Minimum Variance in Biased Estimation With Singular Fisher Information Matrix

This paper extends the work of Y. C. Eldar, ldquoMinimum variance in biased estimation: Bounds and asymptotically optimal estimators,rdquo in IEEE Trans. Signal Process., vol. 52, pp. 1915-1929, Jul. 2004, which deals with only nonsingular Fisher information matrix. In order to guarantee the uniform Cramer-Rao bound to be a finite lower bound and also to have a feasible solution to the optimization problem in the work of Y. C. Eldar, it is proved that the norms of bias gradient matrices of all biased estimators must have a nonzero exact lower bound, which mainly depends on the rank of the singular Fisher information matrix. The smaller the rank of the singular Fisher information matrix is, the larger the lower bound of norms of bias gradient matrices of all biased estimators is. For a specific Frobenius norm, the exact lower bound is simply the difference between the parameter dimension and the rank of the singular Fisher information matrix.