# Optimal Allocation for Estimating the Mean
of a Bivariate Polynomial

### Statistics Department, University of Michigan

This research has been funded by the
National Science Foundation, and has
been conducted jointly with
Quentin F. Stout,
EECS Department, University of Michigan.

## Overview

This seminar describes our work on allocation to
estimate the mean value of a bivariate polynomial by
estimating its components. Such problems arise in statistical estimation
problems such as estimating the fault tolerance of a system with two
types of components. The goal is to sample between the two populations
to minimize the mean squared error of the estimate, given a fixed total
sample size.
We first examine the optimal fully sequential allocation, where the
results of all previous tests are known before we decide which population
to sample next. This is compared to various ad-hoc rules which are
easier to determine. We then consider optimal 1-, 2- and 3-stage
designs, where we are only given 0, 1, or 2 midpoints in which to examine
the results and adjust our allocation for the next stage. Such designs
are often prefered in practice, and may be required if there is significant
delay in obtaining results.
We find that optimal 2- and 3-stage designs are very close to the
fully sequential ones in terms of minimizing the objective function.
However, the optimal stage sizes are
poorly predicted by the asymptotic theory, and instead need to be
found computationally.
Robustness of the designs is also examined computationally, and it
is shown that the adaptive designs are especially robust.

Throughout, efficient
algorithms play an important role in enabling us to perform the required
optimizations and evaluations for problems of useful size.
Algorithmic techniques employed include
variations of dynamic programming and forward induction.

While our
focus is on evaluating polynomials, the techniques and algorithms can
be applied to other problems involving a few populations with discrete
outcomes. Examples of such problems include clinical trials and experiments
to estimate the prevelance of disease.

An example

Types of sequential allocation rules

Some results

Robustness

Few-stage allocation rules

Points of interest

Future research

References

Example