Quadrant Dependence Research Articles

Conditional limit theorems for conditionally linearly negative quadrant dependent random variables

From the ordinary notion of linearly negative quadrant dependence for a sequence of random variables, a new concept called conditionally linearly negative quadrant dependence is introduced. The relation between the two kinds of dependence is answered by examples, that is, the linearly negative quadrant dependence does not imply the conditionally linearly negative quadrant dependence, and vice versa. The fundamental properties of conditionally linearly negative quadrant dependence are developed, which extend the corresponding ones under the non-conditioning setup. By means of these properties, some conditional exponential inequalities, conditionally complete convergence results and a conditional central limit theorem stated in terms of conditional characteristic functions are established.

Constructing archimedean copulas from diagonal sections

We introduce a family ℱ of functions called diagonal generators. These are convex functions with the properties of diagonal sections of archimedean copulas. We show that to each diagonal generator f there corresponds an archimedean copula Cf with the asymptotic representation Cf(u1,u2)=limk→∞fk[f−k(u1)+f−k(u2)−1]. Moreover, the diagonal section of Cf equals f.We characterize archimedean copulas in terms of their asymptotic form. We construct a family ℱF of diagonal generators, induced by a regular distribution function F. We study a differential equation (depending on a function parameter), whose solution is F. We give four applications of diagonal generators: to concordance, quadrant dependence, measures of dependence and convergence of copulas.

Limiting Behavior of the Maximum of the Partial Sum for Linearly Negative Quadrant Dependent Random Variables under Residual Cesàro Alpha‐Integrability Assumption

Linearly negative quadrant dependence is a special dependence structure. By relating such conditions to residual Cesàro alpha‐integrability assumption, as well as to strongly residual Cesàro alpha‐integrability assumption, some Lp‐convergence and complete convergence results of the maximum of the partial sum are derived, respectively.

Grüss-type bounds for covariances and the notion of quadrant dependence in expectation

Abstract We show that Grüss-type probabilistic inequalities for covariances can be considerably sharpened when the underlying random variables are quadrant dependent in expectation (QDE). The herein established covariance bounds not only sharpen the classical Grüss inequality but also improve upon recently derived Grüss-type bounds under the assumption of quadrant dependency (QD), which is stronger than QDE. We illustrate our general results with examples based on specially devised bivariate distributions that are QDE but not QD. Such results play important roles in decision making under uncertainty, and particularly in areas such as economics, finance, and insurance.

Engel's Law Reconsidered

Engel's Law expresses a negative stochastic association of the bivariate distribution of income and food share across a population. Among the many different definitions which can be found in the statistical literature four concepts are discussed and tested: Kendall's τ, quadrant dependence, stochastic decreasing conditional food share distribution function and decreasing regression. Only the last one is used in the economic literature, yet it does not imply useful information of the underlying distribution. For linking Engels's Law to micro-economics, stronger concepts of stochastic association are needed. This motivates the empirical study of the proposed alternative concepts of negative association.

STRONG LAW OF LARGE NUMBERS FOR ASYMPTOTICALLY NEGATIVE DEPENDENT RANDOM VARIABLES WITH APPLICATIONS

In this paper, we obtain the Hjeck-Rnyi type inequality and the strong law of large numbers for asymptotically linear negative quadrant dependent random variables by using this inequality. We also give the strong law of large numbers for the linear process under asymptotically linear negative quadrant dependence assumption.

ON THE LIMIT BEHAVIOR OF EXTENDED NEGATIVE QUADRANT DEPENDENCE

We discuss in this paper the notions of extended negative quadrant dependence and its properties. We study a class of bivariate uniform distributions having extended negative quadrant dependence, which is derived by generalizing the uniform representation of a well-known Farlie-Gumbel-Morgenstern distribution. Finally, we also study the limit behavior on the extended negative quadrant dependence.

Positive quadrant dependence tests for copulas

AbstractIn this paper the interest is in testing the null hypothesis of positive quadrant dependence (PQD) between two random variables. Such a testing problem is important since prior knowledge of PQD is a qualitative restriction that should be taken into account in further statistical analysis, for example, when choosing an appropriate copula function to model the dependence structure. The key methodology of the proposed testing procedures consists of evaluating a “distance” between a nonparametric estimator of a copula and the independence copula, which serves as a reference case in the whole set of copulas having the PQD property. Choices of appropriate distances and nonparametric estimators of copula are discussed, and the proposed methods are compared with testing procedures based on bootstrap and multiplier techniques. The consistency of the testing procedures is established. In a simulation study the authors investigate the finite sample size and power performances of three types of test statistics, Kolmogorov–Smirnov, Cramér–von‐Mises, and Anderson–Darling statistics, together with several nonparametric estimators of a copula, including recently developed kernel type estimators. Finally, they apply the testing procedures on some real data. The Canadian Journal of Statistics 38: 555–581; 2010 © 2010 Statistical Society of Canada

A class of bivariate negative binomial distributions with different index parameters in the marginals

In this paper, we consider a new class of bivariate negative binomial distributions having marginal distributions with different index parameters. This feature is useful in statistical modelling and simulation studies, where different marginal distributions and a specified correlation are required. This feature also makes it more flexible than the existing bivariate generalizations of the negative binomial distribution, which have a common index parameter in the marginal distributions. Various interesting properties, such as canonical expansions and quadrant dependence, are obtained. Potential application of the proposed class of bivariate negative binomial distributions, as a bivariate mixed Poisson distribution, and computer generation of samples are examined. Numerical examples as well as goodness-of-fit to simulated and real data are also given here in order to illustrate the application of this family of bivariate negative binomial distributions.

Gruss-Type Bounds for the Covariance of Transformed Random Variables

A number of problems in economics, finance, information theory, insurance, and generally in decision making under uncertainty rely on estimates of the covariance between (transformed) random variables, which can for example be losses, risks, incomes, financial returns, etc. Several avenues relying on inequalities for analyzing the covariance are available in the literature, bearing the names of Chebyshev, Gruss, Hoeffding, Kantorovich, and others. In the present paper we sharpen the upper bound of a Gruss-type covariance inequality by incorporating a notion of quadrant dependence between random variables and also utilizing the idea of constraining the means of the random variables.

Revisiting Grüss’s Inequality: Covariance Bounds, QDE but not QD Copulas, and Central Moments

Since the pioneering work of Gerhard Gruss dating back to 1935, Gruss’s inequality and, more generally, Gruss-type bounds for covariances have fascinated researchers and found numerous applications in areas such as economics, insurance, reliability, and, more generally, decision making under uncertainly. Gruss-type bounds for covariances have been established mainly under most general dependence structures, meaning no restrictions on the dependence structure between the two underlying random variables. Recent work in the area has revealed a potential for improving Gruss-type bounds, including the original Gruss’s bound, assuming dependence structures such as quadrant dependence (QD). In this paper we demonstrate that the relatively little explored notion of ‘quadrant dependence in expectation’ (QDE) is ideally suited in the context of bounding covariances, especially those that appear in the aforementioned areas of application. We explore this research avenue in detail, establish general Gruss-type bounds, and illustrate them with newly constructed examples of bivariate distributions, which are not QD but, nevertheless, are QDE. The examples rely on specially devised copulas. We supplement the examples with results concerning general copulas and their convex combinations. In the process of deriving Gruss-type bounds, we also establish new bounds for central moments, whose optimality is demonstrated.

Grüss-Type Bounds for the Covariance of Transformed Random Variables

A number of problems in Economics, Finance, Information Theory, Insurance, and generally in decision making under uncertainty rely on estimates of the covariance between (transformed) random variables, which can, for example, be losses, risks, incomes, financial returns, and so forth. Several avenues relying on inequalities for analyzing the covariance are available in the literature, bearing the names of Chebyshev, Grüss, Hoeffding, Kantorovich, and others. In the present paper we sharpen the upper bound of a Grüss-type covariance inequality by incorporating a notion of quadrant dependence between random variables and also utilizing the idea of constraining the means of the random variables.

Measures of the intergenerational transmission of body mass index between mothers and their children in the United States, 1981–2004

This research provides estimates of the intergenerational persistence of body mass index (BMI) between women and their children when both are at similar stages of the lifecycle. Using data from the National Longitudinal Survey of Youth 1979 (NLSY79) and the Young Adults of the NLSY79, associations between the weight status of women and their children are measured when both generations are between the ages of 16 and 24. In the entire sample, the measured intergenerational correlation of BMI is significantly different from zero and equal to 0.35. This result differs by gender with a BMI correlation between female children and their mothers of 0.38, compared to a significantly lower BMI correlation of 0.32 between mothers and their sons. Measures of this relationship across the distribution of BMI using quantile regression and quadrant dependence techniques indicate that the intergenerational persistence of BMI is strongest at higher levels of BMI. Strong dependence across generations is found when categorical outcomes of obesity and overweight are implemented. These results provide evidence of the strong persistence of weight problems across generations which may affect economic mobility within families.

A Kolmogorov-Smirnov Type Test for Positive Quadrant Dependence

We consider a consistent test, that is similar to a Kolmogorov-Smirnov test, of the complete set of restrictions that relate to the copula representation of positive quadrant dependence. For such a test we propose and justify inference relying on a simulation based multiplier method and a bootstrap method. We also explore the finite sample behavior of both methods with Monte Carlo experiments. A first empirical illustration is given for US insurance claim data. A second one examines the presence of positive quadrant dependence in life expectancies at birth of males and females among countries.

Non-exchangeability of negatively dependent random variables

We compute the measure of non-exchangeability (with respect to the L∞-norm) for a pair of identically distributed continuous random variables that satisfy some negative dependence property, namely quadrant dependence or stochastic decreasingness.

On Conditional Independence, Mixing, and Association

This is a follow-up to a recent article by Prakasa Rao [15] on conditional independence, conditional mixing and conditional association. The purpose of this article is to derive rigorously some results following from conditioning. To this end, a brief review is presented of the concepts of conditional independence of events, classes of events, and random variables, followed by a conditional version of a factorization theorem, as well as a first installment of some basic results. Next, the concepts of conditional covariance and variance are introduced, and a second installment of basic results follows. Furthermore, a certain representation of the covariance is established in detail, followed by a conditional version of it, as well as a generalization. The concept of the conditional characteristic function is also recalled, and a certain inequality is established. Finally, the concept of conditional positive (negative) quadrant dependence, as well as that of conditional positive (negative) association are introduced. The article concludes with the derivation of the conditional versions of some known results, regarding positive (negative) association. This is done anticipating that conditional association (and also conditional mixing) will prove to be of significant applicability.

Laws of large numbers for residual Cesàro alpha-integrable sequences under dependence assumptions

Martingale difference, pairwise negative quadrant dependence and L p -mixingale difference are three special kinds of dependence structures for random variables. By relating these dependence conditions to residual Cesàro alpha-integrability, as well as to strongly residual Cesàro alpha-integrability, L p -convergence and strong laws of large numbers are studied, and significant improvements over earlier results are obtained.

Nonparametric Tests for Positive Quadrant Dependence

We consider distributional free inference to test for positive quadrant dependence, i.e. for the probability that two variables are simultaneously small (or large) being at least as great as it would be were they dependent. Tests for its generalisation in higher dimensions, namely positive orthant dependences, are also analysed. We propose two types of testing procedures. The first procedure is based on the specification of the dependence concepts in terms of distribution functions, while the second procedure exploits the copula representation. For each specification a distance test and an intersection-union test for inequality constraints are developed depending on the definition of null and alternative hypotheses. An empirical illustration is given for US and Danish insurance claim data. Practical implications for the design of reinsurance treaties are also discussed.

The law of the iterated logarithm for positively dependent random variables

The Levy's type maximal inequality is a key to establish the law of the iterated logarithm for associated random variables. Unfortunately, this type inequality cannot be obtained for a generalization of association, i.e., linear positive quadrant dependence, because of their special dependence structure. The purpose of this paper is to provide a different approach to obtain a law of the iterated logarithm for a sequence of linear positive quadrant dependent random variables.

Conditions Ensuring the Decomposition of Asset Demand for All Risk-Averse Investors

The paper explores how the demand for a risky asset can be decomposed into an investment effect and a hedging effect by all risk-averse investors. This question has been shown to be complex when considered outside of the mean-variance framework. Dependence among returns on the risky assets is restricted to quadrant dependence and it is found that the demand for one risky asset can be decomposed into an investment component based on the risk premium offered by the asset and a hedging component used against the fluctuations in the return on the other risky asset. The paper also discusses how the class of quadrant-dependent distributions is related to that of two-fund separating distributions. This contribution opens up the search for broader distributional hypotheses suitable to asset demand models. Examples are discussed.