Multivariate extremes, aggregation and dependence in elliptical distributions

Abstract

In this paper, we clarify dependence properties of elliptical distributions by deriving general but explicit formulae for the coefficients of upper and lower tail dependence and spectral measures with respect to different norms. We show that an elliptically distributed random vector is regularly varying if and only if the bivariate marginal distributions have tail dependence. Furthermore, the tail dependence coefficients are fully determined by the tail index of the random vector (or equivalently of its components) and the linear correlation coefficient. Whereas Kendall's tau is invariant in the class of elliptical distributions with continuous marginals and a fixed dispersion matrix, we show that this is not true for Spearman's rho. We also show that sums of elliptically distributed random vectors with the same dispersion matrix (up to a positive constant factor) remain elliptical if they are dependent only through their radial parts.

A mixed effect model for bivariate meta‐analysis of diagnostic test accuracy studies using a copula representation of the random effects distribution

“…To depict the different shapes of the SROC curves, in Figures and , we plot them from the copula representation of the random effects distribution with normal and beta margins, respectively, and BVN, Frank and Clayton by 90 and 270 copulas with the same model parameters { π 1 =0.7, π 2 =0.9, σ 1 =2, σ 2 =1, τ =− 0.5} and { π 1 =0.7, π 2 =0.9, γ 1 =0.2, γ 2 =0.1, τ =− 0.5}, respectively. We convert from τ to the BVN, Frank and rotated Clayton copula parameter θ via the relations in , and respectively.…”

Large covariance estimation through elliptical factor models

“…To see this, for elliptically distributed x , we have $$\mathrm{boldx}-boldnormal\stackrel{\sim }{x}\stackrel{d}{=}\zeta {\mathbf{\Lambda }}_p^{\frac{1}{2}}\mathrm{boldU}-\stackrel{\sim }{\zeta }{\mathbf{\Lambda }}_p^{\frac{1}{2}}boldnormal\stackrel{\sim }{U}\stackrel{d}{=}\overline{\zeta }{\mathbf{\Lambda }}_p^{\frac{1}{2}}\mathrm{boldU},$$ where $boldnormal\stackrel{\sim }{x}$ is an independent copy of x and the characteristic function of $\overline{\zeta }$ is determined by that of ζ . See Hult and Lindskog (2002) for a detailed expression of the characteristic function. Thus, for a multivariate standard normal vector g = ( g 1 ,…, g p )′, $$kfalse(\mathrm{boldx},boldnormal\stackrel{\sim }{x}false)=\frac{false(\mathrm{boldx}-boldnormal\stackrel{\sim }{x}false){false(\mathrm{boldx}-boldnormal\stackrel{\sim }{x}false)}^{\prime }}{{\Vert \mathrm{boldx}-boldnormal\stackrel{\sim }{x}\Vert }_2^2}\stackrel{d}{=}\frac{{\mathbf{\Lambda }}_p^{\frac{1}{2}}boldnormalU{U}^{\prime }{\mathbf{\Lambda }}_p^{\frac{1}{2}}}{boldnormal{U}^{\prime }{\mathbf{\Lambda }}_p\mathrm{boldU}}\stackrel{d}{=}\frac{{\mathbf{\Lambda }}_p^{\frac{1}{2}}boldnormalg{g}^{\prime }{\mathbf{\Lambda }}_p^{\frac{1}{2}}}{boldnormal{g}^{\prime }{\mathbf{\Lambda }}_p\mathrm{boldg}},$$ which depends only on g .…”

Coupling Couples With Copulas: Analysis of Assortative Matching on Risk Attitude

“…For the model we allow three different copula families, one for the male time‐series, one for the female time‐series, and one to join them. To make it easier to compare the dependence parameters, we convert the estimated parameters to Kendall's τ 's in (0, 1) via the relations , and τ = 1 − θ −1 for elliptical, Frank and Gumbel copulas in Hult and Lindskog (), Genest (), and Genest and MacKay (), respectively. Note that Kendall's τ only accounts for the dependence dominated by the middle of the data, and it is expected to be similar amongst different families of copulas.…”