On univariate statistical inference of multidimensional extremes

Extreme value theory is concerned with statistical analysis of rare, and often catastrophic, events. Thus, extreme value theory provides risk analysis tools for a wide range of fields such as finance, insurance, telecommunications and climate modeling. A classical approach to multivariate and infinite-dimensional extreme value theory is to generalize the maximum domain of attraction condition by considering the componentwise sample maximum. This thesis considers a different approach. We reduce the multivariate or infinite-dimensional case to a univariate one by mapping the original observations to the positive real line with a suitable functional. The desired functional depends on the context, as the map should measure the extremity of an observation. Our approach to extremes is complicated by the fact that, often in practical settings, only an approximation of the chosen functional is available. Consequently, the contributions of the thesis are two-fold. Firstly, we derive various sufficient conditions for the approximation error such that the standard asymptotic results hold for the selected extreme value index and extreme quantile estimators computed with approximations instead of the true observations. Secondly, the univariate extreme value theory framework under approximations is applied in chosen multivariate and infinitedimensional settings. As a multivariate application, we consider extreme quantile region estimation for elliptical distributions. Under ellipticity, it turns out that the proper functional measuring extremity is the Mahalanobis distance. In asymptotics, the effect of estimating the unknown location–scatter pair must be taken into account. We give two different extreme quantile region estimators for elliptical distributions—the first estimator is adapted for the heavy-tailed case, and the second one is suitable in a more general framework. As an infinite-dimensional application, we consider extreme quantile estimation for an Lp-norm of a random function. The Lp-norms corresponding to the realized sample paths must be approximated. As the approximation methods, we consider Riemann sums and Monte Carlo integration. For both methods, we give sufficient conditions such that the standard asymptotic results hold for the Hill estimator and the corresponding extreme quantile estimator computed with the approximated norms. As an interesting excursion, we give a new Chernoff-type bound for the intermediate order statistics. The bound for the intermediate order statistics is used in the derivation of concentration inequalities for the deviation between the Hill estimators computed with the approximations and the true Lp-norms, respectively.