Electronic Communications in Probability

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

Amir Dembo and Qi-Man Shao

Full-text: Open access


Let $\mathbf{X}, \mathbf{X}_1, \mathbf{X}_2, ...$ be i.i.d. $\mathbb{R}^d$-valued random variables. We prove large and moderate deviations for Hotelling's $T^2$-statistic when $\mathbf{X}$ is in the generalized domain of attraction of the normal law.

Article information

Electron. Commun. Probab., Volume 11 (2006), paper no. 15, 149-159.

Accepted: 7 August 2006
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: Primary 60F10
Secondary: 60F15: Strong theorems Secondary 62E20 60G50: Sums of independent random variables; random walks

large deviation moderate deviation self-normalized partial sums law of the iterated logarithm $T^2$ statistic

This work is licensed under aCreative Commons Attribution 3.0 License.


Dembo, Amir; Shao, Qi-Man. Large and Moderate Deviations for Hotelling's $T^2$-Statistics. Electron. Commun. Probab. 11 (2006), paper no. 15, 149--159. doi:10.1214/ECP.v11-1209. https://projecteuclid.org/euclid.ecp/1465058858

Export citation


  • Anderson, T.W. (1984). An introduction to Multivariate Analysis (2nd ed.). Wiley, New York.
  • Bercu, B., Gassiat, E. and Rio, E. (2002). Concentration inequalities, large and moderate deviations for self-normalized empirical processes. Ann. Probab. 30, 1576–1604.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular variation. Cambridge University Press, Cambridge.
  • 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.
  • Chistyakov, G.P. and Gotze, F. (2004a). On bounds for moderate deviations for Student's statistic. Theory Probab. Appl. 48, 528-535.
  • Chistyakov, G.P. and Gotze, F. (2004b). Limit distributions of Studentized means. Ann. Probab. 32 (2004), 28-77.
  • Dembo, A. and Shao, Q.M. (1998a). Self-normalized moderate deviations and lils. Stoch. Proc. and Appl. 75, 51-65.
  • 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.
  • Faure, M. (2002). Self-normalized large deviations for Markov chains. Electronic J. Probab. 7, 1-31.
  • Fujikoshi, Y. (1997). An asymptotic expansion for the distribution of Hotelling's $T^2$-statistic under nonnormality. J. Multivariate Anal. 61, 187-193.
  • Griffin, P. and Kuelbs, J. (1989). Self-normalized laws of the iterated logarithm. Ann. Probab. 17, 1571–1601.
  • 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.
  • He, X. and Shao, Q. M. (1996). Bahadur efficiency and robustness of studentized score tests. Ann. Inst. Statist. Math. 48, 295-314.
  • 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.
  • 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.
  • 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.
  • Kariya, T. (1981). A robustness property of Hotelling's $T^2$-test. Ann. Statist. 9, 210-213.
  • 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.
  • Muirhead, R.J. (1982). Aspects of Multivariate Statistical Theory. John Wiley, New York.
  • Robinson, J. and Wang, Q.Y. (2005). On the self-normalized Cramer-type large deviation. J. Theoretic Probab. 18, 891-909.
  • Sepanski, S. (1994). Asymptotics for Multivariate t-statistic and Hotelling's $T^2$-statistic under infinite second moments via bootstrapping. J. Multivariate Anal. 49, 41-54.
  • Shao, Q.M. (1997). Self-normalized large deviations. Ann. Probab. 25, 285–328.
  • 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.
  • 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.
  • Simaika, J.B. (1941). On an optimal property of two important statistical tests. Biometrika 32, 70-80.
  • Stein, C.(1956). The admissibility of Hotelling's T-test. Ann. Math. Statist. 27, 616-623.
  • Wang, Q.Y. (2005). Limit theorems for self-normalized large deviation. Electronic J. Probab. 10, 1260-1285.