Non-Parametric Estimation of the Limit Dependence Function of Multivariate Extremes (original) (raw)

Nonparametric estimation of multivariate extreme-value copulas

Journal of Statistical Planning and Inference, 2012

Extreme-value copulas arise in the asymptotic theory for componentwise maxima of independent random samples. An extreme-value copula is determined by its Pickands dependence function, which is a function on the unit simplex subject to certain shape constraints that arise from an integral transform of an underlying measure called spectral measure. Multivariate extensions are provided of certain rank-based nonparametric estimators of the Pickands dependence function. The shape constraint that the estimator should itself be a Pickands dependence function is enforced by replacing an initial estimator by its best least-squares approximation in the set of Pickands dependence functions having a discrete spectral measure supported on a sufficiently fine grid. Weak convergence of the standardized estimators is demonstrated and the finite-sample performance of the estimators is investigated by means of a simulation experiment.

Dependence of Multivariate Extremes

Advances in Regression, Survival Analysis, Extreme Values, Markov Processes and Other Statistical Applications, 2013

We give necessary and sufficient conditions for two sub-vectors of a random vector with a multivariate extreme value distribution, corresponding to the limit distribution of the maximum of a multidimensional stationary sequence with extremal index, to be independent or totally dependent. Those conditions involve first relations between the multivariate extremal indexes of the sequences and secondly a coefficient that measure the strength of dependence between both sub-vectors. The main results are illustrated with an auto-regressive sequence and a 3-dependent sequence.

Non-parametric estimators of multivariate extreme dependence functions

Journal of Nonparametric Statistics, 2005

This article reviews various characterizations of a multivariate extreme dependence function A(·). The most important estimators derived from these characterizations are also sketched. Then, a unifying approach, which puts all these estimators under the same framework, is presented. This unifying approach enables us to construct new estimators and, most importantly, to propose an automatic selection method for an unknown parameter on which all the existing non-parametric estimators of A(·) depend. Constrained smoothing splines and convex hull techniques are used to force the obtained estimators to be extreme dependence functions. A simulation study comparing these estimators on a wide range of extreme dependence functions is provided.

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.

Dependence in multivariate extreme values

Institute of Mathematical Statistics Lecture Notes - Monograph Series, 1990

We review the limiting behavior of extreme values of sequences of random vectors in R d by considering mainly the dependence properties of its nondegenerate limit laws. We treat separately the i.i.d. case, the stationary case, the independent non-identically distributed case, and the general nonstationary case. As dependence concepts we discuss total dependence, association, positive lower orthant dependence, and independence.

Large-sample tests of extreme-value dependence for multivariate copulas

Canadian Journal of Statistics, 2011

Starting from the characterization of extreme-value copulas based on maxstability, large-sample tests of extreme-value dependence for multivariate copulas are studied. The two key ingredients of the proposed tests are the empirical copula of the data and a multiplier technique for obtaining approximate p-values for the derived statistics. The asymptotic validity of the multiplier approach is established, and the finite-sample performance of a large number of candidate test statistics is studied through extensive Monte Carlo experiments for data sets of dimension two to five. In the bivariate case, the rejection rates of the best versions of the tests are compared with those of the test of recently revisited by Ben . The proposed procedures are illustrated on bivariate financial data and trivariate geological data.

Estimating strategies for multiparameter Multivariate Extreme Value copulas

Hydrology and Earth System Sciences, 2011

Multivariate Extreme Value models are a fundamental tool in order to assess potentially dangerous events. Exploiting recent theoretical developments in the theory of Copulas, new multiparameter models can be easily constructed. In this paper we suggest several strategies in order to estimate the parameters of the selected copula, according 5 to different criteria: these may use either a nearest neighbor or a nearest cluster approach, or exploit all the pair-wise relationships between the available gauge stations. An application to flood data is also illustrated and discussed.

Estimation of extreme quantiles for functions of dependent random variables

Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2014

We propose a new method for estimating the extreme quantiles for a function of several dependent random variables. In contrast to the conventional approach based on extreme value theory, we do not impose the condition that the tail of the underlying distribution admits an approximate parametric form, and, furthermore, our estimation makes use of the full observed data. The proposed method is semiparametric as no parametric forms are assumed on all the marginal distributions. But we select appropriate bivariate copulas to model the joint dependence structure by taking the advantage of the recent development in constructing large dimensional vine copulas. Consequently a sample quantile resulted from a large bootstrap sample drawn from the fitted joint distribution is taken as the estimator for the extreme quantile. This estimator is proved to be consistent. The reliable and robust performance of the proposed method is further illustrated by simulation.