Identifiability of the multinormal and other distributions under competing risks model

Abstract

Let X₁, X₂ ,..., X_p be p random variables with joint distribution function F(x₁ ,..., x_p). Let Z = min(X₁, X₂,.., X_p) and I = i if Z = X_i. In this paper the problem of identifying the distribution function F(x₁ ,..., x_p), given the distribution Z or that of the identified minimum (Z, I), has been considered when F is a multivariate normal distribution. For the case p = 2, the problem is completely solved. If p = 3 and the distribution of (Z, I) is given, we get a partial solution allowing us to identify the independent case. These results seem to be highly nontrivial and depend upon Liouville's result that the (univariate) normal distribution function is a nonelementary function. Some other examples are given including the bivariate exponential distribution of Marshall and Olkin, Gumbel, and the absolutely continuous bivariate exponential extension of Block and Basu.