Distribution free testing for conditional distributions given covariates

Given a sample (ξi,Xi)i=1n of i.i.d. pairs, consider Fx and Qx as two different models for conditional distribution of ξi given Xi=x. The paper shows how the empirical process for testing Fx can be transformed into a process with the same asymptotic behavior as the empirical process for testing Qx thereby rendering the two testing problems equivalent.