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

Abstract

Let (X 1 , X 2 , X 3 ) be a 3-variate normal vector with zero means and a non-singular co-variance matrix , where for i = j , ij 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 {X 1 , X 2 , X 3 }. Our method consists of careful determination and analysis of the asymptotic orders of various bivariate tail probabilities.

Recovering a Gaussian distribution from its minimum

“…This allows us to significantly extend previous results. Moreover, from this approach much intuition is gained regarding the 'backstage' geometric difficulty implicit in this and similar recovery problems, particularly improving previous results established in [9] and [10].…”

“…Additionally, a similar set of problems were previously known in the context of competing and complimentary risks [4,5,14]. These kind of problems of identification have been the subject of interest to many authors, including [3,[6][7][8][10][11][12][13]. Particularly, in the same setting established above, the authors in [4] studied this recovery problem under the hypothesis ρ ij σ i < σ j .…”

“…In other words, is it possible to recover the matrix Σ from the distribution of X min ? The purpose of this paper is to solve this problem affirmatively, under the assumption that 1 t Σ −1 > 0, which in particular covers the case where the correlations are negative [9,10].…”

“…In [8,11] it was studied the case of common correlations. And, in [9,10] it was studied the case of non-negative correlations. The novelty of our approach consists in tackling the problem via a geometric approach, reducing it to a recovering problem in the context of circular radon transform.…”