Nonparametric lack-of-fit testing and consistent variable selection

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Let X be a d dimensional vector of covariates and Y be the response variable. Under the nonparametric model Y = m(X) + σ(X) we develop an ANOVA-type test for the null hypothesis that a particular coordinate of X has no influence on the regression function. The asymptotic distribution of the test statistic, using residuals based on local polynomial regression, is established under the null hypothesis and local alternatives. Simulations suggest that the test outperforms existing procedures in heteroscedastic settings. Using p-values from this test, a variable selection method based on False Discovery Rate corrections is proposed, and proved to be consistent in estimating the set of indices corresponding to the significant covariates. Simulations suggest that, under a sparse model, dimension reduction techniques can help avoid the curse of dimensionality. We also propose a backward elimination version of this procedure, called BEAMS (Backward Elimination ANOVA-type Model Selection), which performs competitively against well established procedures in linear regression settings, and outperforms them in nonparametric settings. A real data set is analyzed.