Rank-based estimation under asymptotic dependence and independence, with applications to spatial extremes (original) (raw)

Bivariate tail estimation: dependence in asymptotic independence

Bernoulli, 2004

In the classical setting of bivariate extreme value theory, the procedures for estimating the probability of an extreme event are not applicable if the componentwise maxima of the observations are asymptotically independent. To cope with this problem, Ledford and Tawn proposed a submodel in which the penultimate dependence is characterized by an additional parameter. We discuss the asymptotic properties of two estimators for this parameter in an extended model. Moreover, we develop an estimator for the probability of an extreme event that works in the case of asymptotic independence as well as in the case of asymptotic dependence, and prove its consistency.

Dense classes of multivariate extreme value distributions

Journal of Multivariate Analysis, 2013

In this paper, we explore tail dependence modelling in multivariate extreme value distributions. The measure of dependence chosen is the scale function, which allows combinations of distributions in a very flexible way. The correspondences between the scale function and the spectral measure or the stable tail dependence function are given. Combining scale functions by simple operations, three parametric classes of laws are (re)constructed and analyzed, and resulting nested and structured models are discussed. Finally, the denseness of each of these classes is shown.

Multivariate Tail Estimation: Conditioning on an extreme event

2015

We consider regularly varying random vectors. Our goal is to estimate in a non-parametric way some characteristics related to conditioning on an extreme event, like the tail dependence coefficient. We introduce a quasi-spectral decomposition that allow to improve efficiency of estimators. Asymptotic normality of estimators is based on weak convergence of tail empirical processes. Theoretical results are supported by simulation studies.

Non-Parametric Estimation of the Limit Dependence Function of Multivariate Extremes

1999

This paper presents a new estimation procedure for the limit distribution of the maximum of a multivariate random sample. This procedure relies on a new and simple relationship between the copula of the underlying multivariate distribution function and the dependence function of its maximum attractor. The obtained characterization is then used to de®ne a class of kernel-based estimates for the dependence function of the maximum attractor. The consistency and the asymptotic distribution of these estimates are considered.

An M-estimator for tail dependence in arbitrary dimensions

The Annals of Statistics, 2012

Consider a random sample in the max-domain of attraction of a multivariate extreme value distribution such that the dependence structure of the attractor belongs to a parametric model. A new estimator for the unknown parameter is defined as the value that minimises the distance between a vector of weighted integrals of the tail dependence function and their empirical counterparts. The minimisation problem has, with probability tending to one, a unique, global solution. The estimator is consistent and asymptotically normal. The spectral measures of the tail dependence models to which the method applies can be discrete or continuous. Examples demonstrate the applicability and the performance of the method.

On tail index estimation based on multivariate data

Journal of Nonparametric Statistics, 2015

This article is devoted to the study of tail index estimation based on i.i.d. multivariate observations, drawn from a standard heavy-tailed distribution, i.e. of which Pareto-like marginals share the same tail index. A multivariate Central Limit Theorem for a random vector, whose components correspond to (possibly dependent) Hill estimators of the common tail index α, is established under mild conditions. Motivated by the statistical analysis of extremal spatial data in particular, we introduce the concept of (standard) heavy-tailed random field of tail index α and show how this limit result can be used in order to build an estimator of α with small asymptotic mean squared error, through a proper convex linear combination of the coordinates. Beyond asymptotic results, simulation experiments illustrating the relevance of the approach promoted are also presented.

A new class of models for bivariate joint tails

Journal of The Royal Statistical Society Series B-statistical Methodology, 2009

Summary. A fundamental issue in applied multivariate extreme value analysis is modelling dependence within joint tail regions. The primary focus of this work is to extend the classical pseudopolar treatment of multivariate extremes to develop an asymptotically motivated representation of extremal dependence that also encompasses asymptotic independence. Starting with the usual mild bivariate regular variation assumptions that underpin the coefficient of tail dependence as a measure of extremal dependence, our main result is a characterization of the limiting structure of the joint survivor function in terms of an essentially arbitrary non-negative measure that must satisfy some mild constraints. We then construct parametric models from this new class and study in detail one example that accommodates asymptotic dependence, asymptotic independence and asymmetry within a straightforward parsimonious parameterization. We provide a fast simulation algorithm for this example and detail likelihood-based inference including tests for asymptotic dependence and symmetry which are useful for submodel selection. We illustrate this model by application to both simulated and real data. In contrast with the classical multivariate extreme value approach, which concentrates on the limiting distribution of normalized componentwise maxima, our framework focuses directly on the structure of the limiting joint survivor function and provides significant extensions of both the theoretical and the practical tools that are available for joint tail modelling.

Estimation of the tail-index in a conditional location-scale family of heavy-tailed distributions

Dependence Modeling, 2019

We introduce a location-scale model for conditional heavy-tailed distributions when the covariate is deterministic. First, nonparametric estimators of the location and scale functions are introduced. Second, an estimator of the conditional extreme-value index is derived. The asymptotic properties of the estimators are established under mild assumptions and their finite sample properties are illustrated both on simulated and real data.

Nonparametric Estimation of the Dependence Function in Bivariate Extreme Value Distributions

Journal of Multivariate Analysis, 2001

The paper considers the problem of estimating the dependence function of a bivariate extreme survival function with standard exponential marginals. Nonparametric estimators for the dependence function are proposed and their strong uniform convergence under suitable conditions is demonstrated. Comparisons of the proposed estimators with other estimators are made in terms of bias and mean squared error. Several real data sets from various applications are used to illustrate the procedures.