均值非零的多元正态分布函数矩形区域的概率质量单调性(英文)

Monotonicity of the Probability of a Rectangular Region Under a Multinormal Distribution With Nonzero Means

设X=(X_1,X_2,…,X_n)′～N_n(∑β,∑),其中β∈R^n,∑=(σ_(ij))。如果β的所有分量同时非负或同时非正,且存在对角元取值±1的对角矩阵D使得-DΣ^(-1)D的非对角元元素非负,则P[|X_j|<c_j,j=1,2,…,n]关于所有|σ_(ij)|,i≠j为非降的,其中c=(c_1,c_2,…,c_n)′为任意的。此结论推广了Bolviken和Joag—dev(1982)的结果。

Let random vecter X=(X_1, X_2,…, X_n)' follow the distribution N_n (∑γ,∑), where β∈R^n and ∑=(σ_(ij)). If all elements of β are nonnegative or nonpositive, and if there exists a diagonal matrix D with diagonal elements ±1 such that the off-diagonal elements of-D∑^(-1) D are all nonnegative, then P[|X_1|<c_1,i=1,…,n] is shown to be nondecreasing in |σ_(ij)|, i≠j, for every c=(c_,c_2,…,c_n)'. This generalizes the result of Bolviken and Joag-dev (1982).