A Dynamic Programming Algorithm to Compute Joint Distribution of Order Statistics on Graphs

TL;DR

This paper introduces a more efficient dynamic programming algorithm for computing the exact joint distribution of order statistics for dependent variables, significantly reducing computational complexity.

Contribution

It presents a novel, direct algorithm that improves upon existing methods by applying dimensionality reduction, enhancing efficiency in calculating joint distributions.

Findings

Achieves a $O(rac{d^{d-1}}{n})$ time complexity improvement.

Achieves a $O(d^{d})$ space complexity improvement.

Provides exact joint distribution calculations for dependent variables.

Abstract

Order statistics play a fundamental role in statistical procedures such as risk estimation, outlier detection, and multiple hypothesis testing as well as in the analyses of mechanism design, queues, load balancing, and various other logistical processes involving ranks. In some of these cases, it may be desirable to compute the \textit{exact} values from the joint distribution of $d$ order statistics. While this problem is already computationally difficult even in the case of $n$ independent random variables, the random variables often have no such independence guarantees. Existing methods obtain the cumulative distribution indirectly by first computing and then aggregating over the marginal distributions. In this paper, we provide a more direct, efficient algorithm to compute cumulative joint order statistic distributions of dependent random variables that improves an existing dynamic programming solution via dimensionality reduction techniques. Our solution guarantees a $O(\frac{d^{d-1}}{n})$ and $O(d^{d})$ factor of improvement in both time and space complexity respectively over previous methods.