Confidence intervals for the length of a vector mean

Abstract

Suppose we have a random sample of size n from a multivariate distribution with finite moments, for which a parametric form is not available. We wish to obtain a confidence interval (CI) for the length of its mean. The usual method is to Studentize. The resulting CIs are not exact. The error in their nominal levels is ∼n ^−1/2 and ∼n ⁻¹ in the one-sided and two-sided cases. We show how to reduce these errors to ∼n ^−3/2 and ∼n ⁻².

Keywords:

• confidence intervals
• length
• simulation

AMS 2000 Subject Classification :

• Primary: 60F05
• Secondary: 62E20

Acknowledgements

The authors would like to thank the Editor-in-Chief and the Associate Editor for carefully reading the paper and for their comments which greatly improved the paper.