A Kolmogorov-Smirnov type test for two inter-dependent random variables

Abstract

Consider $n$ iid random variables, where ${\xi }_1,\dots ,{\xi }_n$ are $n$ realisations of a random variable $\xi$ and ${\zeta }_1,\dots ,{\zeta }_n$ are $n$ realisations of a random variable $\zeta$. The distribution of each realisation of $\xi$, that is the distribution of \emph{one} ${\xi }_i$, depends on the value of the corresponding ${\zeta }_i$, that is the probability $P\left({\xi }_i\le x\right)=F(x,{\zeta }_i)$. We develop a statistical test to see if the ${\xi }_1,\dots ,{\xi }_n$ are distributed according to the distribution function $F(x,{\zeta }_i)$. We call this new statistical test the condition Kolmogorov-Smirnov test.