The problem of identification of parameters by the distribution of the maximum random variable (Q1074213)

Suppose the equation \(\prod^{n}_{i=1}F(a_ ix)=\prod^{m}_{j=1}F(b_ jx)\) holds, where F is a df and \(a_ i\), \(b_ j\) \((i=1,2,...,n\); \(j=1,2,...,m)\) are some parameters. It is shown that \(\{a_ 1,a_ 2,...,a_ n\}\) is a permutation of \(\{b_ 1,b_ 2,...,b_ m\}\) under two different conditions, each of which holds for the case when F is the Cauchy distribution. The following theorem is also proved. Suppose \(F_ 1,F_ 2,...,F_ n\) are non-singular bivariate normal distributions with zero means and that at least one of them has a nonzero correlation. Also suppose that \(\prod^{n}_{i=1}F_ i^{c_ i}=1\), where each \(c_ i\) is 1 or -1. Then this equation can always be reduced to an equality where the number of factors on the left side is n-2s, \(s\geq 1\); moreover, for the factors that simplify out, \(\sum c_ i=0\), the summation being over these factors, and the parameters are the same.