Positive dependence of random variables with a common marginal distribution

In this paper we review some notions of positive dependence of random variables with a common univariate marginal distribution and describe the related moment and probability inequalities. We first present a comparison between i.i.d. random variables and exchangeable random variables via an application of de Finetti's theorem, then describe some useful probability inequalities via partial orderings of the strength of their positive dependence. Finally, we state a result for random variables which are not necessarily exchangeable. Special applications to the multivariate normal distribution will be discussed, and the results involve only the correlation matrix of the distribution.     

Circular consecutive k-out-of-n systems with exchangeable dependent components

In this paper we study circular consecutive k-out-of-n systems consisting of exchangeable components. We derive explicit expressions for both unconditional and conditional survival functions for 2k+1≥n, while signature based mixture representations for general k are obtained. The applications and computational results concerned with mean residual life function and stochastic ordering are presented.     

Invariant co-ordinate selection

Summary.  A general method for exploring multivariate data by comparing different estimates of multivariate scatter is presented. The method is based on the eigenvalue–eigenvector decomposition of one scatter matrix relative to another. In particular, it is shown that the eigenvectors can be used to generate an affine invariant co-ordinate system for the multivariate data. Consequently, we view this method as a method for invariant co-ordinate selection . By plotting the data with respect to this new invariant co-ordinate system, various data structures can be revealed. For example, under certain independent components models, it is shown that the invariant co- ordinates correspond to the independent components. Another example pertains to mixtures of elliptical distributions. In this case, it is shown that a subset of the invariant co-ordinates corresponds to Fisher's linear discriminant subspace, even though the class identifications of the data points are unknown. Some illustrative examples are given.     

A new class of models for bivariate joint tails

Summary.  A fundamental issue in applied multivariate extreme value analysis is modelling dependence within joint tail regions. The primary focus of this work is to extend the classical pseudopolar treatment of multivariate extremes to develop an asymptotically motivated representation of extremal dependence that also encompasses asymptotic independence. Starting with the usual mild bivariate regular variation assumptions that underpin the coefficient of tail dependence as a measure of extremal dependence, our main result is a characterization of the limiting structure of the joint survivor function in terms of an essentially arbitrary non-negative measure that must satisfy some mild constraints. We then construct parametric models from this new class and study in detail one example that accommodates asymptotic dependence, asymptotic independence and asymmetry within a straightforward parsimonious parameterization. We provide a fast simulation algorithm for this example and detail likelihood-based inference including tests for asymptotic dependence and symmetry which are useful for submodel selection. We illustrate this model by application to both simulated and real data. In contrast with the classical multivariate extreme value approach, which concentrates on the limiting distribution of normalized componentwise maxima, our framework focuses directly on the structure of the limiting joint survivor function and provides significant extensions of both the theoretical and the practical tools that are available for joint tail modelling.     

Linear Models with Doubly Exchangeable Distributed Errors

We study the general linear model (GLM) with doubly exchangeable distributed error for m observed random variables. The doubly exchangeable general linear model (DEGLM) arises when the m-dimensional error vectors are “doubly exchangeable,” jointly normally distributed, which is a much weaker assumption than the independent and identically distributed error vectors as in the case of GLM or classical GLM (CGLM). We estimate the parameters in the model and also find their distributions. We show that the tests of intercept and slope are possible in DEGLM as a particular case using parametric bootstrap as well as multivariate Satterthwaite approximation.     

Some new classes of multivariate survival distribution functions

The concept of the univariate mean remaining life (m.r.l.) function is generalized to the multivariate case. The multivariate mean remaining life (m.m.r.l.) function is utilized to introduce four new classes of multivariate survival distribution functions (s.d.f.'s). Each of these classes is a new generalization of the univariate decreasing mean remaining life (DMRL) class of s.d.f.'s. The duals of these classes are introduced. Some properties, physical interpretation, and relationships among these classes are investigated. Also for each case, the class of s.d.f.'s common in a class and its dual is characterized.     

A Bayesian nonparametric estimator of a multivariate survival function

Using reinforced processes related to beta-Stacy process and generalized Pólya urn scheme jointly with a structure assumption about dependence, a Bayesian nonparametric prior and a predictive estimator for a multivariate survival function are provided. This estimator can be computed through an easy implementation of a Gibbs sampler algorithm. Moreover consistency of the estimator is studied.     

The Extremal Dependence Measure and Asymptotic Independence

Abstract

Extremal dependence analysis assesses the tendency of large values of components of a random vector to occur simultaneously. This kind of dependence information can be qualitatively different than what is given by correlation which averages over the total body of the joint distribution. Also, correlation may be completely inappropriate for heavy tailed data. We study the extremal dependence measure (EDM), a measure of the tendency of large values of components of a random vector to occur simultaneously and show consistency of an estimator of the EDM. We also show asymptotic normality of an idealized estimator in a restricted case of multivariate regular variation where scaling functions do not have to be estimated.     

Efficiency of generalized estimating equations for binary responses

Summary.  Using standard correlation bounds, we show that in generalized estimation equations (GEEs) the so-called 'working correlation matrix' R ( α ) for analysing binary data cannot in general be the true correlation matrix of the data. Methods for estimating the correlation param-eter in current GEE software for binary responses disregard these bounds. To show that the GEE applied on binary data has high efficiency, we use a multivariate binary model so that the covariance matrix from estimating equation theory can be compared with the inverse Fisher information matrix. But R ( α ) should be viewed as the weight matrix, and it should not be confused with the correlation matrix of the binary responses. We also do a comparison with more general weighted estimating equations by using a matrix Cauchy–Schwarz inequality. Our analysis leads to simple rules for the choice of α in an exchangeable or autoregressive AR(1) weight matrix R ( α ), based on the strength of dependence between the binary variables. An example is given to illustrate the assessment of dependence and choice of α .     

Duality of the Multivariate Distributions of Marshall–Olkin Type and Tail Dependence

A dual class of the multivariate distributions of Marshall–Olkin type is introduced, and their copulas are presented and utilized to derive explicit expressions of the distributional tail dependencies, which describe the amount of dependence in the upper-orthant tail or lower-orthant tail of a multivariate distribution and can be used in the study of dependence among extreme values. A sufficient condition under which tail dependencies of two such distributions can be compared are obtained. Some examples are also presented to illustrate our results.     

Flexible Bivariate Count Data Regression Models

The article develops a semiparametric estimation method for the bivariate count data regression model. We develop a series expansion approach in which dependence between count variables is introduced by means of stochastically related unobserved heterogeneity components, and in which, unlike existing commonly used models, positive as well as negative correlations are allowed. Extensions that accommodate excess zeros, censored data, and multivariate generalizations are also given. Monte Carlo experiments and an empirical application to tobacco use confirms that the model performs well relative to existing bivariate models, in terms of various statistical criteria and in capturing the range of correlation among dependent variables. This article has supplementary materials online.     

Properties of Coherent Systems with Dependent Components

We introduce two new families of univariate distributions that we call hyperminimal and hypermaximal distributions. These families have interesting applications in the context of reliability theory in that they contain that of coherent system lifetime distributions. For these families, we obtain distributions, bounds, and moments. We also define the minimal and maximal signatures of a coherent system with exchangeable components which allow us to represent the system distribution as generalized mixtures (i.e., mixtures with possibly negative weights) of series and parallel systems. These results can also be applied to order statistics (k-out-of-n systems). Finally, we give some applications studying coherent systems with different multivariate exponential joint distributions.     

Riesz inverse Gaussian distributions

The notion of generalized power of a positive definite symmetric matrix and a related notion of generalized Bessel function are used to introduce an extension of the class of matrix generalized inverse Gaussian distributions. The new distributions are shown to arise as conditional distributions of Peirce components of Riesz random matrices. Things are explained in the modern framework of symmetric cones and simple Euclidean Jordan algebra.     

Some contributions on the multivariate Poisson–Skellam probability distribution

In this article, we introduce a new form of distribution whose components have the Poisson or Skellam marginal distributions. This new specification allows the incorporation of relevant information on the nature of the correlations between every component. In addition, we present some properties of this distribution. Unlike the multivariate Poisson distribution, it can handle variables with positive and negative correlations. It should be noted that we are only interested in modeling covariances of order 2, which means between all pairs of variables. Some simulations are presented to illustrate the estimation methods. Finally, an application of soccer teams data will highlight the relationship between number of points per season and the goal differential by some covariates.     

A new approach to use multivariate auxiliary information in sample surveys

A new approach to form multivariate difference estimator is suggested which does not require the knowledge of unknown population parameters as such. It gives minimum variance among the class of multivariate difference estimators. The performance of this estimator with respect to Des Raj's (J. Amer. Statist. Assoc. 60 (1965), 270–277) multivariate difference estimator is illustrated. Using the information on two auxiliary variates, the robustness of Des Raj's estimator y_d is studied empirically. Two new estimators to estimate population mean/total are developed on the same lines as that of y_d. The performance of these estimators is studied for a wide variety of populations.     

Institutions and Economic Outcomes: A Dominance-Based Analysis

An important issue in both welfare and development economics is the interaction between institutions and economic outcomes. While welfarists are typically concerned with how these variables contribute to overall wellbeing, empirical assessments of their joint contribution are limited. Development economists, on the other hand, have focused extensively on whether institutions cause or are caused by growth yet the relevant literature is still rife with debate. In this article, we use a notion of distributional dominance to tackle both the measurement of multivariate welfare and the evaluation of inter-temporal dependence without hindrance from the mix of discrete (political) and continuous (economic) variables in our data set. On the causality front, our results support the view that institutions promote growth more than growth promotes institutions. On the welfare front, we find that economic growth had a positive impact from 1960 to 2000 but declines in institutional quality over the earlier part of this period were sufficient to produce a decline in overall wellbeing until the mid-1970s. Subsequent improvements in institutions then reversed the trend and, ultimately, wellbeing in 2000 was higher than that in 1960.     

Optimal Weight Functions for Marginal Proportional Hazards Analysis of Clustered Failure Time Data

The choice of weights in estimating equations for multivariate survival data is considered. Specifically, we consider families of weight functions which are constant on fixed time intervals, including the special case of time-constant weights. For a fixed set of time intervals, the optimal weights are identified as the solution to a system of linear equations. The optimal weights are computed for several scenarios. It is found that for the scenarios examined, the gains in efficiency using the optimal weights are quite small relative to simpler approaches except under extreme dependence, and that a simple estimator of an exchangeable approximation to the weights also performs well.     

Effect of Dependency in Systems for Multivariate Surveillance

Systems for multivariate on-line surveillance (e.g., outbreak detection) are investigated. Optimal systems for statistical surveillance are based on likelihood ratios. Three systems are compared: based on each marginal density, based on the joint density, and based on the Hotelling's T2. The effect of dependency between the monitored processes is investigated, and the effect of correlation between the change times. When the first change occurs immediately, the three methods give similar delay of an alarm, in the situation with independency. For late changes, T2 has the longest delay, both for independent processes and for processes with a positive covariance.     

IDENTIFYING THE OUT OF CONTROL VARIABLE IN A MULTIVARIATE CONTROL CHART

ABSTRACT

The identification of the out of control variable, or variables, after a multivariate control chart signals, is an appealing subject for many researchers in the last years. In this paper we propose a new method for approaching this problem based on principal components analysis. Theoretical control limits are derived and a detailed investigation of the properties and the limitations of the new method is given. A graphical technique which can be applied in some of these limiting situations is also provided.     

Supervised classifiers for high-dimensional higher-order data with locally doubly exchangeable covariance structure

We explore the performance accuracy of the linear and quadratic classifiers for high-dimensional higher-order data, assuming that the class conditional distributions are multivariate normal with locally doubly exchangeable covariance structure. We derive a two-stage procedure for estimating the covariance matrix: at the first stage, the Lasso-based structure learning is applied to sparsifying the block components within the covariance matrix. At the second stage, the maximum-likelihood estimators of all block-wise parameters are derived assuming the doubly exchangeable within block covariance structure and a Kronecker product structured mean vector. We also study the effect of the block size on the classification performance in the high-dimensional setting and derive a class of asymptotically equivalent block structure approximations, in a sense that the choice of the block size is asymptotically negligible.