The dispersive effect of cross-aging with archimedean copulas

In this work, we compare conditional distributions derived from bivariate archimedean copulas in terms of their respective variabilities using the dispersive stochastic order. Specifically, we fix the underlying copula and we consider the effect of increasing the second component on the variability of the conditional distribution of the first component. Characterizations are provided in terms of the generator and of the marginal distributions. Several examples involving standard parametric copulas such as Clayton and Frank ones are discussed.     

Concordance measures for multivariate non-continuous random vectors

A notion of multivariate concordance suitable for non-continuous random variables is defined and many of its properties are established. This allows the definition of multivariate, non-continuous versions of Kendall’s tau, Spearman’s rho and Spearman’s footrule, which are concordance measures. Since the maximum values of these association measures are not +1 in general, a special attention is given to the computation of upper bounds. The latter turn out to be multivariate generalizations of earlier findings made by Nešlehová (2007) [9] and Denuit and Lambert (2005) [2]. They are easy to compute and can be estimated from a data set of (possibly) discontinuous random vectors. Corrected versions are considered as well.     

Vector-Valued Tail Value-at-Risk and Capital Allocation

Enterprise risk management, actuarial science or finance are practice areas in which risk measures are important to evaluate for heterogeneous classes of homogeneous risks. We present new measures: bivariate lower and upper orthant Tail Value-at-Risk. They are based on bivariate lower and upper orthant Value-at-Risk, introduced in Cossette et al. (Insurance: Math Econ 50(2):247–256, 2012). Many properties and applications are derived. Notably, they are shown to be positive homogeneous, invariant under translation and subadditive in distribution. Capital allocation criteria are suggested. Moreover, results on the sum of random pairs are presented, allowing to use a more accurate model for dependent classes of homogeneous risks.     

Dependence structure of conditional Archimedean copulas

In this article, copulas associated to multivariate conditional distributions in an Archimedean model are characterized. It is shown that this popular class of dependence structures is closed under the operation of conditioning, but that the associated conditional copula has a different analytical form in general. It is also demonstrated that the extremal copula for conditional Archimedean distributions is no longer the Fréchet upper bound, but rather a member of the Clayton family. Properties of these conditional distributions as well as conditional versions of tail dependence indices are also considered.     

Ordering Functions of Random Vectors, with Application to Partial Sums

It is known that the sums of the components of two random vectors (X ₁,X ₂,…,X _n ) and (Y ₁,Y ₂,…,Y _n ) ordered in the multivariate (s ₁,s ₂,…,s _n )-increasing convex order are ordered in the univariate (s ₁+s ₂+?+s _n )-increasing convex order. More generally, real-valued functions of (X ₁,X ₂,…,X _n ) and (Y ₁,Y ₂,…,Y _n ) are ordered in the same sense as long as these functions possess some specified non-negative cross-derivatives. This note extends these results to multivariate functions. In particular, we consider vectors of partial sums (S ₁,S ₂,…,S _n ) and (T ₁,T ₂,…,T _n ) where S _j =X ₁+?+X _j and T _j =Y ₁+?+Y _j and we show that these random vectors are ordered in the multivariate (s ₁,s ₁+s ₂,…,s ₁+?+s _n )-increasing convex order. The consequences of these general results for the upper orthant order and the orthant convex order are discussed.     

Bounds on Multivariate Kendall’s Tau and Spearman’s Rho for Zero-Inflated Continuous Variables and their Application to Insurance

Methodology and Computing in Applied Probability - In this note, we derive upper bounds on Kendall’s tau and Spearman’s rho for multivariate zero-inflated continuous variables often...     

Multivariate Higher-Degree Stochastic Increasing Convexity

Building on the seminal work by Shaked and Shanthikumar (Adv Appl Probab 20:427–446, 1988a; Stoch Process Appl 27:1–20, 1988b), Denuit et al. (Eng Inf Sci 13:275–291, 1999; Methodol Comput Appl Probab 2:231–254, 2000; 2001) studied the stochastic s-increasing convexity properties of standard parametric families of distributions. However, the analysis is restricted there to a single parameter. As many standard families of distributions involve several parameters, multivariate higher-order stochastic convexity properties also deserve consideration for applications. This is precisely the topic of the present paper, devoted to stochastic \((s_1,s_2,\ldots ,s_d)\)-increasing convexity of distribution families indexed by a vector \((\theta _1,\theta _2,\ldots ,\theta _d)\) of parameters. This approach accounts for possible correlation in multivariate mixture models.