Nonparametric Prediction in Measurement Error Models.
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Predicting the value of a 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|X). When the observed X(i)s 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|T). In practice, estimating E(Y|T) can be quite challenging, as data may be collected under different conditions, making the measurement errors on X(i) and X non-identically 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.