Maximum set estimators with bounded estimation error

We consider the linear regression problem of estimating a deterministic parameter vector x from observations y = Hx + w, where H is known, and w is additive noise. We seek an estimator whose estimation error does not exceed a given maximum error for as wide a range of conditions as possible. The maximum error is a design choice and is generally lower than the error provided by the well-known least-squares (LS) estimator. We develop estimators guaranteeing the required error for as large a parameter set as possible and for as large a noise level as possible. We discuss methods for finding these estimators and demonstrate that in many cases, the proposed estimators outperform the LS estimator.