Geodesic
Some inequalities related to the Stam inequality
Applications of Mathematics, Tome 53 (2008) no. 3, pp. 195-205.

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Zamir showed in 1998 that the Stam classical inequality for the Fisher information (about a location parameter) $$ 1/I(X+Y)\geq 1/I(X)+1/I(Y) $$ 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 parameter and for the variance of the Pitman estimator of the latter.
DOI : 10.1007/s10492-008-0004-2
Classification : 60E15, 62B10, 62F11
Keywords: Fisher information; location parameter; Pitman estimators 
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Kagan, Abram; Yu, Tinghui. Some inequalities related to the Stam inequality. Applications of Mathematics, Tome 53 (2008) no. 3, pp. 195-205. doi : 10.1007/s10492-008-0004-2. http://geodesic.mathdoc.fr/articles/10.1007/s10492-008-0004-2/

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