials are used in this paper due to their ease of implementation, continuous differentiability, and theoretical properties. In this work, we demonstrate a connection between the monotonic regression problem and the variable selection problem in the linear model. We develop a Bayesian procedure for fitting the monotonic regression model by adapting currently available variable selection procedures. We demonstrate the effectiveness of our method through simulations and the analysis of real data.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:japsta:v:38:y:2011:i:5:p:961-976
    DOI: 10.1080/02664761003692423
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    Citations

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    Cited by:

    1. Ander Wilson & Jessica Tryner & Christian L'Orange & John Volckens, 2020. "Bayesian nonparametric monotone regression," Environmetrics, John Wiley & Sons, Ltd., vol. 31(8), December.
    2. Ouimet, Frédéric, 2021. "Asymptotic properties of Bernstein estimators on the simplex," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    3. Kevin Murray & Samuel Müller & Berwin Turlach, 2013. "Revisiting fitting monotone polynomials to data," Computational Statistics, Springer, vol. 28(5), pages 1989-2005, October.
    4. Yang Liu & Xiaojing Wang, 2020. "Bayesian Nonparametric Monotone Regression of Dynamic Latent Traits in Item Response Theory Models," Journal of Educational and Behavioral Statistics, , vol. 45(3), pages 274-296, June.
    5. Ander Wilson & David M. Reif & Brian J. Reich, 2014. "Hierarchical dose–response modeling for high-throughput toxicity screening of environmental chemicals," Biometrics, The International Biometric Society, vol. 70(1), pages 237-246, March.
    6. 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.
    7. 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.
    8. Taeryon Choi & Hea-Jung Kim & Seongil Jo, 2016. "Bayesian variable selection approach to a Bernstein polynomial regression model with stochastic constraints," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(15), pages 2751-2771, November.
    9. Nadja Klein & Torsten Hothorn & Luisa Barbanti & Thomas Kneib, 2022. "Multivariate conditional transformation models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 116-142, March.
    10. Frédéric Ouimet, 2021. "General Formulas for the Central and Non-Central Moments of the Multinomial Distribution," Stats, MDPI, vol. 4(1), pages 1-10, January.
    11. Wang, J. & Ghosh, S.K., 2012. "Shape restricted nonparametric regression with Bernstein polynomials," Computational Statistics & Data Analysis, Elsevier, vol. 56(9), pages 2729-2741.
    12. Ghosal, Rahul & Ghosh, Sujit & Urbanek, Jacek & Schrack, Jennifer A. & Zipunnikov, Vadim, 2023. "Shape-constrained estimation in functional regression with Bernstein polynomials," Computational Statistics & Data Analysis, Elsevier, vol. 178(C).

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