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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big STATISTICS SEMINAR}
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\centerline{4:15 p.m., Tuesday, May 4, 2004}
\centerline{Sequoia Hall Room 200}
\centerline{(Cookies at 3:45 in the 1st Floor Lounge)}


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\centerline{\sl Ed George}
\centerline{\sl University of Pennsylvania}
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\centerline{\bf Improved Minimax Prediction Under Kullback-Leibler Loss}
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Abstract:

Let $X | \mu \sim N_p(\mu, v_x I)$ and $Y | \mu \sim N_p(\mu, v_y I)$ be
independent $p$-dimensional multivariate normal vectors with common
unknown mean $\mu$, and let $p(x |  \mu)$  and $p(y |  \mu)$ denote the
conditional densities of $X$ and $Y$.  Based on only observing $X = x$,
we consider the problem of obtaining a predictive distribution $\hat p(y
| x)$ for $Y$ that is close to $p(y |  \mu)$ as measured by
Kullback-Leibler loss.   The natural straw man for this problem is the
best invariant predictive distribution, the Bayes rule $p_U(y | x)$
under the uniform prior $\pi_U(\mu) \equiv 1$, which is seen to be
minimax.  We show that $p_U(y | x)$ is dominated by any Bayes rules for
which the square root of the marginal distribution is superharmonic.
This yields wide classes of dominating predictive distributions
including Bayes rules under superharmonic priors.  These dominating
predictive shrinkage distributions can be constructed to adaptively
shrink $p_U(y | x)$ towards arbitrary points or subspaces.  Those
procedures corresponding to superharmonic priors can be further combined
to obtain minimax multiple shrinkage predictive distributions that
adaptively shrink $p_U(y | x)$ towards an arbitrary number of points or
subspaces.  Fundamental similarities and differences with the parallel
theory of estimating a multivariate normal mean under quadratic loss are
described throughout.  (This is joint work with Feng Liang and Xinyi
Xu).



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