Some positive dependence stochastic orders

In this paper we study some stochastic orders of positive dependence that arise when the underlying random vectors are ordered with respect to some multivariate hazard rate stochastic orders, and have the same univariate marginal distributions. We show how the orders can be studied by restricting them to copulae, we give a number of examples, and we study some positive dependence concepts that arise from the new positive dependence orders. We also discuss the relationship of the new orders to other positive dependence orders that have appeared in the literature.     

Bivariate interpolating polynomials and splines (I)

The multivariate splines which were first presented by de Boor as a complete theoretical system have intrigued many mathematicians who have devoted many works in this field which is still in the process of development. The author of this paper is interested in the area of interpolation with special emphasis on the interpolation methods and their approximation orders. But such B-splines (both univariate and multivariate) do not interpolated directly, so I approached this problem in another way which is to extend my interpolating spline of degree 2n-1 in univariate case (See[7]) to multivariate case. I selected triangulated region which is inspired by other mathematician’s works (e.g. [2] and [3]) and extend the interpolating polynomials from univariate to m-variate case (See [10])In this paper some results in the case m=2 are discussed and proved in more concrete details. Based on these polynomials, the interpolating splines (it is defined by me as piecewise polynomials in which the unknown partial derivatives are determined under certain continuous conditions) are also discussed. The approximation orders of interpolating polynomials and of cubic interpolating splines are inverstigated. We limited our discussion on the rectangular domain which is partitioned into equal right triangles. As to the case in which the rectangular domain is partitioned into unequal right triangles as well as the case of more complicated domains, we will discuss in the next paper.     

Conditional orderings and positive dependence

Every univariate random variable is smaller, with respect to the ordinary stochastic order and with respect to the hazard rate order, than a right censored version of it. In this paper we attempt to generalize these facts to the multivariate setting. It turns out that in general such comparisons do not hold in the multivariate case, but they do under some assumptions of positive dependence. First we obtain results that compare the underlying random vectors with respect to the usual multivariate stochastic order. A larger slew of results, that yield comparisons of the underlying random vectors with respect to various multivariate hazard rate orders, is given next. Some comparisons with respect to the orthant orders are also discussed.     

Analysis and Construction of Multivariate Interpolating Refinable Function Vectors

In this paper, we shall introduce and study a family of multivariate interpolating refinable function vectors with some prescribed interpolation property. Such interpolating refinable function vectors are of interest in approximation theory, sampling theorems, and wavelet analysis. In this paper, we characterize a multivariate interpolating refinable function vector in terms of its mask and analyze the underlying sum rule structure of its generalized interpolatory matrix mask. We also discuss the symmetry property of multivariate interpolating refinable function vectors. Based on these results, we construct a family of univariate generalized interpolatory matrix masks with increasing orders of sum rules and with symmetry for interpolating refinable function vectors. Such a family includes several known important families of univariate refinable function vectors as special cases. Several examples of bivariate interpolating refinable function vectors with symmetry will also be presented.     

Singular rules for the calculation of non-normal multivariate Padé approximants

Section 1 describes the univariate situation in the case of non-normal Padé approximants and Cordellier's extension of the famous five-star identity of Wynn. Section 2 repeats our definition of multivariate Padé approximants and proves a number of theorems that remain valid when going from the univariate to the multivariate case. These theorems and more new results given in Section 3, will finally also copy Cordellier's extension from the univariate to the multivariate case.     

Application de la théorie des fonctions caractéristiques de plusieurs variables á l'étude de certains problémes de décomposition des fonctions caractéristiques d'une variable

Summary We describe here a method showing the equivalency between the problems of decomposition for some univariate characteristic functions and some associated multivariate characteristic functions. Using this method, some new results on univariate characteristic functions are deduced from known theorems on multivariate characteristic functions (it seems very difficult to obtain a direct proof of these results). For example, we give a characterization of finite products of Poisson laws without indecomposable factors, generalizing a Paul Lévy's result on lattice characteristic functions. Finally, modifying a method due to B. Ramachandran, we extend a result of I. V. Ostrovskiy on decompositions of infinitely divisible characteristic functions with independent Poisson spectrum.     

A characterization of the multivariate excess wealth ordering

In this paper, some new properties of the upper-corrected orthant of a random vector are proved. The univariate right-spread or excess wealth function, introduced by Fernández-Ponce et al. (1996), is extended to multivariate random vectors, and some properties of this multivariate function are studied. Later, this function was used to define the excess wealth ordering by Shaked and Shanthikumar (1998) and Fernández-Ponce et al. (1998). The multivariate excess wealth function enable us to define a new stochastic comparison which is weaker than the multivariate dispersion orderings. Also, some properties relating the multivariate excess wealth order with stochastic dependence are described.     

An approximation method to model multivariate interpolation problems: Indexing HDMR

Dealing with univariate or bivariate data sets instead of a multivariate data set is an important concern in interpolation problems and computer-based applications. This paper presents a new data partitioning method that partitions the given multivariate data set into univariate and bivariate data sets and constructs an approximate analytical structure that interpolates function values at arbitrarily distributed points of the given grid. A number of numerical implementations are also given to show the performance of this new method.     

On a new multivariate IFR ageing notion based on the standard construction

Many criteria of ageing for random variables or vectors have been proposed in the literature over many years. For instance, a random variable is increasing in failure rate (IFR) if, and only if, it can be ordered with an exponentially distributed random variable in the classical univariate convex transform order. A new multivariate generalization of the convex transform order has recently been proposed in the literature. In this work, we propose a new multivariate IFR notion for multivariate distributions based on comparisons in this new order with a properly defined exponentially distributed random vector. Properties, applications, and illustrations of this new notion are given as well. Copyright © 2016 John Wiley & Sons, Ltd.     

Multivariate Stochastic Orders Induced by Case-Control Sampling

It is shown that retrospective sampling induces stochastic order relations in case-control studies. More specifically if the regression function is increasing and the covariates are positively dependent, then the covariates for cases are larger, with respect to some multivariate stochastic order, than the covariates of the controls. Strong dependence concepts yield strong multivariate stochastic orders. Conversely, different multivariate stochastic orders imply different monotonicity properties on the regression function. The results carry over to marginal models, transformed models and to problems involving confounders. The results set forth a new theoretical foundation for the analysis of case-control studies.     

Dependence structures of multivariate Bernoulli random vectors

In some situations, it is difficult and tedious to check notions of dependence properties and dependence orders for multivariate distributions supported on a finite lattice. The purpose of this paper is to utilize a newly developed tool, majorization with respect to weighted trees, to lay out some general results that can be used to identify some dependence properties and dependence orders for multivariate Bernoulli random vectors. Such a study gives us some new insight into the relations between the concepts of dependence.     

On Multivariate Interpolation

A new approach to interpolation theory for functions of several variables is proposed. We develop a multivariate divided difference calculus based on the theory of noncommutative quasi-determinants. In addition, intriguing explicit formulae that connect the classical finite difference interpolation coefficients for univariate curves with multivariate interpolation coefficients for higher dimensional submanifolds are established.     

Multivariate hazard rate orders

Two multivariate hazard rate stochastic orders are introduced and studied. Their meaning, properties, and relationship to other common stochastic orders are examined and investigated. Some examples that illustrate the theory are detailed. Finally, some applications of the new orders in reliability theory and in actuarial science are described.     

Regression models for multivariate ordered responses via the Plackett distribution

We investigate the properties of a class of discrete multivariate distributions whose univariate marginals have ordered categories, all the bivariate marginals, like in the Plackett distribution, have log-odds ratios which do not depend on cut points and all higher-order interactions are constrained to 0. We show that this class of distributions may be interpreted as a discretized version of a multivariate continuous distribution having univariate logistic marginals. Convenient features of this class relative to the class of ordered probit models (the discretized version of the multivariate normal) are highlighted. Relevant properties of this distribution like quadratic log-linear expansion, invariance to collapsing of adjacent categories, properties related to positive dependence, marginalization and conditioning are discussed briefly. When continuous explanatory variables are available, regression models may be fitted to relate the univariate logits (as in a proportional odds model) and the log-odds ratios to covariates.     

On Hsu's theorem in multivariate regression

The paper deals with optimal quadratic unbiased estimation of the unknown dispersion matrix in multivariate regression models without assuming normality of the errors. We show that Hsu's theorem for univariate regression models continues to multivariate models with no additional assumptions. Furthermore optimal quadratic plus linear estimating functions for regression coefficients are considered, and we investigate whether the ordinary linear estimates are the best. This leads to a new theorem which is similar to that of Hsu.     

A multivariate IFR notion based on the multivariate dispersive ordering

In this paper we present a definition of multivariate increasing failure rate based on the concept of multivariate dispersion. This new definition is an extension of the univariate characterization of increasing failure rate distributions under dispersive ordering of the residual lives. We study this definition in the Clayton–Oakes model and the family of generalized order statistics. Copyright © 2009 John Wiley & Sons, Ltd.     

Dependence between two multivariate extremes

We extend the characterizations given by Takahashi (1988) for the independence and the total dependence of the univariate marginals of a multivariate extreme value distribution to its multivariate marginals. We also deal with the problem of how to measure the strength of the dependence among multivariate extremes. By presenting new definitions for the extremal coefficient, we propose measures that summarize the dependence between two multivariate extreme value distributions and preserve the main properties of the known bivariate coefficient for two univariate extreme value distributions. Finally, we illustrate these contributions to model the dependence among multivariate marginals with examples.     

On generating multivariate Poisson data in management science applications

Generating multivariate Poisson random variables is essential in many applications, such as multi echelon supply chain systems, multi‐item/multi‐period pricing models, accident monitoring systems, etc. Current simulation methods suffer from limitations ranging from computational complexity to restrictions on the structure of the correlation matrix, and therefore are rarely used in management science. Instead, multivariate Poisson data are commonly approximated by either univariate Poisson or multivariate Normal data. However, these approximations are often not adequate in practice. In this paper, we propose a conceptually appealing correction for NORTA (NORmal To Anything) for generating multivariate Poisson data with a flexible correlation structure and rates. NORTA is based on simulating data from a multivariate Normal distribution and converting it into an arbitrary continuous distribution with a specific correlation matrix. We show that our method is both highly accurate and computationally efficient. We also show the managerial advantages of generating multivariate Poisson data over univariate Poisson or multivariate Normal data. Copyright © 2011 John Wiley & Sons, Ltd.     

Multivariate θ-generalized normal distributions

A new family of continuous multivariate distributions is introduced, generalizing the canonical form of the multivariate normal distribution. The well-known univariate version of this family, as developed by Box, Tiao and Lund, among others, has proven a valuable tool in Bayesian analysis and robustness studies, as well as serving as a unified model for least θ's and maximum likelihood estimates. The purpose of the family introduced here is to extend, to a degree of generality which will permit practical applications, the useful role played by the univariate family to a multidimensional setting.     

A new characterization of convergent multivariate subdivision schemes with nonnegative masks

We present in this paper new necessary and sufficient conditions for convergence of multivariate subdivision schemes with nonnegative finite masks, which simplify the assertion obtained by Wang in (J Approx Theory 113:207–220, 2001). Moreover, we construct an example, which shows that the convergence behaviors for univariate subdivision schemes and multivariate ones are essentially different.