Parameter estimation with expected and residual-at-risk criteria

In this paper we study a class of uncertain linear estimation problems in which the data are affected by random uncertainty. We consider two estimation criteria, one based on minimization of the expected ℓ1ℓ1 or ℓ2ℓ2 norm residual and one based on minimization of the level within which the ℓ1ℓ1 or ℓ2ℓ2 norm residual is guaranteed to lie with an a-priori fixed probability (residual at risk). The random uncertainty affecting the data is characterized by means of its first two statistical moments, and the above criteria are intended in a worst-case probabilistic sense, that is worst-case expectations and probabilities over all possible distribution having the specified moments are considered. The ensuing estimation problems can be solved efficiently via convex programming, yielding exact solutions in the ℓ2ℓ2 norm case and upper-bounds on the optimal solutions in the ℓ1ℓ1 case.