HISTOGRAM REGRESSION ESTIMATION USING DATA-DEPENDENT PARTITIONS

We establish general sufficient conditions for the L(2)-consistency of multivariate histogram regression estimates based on data-dependent p artitions. These same conditions insure the consistency of partitionin g regression estimates based on local polynomial fits, and, with an ad ditional regularity assumption, the consistency of histogram estimates for conditional medians. Our conditions require shrinking cells, sube xponential growth of a combinatorial complexity measure and sublinear growth of restricted cell. counts. It is not assumed that the cells of every partition be rectangles with sides parallel to the coordinate a xis or that each cell contain a minimum number of points. Response var iables me assumed to be bounded throughout. Our results may be applied to a variety of partitioning schemes. We established the consistency of histograms regression estimates based on cubic partitions with data -dependent offsets, k-thresholding in one dimension and empirically op timal nearest-neighbor clustering schemes. In addition, it is shown th at empirically optimal regression trees are consistent when the size o f the trees grows with the number of samples at an appropriate rate.