Statistics of Heteroscedastic Extremes

Abstract: We extend classical extreme value theory to non-identically distributed observations. When the distribution tails are proportional much of extreme value statistics remains valid. The proportionality function for the tails can be estimated nonparametrically along with the (common) extreme value index. Joint asymptotic normality of both estimators is shown; they are asymptotically independent. We develop tests for the proportionality function and for the validity of the model. We show through simulations the good performance of tests for tail homoscedasticity. The results are applied to stock market returns. A main tool is the weak convergence of a weighted sequential tail empirical process.