Accurate inference for a one parameter distribution based on the mean of a transformed sample

Abstract

A great deal of inference in statistics is based on making the approximation that a statistic is normally distributed. The error in doing so is generally $O(n^{-1/2})$ and can be very considerable when the distribution is heavily biased or skew. This note shows how one may reduce this error to $O(n^{-(j+1)/2})$, where $j$ is a given integer. The case considered is when the statistic is the mean of the sample values from a continuous one-parameter distribution, after the sample has undergone an initial transformation.