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

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.