Lower Bounds for the Integrated and Minimax Risks in Intrinsic Statistical Estimation: A Geometric Approach

In parametric statistics, it is well established that the canonical measures of estimator performance—such as bias, variance, and mean squared error—are inherently dependent on the parameterization of the model. Consequently, these quantities describe the behavior of an estimator only relative to a particular parameterization, rather than representing intrinsic properties of either the estimator itself or the underlying probability distribution it seeks to estimate. Some years ago, the authors introduced a framework, termed the intrinsic analysis of point estimation, in which tools from information geometry were employed to construct analogues of classical statistical notions that are intrinsic to both the estimator and the associated probability measure. Within this framework, a contravariant vector field was introduced to define the intrinsic bias, while the squared Riemannian distance naturally emerged as the intrinsic analogue of the classical squared distance. Intrinsic counterparts of the Cramér–Rao inequalities, as well as the Rao–Blackwell and Lehmann–Scheffé theorems, were also established. The present work extends the intrinsic analysis—originally founded on the concept of intrinsic risk, a fundamentally local measure of estimator performance—to an approach that characterizes the estimator over an entire region of the parameter space, thereby yielding an intrinsically global perspective. Building upon intrinsic risk, two indices are proposed to evaluate estimator performance within a bounded region: (i) the integral of the intrinsic risk with respect to the Riemannian volume over the specified region, and (ii) the maximum intrinsic risk attained within that region. The Riemannian volume induced by the Fisher information metric on the manifold associated with the parametric model provides a natural means of averaging the intrinsic risk. Using variational methods, integral inequalities of the Cramér–Rao type are derived for the mean squared integrated Rao distance of the estimators, thereby extending previous contributions by several authors. Furthermore, lower bounds for the maximum intrinsic risk are obtained through corresponding integral formulations.