## The Annals of Statistics

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

#### Abstract

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

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

Dates
First available in Project Euclid: 7 February 2005

https://projecteuclid.org/euclid.aos/1107794883

Digital Object Identifier
doi:10.1214/009053604000000742

Mathematical Reviews number (MathSciNet)
MR2153999

Zentralblatt MATH identifier
1068.62016

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1107794883

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