# Efficient Calculation of the Joint Distribution of Order Statistics

## TL;DR

This paper introduces novel recursive methods for efficiently computing the joint distribution of order statistics of independent variables, enabling exact and rounded calculations, with applications in multiple hypothesis testing.

## Contribution

It generalizes existing recursive formulas to improve numerical computation of joint distributions of order statistics, both exactly and approximately.

## Key findings

- Developed generalized recursive formulas for joint distribution calculation.
- Achieved exact and faithfully rounded numerical results.
- Discussed applications in stepwise multiple hypothesis testing.

## Abstract

We consider the problem of computing the joint distribution of order statistics of stochastically independent random variables in one- and two-group models. While recursive formulas for evaluating the joint cumulative distribution function of such order statistics exist in the literature for a longer time, their numerical implementation remains a challenging task. We tackle this task by presenting novel generalizations of known recursions which we utilize to obtain exact results (calculated in rational arithmetic) as well as faithfully rounded results. Finally, some applications in stepwise multiple hypothesis testing are discussed.