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Zamir showed in 1998 that the Stam classical inequality for the Fisher information (about a location parameter) $$1/I(X+Y)\ge 1/I\left(X\right)+1/I\left(Y\right)$$ for independent random variables $X$, $Y$ is a simple corollary of basic properties of the Fisher information (monotonicity, additivity and a reparametrization formula). The idea of his proof works for a special case of a general (not necessarily location) parameter. Stam type inequalities are obtained for the Fisher information in a multivariate observation depending on a univariate location...