Nonparametric regression to the mean.
Available data may reflect a true but unknown random variable of interest plus an additive error, which is a nuisance. The problem in predicting the unknown random variable arises in many applied situations where measurements are contaminated with errors; it is known as the regression-to-the-mean problem. There exists a well known solution when both the distributions of the true underlying random variable and the contaminating errors are normal. This solution is given by the classical regression-to-the-mean formula, which has a data-shrinkage interpretation. We discuss the extension of this solution to cases where one or both of these distributions are unknown and demonstrate that the fully nonparametric case can be solved for the case of small contaminating errors. The resulting nonparametric regression-to-the-mean paradigm can be implemented by a straightforward data-sharpening algorithm that is based on local sample means. Asymptotic justifications and practical illustrations are provided.