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

The Annals of Statistics · 2004 · Volume 32 · Issue 6

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.

Keywords

60G50, 62E20, absolute error, asymptotic normality, Edgeworth expansion, large deviation, relative error, saddlepoint approximation, self-normalized sum, Student’s t-statistic

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