Estimation of constrained parameters in a linear model with multiplicative and additive noise

Estimation of a Hilbert-space valued parameter in a linear model with compact linear transformation is considered with both multiplicative and additive noise present. The unknown parameter is assumed a priori to lie in a compact rectangular parallelepiped oriented in a certain way in the Hilbert space. Linear estimators are devised that minimize reasonable upper bounds on mean-squared error depending on conditions on the noise. Under prescribed conditions the estimators are minimax in the class of linear estimators. With the prior constraint on the unknown parameter removed, the estimation problem is ill-posed. Restricting the unknown provides a regularization of the basically ill-posed estimation. It turns out the estimators developed here belong to a well-known class of regularized estimators. With the interpretation that the constraint is soft, the procedure is applicable to many signal-processing problems