Geodesic
A note concerning the Kram\'er-Rao and Bhattacharyya inequalities in statistical estimation theory
Matematičeskie zametki, Tome 8 (1970) no. 1, pp. 3-7

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Explicit necessary conditions are derived under which the inequalities referred to in the title become equalities for functions of a parameter. 
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     title = {A note concerning the {Kram\'er-Rao} and {Bhattacharyya} inequalities in statistical estimation theory},
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Yu. V. Linnik. A note concerning the Kram\'er-Rao and Bhattacharyya inequalities in statistical estimation theory. Matematičeskie zametki, Tome 8 (1970) no. 1, pp. 3-7. http://geodesic.mathdoc.fr/item/MZM_1970_8_1_a0/