Identification of parameters by the distribution of the minimum: The tri-variate normal case with negative correlations

Let (X1,X2,X3) be a 3-variate normal vector with zero means and a non-singular co-variance matrix Σ, where for i≠j, Σij≤0. It is shown here that it is then possible to determine the three variances and the three correlations based only on the knowledge of the density of the minimum {X1,X2,X3}. Our method consists of careful determination and analysis of the asymptotic orders of various bivariate tail probabilities.