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Title: Some inequalities related to the Stam inequality (English)
Author: Kagan, Abram
Author: Yu, Tinghui
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 53
Issue: 3
Year: 2008
Pages: 195-205
Summary lang: English
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Category: math
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Summary: Zamir showed in 1998 that the Stam classical inequality for the Fisher information (about a location parameter) $$ 1/I(X+Y)\geq 1/I(X)+1/I(Y) $$ for independent random variables $X$, $Y$ is a simple corollary of basic properties of the Fisher information (monotonicity, additivity and a reparametrization formula). The idea of his proof works for a special case of a general (not necessarily location) parameter. Stam type inequalities are obtained for the Fisher information in a multivariate observation depending on a univariate location parameter and for the variance of the Pitman estimator of the latter. (English)
Keyword: Fisher information
Keyword: location parameter
Keyword: Pitman estimators
MSC: 60E15
MSC: 62B10
MSC: 62F11
idZBL: Zbl 1186.62009
idMR: MR2411124
DOI: 10.1007/s10492-008-0004-2
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Date available: 2010-07-20T12:15:54Z
Last updated: 2020-07-02
Stable URL: http://hdl.handle.net/10338.dmlcz/140315
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Reference: [1] Carlen, E. A.: Superadditivity of Fisher's information and logarithmic Sobolev inequalities.J. Funct. Anal. 101 (1991), 194-211. Zbl 0732.60020, MR 1132315, 10.1016/0022-1236(91)90155-X
Reference: [2] Ibragimov, I. A., Khas'minskij, R. Z.: Statistical Estimation. Asymptotic Theory.Springer New York (1981). Zbl 0467.62026, MR 0620321
Reference: [3] Kagan, A., Landsman, Z.: Statistical meaning of Carlen's superadditivity of the Fisher information.Statist. Probab. Lett. 32 (1997), 175-179. Zbl 0874.60002, MR 1436863, 10.1016/S0167-7152(96)00070-3
Reference: [4] Kagan, A.: An inequality for the Pitman estimators related to the Stam inequality.Sankhya A64 (2002), 282-292. Zbl 1192.62099, MR 1981759
Reference: [5] Kagan, A., Shepp, L. A.: A sufficiency paradox: an insufficient statistic preserving the Fisher information.Amer. Statist. 59 (2005), 54-56. MR 2113195, 10.1198/000313005X21041
Reference: [6] Kagan, A., Yu, T., Barron, A., Madiman, M.: Contribution to the theory of Pitman estimators.Submitted.
Reference: [7] Madiman, M., Barron, A.: The monotonicity of information in the central limit theorem and entropy power inequalities.Preprint Dept. of Statistics, Yale University (2006). MR 2128239
Reference: [8] Shao, J.: Mathematical Statistics, 2nd ed.Springer New York (2003). Zbl 1018.62001, MR 2002723
Reference: [9] Stam, A. J.: Some inequalities satisfied by the quantities of information of Fisher and Shannon.Inform. and Control 2 (1959), 101-112. Zbl 0085.34701, MR 0109101, 10.1016/S0019-9958(59)90348-1
Reference: [10] Zamir, R.: A proof of the Fisher information inequality via a data processing argument.IEEE Trans. Inf. Theory 44 (1998), 1246-1250. Zbl 0901.62005, MR 1616672, 10.1109/18.669301
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