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

TL;DR

This paper introduces a new statistical test, called the conditional Kolmogorov-Smirnov test, designed to determine if a set of dependent random variables follow a specified distribution conditioned on another variable.

Contribution

The paper develops a novel test for assessing the distribution of dependent variables conditioned on another variable, extending the classical Kolmogorov-Smirnov test to dependent data.

Findings

The test effectively detects deviations from the specified conditional distribution.

Simulation studies demonstrate the test's good size and power properties.

The method is applicable to various dependent data scenarios.

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.