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

Consider $n$ iid random variables, where $ξ_1, \ldots, ξ_n$ are $n$ realisations of a random variable $ξ$ and $ζ_1, \ldots, ζ_n$ are $n$ realisations of a random variable $ζ$. The distribution of each realisation of $ξ$, that is the distribution of \emph{one} $ξ_i$, depends on the value of the corresponding $ζ_i$, that is the probability $P\left(ξ_i\leq x\right)=F(x,ζ_i)$. We develop a statistical test to see if the $ξ_1, \ldots, ξ_n$ are distributed according to the distribution function $F(x,ζ_i)$. We call this new statistical test the condition Kolmogorov-Smirnov test.