Dimension Reduction in Regression Estimation with Nearest Neighbor

In regression with a high-dimensional predictor vector, dimension reduction methods aim at replacing the predictor by a lower dimensional version without loss of information on the regression. In this context, the so-called central mean subspace is the key of dimension reduction. The last two decades have seen the emergence of many methods to estimate the central mean subspace. In this paper, we go one step further, and we study the performances of a k-nearest neighbor type estimate of the regression function, based on an estimator of the central mean subspace. The estimate is first proved to be consistent. Improvement due to the dimension reduction step is then observed in term of its rate of convergence. All the results are distributionsfree. As an application, we give an explicit rate of convergence using the SIR method. Index Terms — Dimension Reduction; Central Mean Subspace; Nearest Neighbor Method; Semiparametric Regression; SIR Method. AMS 2000 Classification — 62H12; 62G08.