The Annals of Statistics

Saddlepoint approximation for Student’s t-statistic with no moment conditions

Bing-Yi Jing, Qi-Man Shao, and Wang Zhou

Full-text: Open access


A saddlepoint approximation of the Student’s t-statistic was derived by Daniels and Young [Biometrika 78 (1991) 169–179] under the very stringent exponential moment condition that requires that the underlying density function go down at least as fast as a Normal density in the tails. This is a severe restriction on the approximation’s applicability. In this paper we show that this strong exponential moment restriction can be completely dispensed with, that is, saddlepoint approximation of the Student’s t-statistic remains valid without any moment condition. This confirms the folklore that the Student’s t-statistic is robust against outliers. The saddlepoint approximation not only provides a very accurate approximation for the Student’s t-statistic, but it also can be applied much more widely in statistical inference. As a result, saddlepoint approximations should always be used whenever possible. Some numerical work will be given to illustrate these points.

Article information

Ann. Statist., Volume 32, Number 6 (2004), 2679-2711.

First available in Project Euclid: 7 February 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory
Secondary: 60G50: Sums of independent random variables; random walks

Saddlepoint approximation large deviation asymptotic normality Edgeworth expansion self-normalized sum Student’s t-statistic absolute error relative error


Jing, Bing-Yi; Shao, Qi-Man; Zhou, Wang. Saddlepoint approximation for Student’s t -statistic with no moment conditions. Ann. Statist. 32 (2004), no. 6, 2679--2711. doi:10.1214/009053604000000742.

Export citation


  • Daniels, H. E. and Young, G. A. (1991). Saddlepoint approximation for the Studentized mean, with an application to the bootstrap. Biometrika 78 169–179.
  • Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge Univ. Press.
  • Dembo, A. and Shao, Q.-M. (1998). Self-normalized large deviations in vector spaces. In High-Dimensional Probability (E. Eberlein, M. Hahn and M. Talagrand, eds.) 27–32. Birkhäuser, Basel.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
  • Field, C. and Ronchetti, E. (1990). Small Sample Asymptotics. IMS, Hayward, CA.
  • Giné, E., Götze, F. and Mason, D. M. (1997). When is the Student $t$-statistic asymptotically standard normal? Ann. Probab. 25 1514–1531.
  • Griffin, P. S. and Kuelbs, J. (1989). Self-normalized laws of the iterated logarithm. Ann. Probab. 17 1571–1601.
  • Griffin, P. S. and Kuelbs, J. (1991). Some extensions of the LIL via self-normalizations. Ann. Probab. 19 380–395.
  • Hall, P. (1987). Edgeworth expansion for Student's $t$ statistic under minimal moment conditions. Ann. Probab. 15 920–931.
  • Jensen, J. L. (1995). Saddlepoint Approximations. Oxford Univ. Press.
  • Kolassa, J. E. (1997). Series Approximation Methods in Statistics, 2nd ed. Lecture Notes in Statist. 88. Springer, New York.
  • Logan, B. F., Mallows, C. L., Rice, S. O. and Shepp, L. A. (1973). Limit distributions of self-normalized sums. Ann. Probab. 1 788–809.
  • Lugannani, R. and Rice, S. (1980). Saddlepoint approximation for the distribution of the sum of independent random variables. Adv. in Appl. Probab. 12 475–490.
  • Reid, N. (1988). Saddlepoint methods and statistical inference (with discussion). Statist. Sci. 3 213–238.
  • Shao, Q.-M. (1997) Self-normalized large deviations. Ann. Probab. 25 285–328.\goodbreak
  • Wang, Q. Y. and Jing, B.-Y. (1999). An exponential nonuniform Berry–Esseen bound for self-normalized sums. Ann. Probab. 27 2068–2088.