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 ℓ 1 or ℓ 2 norm residual and one based on minimization of the level within which the ℓ 1 or ℓ 2 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 ℓ 2 norm case and upper-bounds on the optimal solutions in the ℓ 1 case.