  In the present work, we show how the generalized Cram\'er-Rao inequality for
the estimation of a parameter, presented in a recent paper, can be extended to
the mutidimensional case with general norms on $\mathbb{R}^{n}$, and to a wider
context. As a particular case, we obtain a new multidimensional Cram\'er-Rao
inequality which is saturated by generalized $q$-Gaussian distributions. We
also give another related Cram\'er-Rao inequality, for a general norm, which is
saturated as well by these distributions. Finally, we derive uncertainty
relations from these Cram\'er-Rao inequalities. These uncertainty relations
involve moments computed with respect to escort distributions, and we show that
some of these relations are saturated by generalized $q$-Gaussian
distributions. These results introduce extended versions of Fisher information,
new Cram\'er-Rao inequalities, and new characterizations of generalized
$q$-Gaussian distributions which are important in several areas of physics and
mathematics.
