The RKHS Approach to Minimum Variance Estimation Revisited: Variance Bounds, Sufficient Statistics, and Exponential Families

摘要

The mathematical theory of reproducing kernel Hilbert spaces (RKHS) providespowerful tools for minimum variance estimation (MVE) problems. Here, we extendthe classical RKHS based analysis of MVE in several directions. We develop ageometric formulation of five known lower bounds on the estimator variance(Barankin bound, Cramer-Rao bound, constrained Cramer-Rao bound, Bhattacharyyabound, and Hammersley-Chapman-Robbins bound) in terms of orthogonal projectionsonto a subspace of the RKHS associated with a given MVE problem. We show that,under mild conditions, the Barankin bound (the tightest possible lower bound onthe estimator variance) is a lower semicontinuous function of the parametervector. We also show that the RKHS associated with an MVE problem remainsunchanged if the observation is replaced by a sufficient statistic. Finally,for MVE problems conforming to an exponential family of distributions, wederive novel closed-form lower bound on the estimator variance and show that areduction of the parameter set leaves the minimum achievable varianceunchanged.