Conditional marginal expected shortfall

Extremes - In the context of bivariate random variables $\left (Y^{(1)},Y^{(2)}\right )$ , the marginal expected shortfall, defined as $\mathbb {E}\left (Y^{(1)}|Y^{(2)} \ge Q_{2}(1-p)\right )$ for...     

Tail Index Estimation and an Exponential Regression Model

One of the most important problems involved in the estimation of Pareto indices is the reduction of bias in case the slowly varying part of the Pareto type model disappears at a very slow rate. In other cases, when the bias problem is not so severe, the application of well-known estimators such as the Hill (1975) and the moment estimator (Dekkers et al. (1989)) still asks for an adaptive selection of the sample fraction to be used in such estimation procedures. We show that in both circumstances, solutions can be constructed for the given problems using maximum likelihood estimators based on a regression model for upper order statistics. Via this technique one can also infer about the bias-variance trade-off for a given data set. The behavior of the new maximum likelihood estimator is illustrated through simulation experiments, among others for ARCH processes.     

Dispersion models for extremes

We propose extreme value analogues of natural exponential families and exponential dispersion models, and introduce the slope function as an analogue of the variance function. A class of extreme generalized linear regression models for analysis of extremes and lifetime data is introduced. The set of quadratic and power slope functions characterize well-known families such as the Rayleigh, Gumbel, power, Pareto, logistic, negative exponential, Weibull and Fréchet. We show a convergence theorem for slope functions, by which we may express the classical extreme value convergence results in terms of asymptotics for extreme dispersion models. The key idea is to explore the parallels between location families and natural exponential families, and between the convolution and minimum operations.