Nonparametric Prediction in Measurement Error Models

Predicting the value of it variable Y corresponding to a future value of an explanatory variable X, based on a sample of previously observed independent data pairs (X-1, Y-1),...,(X-n, Y-n,) distributed like (X, Y), is very important in statistics. In the error-free case, where X is observed accurately, this problem is strongly related to that of standard regression estimation, since prediction of Y can be achieved via estimation of the regression Curve E(Y vertical bar X). When the observed Xis and the future observation of X are measured with error, prediction is of a quite different nature. Here, if T denotes the future (contaminated) available version of X, prediction of Y can be achieved via estimation of E(Y vertical bar T). In practice, estimating E(Y vertical bar T) can be quite challenging, as data may be collected under different conditions, making the measurement errors on X-i and X nonidentically distributed. We take up this problem in the nonparametric setting and introduce estimators which allow a highly adaptive approach to smoothing. Reflecting the complexity of the problem, optimal rates of convergence of estimators can vary from the semiparametric n(-1/2) rate to much slower rates that are characteristic of nonparametric problems. Nevertheless, we are able to develop highly adaptive, data-driven methods that achieve very good performance in practice. This article has the supplementary materials online.