Testing Extreme Value Models

We consider the full statistical families of extreme value distributions and generalized Pareto distributions , where , and denote the shape, scale and location parameters, respectively. We consider the testing problems against and = 0 against 0, where and are treated as nuisance parameters. Showing local asymptotic normality (LAN), we derive asymptotic envelope power functions for test sequences and establish tests which attain these upper bounds. The finite sample size behavior is studied by simulations.     

Local Asymptotic Normality in Extreme Value Index Estimation

This paper deals with the estimation of the extreme value index in local extreme value models. We establish local asymptotic normality (LAN) under certain extreme value alternatives. It turns out that the central sequence occurring in the LAN expansion of the likelihood process is up to a rescaling procedure the Hill estimator. The central sequence plays a crucial role for the construction of asymptotic optimal statistical procedures. In particular, the Hill estimator is asymptotically minimax.     

LAN of extreme order statistics

Consider an iid sampleZ ₁,...,Z _n with common distribution functionF on the real line, whose upper tail belongs to a parametric family {F : }. We establish local asymptotic normality (LAN) of the loglikelihood process pertaining to the vector(Z _n–i+1n ) _i=1 ^k of the upperk=k(n) _n order statistics in the sample, if the family {F :} is in a neighborhood of the family of generalized Pareto distributions. It turns out that, except in one particular location case, thekth-largest order statisticZ _n–k+1n is the central sequence generating LAN. This implies thatZ _n–k+1n is asymptotically sufficient and that asymptotically optimal tests for the underlying parameter can be based on the single order statisticZ _n–k+1n . The rate at whichZ _n–k+1n becomes asymptotically sufficient is however quite poor.     

A Class of Tests on the Tail Index

In the family of distribution functions with nondegenerate right tail, we test the hypothesis with a hypothetical m ₀ > 0 and with some x ₀ 0. The proposed test is fully nonparametric and is based on splitting the set of observations into N subsamples of size n and on the empirical distribution function of the extremes of the subsamples; the asymptotics is for N and fixed n (eventually small), and the asymptotic null distribution of the test criterion is normal. The test is consistent against exponentially tailed alternatives, as well as against heavy tailed alternatives with index m > m ₀, and is asymptotically unbiased for the broad family of distributions represented by and its alternative. It may be used as a supplement to the usual tests of the Gumbel hypothesis m = against m < , namely in the situation that the latter tests reject the hypothesis of exponentiality, and we need to know how heavy-tailed F really can be. This knowledge may be very important in the applications. The performance of the proposed test is illustrated on simulated data; we see that it distinguishes well the tails even for moderate samples. For an illustration, the proposed (nonparametric) test is numerically compared with the (parametric) likelihood ratio test for the class of generalized Pareto distributions. As it can be expected, the parametric test behaves well, provided F is exactly generalized Pareto, while the nonparametric test performs better for all other considered distribution shapes.     

Efficient Estimation of the Canonical Dependence Function

The canonical dependence function (z), z [0,1], is introduced and studied in detail for distributions, which belong to the -neighborhood of a bivariate generalized Pareto distribution. We establish local asymptotic normality (LAN) of the loglikelihood function of a 2×2 table sorting of n i.i.d. observations and derive efficient estimators of (z) from the Hájek-LeCam Convolution Theorem. These results extend results by Falk and Reiss (2003) for the canonical dependence parameter (1/2) to arbitrary z (0,1).