Nonparametric estimation of global functionals and a measure of the explanatory power of covariates in regression. (English) Zbl 0843.62045

Doksum, Kjell; Samarov, Alexander

Ann. Stat. 23, No. 5, 1443-1473 (1995).

Summary: 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.

MSC:

62G07	Density estimation
62J02	General nonlinear regression
62E20	Asymptotic distribution theory in statistics
62G20	Asymptotic properties of nonparametric inference

Keywords:

nonparametric R-squared; measure of subset importance; index of nonlinearity; cross-validation; nonparametric regression; multiple random predictor variables; estimators of global integral functionals; regression surface; coefficient of determination; Pearson’s correlation ratio; total variability; covariates; measures of nonlinearity; data-based bandwidth selection rules; Monte Carlo simulation; finite sample properties