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  \textbf{\textsc{STANFORD UNIVERSITY}}\\[5pt]
  \textbf{\textsc{DEPARTMENT OF STATISTICS}}\\[5pt]
  \Large{\textbf\textsc{{DEPARTMENTAL SEMINAR}}}
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\begin{center}
  4:15 p.m., Tuesday, July 12, 2005\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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\begin{center}
  \textsl{Johan Lim} \\
  Texas A\&M University? \\
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\begin{center}
  \textbf{ESTIMATION OF SHAPE-RESTRICTED FUNCTIONS: SHAPE MODIFICATION 
VIA CONSTRAINED UNIFORM APPROXIMATION}
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In this talk, we describe an estimator for shape-restricted functions
that consists of: (i) nonparametric function estimation without taking
into account the shape constraint and (ii) shape modification of the
nonparametric estimate by solving a related constrained uniform
approximation problem.  We consider the three shape
constraints--monotonicity, convexity/concavity, and monotone
convexity/concavity which occur commonly in practical applications.
The main motivation behind the two-stage estimator is that, for these
constraints, it is relatively inexpensive to modify the shape of the
first-stage nonparametric estimate via constrained uniform
approximation so that the shape-modified one always has a uniform
approximation error smaller than or equal to that of the first-stage
one.  As a result, the shape-modified estimate converges uniformly to
the true function at least as fast as the first-stage. However, the
performance is shown to be asymptotically dominated by the first-stage
nonparametric estimate. This is joint work with S.J. Kim at EE,
Stanford Univ.

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