Accurate inference for scale and location families

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), where n is the sample size and can be considered when the distribution of the statistic is heavily biased or skewed. This note shows how one may reduce the 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 of a continuous distribution with a scale or location change after the sample has undergone an initial transformation, which may depend on an unknown parameter. The transformation corresponding to Fisher's score function yields an asymptotically efficient procedure.

Keywords:

• accurate inference
• confidence interval
• Edgeworth expansion
• percentiles
• scale and location parameters
• transformation

Acknowledgements

The authors would like to thank the referee, the Associate Editor and the Editor for careful reading and for their comments which greatly improved the paper.