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We investigate the estimation of the extreme value index when the data are subject to random censorship. We prove, in a unified way, detailed asymptotic normality results for various estimators of the extreme value index and use these estimators as the main building block for estimators of extreme quantiles. We illustrate the quality of these methods by a small simulation study and apply the estimators to medical data.

Summary We extend classical extreme value theory to non‐identically distributed observations. When the tails of the distribution are proportional much of extreme value statistics remains valid. The proportionality function for the tails can be estimated non‐parametrically along with the (common) extreme value index. For a positive extreme value index, joint asymptotic normality of both estimators is shown; they are asymptotically independent. We also establish asymptotic normality of a forecasted high quantile and develop tests for the proportionality function and for the validity of the model. We show through simulations the good performance of the procedures and also present an application to stock market returns. A main tool is the weak convergence of a weighted sequential tail empirical process.

Abstract. Denote the loss return on the equity of a financial institution as X and that of the entire market as Y . For a given very small value of p > 0, the marginal expected shortfall (MES)The MES is an important factor when measuring the systemic risk of financial institutions.For a wide nonparametric class of bivariate distributions, we construct an estimator of the MES and establish the asymptotic normality of the estimator when p ↓ 0, as the sample size n → ∞.Since we are in particular interested in the case p = O(1/n), we use extreme value techniques for deriving the estimator and its asymptotic behavior. The finite sample performance of the estimator and the adequacy of the limit theorem are shown in a detailed simulation study. We also apply our method to estimate the MES of three large U.S. investment banks. Running title. Marginal expected shortfall.JEL codes. C13 C14.

Consider a random sample from a bivariate distribution function F in the max-domain of attraction of an extreme-value distribution function G. This G is characterized by two extreme-value indices and a spectral measure, the latter determining the tail dependence structure of F . A major issue in multivariate extreme-value theory is the estimation of the spectral measure Φp with respect to the Lp norm. For every p ∈ [1, ∞], a nonparametric maximum empirical likelihood estimator is proposed for Φp. The main novelty is that these estimators are guaranteed to satisfy the moment constraints by which spectral measures are characterized. Asymptotic normality of the estimators is proved under conditions that allow for tail independence. Moreover, the conditions are easily verifiable as we demonstrate through a number of theoretical examples. A simulation study shows a substantially improved performance of the new estimators. Two case studies illustrate how to implement the methods in practice.