My question is motivated by a problem with estimation by maximum likelihood. In simple terms, I want to estimate a parameter \(\beta \) and have three variables, \(X\), \(Y\) and \(Z\), such that \(Z = X+Y\). Using all three variables is pointless and the joint pdf is singular. Now, suppose that instead of \(Z\) I have \(U\) such that \(U = Z + \epsilon\), where \(\epsilon\) is independent from \(X\) and \(Y\) and its distribution does not depend on \(\beta\). The joint pdf of \(X\), \(Y\) and \(U\) is not singular. Seems to me obvious that \(U\) does not contain any additional information about \(\beta\), given \(X\) and \(Y\). So, I think it must be true that, in terms of the likelihood function, \(L(\beta;X,Y,U)=L(\beta;X,Y)*const\). If I am correct, what would be the best way to prove it? Thanks.