Some Uniform Limit Results in Additive Regression Model

For 0 < p < ∞, the optimal Lp rate of convergence of a nonparametric estimate of m is of order n −k 2k+d when m is assumed to be a k-times differentiable function and for p = ∞, the optimal rate is (n−1 log n) k 2k+d (see, Stone (1985)). This rate of convergence which depends on the dimension d of the covariable X becomes worse as the dimensionality of the problem increases. In the literature, this phenomena is known under the name of “curse of dimensionality”. To reduce the dimension impact upon the estimates, Stone (1985) proposed several sub-models of model (1). More particularly, he studied the nonparametric additive regression model in which the multivariate regression function is written as the sum of univariate functions, i.e,