Computing Boundary Crossing Probabilities of General Empirical Processes
Rigel Galgana, Takehiro Oyakawa, Amy Greenwald · Society for Industrial and Applied Mathematics eBooks · 2023
Order statistics play a fundamental role in the construction of empirical distribution functions, which are fundamental to empirical process theory and non- parametric statistics. In some applications, it is desirable to compute the joint cumulative distribution function (cdf) of d order statistics exactly. Efficient algorithms to compute this quantity are known when the data are i.i.d.; however, the task becomes significantly more challenging when relaxing either the identically distributed or the independence assumption. Existing methods for the non-i.i.d. setting obtain the joint cdf indirectly, by first computing and then aggregating over the marginal distributions. In this paper, we simplify an existing dynamic programming solution to achieve an exponential-in-d factor improvement in both time and space complexity over known methods. We detail the independent, non-identical setting, and then outline how our method extends to more general settings (e.g., dependent random variables) in an online appendix.