A note on some negative dependence notions

A random vector X = (X ₁,…,X _n ) is negatively associated if and only if for every pair of partitions X ₁ = (X _π(1),…,X _π(k)), X ₂ = (X _π(k+1),…,X _π(n)) of X , P( X ₁ ? A, X ₂ ? B) ≤ P( X ₁ ? A)P( X ₂ ? B) whenever A and B are open upper sets and π is any permutation of {1,…,n}. In this paper, we develop some of concepts of negative dependence, which are weaker than negative association but stronger than negative orthant dependence by requiring the above inequality to hold only for some upper sets A and B and applying the arguments in Shaked.     

On bivariate ageing properties of exchangeable Farlie–Gumbel–Morgenstern distributions

This paper deals with bivariate Farlie–Gumbel–Morgenstern distributions. We build the TP2 (RR2) property of the residual lifetime and study the evolution of the dependence of the residual lifetime at large age. Also, we derive the aging property in the sense of the upper orthant order. Furthermore, in the context of IFR (DFR) marginals the residual lifetime at the age above the median is found to be increasing (decreasing) in the upper orthant order.     

A representation for quadrivariate normal positive orthant probabilities

For four variables x₁,x₂, x₃ and x₄, which have a quadrivariate normal distribution with means equal to zero, the positive ortrhant probability is the probability that all of the x.'s are simultaneously positive. A representation for the quadrivariate normal positive orthant probability is obtained and it is a function of no more than three integrals over a single variable. Extensive testing has shown this representation to be very efficient on a computational basis.     

Adaptive ridge estimator in a linear regression model with spherically symmetric error under constraint

Abstract

We consider adaptive ridge regression estimators in the general linear model with homogeneous spherically symmetric errors. A restriction on the parameter of regression is considered. We assume that all components are non negative (i.e. on the positive orthant). For this setting, we produce under general quadratic loss such estimators whose risk function dominates that of the least squares provided the number of regressors in the least fore.     

Measuring serial dependence in categorical time series

The analysis of time-indexed categorical data is important in many fields, e.g., in telecommunication network monitoring, manufacturing process control, ecology, etc. Primary interest is in detecting and measuring serial associations and dependencies in such data. For cardinal time series analysis, autocorrelation is a convenient and informative measure of serial association. Yet, for categorical time series analysis an analogous convenient measure and corresponding concepts of weak stationarity have not been provided. For two categorical variables, several ways of measuring association have been suggested. This paper reviews such measures and investigates their properties in a serial context. We discuss concepts of weak stationarity of a categorical time series, in particular of stationarity in association measures. Serial association and weak stationarity are studied in the class of discrete ARMA processes introduced by Jacobs and Lewis (J. Time Ser. Anal. 4(1):19–36, 1983). An intrinsic feature of a time series is that, typically, adjacent observations are dependent. The nature of this dependence among observations of a time series is of considerable practical interest. Time series analysis is concerned with techniques for the analysis of this dependence. (Box et al. 1994p. 1)     

Large deviations for the stochastic present value of aggregate claims in the nonstandard compound renewal risk model with widely upper Orthant dependent claims

Abstract

Recently, Jiang et al. (Statist. Probab. Lett. 101, 83–91) obtained the asymptotic formulas for the large deviations for the stochastic present value of aggregate claims in the renewal risk model with Pareto-type claims and stochastic return on investments, where the price process of the investment portfolio is described as a geometric Lévy process. In the paper, we extend the above results to a nonstandard compound renewal risk model with widely upper orthant dependent and dominatedly-varying-tailed claims.     

Functional Estimation of a Density Under a New Weak Dependence Condition

The purpose of this paper is to prove, through the analysis of the behaviour of a standard kernel density estimator, that the notion of weak dependence defined in a previous paper (cf. Doukhan & Louhichi, 1999) has sufficiently sharp properties to be used in various situations. More precisely we investigate the asymptotics of high order losses, asymptotic distributions and uniform almost sure behaviour of kernel density estimates. We prove that they are the same as for independent samples (with some restrictions for a.s. behaviours). Recall finally that this weak dependence condition extends on the previously defined ones such as mixing, association and it allows considerations of new classes such as weak shifts processes based on independent sequences as well as some non-mixing Markov processes.     

A general reduction method for n–variate normal orthant probability

Let X₁,X₂,…,X_n be n normal variates with zero means, unit variances and correlation matrix {p_ij). The orthant probability is the probability that all of the X₁'s are simultaneously positive. This paper presents a general reduction method by extending the method of Childs (1967), and shows that the probability can be represented by a linear combination of some multivariate integrals of order([n/2]?1). As illustrations, we apply the proposed method to the quadrivariate and six–variate cases. Some numerical results are also given.     

A note on measuring the degree of dependence between two discrete random variables measured on a nominal scale

This paper discusses some of the problematic aspects of the term “dependence” and suggests a measure of a special kind of association - complete dependence - between two discrete random variables measured on a nominal scale.     

Precise large deviations for sums of WUOD and φ-mixing random variables with dominated variation

In this paper, we investigate the precise large deviations for sums of not identical distributed random variables which are widely upper orthant dependent (WUOD, in short) and φ-mixing. The asymptotic relations for sums of random variables with dominated variation are obtained.     

A bayesian method for inferring the degree fo dependence for a positively quadrant dependent distribution

A Bayesian method for making inferences on the degree of dependence for a positively quadrant dependent distribution is developed. This method is based on a parametric model, where the parameter measures the degree of association. The parametric model is a natural simplification of the original one: it is a mixture of the two extreme cases corresponding to independence and positive extremal dependence.     

Concepts de dépendance et ordres stochastiques pour des lois bidimensionnelles

Yanagimoto and Okamoto (1969) introduced a stochastic ordering that generalizes a concept of monotone regression dependence introduced by Lehmann (1966). In this paper, we define and examine the properties of three new orderings which imply that of Yanagimoto and Okamoto. One of these orderings is seen to extend Shaked's (1977) notion of DTP(0, 1), and another includes Lehmann's concept of positive likelihood-ratio dependence as a special case. The proposed orderings are also compared with the TP₂ positive-dependence ordering defined by Kimeldorf and Sampson (1987).     

Modeling Dependence in High Dimensions With Factor Copulas

This article presents flexible new models for the dependence structure, or copula, of economic variables based on a latent factor structure. The proposed models are particularly attractive for relatively high-dimensional applications, involving 50 or more variables, and can be combined with semiparametric marginal distributions to obtain flexible multivariate distributions. Factor copulas generally lack a closed-form density, but we obtain analytical results for the implied tail dependence using extreme value theory, and we verify that simulation-based estimation using rank statistics is reliable even in high dimensions. We consider “scree” plots to aid the choice of the number of factors in the model. The model is applied to daily returns on all 100 constituents of the S&P 100 index, and we find significant evidence of tail dependence, heterogeneous dependence, and asymmetric dependence, with dependence being stronger in crashes than in booms. We also show that factor copula models provide superior estimates of some measures of systemic risk. Supplementary materials for this article are available online.     

Cross-Sectional Dependence in Panel Data Analysis

This article provides an overview of the existing literature on panel data models with error cross-sectional dependence (CSD). We distinguish between weak and strong CSD and link these concepts to the spatial and factor structure approaches. We consider estimation under strong and weak exogeneity of the regressors for both T fixed and T large cases. Available tests for CSD and methods for determining the number of factors are discussed in detail. The finite-sample properties of some estimators and statistics are investigated using Monte Carlo experiments.     

Bootstrapping the conditional copula

This paper is concerned with inference about the dependence or association between two random variables conditionally upon the given value of a covariate. A way to describe such a conditional dependence is via a conditional copula function. Nonparametric estimators for a conditional copula then lead to nonparametric estimates of conditional association measures such as a conditional Kendall's tau. The limiting distributions of nonparametric conditional copula estimators are rather involved. In this paper we propose a bootstrap procedure for approximating these distributions and their characteristics, and establish its consistency. We apply the proposed bootstrap procedure for constructing confidence intervals for conditional association measures, such as a conditional Blomqvist beta and a conditional Kendall's tau. The performances of the proposed methods are investigated via a simulation study involving a variety of models, ranging from models in which the dependence (weak or strong) on the covariate is only through the copula and not through the marginals, to models in which this dependence appears in both the copula and the marginal distributions. As a conclusion we provide practical recommendations for constructing bootstrap-based confidence intervals for the discussed conditional association measures.     

Semi-polynomial copulas

In this paper, we characterise a family of bivariate copulas whose sections between the main diagonal and the border of the unit square are polynomial, generalising several families of copulas, including those with quadratic and cubic sections. We also study a measure of association and the tail dependence for this class, illustrating our results with several examples.     

Some concepts of mult1variate dependence

Several types of positive dependence are shown to be equiv-alent to the concept of a distribution with a density which is TP₂ in pairs. Among these is the concept of m^?-positive depend-ence of Alam and Wallenius. Using this result, all relationships among many of the most important concepts of positive dependence are determined. Furthermore, an application of the equivalences of these types of positive dependence yields a results of Ahmed, Langberg, Leon and Proschan.     

Some results on dependent random variables and a connection with the multivariate s-increasing convex order

In this paper, some new concepts of dependence are introduced that generalize the concepts of positive and negative association. The new concepts of dependence are linked to the multivariate s-increasing convex order. Furthermore, a Kolmogorov-type inequality and a Hàjek–Rènyi inequality are proven that lead to an asymptotic result for these new random objects.     

Distribution-free tests for independence against positive quadrant dependence: A generalization

A class of distribution-free tests based on two matched pairs is considered for testing independence against positive quadrant dependence. The class of tests proposed by Kochar and Gupta (1990) is a member of the proposed class. The performance of the proposed class is evaluated in terms of Pitman asymptotic relative efficiency for Block-Basu (1974) model and Woodworth family of distributions. The small sample performance of few members of the proposed class is also studied by finding empirical powers. Unbiasedness and consistency of the proposed class of tests have been established.     

Dependence structures and asymptotic properties of Baker's distributions with fixed marginals

We investigate the properties of Baker's (2008) bivariate distributions with fixed marginals and their multivariate extensions. The properties include the weak convergence to the Fréchet–Hoeffding upper bound, the product-moment convergence, as well as the dependence structures TP₂ (totally positive of order 2), or MTP₂ (multivariate TP₂). In proving the weak convergence, a generalized local limit theorem for binomial distribution is provided.