Intrinsic distance lower bound for unbiased estimators on Riemannian manifolds

Abstract

We consider statistical models parameterized over connected Rie mannian manifolds. We present a lower bound on the mean-square distance of unbiased estimators about their mean values. The de rived bound depends both on the curvature of the parameter man ifold and a coordinate-free extension of the classical Fisher in formation matrix. Our study can be applied in estimation prob lems with smooth parametric constraints, and in statistical models indexed over coset spaces. Illustrative examples concerning in ferences on the unit-sphere and the complex projective space are worked out

Intrinsic Variance Lower Bound (IVLB): An Extension of the Cramer-Rao Bound to Riemannian Manifolds

“…In this paper, we improve our previous results in [9] by presenting a much tighter version of the IVLB. Contrary to the IVLB discussed in [9], this new version now coincides with the CRB for flat Riemannian manifolds.…”

“…The existence of the Riemannian structure on Θ turns it into a metric space (the distance is induced from the Riemannian layer) and this canonical Riemannian distance can be used to measure the precision of estimators taking values in Θ: the magnitude of an estimaton error corresponds to the Riemannian distance between the "true" (unknown) family member θ ∈ Θ and the estimated point θ(y) ∈ Θ (here, θ(y) denotes a realization of an estimate of θ through the estimator θ : R n → Θ). The preliminary version of the IVLB discussed in [9] establishes a lower-bound for the mean-square distance of unbiased estimators taking values in Θ. A clear drawback of the IVLB in [9] is the presence of an information gap relative to the CRB.…”

“…The preliminary version of the IVLB discussed in [9] establishes a lower-bound for the mean-square distance of unbiased estimators taking values in Θ. A clear drawback of the IVLB in [9] is the presence of an information gap relative to the CRB. More precisely, when applied to the special case of flat Riemannian manifolds, e.g., Euclidean spaces, the IVLB does not coincide with the Cramér-Rao bound: the CRB is more informative.…”

“…However, no metric structure on Θ is assumed in [8]: the covariance inequality derived therein does not translate in a quantitative statement concerning the accuracy (measured in terms of some distance on Θ) of unbiased estimators. We introduced a preliminary version of the Intrinsic Variance Lower Bound (IVLB) in [9]. Similar to [8], the work in [9] assumes Θ to be a generic manifold and thus can be applied to cases (i) and (ii).…”

“…We introduced a preliminary version of the Intrinsic Variance Lower Bound (IVLB) in [9]. Similar to [8], the work in [9] assumes Θ to be a generic manifold and thus can be applied to cases (i) and (ii). However, it assumes Θ to be endowed with an extra piece of structure: a Riemannian layer.…”