On Kolmogorovʹs representation of functions of several variables by functions of one variable

This paper proposes a multivariate nonparametric regression estimator. Motivated by Kolmogorov (Doklady 114 (1957) 679), we consider functions of the form ∑k=1t gk(λk,1φk(x1)+⋯+λk,dφk(xd)), 1⩽t<∞, where gk(·) and φk(·) are three times continuously differentiable and φk(·) is nondecreasing. These functions are approximated by cubic B-splines. Optimal univariate convergence rates are obtained. A small number of necessary and sufficient monotonicity constraints are developed. Furthermore, we directly bound the bias of this estimator when the underlying function is smooth, and not necessarily taking the above form. This bias can be made arbitrarily small. These contributions allow for a better understanding of the approximating properties of the estimator.