CLASSES OF ORDERINGS OF MEASURES AND RELATED CORRELATION INEQUALITIES .1. MULTIVARIATE TOTALLY POSITIVE DISTRIBUTIONS
CLASSES OF ORDERINGS OF MEASURES AND RELATED CORRELATION INEQUALITIES .1. MULTIVARIATE TOTALLY POSITIVE DISTRIBUTIONS
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DOI：
10.1016/0047-259x(80)90065-2
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发表时间：
1980-01-01
影响因子：
1.6
通讯作者：
RINOTT, Y
中科院分区：
文献类型：
--
作者：
KARLIN, S;RINOTT, Y
定义在X= X1 × X2 ×... × Xn上的函数f（x），其中每个Xi都是全序的，满足f（x <$y）f（x <$y）≥ f（x）f（y），其中格运算<$和<$是指X上的通常序，称为2阶多元全正（MTP 2）。一个随机向量Z=（Z1，Z2，.，Zn）的n个实分量是MTP 2，如果它的密度是MTP 2。这类例子包括独立随机变量、协方差矩阵满足− D <$− 1 D且对角矩阵D具有非负非对角元素的绝对值多正态、随机Wishart矩阵的特征根、多元逻辑分布、伽玛分布和F分布等。复合和边际运算保留了MTP 2的性质. MTP 2属性有利于表征的界限置信集，计算覆盖概率，安全估计的多变量排名，在建立一个层次的相关不等式，并在研究单调马尔可夫过程。MTP 2内核的理论扩展，并通过各种各样的应用程序放大。
A function f (x) defined on X= X 1× X 2×…× X n where each X i is totally ordered satisfying f (x∨ y) f (x∧ y)≥ f (x) f (y), where the lattice operations∨ and∧ refer to the usual ordering on X, is said to be multivariate totally positive of order 2 (MTP 2). A random vector Z=(Z 1, Z 2,…, Z n) of n-real components is MTP 2 if its density is MTP 2. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies− DΣ− 1 D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP 2 properties. The MTP 2 property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP 2 kernels are presented and amplified by a wide variety of applications.