On asymptotic Boronkov-Sakhanenko inequality with unbounded parameter set

Integral analogues of Cramér-Rao's inequalities for Bayesian parameter estimators proposed initially by Schützenberger (1958) and later by van Trees (1968) were further developed by Borovkov and Sakhanenko (1980). In this paper, new asymptotic versions of such inequalities are established under ultimately relaxed regularity assumptions and under a locally uniform nonvanishing of the prior density and with R1 as a parameter set. Optimality of Borovkov-Sakhanenko's asymptotic lower bound functional is established.