Identity Reproducing Multivariate Nonparametric Regression

Nonparametric kernel regression estimators of the Nadaraya-Watson type are known to have an undesirable bias behavior. We propose a general technique to improve the bias of any given multivariate nonparametric regression estimator based on the requirement that the identity function should be reproduced, which is achieved by means of an identity reproducing transformation of the predictor variable. The asymptotic distribution of the identity reproducing version of the Nadaraya-Watson estimator is derived and is compared with that of the untransformed Nadaraya-Watson estimator. It is demonstrated by means of a Monte Carlo study that the asymptotic improvements are noticeable already for small sample sizes.