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Nonparametric and semiparametric regression have been suggested as alternatives to parametric models for growth curve analysis. In this article we demonstrate that empirical Bayes estimation can be used to improve linear smoothing estimates when multiple curves are available.

This article proposes some simplifications of the residual variance estimator of Gasset, Sroka, and Jeneen-Steinmetz (GSJ, 1986) which is often used in conjunction with non parametric regression. The GSJ estimator is a quadratic form of the data, which depends on the relative spacings of the design points. When the errors are independent, identically distributed Gaussian variables, and the true regression curve is flat, the estimate is distributed as a weighted sum of x2 variables. By matching the first two moments, the distribution can be approximated by a x2 with degrees of freedom determined by the coefficients of the. quadratic form. Computation of the estimated degrees of freedom requires computing the trace of the square of an n x n matrix, where n is the number of design points. In this article, (n-2)/3 is shown to be a conservative estimate of the approximate degrees of freedom, and (n-2)/2 is shown to be conservative for many designs. In addition, a simplified version of the estimator is shown to...

Nonparametric regression is a set of techniques for estimating a regression curve without making strong assumptions about the shape of the true regression function. These techniques are therefore useful for building and checking parametric models, as well as for data description. Kernel and nearest-neighbor regression estimators are local versions of univariate location estimators, and so they can readily be introduced to beginning students and consulting clients who are familiar with such summaries as the sample mean and median.

Altman (1990) and Hart (1991) have shown that kernel regression can be an effective method for estimating an unknown mean function when the errors are correlated. However, the optimal bandwidth for kernel smoothing depends strongly on the correlation function, as do confidence bands for the regression curve. In this paper, the simultaneous estimation of the regression and correlation functions is explored. An iterative technique analogous to the iterated Cochrane-Orcutt method for linear regression (Cochrane and Orcutt, 1949) is shown to perform well. However, for moderate sample sizes, stopping after the first iteration produces better results. An interesting feature of the simultaneous method is that it performs best when different kernels are used to estimate the regression and correlation functions. For the regression function, unimodal kernels are known to be optimal. However, examination of the mean squared error of the correlation estimator suggests that a bimodal kernel will perform better for est...

Kernel smoothing is a common method of estimating the mean function in the nonparametric regression model y = f(x) + e, where f(x) is a smooth deterministic mean function and e 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 functi...