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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENT SEMINAR}
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\centerline{4:15 p.m., Wednesday, July 24, 2002}
\centerline{Sequoia Hall Room 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Bhramar Muhkerjee}
\centerline{\sl University of Florida}

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\centerline{\bf Optimal designs for estimating the path of a stochastic process }
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A  second-order random process $Y(t)$, with $E(Y(t))\equiv 0$, is sampled at a 
finite number of design points $t_1, t_2,...,t_n$. On the basis of these 
observations, one wants to estimate the values of the process at unsampled 
points using the best linear unbiased estimator (BLUE). The performance of 
the estimator is measured by a weighted integrated mean square error. The goal 
is to find  $t_1,t_2,....,t_n$, such that this integrated mean square error 
(IMSE) is minimized for a fixed $n$.   


This optimization problem depends on the stochastic process
only through its covariance structure.
For processes with a product type covariance structure, i.e., for
$Cov(Y(s),Y(t))=u(s) v(t)$, $s<t$,  a set of necessary and sufficient 
conditions for a design to be exactly optimal will be
presented. Starting from the set of exact optimality conditions for 
a fixed $n$, an asymptotic result yielding the density 
whose percentile points furnish a set of asymptotically optimal 
design points (in some suitable sense) will be 
described.  Results on the problem when one tries to estimate the
integral of $Y(t)$ instead of the path will be discussed briefly. The integral
estimation problem is related to certain regression design problems with
correlated errors discussed by Sacks and Ylvisaker in a series of
papers (1966,1968,1970).


For a more general covariance structure, satisfying Sacks-Ylvisaker 
regularity conditions, it is noted that a much simpler estimator is
asymptotically equivalent to the BLUE. This leads
to an intuitively appealing argument in establishing the asymptotic behaviour 
of the BLUE and also in deriving an analytical expression for the 
asymptotically optimal design density. 

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