Shape Restricted Nonparametric Regression Based on Multivariate Bernstein Polynomials

There has been increasing interest in estimating a multivariate regression function subject to shape restrictions, such as nonnegativity, isotonicity, convexity and concavity. The estimation of such shape-restricted regression curves is more challenging for multivariate predictors, especially for functions with compact support. Most of the currently available statistical estimation methods for shape restricted regression functions are generally computationally very intensive. Some of the existing methods have perceptible boundary biases. This article considers a suitable class of multivariate Bernstein polynomials and proposes a sieved estimator obtained from a nested sequence of shape-restricted multivariate Bernstein polynomials. The proposed nonparametric estimator is shown to be: (i) the regression function estimate is shown to be the solution of a quadratic programming problem; making it computationally attractive (ii) the nonparametric estimator is shown to be universally consistent under some mild regularity conditions and (iii) the estimation methodology is flexible in the sense that it can be easily adapted to accommodate many popular multivariate shape restrictions. Numerical results derived from simulated data sets and real data analysis are used to illustrate the superior performance of the proposed estimator compared to an existing estimator in terms of various goodness of fit metrics.