Tail approximations for samples from a finite population with applications to permutation tests

¹ Department of Statistics and Finance, University of Science and Technology of China, Hefei 230026, Anhui, P.R. China
huzs@ustc.edu.cn
² School of Mathematics and Statistics The University of Sydney, NSW, 2006, Australia
johnr@maths.usyd.edu.au; qiying@maths.usyd.edu.au

Received: 11 September 2009
Revised: 17 June 2010

Abstract

This paper derives an explicit approximation for the tail probability of a sum of sample values taken without replacement from an unrestricted finite population. The approximation is shown to hold under no conditions in a wide range with relative error given in terms of the standardized absolute third moment of the population, β_3N. This approximation is used to obtain a result comparable to the well-known Cramér large deviation result in the independent case, but with no restrictions on the sampled population and an error term depending only on β_3N. Application to permutation tests is investigated giving a new limit result for the tail conditional probability of the statistic given order statistics under mild conditions. Some numerical results are given to illustrate the accuracy of the approximation by comparing our results to saddlepoint approximations requiring strong conditions.