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ISSN 1547-7363(online) ISSN 0094-9000(print)
On asymptotic Borovkov–Sakhanenko inequality with unbounded parameter set
Authors:
R. Abu-Shanab and A. Yu. Veretennikov
Journal:
Theor. Probability and Math. Statist. 90 (2015), 1-12
MSC (2010):
Primary 62F12, 62F15
DOI:
https://doi.org/10.1090/tpms/945
Published electronically:
August 6, 2015
MathSciNet review:
3241856
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 $\mathbf {R}^1$ as a parameter set. Optimality of Borovkov–Sakhanenko’s asymptotic lower bound functional is established.
- A. A. Borovkov, Mathematical statistics, Gordon and Breach Science Publishers, Amsterdam, 1998. Translated from the Russian by A. Moullagaliev and revised by the author. MR 1712750
- A. A. Borovkov and A. I. Sakhanenko, Estimates for averaged quadratic risk, Probab. Math. Statist. 1 (1980), no. 2, 185–195 (1981) (Russian, with English summary). MR 626310
- M. Fréchet, Sur l’extension de certaines évaluations statistiques au cas de petits échantillons, Rev. Inst. Internat. Statist. 11 (1943), 182–205.
- Richard D. Gill and Boris Y. Levit, Applications of the Van Trees inequality: a Bayesian Cramér-Rao bound, Bernoulli 1 (1995), no. 1-2, 59–79. MR 1354456, DOI https://doi.org/10.2307/3318681
- Reman Abu-Shanab, Information Inequalities and Parameter Estimation, PhD Thesis, University of Leeds, UK, 2009.
- A. E. Shemyakin, Rao–Cramér type integral inequalities for the estimates of a vector parameter, Theory Probab. Appl. 33 (1985) no. 3, 426–434.
- A. E. Shemyakin, Multidimensional integral inequalities of Rao-Cramér type for parametric families with singularities, Sibirsk. Mat. Zh. 32 (1991), no. 4, 204–215, 230 (Russian); English transl., Siberian Math. J. 32 (1991), no. 4, 706–715 (1992). MR 1142080, DOI https://doi.org/10.1007/BF00972988
- Marcel-Paul Schützenberger, A propos de l’inégalité de Fréchet-Cramér, Publ. Inst. Statist. Univ. Paris 7 (1958), no. 3-4, 3–6 (French). MR 105762
- H. van Trees, Detection, Estimation and Modulation Theory, vol. I, Wiley, New York, 1968.
- Alexander Veretennikov, On asymptotic information integral inequalities, Theory Stoch. Process. 13 (2007), no. 1-2, 294–307. MR 2343831
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Additional Information
R. Abu-Shanab
Affiliation:
P.O. Box 32038, Department of Mathematics, College of Science, University of Bahrain, Kingdom of Bahrain
Email:
raboshanab@uob.edu.bh
A. Yu. Veretennikov
Affiliation:
School of Mathematics, University of Leeds, LS2 9JT, United Kingdom & Institute for Information Transmission Problems, Moscow, Russia – and – National Research University Higher School of Economics, Moscow, Russia
Email:
a.veretennikov@leeds.ac.uk
Keywords:
Cramér–Rao bounds,
Borovkov–Sakhanenko bounds,
integral information inequalities,
asymptotic efficiency
Received by editor(s):
November 1, 2013
Published electronically:
August 6, 2015
Additional Notes:
The paper is based on a chapter in the first author’s PhD Thesis \cite{Reman}. Both authors are grateful to Professor Sakhanenko for his comments. The second author’s work was partially supported by RFBR grant 13-01-12447 ofi_m2.
Article copyright:
© Copyright 2015
American Mathematical Society