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We extend classical extreme value theory to non-identically distributed observations. If the tails of the distribution are proportional, then the extreme value index remains unchanged and 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 further investigate heteroscedasticiy in the extreme value indices. The varying extreme value indices can be estimated using local observations. The asymptotic normality for both local and global estimators are shown.
The main tool in this work is the extension of the tail empirical process. Firstly, we derive the weak convergence of a weighted sequential tail empirical process. Secondly, we derive asymptotic properties of the moments of the error terms in tail empirical processes.
We establish tests for constant extreme value index across non-stationary observations. We show through simulations the good performance of the procedures and also present an application to stock market returns.
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