Identification of the parameters of a trivariate normal vector by the distribution of the minimum
In competing risks theory and reliability of multicomponent systems occurs the following type of problem (P): Let \(X_1\), \(X_2\), \(X_3\) be three random variables with joint distribution function \(F\), and \(Y_1\), \(Y_2\), \(Y_3\) another three random variables with joint distribution function \(G\). If \(\min \{X_1, X_2, X_3\}\) and \(\min \{Y_1, Y_2, Y_3\}\) have the same distribution, then how \(F\) and \(G\) should be related, and must they be the same? The main result of the present paper is to solve (P) when \(F\) and \(G\) are multivariate normal distributions such that the correlation \(\rho_{ij}\) between \(X_i\) and \(X_j\), and the corelation \(\rho_{ij}'\) between \(Y_i\) and \(Y_j\) are both independent of \(i\) and \(j\). The problem (P) is solved by identifying the parameters of \(F\) uniquely in terms of the density function for the random variable minimum.
scientific article; zbMATH DE number 1331383