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Shape-constrained estimation in functional regression with Bernstein polynomials

Author

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  • Ghosal, Rahul
  • Ghosh, Sujit
  • Urbanek, Jacek
  • Schrack, Jennifer A.
  • Zipunnikov, Vadim

Abstract

Shape restrictions on functional regression coefficients such as non-negativity, monotonicity, convexity or concavity are often available in the form of a prior knowledge or required to maintain a structural consistency in functional regression models. A new estimation method is developed in shape-constrained functional regression models using Bernstein polynomials. Specifically, estimation approaches from nonparametric regression are extended to functional data, properly accounting for shape-constraints in a large class of functional regression models such as scalar-on-function regression (SOFR), function-on-scalar regression (FOSR), and function-on-function regression (FOFR). Theoretical results establish the asymptotic consistency of the constrained estimators under standard regularity conditions. A projection based approach provides point-wise asymptotic confidence intervals for the constrained estimators. A bootstrap test is developed facilitating testing of the shape constraints. Numerical analysis using simulations illustrates improvement in efficiency of the estimators from the use of the proposed method under shape constraints. Two applications include i) modeling a drug effect in a mental health study via shape-restricted FOSR and ii) modeling subject-specific quantile functions of accelerometry-estimated physical activity in the Baltimore Longitudinal Study of Aging (BLSA) as outcomes via shape-restricted quantile-function on scalar regression (QFOSR). R software implementation and illustration of the proposed estimation method and the test is provided.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:csdana:v:178:y:2023:i:c:s0167947322001943
    DOI: 10.1016/j.csda.2022.107614
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    References listed on IDEAS

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    1. Wang, J. & Ghosh, S.K., 2012. "Shape restricted nonparametric regression with Bernstein polynomials," Computational Statistics & Data Analysis, Elsevier, vol. 56(9), pages 2729-2741.
    2. Philip T. Reiss & Jeff Goldsmith & Han Lin Shang & R. Todd Ogden, 2017. "Methods for Scalar-on-Function Regression," International Statistical Review, International Statistical Institute, vol. 85(2), pages 228-249, August.
    3. Davidson, Russell & Flachaire, Emmanuel, 2008. "The wild bootstrap, tamed at last," Journal of Econometrics, Elsevier, vol. 146(1), pages 162-169, September.
    4. Reiss Philip T. & Huang Lei & Mennes Maarten, 2010. "Fast Function-on-Scalar Regression with Penalized Basis Expansions," The International Journal of Biostatistics, De Gruyter, vol. 6(1), pages 1-30, August.
    5. Hojin Yang & Veerabhadran Baladandayuthapani & Arvind U.K. Rao & Jeffrey S. Morris, 2020. "Quantile Function on Scalar Regression Analysis for Distributional Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(529), pages 90-106, January.
    6. Fan, Zhaohu & Reimherr, Matthew, 2017. "High-dimensional adaptive function-on-scalar regression," Econometrics and Statistics, Elsevier, vol. 1(C), pages 167-183.
    7. Daisuke Yagi & Yining Chen & Andrew L. Johnson & Timo Kuosmanen, 2020. "Shape-Constrained Kernel-Weighted Least Squares: Estimating Production Functions for Chilean Manufacturing Industries," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 38(1), pages 43-54, January.
    8. Fang Yao & Hans-Georg Müller, 2010. "Functional quadratic regression," Biometrika, Biometrika Trust, vol. 97(1), pages 49-64.
    9. Rahul Ghosal & Arnab Maity & Timothy Clark & Stefano B. Longo, 2020. "Variable selection in functional linear concurrent regression," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 69(3), pages 565-587, June.
    10. Ghosal, Rahul & Ghosh, Sujit K., 2022. "Bayesian inference for generalized linear model with linear inequality constraints," Computational Statistics & Data Analysis, Elsevier, vol. 166(C).
    11. Yao, Fang & Muller, Hans-Georg & Wang, Jane-Ling, 2005. "Functional Data Analysis for Sparse Longitudinal Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 577-590, June.
    12. I. Gijbels & M. A. Ibrahim & A. Verhasselt, 2017. "Shape testing in quantile varying coefficient models with heteroscedastic error," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(2), pages 391-406, April.
    13. 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.
    14. Eunji Lim & Peter W. Glynn, 2012. "Consistency of Multidimensional Convex Regression," Operations Research, INFORMS, vol. 60(1), pages 196-208, February.
    15. 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, September.
    16. Gareth M. James & Courtney Paulson & Paat Rusmevichientong, 2020. "Penalized and Constrained Optimization: An Application to High-Dimensional Website Advertising," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(529), pages 107-122, January.
    17. 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, June.
    18. M. Ahkim & I. Gijbels & A. Verhasselt, 2017. "Shape testing in varying coefficient models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(2), pages 429-450, June.
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