no. G10
Article de recherche - Statistiques
Some puzzles appearing in statistical inference
[Quelques énigmes apparaissant dans l’inférence statistique]
Alemam, Seyf1 ; Homei, Hazhir1 ; Nadarajah, Saralees2 
1Department of Statistics, University of Tabriz, P. O. Box 51666-17766, Tabriz, Iran
2Department of Mathematics, University of Manchester, Manchester M13 9PL, UK
Comptes Rendus. Mathématique, Tome 362 (2024) no. G10, pp. 1243-1252

Rao-Blackwell theorem is widely known to be a mathematically powerful technique that can be used to improve the precision of an estimator. The procedure entails exploiting a sufficient statistic to obtain an improved estimator or a uniformly minimum variance unbiased estimator. A modification of sufficient statistics is introduced here which can be applied for Rao-Blackwell theorem along with some fruitful applications that illustrate its properties. Also some theorems have been rewritten in statistical inference.

Le théorème de Rao-Blackwell est largement connu pour être une technique mathématique puissante qui peut être utilisée pour améliorer la précision d’un estimateur. La procédure consiste à exploiter une statistique suffisante pour obtenir un estimateur amélioré ou un estimateur sans biais à variance uniformément minimale. Nous présentons ici une modification des statistiques suffisantes qui peut être appliquée au théorème de Rao-Blackwell ainsi que quelques applications fructueuses qui illustrent ses propriétés. De plus, certains théorèmes ont été réécrits dans le cadre de l’inférence statistique.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.636
Classification : 62F10, 62F99
Keywords: Complete minimal sufficient statistic, Incomplete minimal sufficient statistic, Minimal sufficient statistic, Rao–Blackwell theorem, Sufficient statistic
Mots-clés : Statistique suffisante minimale complète, statistique suffisante minimale incomplète, statistique suffisante minimale, théorème de Rao–Blackwell, statistique suffisante

Alemam, Seyf  1   ; Homei, Hazhir  1   ; Nadarajah, Saralees  2

1 Department of Statistics, University of Tabriz, P. O. Box 51666-17766, Tabriz, Iran
2 Department of Mathematics, University of Manchester, Manchester M13 9PL, UK
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{CRMATH_2024__362_G10_1243_0,
     author = {Alemam, Seyf and Homei, Hazhir and Nadarajah, Saralees},
     title = {Some puzzles appearing in statistical inference},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1243--1252},
     year = {2024},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     number = {G10},
     doi = {10.5802/crmath.636},
     zbl = {07939455},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/crmath.636/}
}
TY  - JOUR
AU  - Alemam, Seyf
AU  - Homei, Hazhir
AU  - Nadarajah, Saralees
TI  - Some puzzles appearing in statistical inference
JO  - Comptes Rendus. Mathématique
PY  - 2024
SP  - 1243
EP  - 1252
VL  - 362
IS  - G10
PB  - Académie des sciences, Paris
UR  - https://www.numdam.org/articles/10.5802/crmath.636/
DO  - 10.5802/crmath.636
LA  - en
ID  - CRMATH_2024__362_G10_1243_0
ER  - 
%0 Journal Article
%A Alemam, Seyf
%A Homei, Hazhir
%A Nadarajah, Saralees
%T Some puzzles appearing in statistical inference
%J Comptes Rendus. Mathématique
%D 2024
%P 1243-1252
%V 362
%N G10
%I Académie des sciences, Paris
%U https://www.numdam.org/articles/10.5802/crmath.636/
%R 10.5802/crmath.636
%G en
%F CRMATH_2024__362_G10_1243_0
Alemam, Seyf; Homei, Hazhir; Nadarajah, Saralees. Some puzzles appearing in statistical inference. Comptes Rendus. Mathématique, Tome 362 (2024) no. G10, pp. 1243-1252. doi: 10.5802/crmath.636

[1] Bahadur, Raghu R. On unbiased estimates of uniformly minimum variance, Sankhyā, Ser. A, Volume 18 (1957), pp. 211-224 | MR | Zbl

[2] Boos, Dennis D.; Hughes-Oliver, Jacqueline M. Applications of Basu’s theorem, Am. Stat., Volume 52 (1998) no. 3, pp. 218-221 | DOI | MR

[3] Blackwell, David Conditional expectation and unbiased sequential estimation, Ann. Math. Stat., Volume 18 (1947), pp. 105-110 | DOI | MR | Zbl

[4] Bondesson, Lennart On uniformly minimum variance unbiased estimation when no complete sufficient statistics exist, Metrika, Volume 30 (1983) no. 1, pp. 49-54 | DOI | MR | Zbl

[5] Casella, George; Berger, Roger L. Statistical Inference, Duxbury Press, 2002 | MR

[6] Fisher, R. A. A mathematical Examination of the Methods of determining the Accuracy of Observation by the Mean Error,and by the Mean Square Error, Mon. Not. Roy. Astron. Soc., Volume 80 (1920) no. 8, pp. 758-770 | DOI

[7] Kumar, Surinder; Vaish, Mayank; Miyan, Poonam UMVUE of the stress-strength reliability for a class of distributions by using the estimates of reliability, J. Stat. Manag. Sys., Volume 21 (2018) no. 2, pp. 217-223 | DOI

[8] Lehmann, Erich L.; Casella, George Theory of point estimation, Springer Texts in Statistics, Springer, 1998, xxvi+589 pages | MR | Zbl

[9] Lehmann, Erich L. Theory of point estimation, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1983, xii+506 pages | DOI | MR | Zbl

[10] Lehmann, Erich L.; Scheffé, Henry Completeness, similar regions, and unbiased estimation. I, Sankhyā, Ser. A, Volume 10 (1950), pp. 305-340 | DOI | MR | Zbl

[11] Lehmann, Erich L.; Scheffé, Henry Completeness, similar regions, and unbiased estimation. II, Sankhyā, Ser. A, Volume 15 (1955), pp. 219-236 | DOI | MR | Zbl

[12] Meeden, Glen Estimation When Using a Statistic That is Not Sufficient, Am. Stat., Volume 41 (1987) no. 2, pp. 135-136 | MR | DOI

[13] Mood, Alexander; Graybill, Franklin; Boes, Duane An Introduction to Probability Theory of Statistics, McGraw-Hill, 1974

[14] Mukhopadhyay, Nitis Probability and statistical inference, Statistics: Textbooks and Monographs, 162, Marcel Dekker, 2000 | Zbl

[15] Peña, Edsel A.; Rohatgi, Vijay Some comments about sufficiency and unbiased estimation, Am. Stat., Volume 48 (1994) no. 3, pp. 242-243 | DOI | MR

[16] Rao, C. Radhakrishna Information and the accuracy attainable in the estimation of statistical parameters, Bull. Calcutta Math. Soc., Volume 37 (1945), pp. 81-91 | MR | Zbl

[17] Rao, C. Radhakrishna Linear statistical inference and its applications, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1973, xx+625 pages | DOI | MR | Zbl

[18] Rohatgi, Vijay; Ehsanes Saleh, A. K. Md. An introduction to probability and statistics, Wiley Series in Probability and Statistics, John Wiley & Sons, 2015, xviii+689 pages | DOI | MR | Zbl

[19] Roussas, George G. A course in mathematical statistics., Academic Press Inc., 1997 | Zbl

[20] Shao, Jun Mathematical statistics, Springer Texts in Statistics, Springer, 2003, xvi+591 pages | DOI | MR | Zbl

[21] Stigler, Stephen M. Completeness and Unbiased Estimation, Am. Stat., Volume 26 (1972) no. 2, pp. 28-29 | DOI

Cité par Sources :