  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.
