Piecewise-Polynomial Regression Trees∗

A nonparametric function estimation method called SUPPORT (“Smoothed and Unsmoothed Piecewise-Polynomial Regression Trees”) is described. The estimate is typically made up of several pieces, each piece being obtained by fitting a polynomial regression to the observations in a subregion of the data space. Partitioning is carried out recursively as in a tree-structured method. If the estimate is required to be smooth, the polynomial pieces may be glued together by means of weighted averaging. The smoothed estimate is thus obtained in three steps. In the first step, the regressor space is recursively partitioned until the data in each piece are adequately fitted by a polynomial of a fixed order. Partitioning is guided by analysis of the distributions of residuals and cross-validation estimates of prediction mean square error. In the second step, the data within a neighborhood of each partition are fitted by a polynomial. The final estimate of the regression function is obtained by averaging the polynomial pieces, using smooth weight functions each of which diminishes rapidly to zero outside its associated partition. Estimates of derivatives of the regression function may be ∗Chaudhuri’s research was partially supported by funds from the University of Wisconsin Graduate School. Loh’s research was partially supported by NSF grant DMS 88-03271 and ARO grant DAAL03-91G-0111. Division of Theoretical Statistics & Mathematics, Indian Statistical Institute, 203 B. T. Road, Calcutta 700035, India. Department of Statistics, National Cheng Kung University, Tainan, Taiwan. Department of Statistics, University of Wisconsin, Madison, WI 53706, U.S.A.