We consider daily rainfall observations at 32 stations in the province of North Holland (the Netherlands) during 30 years. Let T be the total rainfall in this area on one day. An important question is: what is the amount of rainfall T that is exceeded once in 100 years? This is clearly a problem belonging to extreme value theory. Also, it is a genuinely spatial problem.Recently, a theory of extremes of continuous stochastic processes has been developed. Using the ideas of that theory and much computer power (simulations), we have been able to come up with a reasonable answer to the question above.

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.