TY - JOUR
T1 - Kernel smoothing of data with correlated errors
AU - Altman, N. S.
N1 - Funding Information:
• N. S. Altman is Assistant Professor in the Biometrics Unit, Cornell University, Ithaca, NY 14853. This work was supported by a Natural Sciences and Engineering Research Council of Canada postgraduate scholarship and Hatch Grant 151410NYF. Much of this work was completed while the author was in the Department of Statistics at Stanford University. The author would like to thank lain Johnstone for his valuable guidance in supervising the dissertation from which this article was taken. Discussions with Brad Efron, Jerome Friedman, and Peter Lewis also contributed substantially to the ideas in this article.
PY - 1990/9
Y1 - 1990/9
N2 - Kernel smoothing is a common method of estimating the mean function in the nonparametric regression model y = f(x) + ε, where f(x) is a smooth deterministic mean function and ε is an error process with mean zero. In this article, the mean squared error of kernel estimators is computed for processes with correlated errors, and the estimators are shown to be consistent when the sequence of error processes converges to a mixing sequence. The standard techniques for bandwidth selection, such as cross-validation and generalized cross-validation, are shown to perform very badly when the errors are correlated. Standard selection techniques are shown to favor undersmoothing when the correlations are predominantly positive and oversmoothing when negative. The selection criteria can, however, be adjusted to correct for the effect of correlation. In simulations, the standard selection criteria are shown to behave as predicted. The corrected criteria are shown to be very effective when the correlation function is known. Estimates of correlation based on the data are shown, by simulation, to be sufficiently good for correcting the selection criteria, particularly if the signal to noise ratio is small.
AB - Kernel smoothing is a common method of estimating the mean function in the nonparametric regression model y = f(x) + ε, where f(x) is a smooth deterministic mean function and ε is an error process with mean zero. In this article, the mean squared error of kernel estimators is computed for processes with correlated errors, and the estimators are shown to be consistent when the sequence of error processes converges to a mixing sequence. The standard techniques for bandwidth selection, such as cross-validation and generalized cross-validation, are shown to perform very badly when the errors are correlated. Standard selection techniques are shown to favor undersmoothing when the correlations are predominantly positive and oversmoothing when negative. The selection criteria can, however, be adjusted to correct for the effect of correlation. In simulations, the standard selection criteria are shown to behave as predicted. The corrected criteria are shown to be very effective when the correlation function is known. Estimates of correlation based on the data are shown, by simulation, to be sufficiently good for correcting the selection criteria, particularly if the signal to noise ratio is small.
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U2 - 10.1080/01621459.1990.10474936
DO - 10.1080/01621459.1990.10474936
M3 - Article
AN - SCOPUS:84919172802
SN - 0162-1459
VL - 85
SP - 749
EP - 759
JO - Journal of the American Statistical Association
JF - Journal of the American Statistical Association
IS - 411
ER -