Generalized order statistics : stability and extreme value index estimation

• Verallgemeinerte Ordnungsstatistiken : Stabilität und Extremwertindex-Schätzung

Schmidt, Jacob Peter; Kamps, Udo (Thesis advisor); Cramer, Erhard (Thesis advisor)

Aachen (2017)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2017

Abstract

In this thesis, we study various probabilistic and statistical aspects of extreme value theory for generalized order statistics and Pfeifer's record values. After a short summary of classical univariate extreme value theory and extreme value theory for generalized order statistics in Chapter 2, we deal with the limit behaviour of large generalized order statistics and Pfeifer's record values in the absence of any standardization in Chapter 3. We illustrate that there are parameter cases, where extreme and intermediate generalized order statistics converge in probability to the right endpoint of support of the baseline distribution function and that there are parameter cases, where other limit points occur. Besides, we take a second look at the limit behaviour of Pfeifer's record values. Moreover, we show that for certain small generalized order statistics the left endpoint of support of the baseline distribution function occurs as limit point. In Chapter 4, we study relative and absolute stability of generalized order statistics and Pfeifer's record values. Based on the results of Chapter 3, we derive sufficient conditions for relative and absolute stability of extreme and intermediate generalized order statistics in terms of regular variation. Furthermore, we establish necessary and sufficient conditions for stability in two subclasses of generalized order statistics. Those are the model of m-generalized order statistics and a subclass which, in the distributional sense, contains common record values. In the second part of this chapter, we consider almost sure relative stability of Pfeifer's record values. The extreme value distribution of m-generalized order statistics again depends on the classical extreme value index. Because of this, one is interested in estimating the extreme value index with m-generalized order statistics. Up to now, there is only one such estimator, a generalization of the classical Hill estimator, which is restricted to positive values of the extreme value index. In Chapter 5, we study a generalization of the classical Pickands estimator, an estimator which is applicable for all values of the extreme value index. Following the derivations for the classical Pickands estimator, we establish a consistency and asymptotic normality result for this generalized extreme value index estimator. To illustrate the influence of the model parameters on those results, we provide a simulation study. In addition, we show how the findings of this section can be rephrased to derive a semi-parametric estimator for the model parameter m. It is shown in Chapter 3 that extreme as well as intermediate m-generalized order statistics are consistent estimators for the right endpoint of support of the baseline distribution function. In the second part of Chapter 5, we apply the results of the first part to obtain an alternative consistent estimator based on m-generalized order statistics. In particular, we prove an asymptotic normality result, which enables the construction of asymptotic confidence intervals for the right endpoint of support. In Chapter 6, the asymptotic results established in Chapter 4 and Chapter 5 for the model of m-generalized order statistics are considered in a natural extension of this model.