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February 2007 Tree-structured regression and the differentiation of integrals
Richard A. Olshen
Ann. Statist. 35(1): 1-12 (February 2007). DOI: 10.1214/009053606000001000


This paper provides answers to questions regarding the almost sure limiting behavior of rooted, binary tree-structured rules for regression. Examples show that questions raised by Gordon and Olshen in 1984 have negative answers. For these examples of regression functions and sequences of their associated binary tree-structured approximations, for all regression functions except those in a set of the first category, almost sure consistency fails dramatically on events of full probability. One consequence is that almost sure consistency of binary tree-structured rules such as CART requires conditions beyond requiring that (1) the regression function be in ℒ1, (2) partitions of a Euclidean feature space be into polytopes with sides parallel to coordinate axes, (3) the mesh of the partitions becomes arbitrarily fine almost surely and (4) the empirical learning sample content of each polytope be “large enough.” The material in this paper includes the solution to a problem raised by Dudley in discussions. The main results have a corollary regarding the lack of almost sure consistency of certain Bayes-risk consistent rules for classification.


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Richard A. Olshen. "Tree-structured regression and the differentiation of integrals." Ann. Statist. 35 (1) 1 - 12, February 2007.


Published: February 2007
First available in Project Euclid: 6 June 2007

zbMATH: 1122.62027
MathSciNet: MR2332266
Digital Object Identifier: 10.1214/009053606000001000

Primary: 26B05 , ‎28A15 , 62C12 , 62G08

Keywords: Binary tree-structured partitions , differentiation of integrals , maximal functions , regression

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 1 • February 2007
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