Gumbel Max-domain Of Attraction Research Articles

On Accompanying Measures and Asymptotic Expansions in the B. V. Gnedenko Limit Theorem

We propose a sequence of accompanying laws in the B.,V. Gnedenko limit theorem for maxima of independent random variables with distributions lying in the Gumbel max domain of attraction. We show that this sequence provides a power-law convergence rate, whereas the Gumbel distribution provides only the logarithmic rate. As examples, we consider in detail the classes of Weibull and log-Weibull type distributions. For the entire Gumbel max domain of attraction, we propose a scale of classes of distributions that includes these two classes as a starting point.

Extremes of randomly scaled Gumbel risks

We investigate the product Y1Y2 of two independent positive risks Y1 and Y2. If Y1 has distribution in the Gumbel max-domain of attraction with some auxiliary function which is regularly varying at infinity and Y2 is bounded, then we show that Y1Y2 has also distribution in the Gumbel max-domain of attraction. If both Y1,Y2 have log-Weibullian or Weibullian tail behaviour, we prove that Y1Y2 has log-Weibullian or Weibullian asymptotic tail behaviour, respectively. We present here three theoretical applications concerned with a) the limit of point-wise maxima of randomly scaled Gaussian processes, b) extremes of Gaussian processes over random intervals, and c) the tail of supremum of iterated processes.

Estimation of the finite right endpoint in the Gumbel domain

A simple estimator for the finite right endpoint of a distribution function in the Gumbel max-domain of attraction is proposed. Large sample properties such as consistency and the asymptotic distribution are derived. A simulation study is also presented.

On conditional extreme values of random vectors with polar representation

An effective approach for studying the asymptotics of bivariate random vectors is to search for the limits of conditional probabilities where the conditioning variable becomes large. In this context, elliptical and related distributions have been extensively investigated. A quite general model was presented by Fougeres and Soulier (Limit conditional distributions for bivariate vectors with polar representation in Stochastic Models, 2010), who derived a conditional limit theorem for random vectors (X, Y) with a polar representation R · (u(T), v(T)), where R, T are stochastically independent and R is in the Gumbel max-domain of attraction. We reformulate their assumptions, such that they have a simpler structure, display more clearly the geometry of the curves (u(t), v(t)) and allow us to deduce interesting generalizations into two directions: u has several global maxima instead of only one, the curve (u(t), v(t)) is no longer differentiable, but forms a “cusp”. The latter generalization yields results where only random norming leads to a non-degenerate limit statement. Ideas and results are elucidated by several figures.

Efficient simulation of tail probabilities for sums of log-elliptical risks

In the framework of dependent risks it is a crucial task for risk management purposes to quantify the probability that the aggregated risk exceeds some large value u. Motivated by Asmussen et al. (2011) [1] in this paper we introduce a modified Asmussen–Kroese estimator for simulation of the rare event that the aggregated risk exceeds u. We show that in the framework of log-Gaussian risks our novel estimator has the best possible performance i.e., it has asymptotically vanishing relative error. For the more general class of log-elliptical risks with marginal distributions in the Gumbel max-domain of attraction we propose a modified Rojas-Nandayapa estimator of the rare events of interest, which for specific importance sampling densities has a good logarithmic performance. Our numerical results presented in this paper demonstrate the excellent performance of our novel Asmussen–Kroese algorithm.

A moment estimator for the conditional extreme-value index

In extreme value theory, the so-called extreme-value index is a parameter that controls the behavior of a distribution function in its right tail. Knowing this parameter is thus essential to solve many problems related to extreme events. In this paper, the estimation of the extreme-value index is considered in the presence of a random covariate, whether the conditional distribution of the variable of interest belongs to the Fréchet, Weibull or Gumbel max-domain of attraction. The pointwise weak consistency and asymptotic normality of the proposed estimator are established. We examine the finite sample performance of our estimator in a simulation study and we illustrate its behavior on a real set of fire insurance data.

Discretization of distributions in the maximum domain of attraction

In this paper we discuss the discretization of distributions belonging to some max-domain of attraction. Given a random variable X its discretization is defined as the minimal integer not less than X. Our first interest is on distributions that preserve the max-domain property after discretization. Secondly, we characterize the distributions which are regarded as the discretization of the distribution in the Gumbel max-domain of attraction. Lastly the correspondence of distribution in Gumbel max-domain of attraction is investigated.

Exact tail asymptotics in bivariate scale mixture models

Let (X, Y) = (RU 1, RU 2) be a given bivariate scale mixture random vector, with R > 0 independent of the bivariate random vector (U 1, U 2). In this paper we derive exact asymptotic expansions of the joint survivor probability of (X, Y) assuming that R has distribution function in the Gumbel max-domain of attraction, and (U 1, U 2) has a specific local asymptotic behaviour around some absorbing point. We apply our results to investigate the asymptotic behaviour of joint conditional excess distribution and the asymptotic independence for two models of bivariate scale mixture distributions.

Asymptotics of the convex hull of spherically symmetric samples

In this paper we consider the convex hull of a spherically symmetric sample in Rd. Our main contributions are some new asymptotic results for the expectation of the number of vertices, number of facets, area and the volume of the convex hull assuming that the marginal distributions are in the Gumbel max-domain of attraction. Further, we briefly discuss two other models assuming that the marginal distributions are regularly varying or O-regularly varying.

On the residual dependence index of elliptical distributions

The residual dependence index of bivariate Gaussian distributions is determined by the correlation coefficient. This tail index is of certain statistical importance when extremes and related rare events of bivariate samples with asymptotic independent components are being modeled. In this paper we calculate the partial residual dependence indices of a multivariate elliptical random vector assuming that the associated random radius has distribution function in the Gumbel max-domain of attraction. Furthermore, we discuss the estimation of these indices when the associated random radius possesses a Weibull-tail distribution.

On the strong Kotz approximation of Dirichlet random vectors

Let (X 1, X 2) be a bivariate L p -norm generalized symmetrized Dirichlet (LpGSD) random vector with parameters α1,α2. If p=α1=α2=2, then (X 1, X 2) is a spherical random vector. The estimation of the conditional distribution of Z u *:=X 2 | X 1>u for u large is of some interest in statistical applications. When (X 1, X 2) is a spherical random vector with associated random radius in the Gumbel max-domain of attraction, the distribution of Z u * can be approximated by a Gaussian distribution. Surprisingly, the same Gaussian approximation holds also for Z u :=X 2| X 1=u. In this paper, we are interested in conditional limit results in terms of convergence of the density functions considering a d-dimensional LpGSD random vector. Stating our results for the bivariate setup, we show that the density function of Z u * and Z u can be approximated by the density function of a Kotz type I LpGSD distribution, provided that the associated random radius has distribution function in the Gumbel max-domain of attraction. Further, we present two applications concerning the asymptotic behaviour of concomitants of order statistics of bivariate Dirichlet samples and the estimation of the conditional quantile function.

Limit laws for power sums and norms of i.i.d. samples

We study the limit behavior of power sums and norms of i.i.d. positive samples from the max domain of attraction of an extreme value distribution. To this end, we combine limit theorems for sums and for maxima and use a link between extreme value theory and the Lévy measures of certain infinitely divisible laws, which are limit distributions of power sums. In connection with the von Mises representation of the Gumbel max domain of attraction, this new approach allows us to extend the limit results for power sums found in Ben Arous et al. (Probab Theory Relat Fields 132:579–612, 2005) and Bogachev (J Theor Probab 19:849–873, 2006). Furthermore, our findings shed a new light on the results of Schlather (Ann Probab 29:862–881, 2001) and treat the Gumbel case which is missing there.

Tail asymptotic results for elliptical distributions

Let ( X , Y ) be a bivariate elliptical random vector with associated random radius in the Gumbel max-domain of attraction. In this paper we obtain a second order asymptotic expansion of the joint survival probability P { X > x , Y > y } and the conditional probability P { Y > y | X > x } , for x , y large.

Asymptotic properties of type I elliptical random vectors

Let $ {\boldsymbol X} = A{\boldsymbol S} $ be an elliptical random vector with $ A \in \mathbb{R}^{{k \times k}} ,k \geqslant 2, $ a non-singular square matrix and ${\boldsymbol S}=(S_1, \ldots ,S_k)^\top $ a spherical random vector in $\mathbb{R}^k$ , and let ${\boldsymbol t}_n, n \ge 1$ be a sequence of vectors in $\mathbb{R}^k$ such that $ \lim _{{n \to \infty }} {\boldsymbol P}\{ {\boldsymbol X} > {\boldsymbol t}_{n} \} = 0 $ . We assume in this paper that the associated random radius R k =(S 1 + S 2 +...+S k )1/2 is almost surely positive, and it has distribution function in the Gumbel max-domain of attraction. Relying on extreme value theory we obtain an exact asymptotic expansion of the tail probability ${\boldsymbol P} \{{\boldsymbol X} > {\boldsymbol t}_n\}$ for ${\boldsymbol t}_n$ converging as $ n \to \infty $ to a boundary point. Further we discuss density convergence under a suitable transformation. We apply our results to obtain an asymptotic approximation of the distribution of partial excess above a high threshold, and to derive a conditional limiting result. Further, we investigate the asymptotic behaviour of concomitants of order statistics, and the tail asymptotics of associated random radius for subvectors of ${\boldsymbol X}$ .

Gaussian Approximation of Conditional Elliptical Random Vectors

Let U d = (U 1,…, U d )⊤, d ≥ 2 be a random vector uniformly distributed on the unit sphere of ℝ d , and let A ∈ ℝ d×d be a non-singular matrix. Consider an elliptical random vector X = (X 1,…, X d )⊤ with stochastic representation R A ⊤ U d where the positive random radius R is independent of U d , and let X I = (X i , i ∈ I)⊤, X J = (X i , i ∈ J)⊤ be two vectors with non-empty disjoint index sets I, J, I ∪ J = {1,…, d}. Motivated by the Gaussian approximation of the conditional distribution of bivariate spherical random vectors obtained in Berman[1] we derive in this paper a Gaussian approximation for the conditional distribution X I | X J = u J , u ∈ ℝ d as u J tends to a boundary point provided that the random radius R has distribution function in the Gumbel max-domain of attraction. Further, we generalise Berman's result to the multivariate elliptical setup.

On the multivariate Hüsler–Reiss distribution attracting the maxima of elliptical triangular arrays

Let ( X 1 n ( j ) , … , X dn ( j ) ) , n ⩾ 1 , 1 ⩽ j ⩽ n , be a triangular array of independent elliptical random vectors in R d , d ⩾ 2 . In this paper we investigate the asymptotic behaviour of the multivariate maxima of this triangular array. Generalising previous results for the bivariate set-up, we show that the normalised maxima of this elliptical triangular array is attracted by the multivariate Hüsler–Reiss distribution function provided that the components of the triangular array become asymptotically dependent with a specific rate, and further the random radius pertaining to the elliptical random vectors is in the Gumbel max-domain of attraction.