  We propose a modified $\chi^{\beta}$-divergence, give some of its properties,
and show that this leads to the definition of a generalized Fisher information.
We give generalized Cram\'er-Rao inequalities, involving this Fisher
information, an extension of the Fisher information matrix, and arbitrary norms
and power of the estimation error. In the case of a location parameter, we
obtain new characterizations of the generalized $q$-Gaussians, for instance as
the distribution with a given moment that minimizes the generalized Fisher
information. Finally we indicate how the generalized Fisher information can
lead to new uncertainty relations.
