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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, May 17, 2005\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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\begin{center}
  \textsl{Michael Woodroofe}\\
  Department of Statistics\\
  University of Michigan\\
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  \textbf{Shape Restricted Estimation in the Search for Dark Matter}
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In addition to visible matter, like stars, there is matter that
astronomers cannot see.  They know that it is there from gravitational
effects.  It's nature and even location are not well known at present,
but there is a wide belief that the vast majority of matter in the
Universe is dark.  The talk centers on statistical issues that arise
in the non-parametric estimation of distribution of dark matter in the
dwarf spheroidal galaxies that are satellites of the Milky Way.
Statistically, this is an inverse problem with missing data.  The
distribution of (total) mass together with the law of gravity
determines the positions and velocities of stars, and the statistical
problem is to recover the distribution of mass from observed positions
and velocities.  Complete positions and velocities cannot be observed
here--only the radial velocity and the projection of position on the
plane orthogonal to the line of sight.  The distribution of projected
position can be estimated quite precisely from large (e.g.150,000
stars) photometric studies.  Velocities require some time to measure,
however, and previous studies have been too small (e.g. 30 stars) to
allow estimation in any generality.  Instead, astronomers have been
forced to adopt highly structured parametric models. New
instrumentation will soon make much larger samples (approx 1500 stars)
available and non-parametric methods feasible.  Assuming spherical
symmetry, let $M(r)$ denote the total mass within $r$ of the center of a
dwarf spheroidal galaxy.  Then Jeans' Equation (from classical
potential theory) shows that $M(r) = c\Psi''(r)/f(r)$, where $c$ is a known
constant, $f$ is a (nearly) known non-negative function, and $\Psi$ is a
function for which an unbiased, $\sqrt{n}$-consistent estimator exists.
From Jeans' Equation, $\Psi$ is a convex function, and $\Psi''(r)/f(r)$ is
non-decreasing in $r$.  The unbiased estimator, $\Psi^{\#}$ say, does not
satisfy these shape restrictions and can be improved by imposing them.
The talk centers on methods for accomplishing this.  This is joint
work with Xiao Wang (UM Statistics), Mario Mateo and Matt Walker (UM
Astronomy), and Ed Olshewsky (UAriz Astronomy).

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