Holger Drees

Selecting the optimal sample fraction in univariate extreme value estimation

In general, estimators of the extreme value index of i.i.d. random variables crucially depend on the sample fraction that is used for estimation. In case of the well-known Hill estimator the optimal number knopt of largest order statistics was given by Hall and Welsh (1985) as a function of some parameters of the unknown distribution function F, which was assumed to admit a certain expansion. Moreover, an estimator of knopt was proposed that is consistent if a second-order parameter [rho] of F belongs to a bounded interval. In contrast, we introduce a sequential procedure that yields a consistent estimator of knopt in the full model without requiring prior information about [rho]. Then it is demonstrated that even in a more general setup the resulting adaptive Hill estimator is asymptotically as efficient as the Hill estimator based on the optimal number of order statistics. Finally, it is shown by Monte Carlo simulations that also for moderate sample sizes the procedure shows a reasonable performance, which can be improved further if [rho] is restricted to bounded intervals.

On Smooth Statistical Tail Functionals

Many estimators of the extreme value index of a distribution function F that are based on a certain number kn of largest order statistics can be represented as a statistical tail functional, that is a functional T applied to the empirical tail quantile function Q.. We study the asymptotic behaviour of such estimators with a scale and location invariant functional T under weak second order conditions on F. For that purpose first a new approximation of the empirical tail quantile function is established. As a consequence we obtain weak consistency and asymptotic normality of T(Q.) if T is continuous and Hadamard differentiable, respectively, at the upper quantile function of a generalized Pareto distribution and kn tends to infinity sufficiently slowly. Then we investigate the asymptotic variance and bias. In particular, those functionals T are characterized that lead to an estimator with minimal asymptotic variance. Finally, we introduce a method to construct estimators of the extreme value index with a made-to-order asymptotic behaviour.

A general class of estimators of the extreme value index

Abstract We consider the class of estimators of the extreme value index β that can be represented as a scale invariant functional T applied to the empirical tail quantile function Qn. From an approximation of Qn first asymptotic normality of T(Qn) is derived under quite natural smoothness conditions on T if β is positive. As a consequence, a widely applicable method for the construction of estimators with a prescribed asymptotic behavior is introduced. If β ≤ 0 then either T must be location invariant or it has to satisfy a certain regularity condition on a neighborhood of a constant function to ensure asymptotic normality. It turns out that in this situation location invariant estimators are clearly preferable.

Tail Empirical Processes Under Mixing Conditions

Limit theorems for tail processes of absolutely regular time series are surveyed. First we discuss Rootzen’s [29] result on the uniform tail empirical process and generalizations thereof. Then similar results on the uniform tail quantile process are derived. If the observations come from a distribution function that belongs to the domain of attraction of an extreme value distribution, then one can prove weighted approximations of the linearly standardized tail quantile function. Moreover, asymptotic normality can be deduced for many estimators of interest in extreme value statistics. Finally, we apply the limit theorems to particular linear and nonlinear time series models.

Refined pickands estimators wtth bias correction

Consider an univariate distribution function F that belongs to the weak domain of attraction of an extreme value distribution. In Drees (1994) mixtures of Pickands estimators of the extreme value index β were considered that are based on the kn largest order statistics. It was shown that under certain second order conditions on F the estimator is asymptotically biased if kn grows too fast. Here we introduce a quite general bias correcting procedure that allows to utilize more largest order statistics. Moreover, it is proven that the estimator yields sensible results even if some of the model assumptions are not satisfied. A simulation study demonstrates the robustness against an unsuitable choice of kn.

On large deviation for extremes

Recently, a weighted approximation for the tail empirical distribution function has been developed (Approximations to the tail empirical distribution function with application to testing extreme value conditions. preprint, submitted for publication). We show that the same result can also be used to improve a known uniform approximation of the distribution of the maximum of a random sample. From this a general result about large deviations of this maximum is derived. In addition, the relationship between two second-order conditions used in extreme value theory is clarified.

A stochastic volatility model with flexible extremal dependence structure

Stochastic volatility processes with heavy-tailed innovations are a well-known model for financial time series. In these models, the extremes of the log returns are mainly driven by the extremes of the i.i.d. innovation sequence which leads to a very strong form of asymptotic independence, that is, the coefficient of tail dependence is equal to

Estimating failure probabilities

1/2

Components of the two-sided Kolmogorov-smirnov test in signal detection problems with Gaussian white noise

for all positive lags. We propose an alternative class of stochastic volatility models with heavy-tailed volatilities and examine their extreme value behavior. In particular, it is shown that, while lagged extreme observations are typically asymptotically independent, their coefficient of tail dependence can take on any value between

Conditions for quantile process approximations

1/2