We begin with two possible extensions of Stam's inequality and of de Bruijn's identity. In both cases, a generalized q-Gaussian plays the same role as the standard Gaussian in the classical case. These generalized q-Gaussians are important in several areas of physics and mathematics. A generalized Fisher information also pops up, playing the same role as the classical Fisher information, but for the extended identity and inequality. In the estimation theory context, we give several extensions of the Cramér-Rao inequality in the multivariate case, with matrix versions as well as versions for general norms. We define new forms of Fisher information, that reduce to the classical one in special cases. In the case of a translation parameter, the general Cramér-Rao inequalities lead to an inequality for distributions, which involves the same generalized Fisher information as in the generalized de Bruijn's identity and Stam's inequality. This Cramér-Rao inequality is saturated by generalized q-Gaussian distributions. This shows that the generalized q-Gaussians also minimize the generalized Fisher information among dis-tributions with a fixed moment. Similarly, the generalized q-Gaussians also minimize the generalized Fisher information among distributions with a given q-entropy.