Lower bounds for parametric estimation with constraints
Gorman, J.D.; Hero, A.O.
Information Theory, IEEE Transactions on
Volume 36, Issue 6, Nov 1990 Page(s):1285 - 1301
Digital Object Identifier 10.1109/18.59929
Summary:A Chapman-Robbins form of the Barankin bound is used to derive a
multiparameter Cramer-Rao (CR) type lower bound on estimator error
covariance when the parameter θ∈Rn is
constrained to lie in a subset of the parameter space. A simple form for
the constrained CR bound is obtained when the constraint set
ΘC, can be expressed as a smooth functional inequality
constraint. It is shown that the constrained CR bound is identical to
the unconstrained CR bound at the regular points of ΘC,
i.e. where no equality constraints are active. On the other hand, at
those points θ∈ΘC where pure equality
constraints are active the full-rank Fisher information matrix in the
unconstrained CR bound must be replaced by a rank-reduced Fisher
information matrix obtained as a projection of the full-rank Fisher
matrix onto the tangent hyperplane of the full-rank Fisher matrix onto
the tangent hyperplane of the constraint set at θ. A necessary
and sufficient condition involving the forms of the constraint and the
likelihood function is given for the bound to be achievable, and
examples for which the bound is achieved are presented. In addition to
providing a useful generalization of the CR bound, the results permit
analysis of the gain in achievable MSE performance due to the imposition
of particular constraints on the parameter space without the need for a
global reparameterization
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