Meta densities and the shape of their sample clouds

Abstract

a b s t r a c tThis paper compares the shape of the level sets for two multivariate densities. The densities are positive and continuous, and have the same dependence structure. The density f is heavy-tailed. It decreases at the same rate -up to a positive constant -along all rays. The level sets {f > c} for c ↓ 0, have a limit shape, a bounded convex set. We transform each of the coordinates to obtain a new density g with Gaussian marginals. We shall also consider densities g with Laplace, or symmetric Weibull marginal densities. It will be shown that the level sets of the new light-tailed density g also have a limit shape, a bounded starshaped set. The boundary of this set may be written down explicitly as the solution of a simple equation depending on two positive parameters. The limit shape is of interest in the study of extremes and in risk theory, since it determines how the extreme observations in different directions relate. Although the densities f and g have the same copula -by construction -the shapes of the level sets are not related. Knowledge of the limit shape of the level sets for one density gives no information about the limit shape for the other density.

Estimation of Star-Shaped Distributions

“…More recently, the considerations in [11] opened a new field of financial applications of more general star-shaped asymptotic distributions, where suitably scaled sample clouds converge onto a deterministic set. Figure 3 in [34] represents a sample cloud which might be modeled with a density being star-shaped w.r.t.…”

“…The more flexible star-shaped densities were studied in [10] and later in [11]. The general structure of their normalizing constant given a density generating function was discovered, a geometric measure representation and, based upon it a stochastic representation were derived, and a survey of applications The boundary of K is just the set {(u, v) T : h K ((u, v) T ) = 1}.…”

Asymptotic independence for unimodal densities

“…In fact, the limit set, if it exists, is always star shaped (see Proposition 4.1 of [15]). The following simple criterion is useful for checking convergence (in probability) of scaled sample clouds (see [5] for a proof). The next theorem gives a sufficient condition for asymptotic independence of a distribution in terms of the limit set of the associated sample clouds.…”

“…It has normal marginals, but the copula of the elliptic t distribution. The shape of the level sets of the density g converges to the symmetric subset D = {u 2 1 + u 2 2 + λ > (λ + 2) (u 1 , u 2 ) 2 ∞ } of the square (−1, 1) 2 ; see [5]. Figure 1(b) shows a detail of the limit set D. The set D is not blunt, and the components of X are asymptotically dependent since those of Z are.…”

Simulation of polyhedral convex contoured distributions

“…It is possible to characterize a multivariate density by the geometry of its density level sets. For an overview about this broad field of research, we refer to (Arnold et al 2008;Balkema et al 2010;Fang et al 1990;Fernandez and Osiewalski 1995;Gupta and Song 1997;Kamiya et al 2008;Richter 2009;2013;2014;2015a;2015b;Richter and Schicker 2017;Sarabia and Gomez-Deniz 2008). Convex polyhedral distributions are characterized by contours being the topological boundaries of convex polyhedra and can be considered being a subclass of polyhedral star-shaped distributions that are studied in (Richter and Schicker 2017).…”