A von Mises approximation to the small sample distribution of the trimmed mean

The small sample distribution of the trimmed mean is usually approximated by a Student’s t distribution. But this approximation is valid only when the observations come from a standard normal model and the sample size is not very small. Moreover, until now, there is only empirical justification for this approximation but no formal proof. Although there are some accurate saddlepoint approximations when the sample size is small and the distribution not normal, these are very difficult to apply and the elements involved in it, difficult to interpret. In this paper we propose a new approximation based on the von Mises expansion for the tail probability functional of the trimmed mean, which improves the usual Student’s t approximation in the normal case and which can be applied for other models. This new approximation allows, for instance, an objective choice of the trimming fraction in a context of hypothesis testing problem using the new tool of the p value line.