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, we can obtain a good geometric image of the asymptotic shape of the level sets of the density. In this paper we establish 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.

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There exist several estimators of the regression line in the simple linear regression: Least Squares, Least Absolute Deviation, Right Median, Theil–Sen, Weighted Balance, and Least Trimmed Squares. Their performance for heavy tails is compared below on the basis of a quadratic loss function. The case where the explanatory variable is the inverse of a standard uniform variable and where the error has a Cauchy distribution plays a central role, but heavier and lighter tails are also considered. Tables list the empirical sd and bias for ten batches of one hundred thousand simulations when the explanatory variable has a Pareto distribution and the error has a symmetric Student distribution or a one-sided Pareto distribution for various tail indices. The results in the tables may be used as benchmarks. The sample size is n = 100 but results for n = ∞ are also presented. The error in the estimate of the slope need not be asymptotically normal. For symmetric errors, the symmetric generalized beta prime densities often give a good fit.

Dependence between coordinate extremes is a key factor in any multivariate risk assessment. Hence, it is of interest to know whether the components of a given multivariate random vector exhibit asymptotic independence or asymptotic dependence. In the latter case the structure of the asymptotic dependence has to be clarified. In the multivariate setting it is common to have an explicit form of the density rather than the distribution function. In this paper we therefore give criteria for asymptotic dependence in terms of the density. We consider distributions with light tails and restrict attention to continuous unimodal densities defined on the whole space or on an open convex cone. For simplicity, the density is assumed to be homothetic: all level sets have the same shape. Balkema and Nolde (2010) contains conditions on the shape which guarantee asymptotic independence. The situation for asymptotic dependence, treated in the present paper, is more delicate.

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...