  This paper deals with Cram\'er-Rao inequalities in the context of
nonextensive statistics and in estimation theory. It gives characterizations of
generalized q-Gaussian distributions, and introduces generalized versions of
Fisher information. The contributions of this paper are (i) the derivation of
new extended Cram\'er-Rao inequalities for the estimation of a parameter,
involving general q-moments of the estimation error, (ii) the derivation of
Cram\'er-Rao inequalities saturated by generalized q-Gaussian distributions,
(iii) the definition of generalized Fisher informations, (iv) the
identification and interpretation of some prior results, and finally, (v) the
suggestion of new estimation methods.
