Asymptotic distribution of LR1LR2 under H0 in a copula model

In this section, we assume that the joint survival function of the variables {(Tiji,T2ji) ,i = 1,...,nj} is formulated by a copula model, that is

Fj (xi,x2) = Ca(F3i(xi),F2,(x2)) for any (xi,x2) e [0,t]2. The null hypothesis we wish to test is equality in both groups of the marginal survival distributions, that is H0 : {AA = Af = A1, AA = AB = A2 on [0, t]}. _

Then, under Ho, the {Tkji,i = 1, ...,nj} have marginal survival function Fk and cumulative hazard function Ak for k =1, 2, and the couples {(Tiji, T2ji), i = 1,..., nj} have joint survival function F and density f.

We introduce the following martingales, for k = 1,2, j = A,B and i = 1,...,nj:

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