An invariance property of quadratic forms in random vectors with a selection distribution,with application to sample variogram and covariogram estimators

We study conditions under which an invariance property holds for the class of selection distributions. First, we consider selection distributions arising from two uncorrelated random vectors. In that setting, the invariance holds for the so-called C{\cal{C}} -class and for elliptical distributions. Second, we describe the invariance property for selection distributions arising from two correlated random vectors. The particular case of the distribution of quadratic forms and its invariance, under various selection distributions, is investigated in more details. We describe the application of our invariance results to sample variogram and covariogram estimators used in spatial statistics and provide a small simulation study for illustration. We end with a discussion about other applications, for example such as linear models and indices of temporal/spatial dependence.     

Generalized skew-elliptical distributions and their quadratic forms

This paper introduces generalized skew-elliptical distributions (GSE), which include the multivariate skew-normal, skew-t, skew-Cauchy, and skew-elliptical distributions as special cases. GSE are weighted elliptical distributions but the distribution of any even function in GSE random vectors does not depend on the weight function. In particular, this holds for quadratic forms in GSE random vectors. This property is beneficial for inference from non-random samples. We illustrate the latter point on a data set of Australian athletes.     

The weak tail dependence coefficient of the elliptical generalized hyperbolic distribution

The strong and the weak tail dependence coefficients are measures that quantify the probability of conjoint extreme events of two random variables. Whereas formulas for both tail dependence coefficients exist for the Gaussian and Student t distribution, only the strong tail dependence coefficient is known for their super-model, the elliptical generalized hyperbolic distribution, which is extremely popular in finance (see Schmidt 2003). In this work we derive a simple expression for the corresponding weak tail dependence coefficient using the mixture representation of the elliptical generalized hyperbolic distribution.     

Tails of correlation mixtures of elliptical copulas

Correlation mixtures of elliptical copulas arise when the correlation parameter is driven itself by a latent random process. For such copulas, both penultimate and asymptotic tail dependence are much larger than for ordinary elliptical copulas with the same unconditional correlation. Furthermore, for Gaussian and Student t-copulas, tail dependence at sub-asymptotic levels is generally larger than in the limit, which can have serious consequences for estimation and evaluation of extreme risk. Finally, although correlation mixtures of Gaussian copulas inherit the property of asymptotic independence, at the same time they fall in the newly defined category of near asymptotic dependence. The consequences of these findings for modeling are assessed by means of a simulation study and a case study involving financial time series.     

On the residual dependence index of elliptical distributions

The residual dependence index of bivariate Gaussian distributions is determined by the correlation coefficient. This tail index is of certain statistical importance when extremes and related rare events of bivariate samples with asymptotic independent components are being modeled. In this paper we calculate the partial residual dependence indices of a multivariate elliptical random vector assuming that the associated random radius has distribution function in the Gumbel max-domain of attraction. Furthermore, we discuss the estimation of these indices when the associated random radius possesses a Weibull-tail distribution.     

Multivariate tail conditional expectation for elliptical distributions

In this paper we introduce a novel type of a multivariate tail conditional expectation (MTCE) risk measure and explore its properties. We derive an explicit closed-form expression for this risk measure for the elliptical family of distributions taking into account its variance–covariance dependency structure. As a special case we consider the normal, Student-t and Laplace distributions, important and popular in actuarial science and finance. The motivation behind taking the multivariate TCE for the elliptical family comes from the fact that unlike the traditional tail conditional expectation, the MTCE measure takes into account the covariation between dependent risks, which is the case when we are dealing with real data of losses. We illustrate our results using numerical examples in the case of normal and Student-t distributions.     

Constant Local Dependence

The local dependence function is constant for the bivariate normal distribution. Here we identify all other distributions which also have constant local dependence. The key property is exponential family conditional distributions and a linear conditional mean. When given two marginal distributions only, this characterisation is not very helpful, and numerical solutions are necessary.     

Extreme behavior of bivariate elliptical distributions

This paper exploits a stochastic representation of bivariate elliptical distributions in order to obtain asymptotic results which are determined by the tail behavior of the generator. Under certain specified assumptions, we present the limiting distribution of componentwise maxima, the limiting upper copula, and a bivariate version of the classical peaks over threshold result.     

Portfolio separation properties of the skew-elliptical distributions, with generalizations

The two-fund separation property of the elliptical distributions is extended to the skew-elliptical case by adding a number of funds equaling the rank of the skewness matrix. The singular extended skew-elliptical distributions are covered, as is a further generalization to the case where the set conditioned upon is not an orthant.     

Multivariate negative binomial models for insurance claim counts

It is no longer uncommon these days to find the need in actuarial practice to model claim counts from multiple types of coverage, such as the ratemaking process for bundled insurance contracts. Since different types of claims are conceivably correlated with each other, the multivariate count regression models that emphasize the dependency among claim types are more helpful for inference and prediction purposes. Motivated by the characteristics of an insurance dataset, we investigate alternative approaches to constructing multivariate count models based on the negative binomial distribution. A classical approach to induce correlation is to employ common shock variables. However, this formulation relies on the NB-I distribution which is restrictive for dispersion modeling. To address these issues, we consider two different methods of modeling multivariate claim counts using copulas. The first one works with the discrete count data directly using a mixture of max-id copulas that allows for flexible pair-wise association as well as tail and global dependence. The second one employs elliptical copulas to join continuitized data while preserving the dependence structure of the original counts. The empirical analysis examines a portfolio of auto insurance policies from a Singapore insurer where claim frequency of three types of claims (third party property damage, own damage, and third party bodily injury) are considered. The results demonstrate the superiority of the copula-based approaches over the common shock model. Finally, we implemented the various models in loss predictive applications.     

Asymptotic Results for the Sum of Dependent Non-identically Distributed Random Variables

In this paper we extend some results about the probability that the sum of n dependent subexponential random variables exceeds a given threshold u. In particular, the case of non-identically distributed and not necessarily positive random variables is investigated. Furthermore we establish criteria how far the tail of the marginal distribution of an individual summand may deviate from the others so that it still influences the asymptotic behavior of the sum. Finally we explicitly construct a dependence structure for which, even for regularly varying marginal distributions, no asymptotic limit of the tail of the sum exists. Some explicit calculations for diagonal copulas and t-copulas are given. Dominik Kortschak was supported by the Austrian Science Fund Project P18392.     

A Directory of Coefficients of Tail Dependence

Models characterizing the asymptotic dependence structures of bivariate distributions have been introduced by Ledford and Tawn (1996), among others, and diagnostics for such dependence behavior are presented in Coles et al. (1999). The following pages are intended as a supplement to the papers of Ledford and Tawn and Coles et al. In particular we focus on the coefficient of tail dependence, which we evaluate for a wide range of bivariate distributions. We find that for many commonly employed bivariate distributions there is little flexibility in the range of limiting dependence structure accommodated. Many distributions studied have coefficients of tail dependence corresponding to near independence or a strong form of dependence known as asymptotic dependence.     

Tail dependence between order statistics

In this work, we introduce the s,k-extremal coefficients for studying the tail dependence between the s-th lower and k-th upper order statistics of a normalized random vector. If its margins have tail dependence then so do their order statistics, with the strength of bivariate tail dependence decreasing as two order statistics become farther apart. Some general properties are derived for these dependence measures which can be expressed via copulas of random vectors. Its relations with other extremal dependence measures used in the literature are discussed, such as multivariate tail dependence coefficients, the coefficient η of tail dependence, coefficients based on tail dependence functions, the extremal coefficient ?, the multivariate extremal index and an extremal coefficient for min-stable distributions. Several examples are presented to illustrate the results, including multivariate exponential and multivariate Gumbel distributions widely used in applications.     

Central Regions and Dependency

The paper introduces an approach to the ordering of dependence which is based on central regions. A d-variate probability distribution is described by a nested family of sets, called central regions. Those regions are affine equivariant, compact and starshaped and concentrate about a properly defined center. They can be seen as level sets of a depth function. Special cases are Mahalanobis, zonoid, and likelihood regions. A d-variate distribution is called more dependent than another one if the volume of each central region is smaller with the first distribution. This dependence order is characterized by an inequality between determinants of certain parameter matrices if either (i) F and G are arbitrary distributions and the central regions are Mahalanobis or (ii) F and G belong to an elliptical family of distributions and the central regions are arbitrary. If the regions are zonoid regions, the dependence order implies the ordering of lift zonoid volumes. Alternatively, the dependence order is applied to the copulae of the given distributions. Generalized correlation indices are proposed which are increasing with the dependence orders.     

Extreme value properties of multivariate t copulas

The extremal dependence behavior of t copulas is examined and their extreme value limiting copulas, called the t-EV copulas, are derived explicitly using tail dependence functions. As two special cases, the Hüsler–Reiss and the Marshall–Olkin distributions emerge as limits of the t-EV copula as the degrees of freedom go to infinity and zero respectively. The t copula and its extremal variants attain a wide range in the set of bivariate tail dependence parameters. Supported by NSERC Discovery Grant.     

Archimedean-based Marshall-Olkin Distributions and Related Dependence Structures

In this paper we study the dependence properties of a family of bivariate distributions (that we call Archimedean-based Marshall-Olkin distributions) that extends the class of the Generalized Marshall-Olkin distributions of Li and Pellerey, J Multivar Anal, 102, (10), 1399–1409, 2011 in order to allow for an Archimedean type of dependence among the underlying shocks’ arrival times. The associated family of copulas (that we call Archimedean-based Marshall-Olkin copulas) includes several well known copula functions as specific cases for which we provide a different costruction and represents a particular case of implementation of Morillas, Metrika, 61, (2), 169–184, 2005 construction. It is shown that Archimedean-based copulas are obtained through suitable transformations of bivariate Archimedean copulas: this induces asymmetry, and the corresponding Kendall’s function and Kendall’s tau as well as the tail dependence parameters are studied. The type of dependence so modeled is wide and illustrated through examples and the validity of the weak Lack of memory property (characterizing the Marshall-Olkin distribution) is also investigated and the sub-family of distributions satisfying it identified. Moreover, the main theoretical results are extended to the multidimensional version of the considered distributions and estimation issues discussed.     

Bayesian inference for dependent elliptical measurement error models

In this article we provide a Bayesian analysis for dependent elliptical measurement error models considering nondifferential and differential errors. In both cases we compute posterior distributions for structural parameters by using squared radial prior distributions for the precision parameters. The main result is that the posterior distribution of location parameters, for specific priors, is invariant with respect to changes in the generator function, in agreement with previous results obtained in the literature under different assumptions. Finally, although the results obtained are valid for any elliptical distribution for the error term, we illustrate those results by using the student-t distribution and a real data set.     

The tail probability of the product of dependent random variables from max-domains of attraction

In this article, we investigate the tail probability of the product of finitely many non-negative dependent random variables. They follow distributions from max-domains of attraction of extreme value distributions and their dependence is modeled via a multivariate Farlie–Gumbel–Morgenstern distribution. For each of the Fréchet, Gumbel and Weibull cases, we obtain an explicit asymptotic formula for the tail probability of the product. Our study extends a few known results in the literature.     

Some results on the CTE-based capital allocation rule

Tasche [Tasche, D., 1999. Risk contributions and performance measurement. Working paper, Technische Universität München] introduces a capital allocation principle where the capital allocated to each risk unit can be expressed in terms of its contribution to the conditional tail expectation (CTE) of the aggregate risk. Panjer [Panjer, H.H., 2002. Measurement of risk, solvency requirements and allocation of capital within financial conglomerates. Institute of Insurance and Pension Research, University of Waterloo, Research Report 01-15] derives a closed-form expression for this allocation rule in the multivariate normal case. Landsman and Valdez [Landsman, Z., Valdez, E., 2002. Tail conditional expectations for elliptical distributions. North American Actuarial J. 7 (4)] generalize Panjer’s result to the class of multivariate elliptical distributions.In this paper we provide an alternative and simpler proof for the CTE-based allocation formula in the elliptical case. Furthermore, we derive accurate and easy computable closed-form approximations for this allocation formula for sums that involve normal and lognormal risks.     

Noncentral matrix quadratic forms of the skew elliptical variables

In this paper, the noncentral matrix quadratic forms of the skew elliptical variables are studied. A family of the matrix variate noncentral generalized Dirichlet distributions is introduced as the extension of the noncentral Wishart distributions, the Dirichlet distributions and the noncentral generalized Dirichlet distributions. Main distributional properties are investigated. These include probability density and closure property under linear transformation and marginalization, the joint distribution of the sub-matrices of the matrix quadratic forms in the skew elliptical variables and the moment generating functions and Bartlett's decomposition of the matrix quadratic forms in the skew normal variables. Two versions of the noncentral Cochran's Theorem for the matrix variate skew normal distributions are obtained, providing sufficient and necessary conditions for the quadratic forms in the skew normal variables to have the matrix variate noncentral generalized Dirichlet distributions. Applications include the properties of the least squares estimation in multivariate linear model and the robustness property of the Wilk's likelihood ratio statistic in the family of the matrix variate skew elliptical distributions.