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