On the Exceedances of Exchangeable Random Variables

Abstract

Suppose that X_n = (X₁,…,X_n) have mean 0, and a single-factor covariance Σ = (σ_ij) with σ_ii = 1 and σ_ij = ρ ≥ 0 for i ≠ j. For a threshold c, let S_n be the number of components of X_n that exceed c. We express the distribution of S_n in terms of a single integral, provide the limiting distribution as $n \rightarrow \infty$, and show that the limit resembles the Beta family. We then describe the shape of the exceedance distribution when the underlying distributions of the single-factor model have a certain likelihood ratio criterion with respect to its scale parameter, and we show that it obeys a majorization ordering.