Measure Of Local Dependence Research Articles

Construction of bivariate symmetric and asymmetric copulas and its relationship to ratios of conditional hazard rate functions

ABSTRACT A new method for the construction of both symmetric and asymmetric, absolutely continuous bivariate copulas is introduced. This research was inspired by the well-known cross ratio (CR), studied by Clayton (1978), which is a local dependence measure describing the association between the components of a bivariate random vector, as well as by an alternative risk ratio (RR) proposed recently by Abrams et al. (2022), which unravels the local relation between the two components of the vector. The practical meaning of RR is illustrated in two real data sets by the authors. In this article, various properties of the proposed parametric families of copulas are studied, for example quadrant dependence ordering, tail dependence coefficients and several copula-based measures of dependence. Some relationships of the copulas to RR are discussed and illustrative scatterplots are presented.

Stochastic fixed-point equation and local dependence measure

We study solutions to the stochastic fixed-point equation X=dAX+B where the coefficients A and B are nonnegative random variables. We introduce the “local dependence measure” (LDM) and its Legendre-type transform to analyze the left tail behavior of the distribution of X. We discuss the relationship of LDM with earlier results on the stochastic fixed-point equation and we apply LDM to prove a theorem on a Fleming–Viot-type process.

Portfolio allocation under asymmetric dependence in asset returns using local Gaussian correlations

It is well known that there are asymmetric dependence structures between financial returns. This paper describes a portfolio selection method rooted in the classical mean–variance framework that incorporates such asymmetric dependency structures using a nonparametric measure of local dependence, the local Gaussian correlation (LGC). It is shown that the portfolio optimization process for financial returns with asymmetric dependence structures is straightforward using local covariance matrices. The new method is shown to outperform the equally weighted (“1/N”) portfolio and the classical Markowitz portfolio when applied to historical data on six assets.

A Study of Bivariate Generalized Pareto Distribution and its Dependence Structure Among Model Parameters

Several variants of the classical bivariate and multivariate generalized Pareto distributions have been discussed and studied in the literature (see Arnold (1983, Stat. Prob. Lett. 17: 361–368, 1993, 2015), Arnold and Laguna (1977), Ali and Nadarajah (2007), Rootzen and Tajvidi (2006) and the references cited therein). Ali and Nadarajah (2007) studied a truncated version of the most popular long-tailed generalized bivariate Pareto distribution (GBPD, henceforth, in short) involving six parameters. However, not much discussion exists in the current literature on the structural properties as well as on the dependence structure among the parameters in this model. In this paper we re-visit the GBPD and discuss several other new properties. In addition, we study the shape of GBPD for varying choices of the model parameters and subsequently study their interdependence. Also, we provide copula based construction of GBPD and discuss the associated local dependence measures.

Some Properties of Local Gaussian Correlation and Other Nonlinear Dependence Measures

This article consists of two parts. The first one contains a brief review of global and local dependence measures, including the local Gaussian correlation. For this, the local correlation at a point x is obtained by approximating the given bivariate density f at x by a bivariate Gaussian and then taking the correlation of that bivariate Gaussian as the definition of local Gaussian correlation. The second part consists in a bias study of the local Gaussian correlation. A small simulation experiment is performed. In the Appendix, the possibility of a neighbourhood‐free definition of local Gaussian correlation is explored.

Layer Dependence as a Measure of Local Dependence

A new measure of local dependence called "layer dependence" is proposed and analysed. Layer dependence measures the dependence between two random variables at different percentiles in their joint distribution. Layer dependence satisfies coherence properties similar to Spearman's correlation, such as lying between -1 and 1, with -1, 0 and 1 corresponding to countermonotonicity, independence and comonotonicity, respectively. Spearman's correlation is a weighted average of layer dependence across all percentiles. Alternate overall dependence measures are arrived by varying the weights. Layer dependence is an important input to copula modeling by extracting the dependence structure from past data and incorporating expert opinion if necessary.

Measuring early or late dependence for bivariate lifetimes of twins.

We consider data from the Danish twin registry and aim to study in detail how lifetimes for twin-pairs are correlated. We consider models where we specify the marginals using a regression structure, here Cox's regression model or the additive hazards model. The best known such model is the Clayton-Oakes model. This model can be extended in several directions. One extension is to allow the dependence parameter to depend on covariates. Another extension is to model dependence via piecewise constant cross-hazard ratio models. We show how both these models can be implemented for large sample data, and suggest a computational solution for obtaining standard errors for such models for large registry data. In addition we consider alternative models that have some computational advantages and with different dependence parameters based on odds ratios of the survival function using the Plackett distribution. We also suggest a way of assessing how and if the dependence is changing over time, by considering either truncated or right-censored versions of the data to measure late or early dependence. This can be used for formally testing if the dependence is constant, or decreasing/increasing. The proposed procedures are applied to Danish twin data to describe dependence in the lifetimes of the twins. Here we show that the early deaths are more correlated than the later deaths, and by comparing MZ and DZ associations we suggest that early deaths might be more driven by genetic factors. This conclusion requires models that are able to look at more local dependence measures. We further show that the dependence differs for MZ and DZ twins and appears to be the same for males and females, and that there are indications that the dependence increases over calendar time.

Recognizing and visualizing copulas: An approach using local Gaussian approximation

In this paper we examine the relationship between a newly developed local dependence measure, the local Gaussian correlation, and standard copula theory. We are able to describe characteristics of the dependence structure in different copula models in terms of the local Gaussian correlation. Further, we construct a goodness-of-fit test for bivariate copula models. An essential ingredient of this test is the use of a canonical local Gaussian correlation and Gaussian pseudo-observations which make the test independent of the margins, so that it is a genuine test of the copula structure. A Monte Carlo study reveals that the test performs very well compared to a commonly used alternative test. We also propose two types of diagnostic plots which can be used to investigate the cause of a rejected null. Finally, our methods are applied to a “classical” insurance data set.

Estimation of a Spearman-Type Multivariate Measure of Local Dependence

A multivariate measure of local dependence written in terms of copulas is proposed, which if integrated, coincides with a population version of a multivariate global measure of Spearman's rho. We propose nonparametric estimators of this measure for independent sample data and also for time series data. Some properties of the estimators are derived. Simulations with different copulas and sample sizes were performed to assess the theoretical findings. Empirical applications are given for selected economic indexes of $166$ countries and for the returns of the DAX, CAC40 and FTSE indexes.

Introducinglocalgauss, anRPackage for Estimating and Visualizing Local Gaussian Correlation

Quantifying non-linear dependence structures between two random variables is a challenging task. There exist several bona-fide dependence measures able to capture the strength of the non-linear association, but they typically give little information about how the variables are associated. This problem has been recognized by several authors and has given rise to the concept of local measures of dependence. A local measure of dependence is able to capture the “local” dependence structure in a particular region. The idea is that the global dependence structure is better described by a portfolio of local measures of dependence computed in different regions than a one-number measure of dependence. This paper introduces the R package localgauss which estimates and visualizes a measure of local dependence called local Gaussian correlation. The package provides a function for estimation, a function for local independence testing and corresponding functions for visualization purposes, which are all demonstrated with examples.

Using local Gaussian correlation in a nonlinear re-examination of financial contagion

This paper examines financial contagion, that is, whether the cross-market linkages in financial markets increase after a shock to a country. We use a new measure of local dependence (introduced by Tjøstheim and Hufthammer (2013)) to study the contagion effect. The central idea of the new approach is to approximate an arbitrary bivariate return distribution by a family of Gaussian bivariate distributions. At each point of the return distribution there is a Gaussian distribution that gives a good approximation at that point. The correlation of the approximating Gaussian distribution is taken as the local correlation in that neighbourhood. By examining the local Gaussian correlation before the shock (in a stable period) and after the shock (in the crisis period), we are able to test whether contagion has occurred by a bootstrap testing procedure. The use of local Gaussian correlation is compared to other methods of studying contagion. Further, the bootstrap test is examined in a Monte Carlo study, and shows good level and power properties. We illustrate our approach by re-examining the Mexican crisis of 1994, the Asian crisis of 1997–1998 and the financial crisis of 2007–2009. We find evidence of contagion based on our new procedure and are able to describe the nonlinear dependence structure of these crises.

The Performance of Local Dependence Measures With Psychological Data

The violation of the assumption of local independence when applying item response theory (IRT) models has been shown to have a negative impact on all estimates obtained from the given model. Numerous indices and statistics have been proposed to aid analysts in the detection of local dependence (LD). A Monte Carlo study was conducted to evaluate the relative performance of selected LD measures in conditions considered typical of studies collecting psychological assessment data. Both the Jackknife Slope Index and likelihood ratio statistic G2 are available across the two IRT models used and displayed adequate to good performance in most simulation conditions. The use of these indices together is the final recommendation for applied analysts. Future research areas are discussed.

Recognizing and visualizing departures from independence in bivariate data using local Gaussian correlation

It is well known that the traditional Pearson correlation in many cases fails to capture non-linear dependence structures in bivariate data. Other scalar measures capable of capturing non-linear dependence exist. A common disadvantage of such measures, however, is that they cannot distinguish between negative and positive dependence, and typically the alternative hypothesis of the accompanying test of independence is simply "dependence". This paper discusses how a newly developed local dependence measure, the local Gaussian correlation, can be used to construct local and global tests of independence. A global measure of dependence is constructed by aggregating local Gaussian correlation on subsets of $\mathbb{R}^{2}$ , and an accompanying test of independence is proposed. Choice of bandwidth is based on likelihood cross-validation. Properties of this measure and asymptotics of the corresponding estimate are discussed. A bootstrap version of the test is implemented and tried out on both real and simulated data. The performance of the proposed test is compared to the Brownian distance covariance test. Finally, when the hypothesis of independence is rejected, local independence tests are used to investigate the cause of the rejection.

Estimation of a measure of local correlation for independent samples and time series data

Different from measures of global dependence, measures of local dependence evaluate the dependence along the support of the variables. The aim of this paper is to study a measure of local dependence proposed by Bairamov, Kotz and Kozubowski (2003) in the context of variables not indexed by time and also for stationary time series. We propose similar estimators for both cases. The consistency of the estimators are obtained, and their behavior are studied through simulations. Some empirical illustrations are provided.

Local Gaussian correlation: A new measure of dependence

It is a common view among finance analysts and econometricians that the correlation between financial objects becomes stronger as the market is going down, and that it approaches one when the market crashes, having the effect of destroying the benefit of diversification. The purpose of this paper is to introduce a local dependence measure that gives a precise mathematical description and interpretation of such phenomena. We propose a new local dependence measure, a local correlation function, based on approximating a bivariate density locally by a family of bivariate Gaussian densities using local likelihood. At each point the correlation coefficient of the approximating Gaussian distribution is taken as the local correlation. Existence, uniqueness and limit results are established. A number of properties of the local Gaussian correlation and its estimate are given, along with examples from both simulated and real data. This new method of modelling carries with it the prospect of being able to do locally for a general density what can be done globally for the Gaussian density. In a sense it extends Gaussian analysis from a linear to a non-linear environment.

Bivariate Reversed Hazard Rate, Notions, and Measures of Dependence and their Relationships

In this article, we extend some important results existing in the reliability literature based on the bivariate hazard rate to a case in which the bivariate reversed hazard rate is considered. In particular, we prove the existence of a link between the components of a bivariate reversed hazard rate and notions of dependence based on the left tail of a distribution. Moreover, we use this link to study some properties of a local dependence measure. Finally, we propose some characterizations of bivariate distribution based on the relation between various definitions of reversed hazard.

Measuring Financial Contagion by Local Gaussian Correlation

This paper examines financial contagion, that is, whether the cross-market linkages in financial markets increases after a shock to a country. We introduce the use of a new measure of local dependence (introduced by Hufthammer and Tjostheim (2009)) to study the contagion effect. The central idea of the new approach is to approximate an arbitrary bivariate return distribution by a family of Gaussian bivariate distributions. At each point of the return distribution there is a Gaussian distribution that gives a good approximation at that point. The correlation of the approximating Gaussian distribution is taken as the local correlation in that neighbourhood. By examining the local Gaussian correlation before the shock (in a stable period) and after the shock (in the crisis period), we are able to test whether contagion has occurred by a proposed bootstrap testing procedure. Examining the Mexican crisis of 1994, the Asian crisis of 1997-1998 and the financial crisis of 2007-2009, we find some evidence of contagion based on our new procedure.

Copulas: A Review and Recent Developments

In this review paper we outline some recent contributions to copula theory. Several new author's investigations are presented briefly, namely: order statistics copula, copulas with given multivariate marginals, copula representation via a local dependence measure and applications of extreme value copulas.

On Bivariate Reversed Hazard Rates

In this paper we discuss various definitions of bivariate reversed hazard rate and their properties. An exponential representation of bivariate distribution using reversed hazard rates is given and we also develop a new family of bivariate distributions using bivariate reversed hazard rate. Finally we give a local dependence measure using bivariate reversed hazard rates and study its properties. Various applications of the models are pointed out.

Equivalence and Bifurcations of Finite Order Stochastic Processes

This article presents an equivalence notion of finite order stochastic processes. Local dependence measures are defined in terms of joint and marginal densities. The dependence measures are classified topologically using level sets. The corresponding bifurcation theory is illustrated with some simple examples.