Parameter estimation with expected and residual-at-risk criteria

We study a class of uncertain linear estimation problems in which the data are affected by random uncertainty. In this setting, we consider two estimation criteria, one based on minimization of the expected l₁ or l₂ norm residual and one based on minimization of the level within which the l₁ or l₂ 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 l₂ norm case and upper-bounds on the optimal solutions in the l₁ case.