A. Jung, S. Schmutzhard, F. Hlawatsch:

"The RKHS approach to minimum variance estimation revisited: Variance bounds, sufficient statistics, and exponential families";

IEEE Transactions on Information Theory,60(2014), 7; 4050 - 4065.

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, Cramér-Rao bound, constrained Cramér-Rao bound, Bhattacharyya bound, and Hammersley-Chapman-Robbins 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 semi-continuous 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 closed- form lower bounds on the estimator variance and show that a reduction of the parameter set leaves the minimum achievable variance unchanged.

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, Cramér-Rao bound, constrained Cramér-Rao bound, Bhattacharyya bound, and Hammersley-Chapman-Robbins 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 semi-continuous 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 closed- form lower bounds on the estimator variance and show that a reduction of the parameter set leaves the minimum achievable variance unchanged.

Minimum variance estimation, exponential family, reproducing kernel Hilbert space, RKHS, Cramér- Rao bound, Barankin bound, Hammersley-Chapman-Robbins bound, Bhattacharyya bound, locally minimum variance unbiased estimator

http://publik.tuwien.ac.at/files/PubDat_229646.pdf

Created from the Publication Database of the Vienna University of Technology.