A bias-corrected partial bernstein copula approach for nonparametric regression

Multivariate Bernstein polynomials yield smooth, boundary–bias–free estimators for copulas and their derivatives. The classical partial Bernstein copula derivative of Janssen et al. (2016) suffers from an uncorrected first–order bias that can dominate the mean squared error (MSE) in moderate samples. We introduce a simple double–difference correction that removes the entire leading bias term while increasing the variance by a small fixed multiplicative factor only. When this bias–reduced partial derivative is plugged into the copula based regression integral, the resulting estimator achieves substantially smaller bias and uniformly lower MSE than the classical version. Analytic bias–variance expansions are provided, and a comprehensive Monte Carlo study shows up to 50% reductions in MSE across a range of Bernstein copula degrees. A real data application to the mcycle dataset further demonstrates the practical advantages of the method, with the smoothing parameters selected by least squares cross validation (LSCV).