Nonparametric estimation of global functionals and a measure of the explanatory power of covariates in regression

In a nonparametric regression setting with multiple random predictor variables, we give the asymptotic distributions of estimators of global integral functionals of the regression surface. We apply the results to the problem of obtaining reliable estimators for the nonparametric coefficient of determination. This coefficient, which is also called Pearson's correlation ratio, gives the fraction of the total variability of a response that can be explained by a given set of covariates. It can be used to construct measures of nonlinearity of regression and relative importance of subsets of regressors, and to assess the validity of other model restrictions. In addition to providing asymptotic results, we propose several data-based bandwidth selection rules and carry out a Monte Carlo simulation study of finite sample properties of these rules and associated estimators of explanatory power. We also provide two real data examples.