Smooth Regression Functions (partial Residual Plots).

We assume that the data follow y = X(beta) + f(z) + (epsilon) where E((epsilon)) = 0, cov((epsilon)) = (sigma)('2)I,X and z are known, y is observed, (beta) is unknown and f(z) is a smooth, but unknown, function of z. We discuss existing methods and propose new ones for fitting such data; we demonstrate that the proposed methods are better than the existing ones. Our methods do not assume that f(z) is from a parametric family. Partial residual plots are often used to determine the shape of f(z). A recent study has shown that it may be difficult, or even impossible, to detect the true shape of f(z) from a partial residual plot. We propose a family of generalized weighted least squares estimators and the associated partial residual plots for fitting these data and for estimating f(z). Two particular members of this family are studied in detail. The first, called the non-parametric regression estimator, estimates (beta) with the ordinary least squares estimator when (I-U)y is regressed on (I-U)X, where U is defined below. The second, called the spline estimator, estimates (beta) and f(z) with the minimizers of a weighted combination of the usual sum of squared residuals and a measure of the smoothness of f(z). For both methods the estimators for f(z) can be expressed as U(y-X(beta)) for an appropriate U, but the corresponding estimators for (beta) are intrinsically different. The effects of this difference are studied, and a possible criterion for choosing other estimators for the general family is discussed. Tests of H(,0):f(z) = 0 and H(,0):f(z) = (gamma)z are proposed, based on fitting the data with the spline estimator. The exact distributions of the test statistics are derived, and a computable numerical approximation to the distribution is also given. Several data examples were generated from known models chosen so that the ordinary partial residual plot does not show f(z). The partial residual plots based on the two new estimators do, however, clearly show f(z).PhDPure SciencesStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/127661/2/8412127.pd