On the distribution of the extremes of unequally correlated normal variables with applications to antedependent cluster data (Q1807961)

Let \((Y_1, \ldots, Y_n)\) be multivariate normal with small correlations (under 0.25) between the components. Let \(Y_{(i)}\) be the \(i\)-th order statistic. Approximations to the joint density of all order statistics and the density of \(Y_{(i)}\) are given. For the positive equicorrelated case, the tail probabilities of \(Y_{(n)}\) obtained from this approximation are compared with those of \textit{S.S. Gupta}, \textit{K. Nagel} and \textit{S. Panchapakesan} [Biometrika 60, 403-413 (1973; Zbl 0263.62031)]. Applications to an antedependence model are discussed. In that context, for the unequally correlated case, a simulation study is performed to evaluate the accuracy of the tail probabilities of \(Y_{(n)}\) obtained by this approach, and the proposed approximation is compared with that one obtained by the existing Bonferroni bounds approximation.