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

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