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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Tuesday, July 3, 2001}
\centerline{Sequoia Hall Rm. 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Mary C. Meyer}
\centerline{\sl University of Georgia}
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\centerline{\bf Some Problems in Shape Restriced Inference}

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Nonparametric function estimation with shape restrictions is broadly
applicable.
Suppose we are estimating a function $f$, such as a regression function, a
density function, or probability curve in bioassay, and we want to make only
qualitative assumptions about the shape of $f$.  We might know that $f$ is
increasing and concave, for example, or unimodal.

Estimating the function might involve maximizing a likelihood function over
the
class of shape restricted functions.  If we let $\theta_i=f(x_i)$ for
observations at $x_i$, and restrict our estimator to be piecewise linear with
knots at the $x_i$, then the shape restrictions can be written as a set of
linear inequality constraints.  We find $\hat{\theta}$ to maximize the
likelihood over the set $\Omega=\{\theta:A\theta \geq 0\}$ for an $m \times n$
matrix $A$.

Estimation of $\theta$ is nonlinear programming problem.  Once we have the
estimator, we want to do inference.  There are a lot of opportunities for
research here. For example, suppose we have a sort of semiparametric ANCOVA
model, consisting of a continuous response and two predictor variables, and
{\em
i.i.d.} mean zero normal errors.  One of the predictors is continuous and the
other is categorical.  The interest is in testing the null hypothesis that the
response is the same for each category against the alternative of ``parallel''
responses.  The relationship between the response and predictor is known up to
shape restrictions, such as convex, but the functional form is not specified.
We can find the least-squares parallel convex curves and compare them to
the fit
with a single convex curve using a likelihood ratio test statistic.

Another type of nonparametric regression problem involves the error density.
Suppose we specify a parametric regression function, but the only
assumptions we
make on the errors are that they are {\em i.i.d.} mean zero, symmetric
unimodal.
Simulations show that the regression parameter estimates are robust.
Bootstrap
confidence intervals can be used for inference.

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