Cramer-Rao Type Bounds for Generalized Exponential Families
Abstract
It is well known that the sufficient statistics of an exponential family attain the usual Cramer-Rao Lower Bound (CRLB) for their expected value. In this paper, we extend this result to a broader family of distributions that includes exponential families and power-law families, with Student t-distributions as special case. To derive the generalized Cramer-Rao inequality, we first apply Eguchi's theory to Bregman divergence to obtain a Riemannian metric and then follow the approach of Amari and Nagaoka. Since this approach applies only to models with finite support, we use Cauchy-Schwarz inequality to extend it to models with countably infinite or continuous support. The resulting generalized CRLB provides a finite-sample lower bound related to the asymptotic covariance matrix of the Minimum Bregman Divergence Estimator (MBDE). We show that the sufficient statistic of the generalized family attains the generalized CRLB. Moreover, although the generalized CRLB does not, in general, coincide with the asymptotic covariance matrix of the MBDE, the covariance matrix of an estimator attaining the generalized CRLB coincides with the corresponding matrix for the joint distribution of the sample.
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BibTeX
@misc{dhadumia2026cramerrao,
title = {{Cramer-Rao Type Bounds for Generalized Exponential Families}},
author = {Satyajit Dhadumia and M Ashok Kumar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27679},
eprint = {2610.27679},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27679},
}
RIS
TY - UNPB
TI - Cramer-Rao Type Bounds for Generalized Exponential Families
AU - Satyajit Dhadumia
AU - M Ashok Kumar
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27679
UR - https://arxiv.org/abs/2610.27679
AB - It is well known that the sufficient statistics of an exponential family attain the usual Cramer-Rao Lower Bound (CRLB) for their expected value. In this paper, we extend this result to a broader family of distributions that includes exponential families and power-law families, with Student t-distributions as special case. To derive the generalized Cramer-Rao inequality, we first apply Eguchi's theory to Bregman divergence to obtain a Riemannian metric and then follow the approach of Amari and Nagaoka. Since this approach applies only to models with finite support, we use Cauchy-Schwarz inequality to extend it to models with countably infinite or continuous support. The resulting generalized CRLB provides a finite-sample lower bound related to the asymptotic covariance matrix of the Minimum Bregman Divergence Estimator (MBDE). We show that the sufficient statistic of the generalized family attains the generalized CRLB. Moreover, although the generalized CRLB does not, in general, coincide with the asymptotic covariance matrix of the MBDE, the covariance matrix of an estimator attaining the generalized CRLB coincides with the corresponding matrix for the joint distribution of the sample.
ER -