Optimal Average Case Estimation in Hilbert Norms

Abstract

In contrast to the worst case approach, the average case setting provides less conservative insight into the quality of estimation algorithms. In this paper we consider two local average case error measures of algorithms based on noisy information, in Hilbert norms in the problem element and information spaces. We define the optimal algorithm and provide formulas for its two local errors, which explicitly exhibit the influence of factors such as information, information (measurement) errors, norms in the considered spaces, a subset where approximations are allowed, and “unmodeled dynamics.” Based on the error expression, we formulate in algebraic language the problem of selecting the optimal approximating subspace. The solution is given along with the specific formula for the error, which depends on the eigenvalues of a certain matrix defined by information and norms under consideration.