Bivariate extreme value theory: Models and estimation. (English) Zbl 0653.62045

Tawn, Jonathan A.

Biometrika 75, No. 3, 397-415 (1988).

A bivariate extreme value of a random sample of a bivariate distribution is defined as the vector of the two extreme values (maxima w.l.o.g.) of the sample components. Models for bivariate extreme value distributions are developed which may easily be extended to the multivariate case. Such distributions have to be understood as the limiting distributions of renormalized componentwise maxima. This joint distribution of the bivariate extreme values is constructed by the two univariate marginal distributions and a so-called dependence function. A feature of bivariate extreme value theory is the nonexistence of a finite-dimensional parametric family for the dependence function.
Two new proposals for modeling the dependence function are given. Nonparametric and parametric estimators of the parameters of the models of the dependence function are discussed. Estimators are developed for the case when the parameters of the marginal distributions are known. Later on the problem of joint estimation of the parameters of the marginal distribution and of the model of the dependence function is handled.
In a theorem for generalized extreme value distributions as marginals the asymptotic existence of maximum likelihood estimators is shown. Tests of independence as well as methods for discrimination between models are also presented. The estimation methods and the flexibility of the new models for the dependence function are demonstrated by some sea level data.

Reviewer: D.Rasch

MSC:

62H12	Estimation in multivariate analysis
62F10	Point estimation
62G05	Nonparametric estimation
62N05	Reliability and life testing

Keywords:

bivariate exponential distribution; nonregular estimation; stable distribution; survival data; model discrimination; bivariate extreme value distributions; limiting distributions of renormalized componentwise maxima; marginal distributions; dependence function; asymptotic existence of maximum likelihood estimators; Tests of independence