The univariate Kolmogorov-Smirnov (KS) test is a non-parametric statistical test designed to assess whether two samples come from the same underlying distribution. The versatility of the KS test has made it a cornerstone of statistical analysis across the scientific disciplines. However, the test proposed by Kolmogorov and Smirnov does not naturally extend to multidimensional distributions. Here, we present the fasano.franceschini.test package, an R implementation of the 2-D KS two-sample test as defined by Fasano and Franceschini (Fasano and Franceschini 1987) and provide multiple use cases across the scientific disciplines. The fasano.franceschini.test package provides three improvements over the current 2-D KS test on the Comprehensive R Archive Network (CRAN): (i) the Fasano and Franceschini test has been shown to run in $O(n^2)$ versus the Peacock implementation which runs in $O(n^3)$; (ii) the package implements a procedure for handling ties in the data; and (iii) the package implements a parallelized permutation procedure for improved significance testing. Ultimately, the fasano.franceschini.test package presents a robust statistical test for analyzing random samples defined in 2-dimensions.

翻译：Kolmogorov和Smirnov提出的测试并不自然延伸到多层面分布。在这里,我们介绍Fasano.franceschini.Test 软件包,即Fasano和Francschini(Fasano和Francschini,1987年)定义的2-D KS双模版测试的R执行,并在整个科学学科提供多种使用案例。Fasano.franceschini.Test 软件包为综合档案网络(CRAN)目前的2-DKS测试提供了三项改进:(一) Fasano和Francschini的测试已经显示以美元运行,而Peacocken实施则以美元运行;(二) 软件包实施了数据连接处理程序;Fasano.franceschini.Test 软件包提供了三套件,比目前综合档案网络(CRAN)的2-DKS测试做了三项改进:(一) Fasano和Francschini测试显示Fasano和Francschini测试以美元运行,而Pecocken (n) 3美元运行;(二) 软件安装改进了数据连接连接程序;以及(三) 最终测试标准,按标准进行。