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In the classical two-sample problem, the conventional approach for testingdistributions equality is based on the difference between the two marginalempirical distribution functions, whereas a test for independence is based onthe contrast between the bivariate and the product of the marginal empiricaldistribution functions. In this article we consider the problem of testingindependence and distributions equality when the observer is "colour blind" sohe cannot distinguish the distribution which has generated each of the twomeasurements. Within a nonparametric framework, we propose an empirical processfor this problem and find the linear statistic which is asymptotically optimalfor testing the equality of the marginal distributions against a specific formof contiguous alternatives.

Asymptotic hypothesis testing for the colour blind problem