Robust estimation of time-varying processes

We consider the question of optimally estimating a time-varying multivariate stochastic process, in order to minimize the expected squared estimation error. This is motivated by adaptive control problems under uncertainty in a changing environment. A distinguishing feature of our approach is that we do not need a completely specified model for the stochastic process under consideration. Instead, we merely assume that the process obeys certain predetermined assumptions, and subsequently derive the optimal min-max estimator w.r.t. these assumptions. This can be seen as a stochastically robust method to estimate a time-varying process. We provide tight upper bounds on the expected squared estimation error, and explicitly derive the optimal weighted least-squares estimator in several relevant examples.