  Recently, a framework for application-oriented optimal experiment design has
been introduced. In this context, the distance of the estimated system from the
true one is measured in terms of a particular end-performance metric. This
treatment leads to superior unknown system estimates to classical experiment
designs based on usual pointwise functional distances of the estimated system
from the true one. The separation of the system estimator from the experiment
design is done within this new framework by choosing and fixing the estimation
method to either a maximum likelihood (ML) approach or a Bayesian estimator
such as the minimum mean square error (MMSE). Since the MMSE estimator delivers
a system estimate with lower mean square error (MSE) than the ML estimator for
finite-length experiments, it is usually considered the best choice in practice
in signal processing and control applications. Within the application-oriented
framework a related meaningful question is: Are there end-performance metrics
for which the ML estimator outperforms the MMSE when the experiment is
finite-length? In this paper, we affirmatively answer this question based on a
simple linear Gaussian regression example.
