Asymptotic independence for unimodal densities

Abstract

Asymptotic independence of the components of random vectors is a concept used in many applications. The standard criteria for checking asymptotic independence are given in terms of distribution functions (dfs). Dfs are rarely available in an explicit form, especially in the multivariate case. Often we are given the form of the density or, via the shape of the data clouds, one can obtain a good geometric image of the asymptotic shape of the level sets of the density. This paper establishes a simple sufficient condition for asymptotic independence for light-tailed densities in terms of this asymptotic shape. This condition extends Sibuya's classic result on asymptotic independence for Gaussian densities. MSC: 60G55, 60G70, 62E20The purpose of the present paper is to provide simple sufficient conditions that ensure asymptotic independence of the components of random vectors whose probability distribution is described by a density.Standard criteria for checking asymptotic independence are given in terms of distribution functions (dfs). 1 1 INTRODUCTION 2 However, these are not always available in an explicit form in the multivariate case, and they give little insight in what large samples from a distribution will look like. Often we are given a density in analytic form. For light-tailed densities, the data clouds give a good geometric image of the asymptotic shape of the level sets. Hence it is of interest to have conditions for asymptotic independence in terms of the shape of the level sets of the underlying density, or in terms of a limiting shape for data clouds.For vector valued data it is standard practice to plot the bivariate sample clouds for all component pairs. In our final result it is the asymptotic behaviour of the shape of these bivariate sample clouds, as the size of the data set increases, that determines asymptotic independence of the coordinates for the underlying multivariate distribution.Unimodal densities whose level sets all have the same shape are called homothetic. The decay along any ray then is the same up to a scale constant depending on the direction, and hence the concept of light and heavy tails is well-defined. This remains true if we only assume that the level sets have the same shape asymptotically. Our primary focus here is on light-tailed densities, but for additional insight we also include some results for the heavy-tailed case. Throughout the paper, we assume continuity of dfs and of densities.Intuitively, for bivariate data, asymptotic independence means that large values in one coordinate are unlikely to be accompanied by large values in the other coordinate. Situations with a low chance of simultaneous extremes are often encountered in practice, for example in applications which involve modeling environmental data (e.g. [17]) or network traffic data (e.g. [19]); see [25] for further references.It is a well-known result, dating back to 1959 (see [26]), that the components of a vector with a Gaussian density are asymptotically independent whatever the correlation. Asymptotic ind...

Extremal dependence of random scale constructions

“…Although, differently to Hashorva (2012), we typically assume symmetry, there are nonetheless some connections between the results in our Section 3.5 and that paper. Nolde (2014) provides an interpretation of extremal dependence in terms of a gauge function (see also Balkema and Nolde (2010) and Balkema and Embrechts (2007)), which, loosely speaking, corresponds to level sets of the density in light-tailed margins. The main result of Nolde (2014) (Theorem 2.1) is presented in terms of Weibull-type margins, such that − log F X ∈ RV ∞ δ , δ > 0; in terms of Section 2, this corresponds to − log F R ∈ RV ∞ δ .…”

Extremal attractors of Liouville copulas

“…One option would be to consider triangular arrays as in [23]; extremes of arrays of Liouville vectors can be obtained as a special case of extremes of arrays of weighted Dirichlet distributions developed in [18]. Another avenue worth exploring might be the limits of scaled sample clouds, as in [2] and [32].…”

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