Let us think about the discrete case; that is let us suppose we are interested in determining a probability distribution P on the discrete set
Omega=mathbbZ2n.
Such a probability distribution assigns a nonnegative weight to each (i,j)inOmega. |Omega|=n2, thus P is determined by n2−1 nonzero variables pi,j whose sum is less than 1. To fix the marginals of P means to put 2(n−1) constraints on pi,j. In addition to these constraints, the present question also imposes a distribution on X−Y. These translate into n−1 further constraints on pi,j. Thus, in general, P will be a function of n2−1−3(n−1) free variables.
The special cases you mention (i.e, the clayton and gumbel copulas as well as the normal distribution), however, are determined by the marginal distributions and an additional real parameter. In general, if the given data makes sense, theta can be recovered by first writing an equation that it satisfies and then solving it.
Under any of the above mentioned copulas
the joint distribution equals Phi(F(x),G(y),theta) where F is the X marginal, G is the Y marginal and theta is a real number. The only unknown here is theta. Knowing the distribution of X−Y, means in particular we know the probability that X−Y=n−1[again assuming that we are operating in the discrete setup]. There is only one way this can happen, i.e, if Y=0 and X=n−1. Thus, we know the weight p(n−1,0) of the point
(n−1,0). Then, theta is the solution of
Phi(F(n−1),G(0),theta)−Phi(F(n−2),G(0),theta)=p(n−1,0).
In the case of mathbbR2, one can for example, write the following equation for theta:
int2mathbbR(x−y)dPhi(F(x),G(y),theta)=intzdS(z)
where S is the distribution of X−Y given in the problem.
Edit: more importantly, it seems, one has to check if the given distribution of X−Y is compatible with the copula in question. Because, there are many equations that one can write for theta and all must give the same answer for the solution to be meaningful.
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