Efficiency and Cramér-Rao type lnequalities for convex loss functions

A general method for obtaining inequalities of Cramér-Rao type for convex loss functions is presented. It is shown under rather weak assumptions that there are at least as many such inequalities as best unbiased estimators. More precisely, it is shown that an estimator is efficient with respect to an inequality of Cramér-Rao type if and only if it is the best in the class of unbiased estimators. Moreover, theorems of Blyth and Roberts (“Proceedings Sixth Berkeley Symposium on Math. Statist. Prob.,”) and of Blyth (Ann. Statist.2, 464–473) are extended. We make an use of methods of convex analysis and properties of convex integral functionals on Orlicz spaces.