Asymptotic hypothesis testing for the colour blind problem

TL;DR

This paper develops an asymptotic hypothesis testing method for the colour blind problem, where the observer cannot distinguish which distribution generated each measurement, within a nonparametric framework.

Contribution

It introduces an empirical process and identifies an asymptotically optimal linear statistic for testing distribution equality under colour blindness.

Findings

Proposes a nonparametric empirical process for colour blind testing.

Identifies an asymptotically optimal linear test statistic.

Addresses the problem of testing independence without distribution labels.

Abstract

In the classical two-sample problem, the conventional approach for testing distributions equality is based on the difference between the two marginal empirical distribution functions, whereas a test for independence is based on the contrast between the bivariate and the product of the marginal empirical distribution functions. In this article we consider the problem of testing independence and distributions equality when the observer is "colour blind" so he cannot distinguish the distribution which has generated each of the two measurements. Within a nonparametric framework, we propose an empirical process for this problem and find the linear statistic which is asymptotically optimal for testing the equality of the marginal distributions against a specific form of contiguous alternatives.