TY - JOUR
T1 - Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions
AU - Karlin, Samuel
AU - Rinott, Yosef
PY - 1980/12
Y1 - 1980/12
N2 - A function f(x) defined on X = X1 × X2 × ... × Xn where each Xi 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 (MTP2). A random vector Z = (Z1, Z2,..., Zn) of n-real components is MTP2 if its density is MTP2. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies -DΣ-1D 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 MTP2 properties. The MTP2 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 MTP2 kernels are presented and amplified by a wide variety of applications.
AB - A function f(x) defined on X = X1 × X2 × ... × Xn where each Xi 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 (MTP2). A random vector Z = (Z1, Z2,..., Zn) of n-real components is MTP2 if its density is MTP2. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies -DΣ-1D 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 MTP2 properties. The MTP2 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 MTP2 kernels are presented and amplified by a wide variety of applications.
UR - http://www.scopus.com/inward/record.url?scp=0039766073&partnerID=8YFLogxK
U2 - 10.1016/0047-259X(80)90065-2
DO - 10.1016/0047-259X(80)90065-2
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AN - SCOPUS:0039766073
SN - 0047-259X
VL - 10
SP - 467
EP - 498
JO - Journal of Multivariate Analysis
JF - Journal of Multivariate Analysis
IS - 4
ER -