Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 11 (2006) > Paper 15 open journal systems 


Large and Moderate Deviations for Hotelling's T^2-Statistics

Amir Dembo, Stanford University
Qi-Man Shao, Hong Kong University of Science and Technology


Abstract
Let X , X1, X 2 , ... be i.i.d. R d-valued random variables. We prove large and moderate deviations for Hotelling's T2-statistic when X is in the generalized domain of attraction of the normal law.


Full text: PDF

Pages: 149-159

Published on: August 7, 2006
Bibliography
  1. Anderson, T.W. (1984). An introduction to Multivariate Analysis (2nd ed.). Wiley, New York. MR0771294 (86b:62079)
  2. Bercu, B., Gassiat, E. and Rio, E. (2002). Concentration inequalities, large and moderate deviations for self-normalized empirical processes. Ann. Probab. 30, 1576--1604. MR1944001(2004a:60060)
  3. Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular variation . Cambridge University Press, Cambridge. MR0898871 (88i:26004)
  4. Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat. 23, 493-507. MR0057518 (15,241c)
  5. Chistyakov, G.P. and Gotze, F. (2004a). On bounds for moderate deviations for Student's statistic. Theory Probab. Appl. 48, 528-535. MR2141355 (2005m:60092)
  6. Chistyakov, G.P. and Gotze, F. (2004b). Limit distributions of Studentized means. Ann. Probab. 32 (2004), 28-77. MR2040775 (2005f:60055)
  7. Dembo, A. and Shao, Q.M. (1998a). Self-normalized moderate deviations and lils. Stoch. Proc. and Appl. 75, 51-65. MR1629018 (99i:60053)
  8. Dembo, A. and Shao, Q.M. (1998b). Self-normalized large deviations in vector spaces. In: Progress in Probability (Eberlein, Hahn, Talagrand, eds) Vol. 43, 27-32. MR1652318 (99j:60036)
  9. Faure, M. (2002). Self-normalized large deviations for Markov chains. Electronic J. Probab. 7, 1-31. MR1943896 (2003j:60034)
  10. Fujikoshi, Y. (1997). An asymptotic expansion for the distribution of Hotelling's $T^2$-statistic under nonnormality. J. Multivariate Anal. 61, 187-193. MR1452272 (98h:62091)
  11. Griffin, P. and Kuelbs, J. (1989). Self-normalized laws of the iterated logarithm. Ann. Probab. 17, 1571--1601. MR1048947 (91k:60036)
  12. Hahn, M.G. and Klass, M.J. (1980). Matrix normalization of sums of random vectors in the domain of attraction of the multivariate normal. Ann. Probab. 8, 262-280. MR0566593 (81d:60030)
  13. He, X. and Shao, Q. M. (1996). Bahadur efficiency and robustness of studentized score tests. Ann. Inst. Statist. Math. 48, 295-314. MR1405934 (97m:62016)
  14. Jing, B.Y., Shao, Q.M. and Wang, Q.Y. (2003). Self-normalized Cramer type large deviations for independent random variables. Ann. Probab. 31 , 2167-2215. MR1999791 (2004f:60046)
  15. Jing, B.Y., Shao, Q.M. and Zhou, W. (2004). Saddlepoint approximation for Student's t-statistic with no moment conditions. Ann. Statist. 32, 2679-2711. MR2153999 (2006a:62030)
  16. Kano, Y. (1995). An asymptotic expansion of the distribution of Hotelling's $T^2$-statistic under general condition. Amer. J. Math. Manage. Sci. 15, 317-341. MR1397511 (97h:62044)
  17. Kariya, T. (1981). A robustness property of Hotelling's T 2 -test. Ann. Statist. 9, 210-213. MR0600550 (82c:62052)
  18. Kiefer, J. and Schwartz, R. (1965). Admissible Bayes character of $T^2$- and $R^2$- and other fully invariant tests for classical normal problems. Ann. Math. Statist. 36, 747-760. MR0175245 (30 #5430)
  19. Muirhead, R.J. (1982). Aspects of Multivariate Statistical Theory. John Wiley, New York. MR0652932 (84c:62073)
  20. Robinson, J. and Wang, Q.Y. (2005). On the self-normalized Cramer-type large deviation. J. Theoretic Probab. 18, 891-909.
  21. Sepanski, S. (1994). Asymptotics for Multivariate t-statistic and Hotelling's T2-statistic under infinite second moments via bootstrapping. J. Multivariate Anal. 49, 41-54. MR1275042 (96d:62092)
  22. Shao, Q.M. (1997). Self-normalized large deviations. Ann. Probab. 25, 285--328. MR1428510 (98b:60056)
  23. Shao, Q.M. (1998). Recent developments in self-normalized limit theorems. In Asymptotic Methods in Probability and Statistics (editor B. Szyszkowicz), pp. 467 - 480. Elsevier Science. MR1661499 (99j:60063)
  24. Shao, Q.M. (2004). Recent progress on self-normalized limit theorems. In Probability, finance and insurance (editors Tze Leung Lai, Hailiang Yang and Siu Pang Yung), pp. 50--68, World Sci. Publ., River Edge, NJ, 2004. MR2189198 (2006i:60056)
  25. Simaika, J.B. (1941). On an optimal property of two important statistical tests. Biometrika 32, 70-80. MR0003547 (2,236b)
  26. Stein, C.(1956). The admissibility of Hotelling's T-test. Ann. Math. Statist. 27, 616-623. MR0080413 (18,243c)
  27. Wang, Q.Y. (2005). Limit theorems for self-normalized large deviation. Electronic J. Probab. 10, 1260-1285. MR2176384
















Research
Support Tool
Capture Cite
View Metadata
Supplementary
Files and / or
CORRECTIONS
Printer Friendly
Context
Author Address
Action
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X