Test for homogeneity with unordered paired observations

In some applications, an experimental unit is composed of two distinct but related subunits. The response from such a (X1,X2) we observe only Y1=min{X1,X2} and Y2=max{X1,X2}, i.e., the subunit identities are not observed. We call (Y1,Y2) unordered paired observations. Based on observations {(Y1i,Y2i)}i=1n, interested in whether marginal distributions for X1 X2 identical. Testing methods available literature under assumptions that Var(X1)=Var(X2) COV(X1,X2)=0. However, by extensive simulation studies, when one or both violated, these have inflated type I errors much lower powers. this paper, study likelihood ratio test statistics various scenarios explore their limiting without restrictive assumptions. Furthermore, develop Bartlett correction formulae to enhance precision sample size large. Simulation studies real-data examples used illustrate efficacy proposed methods.