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

Abstract

Consider $n$ iid random variables, where $\xi_1, \ldots, \xi_n$ are $n$ realisations of a random variable $\xi$ and $\zeta_1, \ldots, \zeta_n$ 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 $P\left(\xi_i\leq x\right)=F(x,\zeta_i)$. We develop a statistical test to see if the $\xi_1, \ldots, \xi_n$ are distributed according to the distribution function $F(x,\zeta_i)$. We call this new statistical test the condition Kolmogorov-Smirnov test.