Theory of Probability and Mathematical Statistics
On asymptotic Borovkov–Sakhanenko inequality with unbounded parameter set
R. Abu-Shanab, A. Yu. Veretennikov
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Abstract: 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 R as a parameter set. Optimality of Borovkov-Sakhanenko's asymptotic lower bound functional is established.
Keywords: Cramér-Rao bounds, Borovkov-Sakhanenko bounds, integral information inequalities, asymptotic efficiency
Bibliography: 1. A. A. Borovkov, Mathematical Statistics, Gordon and Breach, Amsterdam, 1998.
2. A. A. Borovkov and A. I. Sakhanenko, Estimates for averaged quadratic risk, Probab. Math. Statist. 1 (1980), no. 2, 185-195. (Russian)
3. M. Fréchet, Sur l'extension de certaines évaluations statistiques au cas de petits échantillons, Rev. Inst. Internat. Statist. 11 (1943), 182-205.
4. R. D. Gill and B. Y. Levit, Applications of the van Trees inequality: a Bayesian Cramér-Rao bound, Bernoulli 1 (1995), 59-79.
5. Reman Abu-Shanab, Information Inequalities and Parameter Estimation, PhD Thesis, University of Leeds, UK, 2009.
6. A. E. Shemyakin, Rao-Cramér type integral inequalities for the estimates of a vector parameter, Theory Probab. Appl. 33 (1985) no. 3, 426-434.
7. A. E. Shemyakin, Rao-Cramér type multidimensional integral inequalities for parametric families with singularities, Sib. Math. J. (1992), 706-715.
8. M. P. Schützenberger, A propos de l'inégalité de Fréchet-Cramér, Publ. Inst. Statist. Univ. Paris 7 (1958), no. 3/4, 3-6; http://igm.univ-mlv.fr/~berstel/Mps/Travaux/A/1958FrechetInstStat.pdf
9. H. van Trees, Detection, Estimation and Modulation Theory, vol. I, Wiley, New York, 1968.
10. A. Yu. Veretennikov, On asymptotic information integral inequalities, Theory Stoch. Process. 13(29) (2007), no. 1-2, 294-307.
