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


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.


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


Published: December 2004
First available in Project Euclid: 7 February 2005

zbMATH: 1068.62016
MathSciNet: MR2153999
Digital Object Identifier: 10.1214/009053604000000742

Primary: 62E20
Secondary: 60G50

Keywords: absolute error , asymptotic normality , Edgeworth expansion , large deviation , relative error , saddlepoint approximation , self-normalized sum , Student’s t-statistic

Rights: Copyright © 2004 Institute of Mathematical Statistics

Vol.32 • No. 6 • December 2004
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