The RKHS Approach to Minimum Variance Estimation Revisited: Variance Bounds, Sufficient Statistics, and Exponential Families
Abstract
The mathematical theory of reproducing kernel Hilbert spaces (RKHS) provides powerful tools for minimum variance estimation (MVE) problems. Here, we extend the classical RKHS based analysis of MVE in several directions. We develop a geometric formulation of five known lower bounds on the estimator variance (Barankin bound, CramerRao bound, constrained CramerRao bound, Bhattacharyya bound, and HammersleyChapmanRobbins bound) in terms of orthogonal projections onto 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 on the estimator variance) is a lower semicontinuous function of the parameter vector. We also show that the RKHS associated with an MVE problem remains unchanged if the observation is replaced by a sufficient statistic. Finally, for MVE problems conforming to an exponential family of distributions, we derive novel closedform lower bound on the estimator variance and show that a reduction of the parameter set leaves the minimum achievable variance unchanged.
 Publication:

arXiv eprints
 Pub Date:
 October 2012
 DOI:
 10.48550/arXiv.1210.6516
 arXiv:
 arXiv:1210.6516
 Bibcode:
 2012arXiv1210.6516J
 Keywords:

 Mathematics  Statistics Theory