Approximation of Kolmogorov-Smirnov Test Statistics

TL;DR

This paper studies the asymptotic behavior of the Kolmogorov-Smirnov test statistics in multivariate settings, providing approximations for large thresholds and exploring various distribution functions.

Contribution

It introduces new asymptotic approximations for the Kolmogorov-Smirnov statistics based on the multivariate Brownian sheet, extending results to general distribution functions.

Findings

Derived asymptotic formulas for large thresholds u

Analyzed the behavior for general distribution functions

Presented specific examples illustrating the results

Abstract

Motivated by the weak limit of the Kolmogorov-Smirnov test statistics, in this contribution, we concern the asymptotics of \begin{align*} \mathbb{P}\left\{\sup_{\boldsymbol{x}\in [0,1]^n}\left(W(\boldsymbol{x})\Big| W(\boldsymbol{1})=w\right)>u\right\}, \ w\in\mathbb{R}, \end{align*} for large $u$ where $W(\boldsymbol{x})$ is the multivariate Brownian sheet based on a distribution function $F$. The results related to general $F$ are investigated and some important examples are also showed.