Nonparametric Prediction in Measurement Error Models

• Published in: Journal of the American Statistical Association Vol. 104; no. 487; pp. 993 - 1003
• Main Authors: Carroll, Raymond J., Delaigle, Aurore, Hall, Peter
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis 01.09.2009; American Statistical Association
• Subjects: Applications; Bandwidth; Calibration; Consistent estimators; Contamination; Deconvolution; Density estimation; Eigenfunctions; Error rates; Errors-in-variables; Estimation methods; Estimators; Exact sciences and technology; General topics; Inference from stochastic processes; time series analysis; Mathematics; Modeling; Parametric rates; Probability and statistics; Probability theory and stochastic processes; Regression; Ridge parameter; Sciences and techniques of general use; Smoothing; Statistical variance; Statistics; Stochastic processes; Theory and Methods; Unbiased estimators; Error estimation; Prediction theory; Non parametric estimation; Optimal estimation; Stochastic process; Adaptive method; Bandwidth; Errors-in-variables; Parametric rates; Convergence rate; Ridge parameter; Measurement error; Smoothing methods; Convergence acceleration; Prediction; Regression; Statistical estimation; Optimal convergence; Semiparametric method; Contamination; Statistical method; Statistical regression; Deconvolution; Numerical analysis; Smoothing; Observation data; Curve estimation; Optimal approximation; Filtering theory; Application
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1198/jasa.2009.tm07543

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 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. Acknowledgments: Carroll's research was supported by grants from the National Cancer Institute (CA57030, CA104620). Delaigle's work was partially supported by a fellowship from the Maurice Belz foundation. Hall's work was partially supported by the Australian Reserach Council and by a grant from the National Science Foundation (DMS 0604698).