| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2010-07-20T12:15:54Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140315 |
| . |
| 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 |
| . |
