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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big JOINT PROBABILITY AND STATISTICS SEMINAR}
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\centerline{4:15 p.m., Tuesday, January 22, 2002}
\centerline{Sequoia Hall Room 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Richard Olshen}
\centerline{\sl Division of Biostatistics}
\centerline{\sl Stanford University}
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\centerline{\bf Tree-structured Regression and the Differentiation of Integrals}
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By regression is meant estimating the conditional expectation of an unknown
numerical outcome Y given a vectorial parameter X. With a vanilla binary
tree-structured approach, at each stage a large tree is grown by
successively cycling through the coordinate axes and ``splitting'' so as
best to reduce some notion of ``node impurity.'' The large tree is then
pruned back to a smaller one on the basis of an estimate of prospective
accuracy. When predictors are Euclidean, the end product is an ``empirical
conditional expectation'' of Y given a finite sigma-field whose atoms are
``boxes'' with sides parallel to the coordinate axes. Almost surely
consistent asymptotic behavior depends upon two issues. One is how close
the empirical conditional expectation is to the expectation of Y given the
finite sigma-field. This involves large deviations for empirical processes.
The other issue concerns how close the latter conditional expectation is to
E(Y|X=x). This part makes contact with the differentiation of integrals.
Interest focuses on a two-dimensional X that is uniformly distributed on
the unit square, a general Y with finite expectation, and a sequence of
sigma-fields whose mesh tends to 0. If the sigma-fields are not almost
surely ultimately nested, almost sure convergence can fail without E[Y
log(1+|Y|)] being finite or control over how oblong the boxes can be.
Nesting is inconsistent with ``most'' sampling distributions of Y. Control
over the geometry of the boxes is precluded by considerations of
equivariance. My understanding of examples comes from conversations and
correspondence with Alexandra Bellow, Izzy Katznelson, and especially Don
Ornstein.

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