Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions

A function f(x) defined on $\text{X}$ = $\text{X}$₁ × $\text{X}$₂ × … × $\text{X}$_n where each $\text{X}$_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 $\text{X}$, is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) 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.     

Multivariate normal distributions,Fisher information and matrix inequalities

Using appropriately parameterized families of multivariate normal distributions and basic properties of the Fisher information matrix for normal random vectors, we provide statistical proofs of the monotonicity of the matrix function A ^-1 in the class of positive definite Hermitian matrices. Similarly, we prove that A ¹¹ < A ^-111, where A ₁₁ is the principal submatrix of A and A ¹¹ is the corresponding submatrix of A ^-1. These results in turn lead to statistical proofs that the the matrix function A ^-1 is convex in the class of positive definite Hermitian matrices and that A ² is convex in the class of all Hermitian matrices. (These results are based on the Loewner ordering of Hermitian matrices, under which A < B if A - B is non-negative definite.) The proofs demonstrate that the Fisher information matrix, a fundamental concept of statistics, deserves attention from a purely mathematical point of view.     

Multivariate Tweedie distributions and some related capital-at-risk analyses

We study a multivariate extension of the univariate exponential dispersion Tweedie family of distributions. The class, referred to as the multivariate Tweedie family (MTwF), on the one hand includes multivariate Poisson, gamma, inverse Gaussian, stable and compound Poisson distributions and on the other hand introduces a high variety of new dependent probabilistic models unstudied so far. We investigate various properties of MTwF and discuss its possible applications to financial risk management.     

Quantization of probability distributions under norm-based distortion measures II: Self-similar distributions

For a probability measure P on Rd and n∈N consider where the infimum is taken over all subsets α of Rd with card(α)?n and V is a nondecreasing function. Under certain conditions on V, we derive the precise n-asymptotics of en for self-similar distributions P and we find the asymptotic performance of optimal quantizers using weighted empirical measures.     

Multivariate conditional versions of Spearman's rho and related measures of tail dependence

A new family of conditional-dependence measures based on Spearman's rho is introduced. The corresponding multidimensional versions are established. Asymptotic distributional results are derived for related estimators which are based on the empirical copula. Particular emphasis is placed on a new type of multidimensional tail-dependence measure and its relationship to other measures of tail dependence is shown. Multivariate tail dependence describes the limiting amount of dependence in the vertices of the copula's domain.     

多元Besov-Wiener类的平均宽度和最优恢复

本文得到了Besov－Wiener类的平均σ-K宽度，平均σ-L宽度，最优恢复的弱渐进结果．     

Maximal regularity via reverse Hölder inequalities for elliptic systems of n-Laplace type involving measures

In this note, we consider the regularity of solutions of the nonlinear elliptic systems of n-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space L ^n,∞. We also obtain the a priori global and local estimates for the L ^n,∞-norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.     

Uniform θ-equivalence of probability distributions based on information and related measures of discrepancy

Summary Some criteria based on K-L information number andW-divergence are presented for a certain type of uniform approximate equivalence of two probability distributions. As applications, some necessary and sufficient conditions are also given for the corresponding uniform asymptotic equivalence of two random sequences. The Institute of Statistical Mathematics     

Relationships between pure tail orderings of lifetime distributions and some concepts of residual life

The concepts of pure-tail orderings as defined by sq- and D-orderings are shown to order the family of reliability life distributions which age smoothly in a natural way. This ordering extends to comparisons regarding the limiting behavior of the residual life, mean residual life, sojourn time between perfect repairs in repairable systems, failure rate and, through the preservation of sq- and D-orderings by various reliability operations, to certain coherent systems of components that age smoothly. Possible applications of the results to the industrial practice of cannibalization are also noted.     

多元 Besov-Wiener 类的无穷维宽度和最优恢复

该文考虑 Besov-Wiener 类S_pqθ^r B(R^d)和 S_pqθ^r B(R^d)在 L_q(R^d) 空间下 (1≤q≤ p <∞ ) 的无穷维σ -宽度和最优恢复问题.通过考虑样条函数逼近和构造一种连续样条算子, 得到了关于无穷维Kolmogorov 宽度、无穷维线性宽度、无穷维 Gel'fand 宽度和最优恢复的弱渐近结果.     

Bounds and inequalities for the Perron root of a nonnegative matrix. II. Circuit bounds and inequalities

The paper suggests two-sided, upper, and lower circuit bounds for the Perron root of a nonnegative matrix, most of which are derived based on an extension of the monotonicity property of the Perron root established by Fiedler and Pták. Bibliography: 9 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 60–88.     

A class of means and related inequalities

The aim of this paper is two-fold: First we describe a certain class of strictly positive, continuous functions u:I defined on certain intervals I]0,+[, and demonstrate that each function u of this class permits the definition of an associated mean M_u (a₁,...,a_n) for any finite number a₁,...,a_n of numbers in I. The arithmetic and the geometric mean are special cases of these u-means M_u. -Thereafter, we improve the classical inequality between the geometric and the arithmetic mean and derive from it a corresponding inequality between certain u-means and the arithmetic mean.     

Generalizations of the Jensen-Steffensen and related inequalities

We present a couple of general inequalities related to the Jensen-Steffensen inequality in its discrete and integral form. The Jensen-Steffensen inequality, Slater’s inequality and a generalization of the counterpart to the Jensen-Steffensen inequality are deduced as special cases from these general inequalities.