### A Projection Approach to Monotonic Regression with Bernstein Polynomials

ZHU Guo, FANG Xiangzhong

1. School of Mathematical Sciences, Peking University, Beijing 100871, China
• Received:2020-12-15 Revised:2021-07-24 Online:2022-10-25 Published:2022-10-12
• Supported by:
This research was supported by Science Challenge Project under Grant No.TZ2018001.

ZHU Guo, FANG Xiangzhong. A Projection Approach to Monotonic Regression with Bernstein Polynomials[J]. Journal of Systems Science and Complexity, 2022, 35(5): 1910-1928.

Monotonic regression problems have been widely seen in many fields like economics and biostatistics.Usually the monotonic parameter space is used by the Bayesian methods using Bernstein polynomials.In this paper the authors extend the usual parameter space to a larger space in which all the proper parameters making the regression function to be monotonic are included.In order to ensure that the problem could be solved in the new parameter space,the authors use a projection posterior method to make inference.The authors show the proposed method has good approximation properties and performs well compared with other competing methods both in simulations and in practical applications.
 [1] Barlow R E, Bartholomew D J, Bremner J M, et al., Statistical inference under order restrictions, The theory and application of isotonic regression, Wiley Series in Probability and Mathematical Stastics, John Wiley&Sons, London-New York-Sydney, 1972.[2] Ramsay and James O, Estimating smooth monotone functions, Journal of the Royal Statistical Society:Series B (Statistical Methodology), 1998, 60(2):365-375.[3] Mammen E, Marron J S, Turlach B A, et al., A general projection framework for constrained smoothing, Statistical Science, 2001, 16(3):232-248.[4] Engebretsen S, Glad I K, et al., Additive monotone regression in high and lower dimensions, Statistics Surveys, 2019, 13:1-51.[5] Shively T S, Sager T W, and Walker S G, A Bayesian approach to non-parametric monotone function estimation, Journal of the Royal Statistical Society:Series B (Statistical Methodology), 2009, 71(1):159-175.[6] Lin L and Dunson D B, Bayesian monotone regression using Gaussian process projection, Biometrika, 2014, 101(2):303-317.[7] Shively T S, Walker S G, and Damien P, Nonparametric function estimation subject to monotonicity, convexity and other shape constraints, Journal of Econometrics, 2011, 161(2):166-181.[8] Wang X and Berger J O, Estimating shape constrained functions using Gaussian processes, SIAM/ASA Journal on Uncertainty Quantification, 2016, 4(1):1-25.[9] Choi T and Lenk P J, Bayesian analysis of shape-restricted functions using Gaussian process priors, Statistica Sinica, 2017, 27(1):43-69.[10] Chang I S, Chien L C, Chao A H, et al., Shape restricted regression with random Bernstein polynomials, Complex Datasets and Inverse Problems, 2007, 54:187-202.[11] McKay Curtis S and Ghosh S K, A variable selection approach to monotonic regression with Bernstein polynomials, Journal of Applied Statistics, 2011, 38(5):961-976.[12] Patra S, Constrained Bayesian inference through posterior projection with applications, PhD thesis, Duke University, 2019.[13] Kwessi E, Double penalized semi-parametric signed-rank regression with adaptive LASSO, Journal of Systems Science&Complexity, 2021, 34(1):381-401.[14] Park T and Casella G, The Bayesian Lasso, Journal of the American Statistical Association, 2008, 103(482):681-686.[15] Yi T, Wang Z, and Yi D, Bayesian sieve methods:Approximation rates and adaptive posterior contraction rates, Journal of Nonparametric Statistics, 2018, 30(3):716-741.[16] Li J, The metric projection and its applications to solving variational inequalities in Banach spaces, Fixed Point Theory, 2004, 5(2):285-298.[17] Stein O, How to solve a semi-infinite optimization problem, European Journal of Operational Research, 2012, 223(2):312-320.[18] Hettich R and Kortanek K O, Semi-infinite programming:Theory, methods, and applications, SIAM Review, 1993, 35(3):380-429.[19] Mishra S K, Singh Y, and Verma R U, Saddle point criteria in nonsmooth semi-infinite minimax fractional programming problems, Journal of Systems Science&Complexity, 2018, 31(2):446-462.[20] Ghosal S, Convergence rates for density estimation with Bernstein polynomials, The Annals of Statistics, 2001, 29(5):1264-1280.[21] Lorentz G G, Bernstein Polynomials, American Mathematical Soc., New York, 2013.[22] Cleveland W S, Grosse E, and Shyu W M, Local regression models, Statistical Models in S, 2017, 309-376.[23] Dette H, Neumeyer N, and Pilz K F, A simple nonparametric estimator of a monotone regression function, Bernoulli, 2003m, 12(3), DOI:10.3150/bj/1151525131.[24] Wang J and Ghosh S K, Shape restricted nonparametric regression with Bernstein polynomials, Computational Statistics&Data Analysis, 2012, 56(9):2729-2741.[25] Wang X and Li F, Isotonic smoothing spline regression, Journal of Computational and Graphical Statistics, 2008, 17(1):21-37.
 No related articles found!
Viewed
Full text

Abstract