PLACE:  319 Snedecor

SPEAKER:
Bhramar Mukherjee
Department of Statistics, Purdue University

TITLE:
Optimal designs for estimating the path of a stochastic process

ABSTRACT:

A second-order random process Y(t), with E(Y(t))  0, is sampled at a finite
number of design points t1, t2, ..., tn.  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 t1,
t2, ...., tn, such that this integrated mean square error (IMSE) is minimized
for a fixed n.  This design problem arises in statistical communication theory
and signal processing as well as in geology and environmental sciences.

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.  Explicit
calculations of optimal designs for any given n for Brownian Motion, Brownian
Bridge and Ornstein-Uhlenbeck process will illustrate the simplicity and
usefulness of these conditions.  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.

For a more general covariance structure, satisfying natural regularity
conditions, some interesting asymptotic results will be presented.  It will be
shown that for processes with no quadratic mean derivative, a much simpler
estimator is asymptotically equivalent to the BLUE.  This will lead 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.
 

COFFEE: 3:45 p.m., 104 Snedecor