A Copula‐Based Non‐parametric Measure of Regression Dependence

Abstract. This article presents a framework for comparing bivariate distributions according to their degree of regression dependence. We introduce the general concept of a regression dependence order (RDO). In addition, we define a new non‐parametric measure of regression dependence and study its properties. Besides being monotone in the new RDOs, the measure takes on its extreme values precisely at independence and almost sure functional dependence, respectively. A consistent non‐parametric estimator of the new measure is constructed and its asymptotic properties are investigated. Finally, the finite sample properties of the estimate are studied by means of a small simulation study.     

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.     

Bivariate Tail Dependence and the Generation of Multivariate Extreme Value Distributions

We define, in a probabilistic way, a parametric family of multivariate extreme value distributions. We derive its copula, which is a mixture of several complete dependent copulas and total independent copulas, and the bivariate tail dependence and extremal coefficients. Based on the obtained results for these coefficients, we propose a method to build multivariate extreme value distributions with prescribed tail/extremal coefficients. We illustrate the results with examples.     

A Sarmanov family with beta and gamma marginal distributions: an application to the Bayes premium in a collective risk model

In this paper we firstly develop a Sarmanov–Lee bivariate family of distributions with the beta and gamma as marginal distributions. We obtain the linear correlation coefficient showing that, although it is not a strong family of correlation, it can be greater than the value of this coefficient in the Farlie–Gumbel–Morgenstern family. We also determine other measures for this family: the coefficient of median concordance and the relative entropy, which are analyzed by comparison with the case of independence. Secondly, we consider the problem of premium calculation in a Poisson–Lindley and exponential collective risk model, where the Sarmanov–Lee family is used as a structure function. We determine the collective and Bayes premiums whose values are analyzed when independence and dependence between the risk profiles are considered, obtaining that notable variations in premiums values are obtained even when low levels of correlation are considered.     

A family of bivariate binomial distributions generated by extreme bernoulli distributions

In this paper, bivariate binomial distributions generated by extreme bivariate Bernoulli distributions are obtained and studied. Representation of the bivariate binomial distribution generated by a convex combination of extreme bivariate Bernoulli distributions as a mixture of distributions in the class of bivariate binomial distribution generated by extreme bivariate Bernoulli distribution is obtained. A subfamily of bivariate binomial distributions exhibiting the property of positive and negative dependence is constructed. Some results on positive dependence notions as it relates to the bivariate binomial distribution generated by extreme bivariate Bernoulli distribution and a linear combination of such distributions are obtained.     

Interval Estimation for the Correlation Coefficient

ABSTRACT

The correlation coefficient (CC) is a standard measure of a possible linear association between two continuous random variables. The CC plays a significant role in many scientific disciplines. For a bivariate normal distribution, there are many types of confidence intervals for the CC, such as z-transformation and maximum likelihood-based intervals. However, when the underlying bivariate distribution is unknown, the construction of confidence intervals for the CC is not well-developed. In this paper, we discuss various interval estimation methods for the CC. We propose a generalized confidence interval for the CC when the underlying bivariate distribution is a normal distribution, and two empirical likelihood-based intervals for the CC when the underlying bivariate distribution is unknown. We also conduct extensive simulation studies to compare the new intervals with existing intervals in terms of coverage probability and interval length. Finally, two real examples are used to demonstrate the application of the proposed methods.     

A bivariate extension of the beta generated distribution derived from copulas

In this paper, we introduce a new class of bivariate distributions whose marginals are beta-generated distributions. Copulas are employed to construct this bivariate extension of the beta-generated distributions. It is shown that when Archimedean copulas and convex beta generators are used in generating bivariate distributions, the copulas of the resulting distributions also belong to the Archimedean family. The dependence of the proposed bivariate distributions is examined. Simulation results for beta generators and an application to financial risk management are presented.     

On a class of bivariate mixed Sarmanov distributions

Multivariate distributions are more and more used to model the dependence encountered in many fields. However, classical multivariate distributions can be restrictive by their nature, while Sarmanov's multivariate distribution, by joining different marginals in a flexible and tractable dependence structure, often provides a valuable alternative. In this paper, we introduce some bivariate mixed Sarmanov distributions with the purpose to extend the class of bivariate Sarmanov distributions and to obtain new dependency structures. Special attention is paid to the bivariate mixed Sarmanov distribution with Poisson marginals and, in particular, to the resulting bivariate Sarmanov distributions with negative binomial and with Poisson‐inverse Gaussian marginals; these particular types of mixed distributions have possible applications in, for example modelling bivariate count data. The extension to higher dimensions is also discussed. Moreover, concerning the dependency structure, we also present some correlation formulas.     

Modeling of censored bivariate extremal events

In this paper we consider the estimation of the coefficient of tail dependence and of small tail probability under a bivariate randomly censoring mechanism. A new class of generalized moment estimators of the coefficient of tail dependence and the estimator of small tail probability are proposed, respectively. Under the bivariate Hall-type conditions, the asymptotic distributions of these estimators are established. Monte Carlo simulations are performed and the new estimators are applied to an insurance data-set.     

On Two General Classes of Discrete Bivariate Distributions

In this article, we develop two general classes of discrete bivariate distributions. We derive general formulas for the joint distributions belonging to the classes. The obtained formulas for the joint distributions are very general in the sense that new families of distributions can be generated just by specifying the “baseline seed distributions.” The dependence structures of the bivariate distributions belonging to the proposed classes, along with basic statistical properties, are also discussed. New families of discrete bivariate distributions are generated from the classes. Furthermore, to assess the usefulness of the proposed classes, two discrete bivariate distributions generated from the classes are applied to analyze a real dataset and the results are compared with those obtained from conventional models.     

Partial derivatives and confidence intervals of bivariate tail dependence functions

Bivariate extreme value theory was used to estimate a rare event (see de Haan and de Ronde [1998. Sea and wind: multivariate extremes at work. Extremes 1, 7–45]). This procedure involves estimating a tail dependence function. There are several estimators for the tail dependence function in the literature, but their limiting distributions depend on partial derivatives of the tail dependence function. In this paper smooth estimators are proposed for estimating partial derivatives of bivariate tail dependence functions and their asymptotic distributions are derived as well. A simulation study is conducted to compare different estimators of partial derivatives in terms of both mean squared errors and coverage accuracy of confidence intervals of the bivariate tail dependence function based on these different estimators of partial derivatives.     

Bivariate Kumaraswamy distribution: properties and a new method to generate bivariate classes

In this paper, we introduce a bivariate Kumaraswamy (BVK) distribution whose marginals are Kumaraswamy distributions. The cumulative distribution function of this bivariate model has absolutely continuous and singular parts. Representations for the cumulative and density functions are presented and properties such as marginal and conditional distributions, product moments and conditional moments are obtained. We show that the BVK model can be obtained from the Marshall and Olkin survival copula and obtain a tail dependence measure. The estimation of the parameters by maximum likelihood is discussed and the Fisher information matrix is determined. We propose an EM algorithm to estimate the parameters. Some simulations are presented to verify the performance of the direct maximum-likelihood estimation and the proposed EM algorithm. We also present a method to generate bivariate distributions from our proposed BVK distribution. Furthermore, we introduce a BVK distribution which has only an absolutely continuous part and discuss some of its properties. Finally, a real data set is analysed for illustrative purposes.     

On the Tukey depth of an atomic measure

This paper gives a relation between the convex Tukey trimmed region (see [J.C. Massé, R. Theodorescu, Halfplane trimming for bivariate distributions, J. Multivariate Anal. 48(2) (1994) 188–202]) of an atomic measure and the support of the measure. It is shown that an atomic measure is concentrated on the extreme points of its Tukey trimmed region. A property concerning the extreme points which have 0 mass is given. As a corollary, we give a new method of proof of the Koshevoy characterization result.     

Canonical expansions,correlation structure,and conditional distributions of bivariate distributions generated by mixtures

A simple result concerning the canonical expansions of mixed bivariate distributions is considered. This result is then applied to analyze the correlation structures of the Bates-Neyman accident proneness model and its generalization, to derive probability inequalities based on the concept of positive dependence, and to construct a bivariate beta distribution with positive correlation coefficient applicable in computer simulation experiments. The mixture formulation of the conditional distribution of this class of mixed bivariate distributions is used to define and generate first-order autoregressive gamma and negative binomial sequences.     

Extremal analysis of short series with outliers: sea-levels and athletics records

The analysis of extreme values is often required from short series which are biasedly sampled or contain outliers. Data for sea-levels at two UK east coast sites and data on athletics records for women's 3000 m track races are shown to exhibit such characteristics. Univariate extreme value methods provide a poor quantification of the extreme values for these data. By using bivariate extreme value methods we analyse jointly these data with related observations, from neighbouring coastal sites and 1500 m races respectively. We show that using bivariate methods provides substantial benefits, both in these applications and more generally with the amount of information gained being determined by the degree of dependence, the lengths and the amount of overlap of the two series, the homogeneity of the marginal characteristics of the variables and the presence and type of the outlier.     

Aspects of Dependence in Cuadras–Auge Family

In this article, we study the dependence structure of Cuadras–Auge (C–A) family of bivariate distributions. We also obtain some association measures and two local dependence functions for this family. In addition, we compare expectation of local dependence function and Pearson's rho via numerical study.     

Bivariate distributions with transmuted conditionals: Models and applications

Abstract

The class of transmuted distributions has received a lot of attention in the recent statistical literature. In this paper, we propose a rich family of bivariate distribution whose conditionals are transmuted distributions. The new family of distributions depends on the two baseline distributions and three dependence parameters. Apart from the general properties, we also study the distribution of the concomitance of order statistics. We study specific bivariate models. Estimation methodologies are proposed. A simulation study is conducted. The usefulness of this family is established by fitting well analyzed real life time data.     

A Method for Multivariate Probability Distributions Construction via Parameter Dependence

The nature of stochastic dependence in the classic bivariate normal density framework is analyzed. In the case of this distribution we stress the way the conditional density of one of the random variables depends on realizations of the other. Typically, in the bivariate normal case this dependence takes the form of a parameter (here the “expected value”) of one probability density depending continuously (here linearly) on realizations of the other random variable. Our point is that such a pattern does not need to be restricted to that classical case of bivariate normal. We show that this paradigm can be generalized and viewed in ways that allows us to extend it far beyond the bivariate normal distributions class.     

Estimating and Testing Nonlinear Local Dependence Between Two Time Series

ABSTRACT

The most common measure of dependence between two time series is the cross-correlation function. This measure gives a complete characterization of dependence for two linear and jointly Gaussian time series, but it often fails for nonlinear and non-Gaussian time series models, such as the ARCH-type models used in finance. The cross-correlation function is a global measure of dependence. In this article, we apply to bivariate time series the nonlinear local measure of dependence called local Gaussian correlation. It generally works well also for nonlinear models, and it can distinguish between positive and negative local dependence. We construct confidence intervals for the local Gaussian correlation and develop a test based on this measure of dependence. Asymptotic properties are derived for the parameter estimates, for the test functional and for a block bootstrap procedure. For both simulated and financial index data, we construct confidence intervals and we compare the proposed test with one based on the ordinary correlation and with one based on the Brownian distance correlation. Financial indexes are examined over a long time period and their local joint behavior, including tail behavior, is analyzed prior to, during and after the financial crisis. Supplementary material for this article is available online.