Saddlepoint Approximation for Student ’ S T - Statistic with No Moment Conditions

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 approxima-tion'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. 1. Introduction. In many statistical applications approximations to the probability that a random variable (r.v.), say T n , exceeds a certain threshold value are important since the exact distribution function (d.f.) of T n may be very difficult or even impossible to obtain in most cases. Such approximations are useful, for example, in constructing confidence intervals and in calculating p-values in hypothesis testing. In those circumstances, we are usually dealing with tail probabilities of the r.v., T n. Since these tail