Multivariate Dependence Measure Research Articles

A New Class of Bivariate Distributions: Properties, Estimation, and Modeling

The development of bivariate distributions is a challenging area of research, and a novel class of bivariate distributions is proposed in this paper. Certain useful mathematical characteristics of the proposed class of bivariate distributions are explored in general. These properties include the joint and conditional and joint survival function moments and measures of dependence. The parameters of the proposed class of distributions are estimated using the maximum likelihood method of estimation. A multivariate version of the proposed class is also proposed, and some necessary properties are discussed. The multivariate dependence measures are obtained for the proposed multivariate class of distributions. A new bivariate power function distribution is proposed using the power function distribution as a baseline distribution in the proposed bivariate class. Some useful properties of the proposed bivariate distribution are studied. The parameters of the proposed bivariate distribution are estimated using the maximum likelihood method. An extensive simulation study has been conducted to see the performance of the estimation method. The new bivariate power function distribution is used to model some real data.

Nonparametric Estimation of Multivariate Copula Using Empirical Bayes Methods

In the fields of finance, insurance, system reliability, etc., it is often of interest to measure the dependence among variables by modeling a multivariate distribution using a copula. The copula models with parametric assumptions are easy to estimate but can be highly biased when such assumptions are false, while the empirical copulas are nonsmooth and often not genuine copulas, making the inference about dependence challenging in practice. As a compromise, the empirical Bernstein copula provides a smooth estimator, but the estimation of tuning parameters remains elusive. The proposed empirical checkerboard copula within a hierarchical empirical Bayes model alleviates the aforementioned issues and provides a smooth estimator based on multivariate Bernstein polynomials that itself is shown to be a genuine copula. Additionally, the proposed copula estimator is shown to provide a more accurate estimate of several multivariate dependence measures. Both theoretical asymptotic properties and finite-sample performances of the proposed estimator based on simulated data are presented and compared with some nonparametric estimators. An application to portfolio risk management is included based on stock prices data.

Concurrent influence of geological parameters on the integrated nano-pore structure and discretized pore families of the petroliferous Cambay shale assessed through multivariate dependence measure

Heterogeneous nanopore structure and distribution regulate the gas trapping, desorption kinetics, and diffusion in shale matrices. In shale, pores range from continuous micro- and mesopore size distributions, varying with organic (total organic matter-TOC) and inorganic constituents (clay content, Fe-bearing minerals, quartz, etc.). Previous research only showed a linear relationship of pore parameters with these intrinsic properties of shale, which limits our understanding of the concurrent influence of multiple intrinsic rock properties. As a result, in this work, we established multivariate dependency of nanopore structure, distribution, and complexity (from low-pressure N2 and CO2 sorption and small-angle scattering; SAXS/MSANS) in the previously little-studied Cambay shales and provided a better tool (partial least square regression) for analyzing the simultaneous effect of intrinsic shale properties on multiply connected pore-parameters. Furthermore, we discretized continuous pore-size distribution into individual pore families using deconvolution to understand the pore space better. Additionally, predicted shale formation environment in terms of deposition probability (P+) and dissolution probability (P−) using a dynamic model of the fractal interface by precipitation and dissolution. Our findings indicate that the Cambay shales have a high potential for future hydrocarbon exploration (S2: 2.42–12.04 mg HC/g rock), “very good” (2-4 wt.%) to “excellent” (>4 wt.%) TOC content, and thermally mature type II–III admixed and type III kerogen. Deconvolution of the micro- and mesopore size distributions reveals that pore width (w) ranges ∼15.30–35 nm occupies greater than 50% of the total pore volume, and its pore volume increases with the presence of quartz, Fe-bearing minerals, and clay content. However, pores with w∼ 3.60–15.30 nm increase exclusively with TOC. In the micro- and early mesopore region, pore volume decreases with TOC from w∼ 0.30–0.75 nm and increases with TOC from w∼ 0.75–3.60 nm. Furthermore, TOC in shale increases the specific surface area and pore volume (micro-, meso-, and total pores), enhancing both sorption and free gas storage capacities. Cambay shales were likely deposited in three distinct environments, with precipitation probability (P+) values of 1, 0.7–0.8, and 0.5, as revealed by a fractal dimension (Ds) analysis of multiple samples.

The Effect of Dependence on European Market Risk. A Nonparametric Time Varying Approach

Multivariate dependence measures are crucial for risk management, where variables usually have heavy tails and non-Gaussian distributions. We propose a multivariate time varying Kendall’s tau estimator in a nonparametric context, allowing for local stationary variables. Consistency and asymptotic normality of the estimator are provided. A simulation study is conducted which supports the idea of better performance than other related methods in many complex scenarios. The proposal is used to draw up a daily estimation of the dependence between European financial market indexes. Nonparametric conditional quantiles are estimated to detect any influence of the degree of dependence on the market returns. That dependence emerges as an important factor in the Euro Stoxx distribution. It is noteworthy that the Kendall’s tau only depends on the multivariate copula, so the effect is not due to hidden effects of the marginals. Local Granger causality is tested and evidence is found that the degree of dependence affects the Euro Stoxx returns in the left tail of the distribution. We believe that these results encourage further research into the effect of diversification in quantiles, linked to the factors behind systemic risk. Additionally, there is a noteworthy increase in dependence following the outbreak of COVID-19.

On a multivariate copula-based dependence measure and its estimation

Working with so-called linkages allows to define a copula-based, [0,1]-valued multivariate dependence measure ζ1(X,Y) quantifying the scale-invariant extent of dependence of a random variable Y on a d-dimensional random vector X=(X1,…,Xd) which exhibits various good and natural properties. In particular, ζ1(X,Y)=0 if and only if X and Y are independent, ζ1(X,Y) is maximal exclusively if Y is a function of X, and ignoring one or several coordinates of X can not increase the resulting dependence value. After introducing and analyzing the metric D1 underlying the construction of the dependence measure and deriving examples showing how much information can be lost by only considering all pairwise dependence values ζ1(X1,Y),…,ζ1(Xd,Y) we derive a so-called checkerboard estimator for ζ1(X,Y) and show that it is strongly consistent in full generality, i.e., without any smoothness restrictions on the underlying copula. Some simulations illustrating the small sample performance of the estimator complement the established theoretical results.

Artificial intelligence in portfolio formation and forecast: Using different variance-covariance matrices

Portfolio optimization relies in three main elements: the utility function, the use of multivariate dependence measure between assets and the returns of the assets. The first problem is well known in finance since one can use several different types of portfolio utility functions depending on her objective function and constraints. The two remaining problems are relevant topics of research: the measure and forecast of the dependence between assets and the forecast of the returns of assets. In this paper, we present a new method with artificial intelligence to estimate the variance and covariance matrices of financial time series data accounting for small sample size, large number of assets and presences of outliers. We compare results using different approaches, e.g., Minimum covariance determinant (MCD), Minimum Volume Ellipsoid (MVE), Minimum Regularized Covariance Determinant (MRCD) and Orthogonalized Gnanadesikan-Kettenring (OGK). The proposed approach is robust under several financial stylized facts and presents good performance with respect to the cumulative returns in the US markets (NASDAQ Stock Exchange) for 95 assets during 2015 to the beginning of 2019. These study contributes to fill in the literature gap in covariance estimation for small sample size, large number of assets and presences of outliers and contributes to investors to making better portfolio management.

Ordinal pattern dependence as a multivariate dependence measure

In this article, we show that the recently introduced ordinal pattern dependence fits into the axiomatic framework of general multivariate dependence measures, i.e., measures of dependence between two multivariate random objects. Furthermore, we consider multivariate generalizations of established univariate dependence measures like Kendall’s τ, Spearman’s ρ and Pearson’s correlation coefficient. Among these, only multivariate Kendall’s τ proves to take the dynamical dependence of random vectors stemming from multidimensional time series into account. Consequently, the article focuses on a comparison of ordinal pattern dependence and multivariate Kendall’s τ in this context. To this end, limit theorems for multivariate Kendall’s τ are established under the assumption of near-epoch dependent data-generating time series. We analyze how ordinal pattern dependence compares to multivariate Kendall’s τ and Pearson’s correlation coefficient on theoretical grounds. Additionally, a simulation study illustrates differences in the kind of dependencies that are revealed by multivariate Kendall’s τ and ordinal pattern dependence.

Ordinal Pattern Dependence in the Context of Long-Range Dependence.

Ordinal pattern dependence is a multivariate dependence measure based on the co-movement of two time series. In strong connection to ordinal time series analysis, the ordinal information is taken into account to derive robust results on the dependence between the two processes. This article deals with ordinal pattern dependence for a long-range dependent time series including mixed cases of short- and long-range dependence. We investigate the limit distributions for estimators of ordinal pattern dependence. In doing so, we point out the differences that arise for the underlying time series having different dependence structures. Depending on these assumptions, central and non-central limit theorems are proven. The limit distributions for the latter ones can be included in the class of multivariate Rosenblatt processes. Finally, a simulation study is provided to illustrate our theoretical findings.

System Identifiability and Structure Identification: Input and Output Variables Selection Based on Consistent Measures of Dependence

This paper analyses the measures of dependence and properties needed to solve problems of identifiability as well as structure identification of systems – input and output variables selection, and constructive approach corresponding to these problems. The approach is based on applying consistent measures of both bivariate and multivariate dependence of random values and vectors; and enables constructing quantitative indexes to select system input and output variables.

Is optimum always optimal? A revisit of the mean‐variance method under nonlinear measures of dependence and non‐normal liquidity constraints

AbstractWe develop a model for optimizing multiple‐asset portfolios with semi‐parametric liquidity‐adjusted value‐at‐risk (LVaR), whereby linear correlations are substituted by the multivariate nonlinear Kendall's tau dependence measure, under multiple credible operational and budget constraints. When considering a diversified large portfolio of international stock markets of both developed and emerging economies and commodities, under both regular and stressed market perspectives, the obtained results consistently confirm the dominance of our modeling algorithms relative to other competing portfolio strategies, including the traditional mean‐variance VaR approach and global minimum‐variance portfolios. The obtained results are robust to different trading scenarios and promising for practical optimization techniques in large multiple‐asset portfolios and operation research models in financial institution management.

On tail dependence matrices

Among bivariate tail dependence measures, the tail dependence coefficient has emerged as the popular choice. Akin to the correlation matrix, a multivariate dependence measure is constructed using these bivariate measures, and this is referred to in the literature as the tail dependence matrix (TDM). While the problem of determining whether a given d × d matrix is a correlation matrix is of the order O(d3) in complexity, determining if a matrix is a TDM (the realization problem) is believed to be non-polynomial in complexity. Using a linear programming (LP) formulation, we show that the combinatorial structure of the constraints is related to the intractable max-cut problem in a weighted graph. This connection provides an avenue for constructing parametric classes admitting a polynomial in d algorithm for determining membership in its constraint polytope. The complexity of the general realization problem is justifiably of much theoretical interest. Since in practice one resorts to lower dimensional parametrization of the TDMs, we posit that it is rather the complexity of the realization problem restricted to parametric classes of TDMs, and algorithms for it, that are more practically relevant. In this paper, we show how the inherent symmetry and sparsity in a parametrization can be exploited to achieve a significant reduction in the LP formulation, which can lead to polynomial complexity of such realization problems - some parametrizations even resulting in the constraint polytope being independent of d. We also explore the use of a probabilistic viewpoint on TDMs to derive the constraint polytopes.

Dependence and dependence structures: estimation and visualization using the unifying concept of distance multivariance

AbstractDistance multivariance is a multivariate dependence measure, which can detect dependencies between an arbitrary number of random vectors each of which can have a distinct dimension. Here we discuss several new aspects, present a concise overview and use it as the basis for several new results and concepts: in particular, we show that distance multivariance unifies (and extends) distance covariance and the Hilbert-Schmidt independence criterion HSIC, moreover also the classical linear dependence measures: covariance, Pearson’s correlation and the RV coefficient appear as limiting cases. Based on distance multivariance several new measures are defined: a multicorrelation which satisfies a natural set of multivariate dependence measure axioms and m-multivariance which is a dependence measure yielding tests for pairwise independence and independence of higher order. These tests are computationally feasible and under very mild moment conditions they are consistent against all alternatives. Moreover, a general visualization scheme for higher order dependencies is proposed, including consistent estimators (based on distance multivariance) for the dependence structure.Many illustrative examples are provided. All functions for the use of distance multivariance in applications are published in the R-package multivariance.

Nonparametric estimation of multivariate tail probabilities and tail dependence coefficients

We propose three methods for estimating the joint tail probabilities based on a d-variate copula with dimension d≥2. For the first two methods, we use two different tail expansions of the copula which are valid under mild regularity conditions. We estimate the coefficients of these expansions using the maximum likelihood approach with appropriate data beyond a threshold in the tail. For the third method, we propose a family of tail-weighted measures of multivariate dependence and use these measures to estimate the coefficients of the second tail expansion using regression. This expansion is then used to estimate the joint tail probabilities when the empirical probabilities cannot be used because of lack of data in the tail. The three proposed methods can also be used to estimate tail dependence coefficients of a multivariate copula. Simulation studies are used to indicate when the methods give more accurate estimates of the tail probabilities and tail dependence coefficients. We apply the proposed methods to analyze tail properties of a data set of financial returns.

Algorithms for Finding Copulas Minimizing Convex Functions of Sums

In this paper, we develop improved rearrangement algorithms to find the dependence structure that minimizes a convex function of the sum of dependent variables with given margins. We propose a new multivariate dependence measure, which can assess the convergence of the rearrangement algorithms and can be used as a stopping rule. We show how to apply these algorithms for example to finding the dependence among variables for which the marginal distributions and the distribution of the sum or the difference are known. As an example, we can find the dependence between two uniformly distributed variables that makes the distribution of the sum of two uniform variables indistinguishable from a normal distribution. Using MCMC techniques, we design an algorithm that converges to the global optimum.

A multivariate dependence measure for aggregating risks

To evaluate the aggregate risk in a financial or insurance portfolio, a risk analyst has to calculate the distribution function of a sum of random variables. As the individual risk factors are often positively dependent, the classical convolution technique will not be sufficient. On the other hand, assuming a comonotonic dependence structure will likely overrate the real aggregate risk. In order to choose between the two approximations, or perhaps use a weighted average, we should have an indication of the accuracy. Clearly this accuracy will depend on the copula between the individual risk factors, but it is also influenced by the marginal distributions. In this paper we introduce a multivariate dependence measure that takes both aspects into account. This new measure differs from other multivariate dependence measures, as it focuses on the aggregate risk rather than on the copula or the joint distribution function itself. We prove several interesting properties of this new measure and discuss its relation to other dependence measures. We also give some comments on the estimation and conclude with examples and numerical results.

Properties of hierarchical Archimedean copulas

Abstract In this paper we analyse the properties of hierarchical Archimedean copulas. This class is a generalisation of the Archimedean copulas and allows for general non-exchangeable dependency structures. We show that the structure of the copula can be uniquely recovered from all bivariate margins. We derive the distribution of the copula values, which is particularly useful for tests and constructing confidence intervals. Furthermore, we analyse dependence orderings, multivariate dependence measures, and extreme value copulas. We pay special attention to the tail dependencies and derive several tail dependence indices for general hierarchical Archimedean copulas.

A Multivariate Dependence Measure for Aggregating Risks

To evaluate the aggregate risk in a financial or insurance portfolio, a risk analyst has to calculate the distribution function of a sum of random variables. As the individual risk factors are often positively dependent, the classical convolution technique will not be sufficient. On the other hand, assuming a comonotonic dependence structure will likely overrate the real aggregate risk. In order to choose between both approximations, or perhaps use a weighted average, we should have an indication on the accuracy. Clearly this accuracy will depend on the copula between the individual risk factors, but it is also influenced by the marginal distributions. In this paper we introduce a multivariate dependence measure that takes both aspects into account. This new measure differs from other multivariate dependence measures, as it focuses on the aggregate risk rather than on the copula or the joint distribution function itself. We prove several interesting properties of this new measure and discuss its relation to other dependencemeasures. We also give some comments on the estimation and conclude with examples and numerical results.

A Multivariate Dependence Measure for Aggregating Risks

A Multivariate Dependence Measure for Aggregating Risks

Measures of multivariate asymptotic dependence and their relation to spectral expansions

Asymptotic dependence can be interpreted as the property that realizations of the single components of a random vector occur simultaneously with a high probability. Information about the asymptotic dependence structure can be captured by dependence measures like the tail dependence parameter or the residual dependence index. We introduce these measures in the bivariate framework and extend them to the multivariate case afterwards. Within the extreme value theory one can model asymptotic dependence structures by Pickands dependence functions and spectral expansions. Both in the bivariate and in the multivariate case we also compute the tail dependence parameter and the residual dependence index on the basis of this statistical model. They take a specific shape then and are related to the Pickands dependence function and the exponent of variation of the underlying density expansion.

Dimensionless Measures of Variability and Dependence for Multivariate Continuous Distributions

In this article, we suggest dimensionless descriptive measures of multivariate variability and dependence which can be used in comparisons of random vectors of different dimensions. Our work generalizes the measures of scatter and linear dependence proposed by Peña and Rodríguez (2003). The measure of variability we introduce is the rth root of the (transformed) entropy and the measure of dependence is based on the mutual information between the components of an r-dimensional random vector, capturing general stochastic dependence instead of merely linear dependence. We further investigate decompositions of the measure of variability into a measure of scale and a measure of stochastic dependence. The decomposition resulting from independent components provides a representation of variability by the scale of independent components and thus generalizes the explanation of covariance by principal components in classical multivariate analysis. We illustrate our ideas for examples of non Gaussian random vectors.