  In this paper, two types of linear estimators are considered for three
related estimation problems involving set-theoretic uncertainty pertaining to
$\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ balls of frequency-responses. The
problems at stake correspond to robust $\mathcal{H}_{2}$ and
$\mathcal{H}_{\infty}$ in the face of non-parametric "channel-model"
uncertainty and to a nominal $\mathcal{H}_{\infty}$ estimation problem. The
estimators considered here are defined by minimizing the worst-case squared
estimation error over the "uncertainty set" and by minimizing an average cost
under the constraint that the worst-case error of any admissible estimator does
not exceed a prescribed value. The main point is to explore the derivation of
estimators which may be viewed as less conservative alternatives to minimax
estimators, or in other words, that allow for trade-offs between worst-case
performance and better performance over "large" subsets of the uncertainty set.
The "average costs" over $\mathcal{H}_{2}-$signal balls are obtained as limits
of averages over sets of finite impulse responses, as their length grows
unbounded. The estimator design problems for the two types of estimators and
the three problems addressed here are recast as semi-definite programming
problems (SDPs, for short). These SDPs are solved in the case of simple
examples to illustrate the potential of the "average cost/worst-case
constraint" estimators to mitigate the inherent conservatism of the minimax
estimators.
