Additive Regression and Other Nonparametric Models

Let $(X, Y)$ be a pair of random variables such that X = (X₁, ⋯, XJ) and let f by a function that depends on the joint distribution of $(X, Y).$ A variety of parametric and nonparametric models for f are discussed in relation to flexibility, dimensionality, and interpretability. It is then supposed that each Xⱼ ∈ 0, 1, that Y is real valued with mean μ and finite variance, and that f is the regression function of Y on $X.$ Let f^, of the form f⁽x₁, ⋯, xJ) = μ + f^₁(x₁) + ⋯ + f^J(xJ), be chosen subject to the constraints Ef^ⱼ = 0 for 1 ≤ j ≤ J to minimize E(f(X) - f⁽X))². Then f^ is the closest additive approximation to $f,$ and f^ = f if f itself is additive. Spline estimates of f^ⱼ and its derivatives are considered based on a random sample from the distribution of $(X, Y).$ Under a common smoothness assumption on f^ⱼ, 1 ≤ j ≤ J, and some mild auxiliary assumptions, these estimates achieve the same (optimal) rate of convergence for general J as they do for $J = 1.$