The problem of identification of parameters by the distribution of the maximum random variable

Suppose that X1, X2,..., Xn are independently distributed according to certain distributions. Does the distribution of the maximum of {X1, X2,..., Xn} uniquely determine their distributions? In the univariate case, a general theorem covering the case of Cauchy random variables is given here. Also given is an affirmative answer to the above question for general bivariate normal random variables with non-zero correlations. Bivariate normal random variables with nonnegative correlations were considered earlier in this context by T. W. Anderson and S. G. Ghurye.