Estimating the spectral measure of a multivariate stable distribution via spherical harmonic analysis

Journal of Multivariate Analysis (Impact Factor: 1.06). 11/2003; 87(2):219-240. DOI: 10.1016/S0047-259X(03)00052-6
Source: RePEc

ABSTRACT A new method is developed for estimating the spectral measure of a multivariate stable probability measure, by representing the measure as a sum of spherical harmonics.

1 Bookmark
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper we consider a variety of procedures for numerical statistical inference in the family of univariate and multivariate stable distributions. In connection with univariate distributions (i) we provide approximations by finite location-scale mixtures and (ii) versions of approximate Bayesian computation (ABC) using the characteristic function and the asymptotic form of the likelihood function. In the context of multivariate stable distributions we propose several ways to perform statistical inference and obtain the spectral measure associated with the distributions, a quantity that has been a major impediment in using them in applied work. We extend the techniques to handle univariate and multivariate stochastic volatility models, static and dynamic factor models with disturbances and factors from general stable distributions, a novel way to model multivariate stochastic volatility through time-varying spectral measures and a novel way to multivariate stable distributions through copulae. The new techniques are applied to artificial as well as real data (ten major currencies, SP100 and individual returns). In connection with ABC special attention is paid to crafting well-performing proposal distributions for MCMC and extensive numerical experiments are conducted to provide critical values of the "closeness" parameter that can be useful for further applied econometric work.
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: It is known that each symmetric stable distribution in Rd is related to a norm on Rd that makes Rd embeddable in Lp([0,1]). In the case of a multivariate Cauchy distribution the unit ball in this norm is the polar set to a convex set in Rd called a zonoid. This work interprets symmetric stable laws using convex or star-shaped sets and exploits recent advances in convex geometry in order to come up with new probabilistic results for multivariate symmetric stable distributions. In particular, it provides expressions for moments of the Euclidean norm of a stable vector, mixed moments and various integrals of the density function. It is shown how to use geometric inequalities in order to bound important parameters of stable laws. Furthermore, covariation, regression and orthogonality concepts for stable laws acquire geometric interpretations.
    Journal of Multivariate Analysis 01/2009; · 1.06 Impact Factor
  • Source

Full-text (2 Sources)

Available from
Jun 5, 2014