Generalized Marshall-Olkin Distributions, and Related Bivariate Aging Properties

National Natural Science Foundation of China [10771090]A class of generalized bivariate Marshall-Olkin distributions, which includes as special cases the Marshall-Olkin bivariate exponential distribution and the Marshall-Olkin type distribution due to Muliere and Scarsini (1987) [19] are examined in this paper. Stochastic comparison results are derived, and bivariate aging properties, together with properties related to evolution of dependence along time, are investigated for this class of distributions. Extensions of results previously presented in the literature are provided as well. (C) 2011 Elsevier Inc. All rights reserved

Copulas in finance and insurance

Copulas provide a potential useful modeling tool to represent the dependence structure among variables and to generate joint distributions by combining given marginal distributions. Simulations play a relevant role in finance and insurance. They are used to replicate efficient frontiers or extremal values, to price options, to estimate joint risks, and so on. Using copulas, it is easy to construct and simulate from multivariate distributions based on almost any choice of marginals and any type of dependence structure. In this paper we outline recent contributions of statistical modeling using copulas in finance and insurance. We review issues related to the notion of copulas, copula families, copula-based dynamic and static dependence structure, copulas and latent factor models and simulation of copulas. Finally, we outline hot topics in copulas with a special focus on model selection and goodness-of-fit testing

The Empirical Beta Copula

Given a sample from a multivariate distribution $F$, the uniform random variates generated independently and rearranged in the order specified by the componentwise ranks of the original sample look like a sample from the copula of $F$. This idea can be regarded as a variant on Baker's [J. Multivariate Anal. 99 (2008) 2312--2327] copula construction and leads to the definition of the empirical beta copula. The latter turns out to be a particular case of the empirical Bernstein copula, the degrees of all Bernstein polynomials being equal to the sample size. Necessary and sufficient conditions are given for a Bernstein polynomial to be a copula. These imply that the empirical beta copula is a genuine copula. Furthermore, the empirical process based on the empirical Bernstein copula is shown to be asymptotically the same as the ordinary empirical copula process under assumptions which are significantly weaker than those given in Janssen, Swanepoel and Veraverbeke [J. Stat. Plan. Infer. 142 (2012) 1189--1197]. A Monte Carlo simulation study shows that the empirical beta copula outperforms the empirical copula and the empirical checkerboard copula in terms of both bias and variance. Compared with the empirical Bernstein copula with the smoothing rate suggested by Janssen et al., its finite-sample performance is still significantly better in several cases, especially in terms of bias.Comment: 23 pages, 3 figure

A characterization of the multivariate excess wealth ordering

In this paper, some new properties of the upper-corrected orthant of a random vector are proved. The univariate right-spread or excess wealth function, introduced by Fernández-Ponce et al. (1996), is extended to multivariate random vectors, and some properties of this multivariate function are studied. Later, this function was used to define the excess wealth ordering by Shaked and Shanthikumar (1998) and Fernández-Ponce et al. (1998). The multivariate excess wealth function enable us to define a new stochastic comparison which is weaker than the multivariate dispersion orderings. Also, some properties relating the multivariate excess wealth order with stochastic dependence are describe

Copula-based orderings of multivariate dependence

In this paper I investigate the problem of defining a multivariate dependence ordering. First, I provide a characterization of the concordance dependence ordering between multivariate random vectors with fixed margins. Central to the characterization is a multivariate generalization of a well-known bivariate elementary dependence increasing rearrangement. Second, to order multivariate random vectors with non- fixed margins, I impose a scale invariance principle which leads to a copula-based concordance dependence ordering. Finally, a wide family of copula-based measures of dependence is characterized to which Spearmanís rank correlation coefficient belongs.copula, concordance ordering, dependence measures, dependence orderings, multivariate stochastic dominance, supermodular ordering

Aging functions and multivariate notions of NBU and IFR

For d≥2, let X=(X1, …, Xd) be a vector of exchangeable continuous lifetimes with joint survival function $\overline{F}$. For such models, we study some properties of multivariate aging of $\overline{F}$ that are described by means of the multivariate aging function $B_{\overline{F}}$, which is a useful tool for describing the level curves of $\overline{F}$. Specifically, the attention is devoted to notions that generalize the univariate concepts of New Better than Used and Increasing Failure Rate. These multivariate notions are satisfied by random vectors whose components are conditionally independent and identically distributed having univariate conditional survival function that is New Better than Used (respectively, Increasing Failure Rate). Furthermore, they also have an interpretation in terms of comparisons among conditional survival functions of residual lifetimes, given a same history of observed survivals