# Saddlepoint Approximations for the Difference of Order Statistics and Studentized Sample Quantiles

## Authors

**Chunsheng Ma** and **John Robinson**

## Status

Research Report 97-36
Date: December 1997

## Abstract

For a sample from a given distribution the difference of two order statistics
and the studentized quantile are statistics whose distribution is needed to
obtain tests and confidence intervals for quantiles and quantile
differences. This paper gives saddlepoint approximations for densities and
saddlepoint approximations of the Lugannani-Rice form for tail probabilities
of these statistics. The relative errors of the approximations are 1/n
uniformly in a neighbourhood of the parameters and this uniformity is global
if the densities are log-concave.

## Key phrases

saddlepoint approximation. Laplace method.
difference of order statistics. Studentized sample quantile.

## AMS Subject Classification (1991)

Primary: 62G09

Secondary: 62G15

## Content

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Sydney Mathematics and Statistics