Thomas Fung

Tailweight, quantiles and kurtosis: A study of competing distributions

This paper studies analytically and numerically the tail behavior of the symmetric variance-gamma (VG), t, and exponential-power (EP) distributions. Special emphasis is on the VG, which is a direct competitor of the t in the financial context of modeling the distribution of log-price increments.

Tail dependence and skew distributions

ð4Þ for x2R. The definition of Kð , Þ will be provided in section 2. l is the location vector, is a symmetric positive definite n n scale matrix and h1⁄4 ( 1, . . . , n) is the vector that controls the asymmetry of the distribution. Its corresponding copula in the special case where is diagonal is the main focus of section 2 of Luciano and Schoutens (2006) in their study of the correlation structure, both in general, and specifically in relation to asymptotic tail dependence. It is sufficient in the study of correlation structure to work, as Luciano and Schoutens (2006) do, with bivariate distributions, so our discussion is also in terms of the components of a random vector X1⁄4 (X1,X2). Of the various concepts of correlation considered by Luciano and Schoutens (2006), we focus on the concept of tail dependence. The coefficient of the lower tail dependence of a random vector X1⁄4 (X1,X2) with marginal inverse distribution function F 1 1 and F 1 2 is defined as L 1⁄4 lim u!0þ LðuÞ, where LðuÞ 1⁄4 PðX1 F 1 1 ðuÞ j X2 F 1 2 ðuÞÞ: ð5Þ X is said to have asymptotic lower tail dependence if L exists and is positive. If L1⁄4 0, then X is said to be asymptotically independent in the lower tail. This quantity provides insight into the tendency for the distribution to generate joint extreme events since it measures the strength of dependence (or association) in the tails of a bivariate distribution. It is a particularly relevant concept in modelling prices and returns, or credit risk modelling, *Corresponding author. Email: eseneta@maths.usyd.edu.au

Extending the multivariate generalised t and generalised VG distributions

The GGH family of multivariate distributions is obtained by scale mixing on the Exponential Power distribution using the Extended Generalised Inverse Gaussian distribution. The resulting GGH family encompasses the multivariate generalised hyperbolic (GH), which itself contains the multivariate t and multivariate Variance-Gamma (VG) distributions as special cases. It also contains the generalised multivariate t distribution [O. Arslan, Family of multivariate generalised t distribution, Journal of Multivariate Analysis 89 (2004) 329-337] and a new generalisation of the VG as special cases. Our approach unifies into a single GH-type family the hitherto separately treated t-type [O. Arslan, A new class of multivariate distribution: Scale mixture of Kotz-type distributions, Statistics and Probability Letters 75 (2005) 18-28; O. Arslan, Variance-mean mixture of Kotz-type distributions, Communications in Statistics-Theory and Methods 38 (2009) 272-284] and VG-type cases. The GGH distribution is dual to the distribution obtained by analogous mixing on the scale parameter of a spherically symmetric stable distribution. Duality between the multivariate t and multivariate VG [S.W. Harrar, E. Seneta, A.K. Gupta, Duality between matrix variate t and matrix variate V.G. distributions, Journal of Multivariate Analysis 97 (2006) 1467-1475] does however extend in one sense to their generalisations.

Convergence rate to a lower tail dependence coefficient of a skew-t distribution

We examine the rate of decay to the limit of the tail dependence coefficient of a bivariate skew-t distribution. This distribution always displays asymptotic tail dependence. It contains as a special case the usual bivariate symmetric t distribution, and hence is an appropriate (skew) extension. The rate is asymptotically a power-law. The second-order structure of the univariate quantile function for such a skew-t distribution is a central issue. Our results generalise those for the bivariate symmetric t.

Tail asymptotics for the bivariate skew normal

We derive the asymptotic rate of decay to zero of the tail dependence of the bivariate skew normal distribution under the equal-skewness condition α 1 = α 2 , = α , say. The rate depends on whether α 0 or α < 0 . For the lower tail, the latter case has rate asymptotically identical with the bivariate normal ( α = 0 ), but has a different multiplicative constant. The case α 0 gives a rate dependent on α . The detailed asymptotic behaviour of the quantile function for the univariate skew normal is a key. This study is partly a sequel to our earlier one on the analogous situation for bivariate skew t .

Contaminated Variance-Mean mixing model

The Generalised Normal Variance-Mean (GNVM) model in which the mixing random variable is Gamma distributed is considered. This model generalises the popular Variance-Gamma (VG) distribution. This GNVM model can be interpreted as the addition of noise to a (skew) VG base. The discussion is based on goodness of fit criteria and on parameter estimation. The conclusion is that the shape of the VG distribution can be adjusted in a favourable way by adding noise.

Rate of Decay of the Tail Dependence Coefficient for the Skew t Distribution

We examine the rate of decay to zero of the tail dependence coefficient of the bivariate skew t distribution which is obtained via normal variance-mean mixture in a case where there is no asymptotic tail dependence. Our result helps to explain the difference in performance in model fitting between this skew t distribution and the one based on the variance mixing of the bivariate skew-normal distribution. This t distribution always displays asymptotic tail dependence, as happens in the symmetric case which is common to both models. DOI: http://dx.doi.org/10.4038/sljastats.v12i0.4966 Sri Lankan Journal of Applied Statistics Vol.12 2011 pp.27-40