Cramér-Rao Lower Bounds Arising from Generalized Csiszár Divergences
This work provides a theoretical extension in information geometry for statisticians and information theorists, though it appears incremental as it builds on existing frameworks like Eguchi's theory and Amari-Nagaoka structures.
The paper tackles the problem of deriving a more widely applicable Cramér-Rao inequality by studying probability distributions with respect to generalized Csiszár divergences, resulting in a lower bound for estimator variance in escort models and enabling the identification of unbiased and efficient estimators.
We study the geometry of probability distributions with respect to a generalized family of Csiszár $f$-divergences. A member of this family is the relative $α$-entropy which is also a Rényi analog of relative entropy in information theory and known as logarithmic or projective power divergence in statistics. We apply Eguchi's theory to derive the Fisher information metric and the dual affine connections arising from these generalized divergence functions. This enables us to arrive at a more widely applicable version of the Cramér-Rao inequality, which provides a lower bound for the variance of an estimator for an escort of the underlying parametric probability distribution. We then extend the Amari-Nagaoka's dually flat structure of the exponential and mixer models to other distributions with respect to the aforementioned generalized metric. We show that these formulations lead us to find unbiased and efficient estimators for the escort model. Finally, we compare our work with prior results on generalized Cramér-Rao inequalities that were derived from non-information-geometric frameworks.