Extreme quantile inference
Thesis (PhD)--Stellenbosch University, 2020.
ENGLISH SUMMARY : A novel approach to performing extreme quantile inference is proposed by applying ridge regression and the saddlepoint approximation to results in extreme value theory. To this end, ridge regression is applied to the log differences of the largest sample quantiles to obtain a bias-reduced estimator of the extreme value index, which is a parameter in extreme value theory that plays a central role in the estimation of extreme quantiles. The utility of the ridge regression estimators for the extreme value index is illustrated by means of simulations results and applications to daily wind speeds. A new pivotal quantity is then proposed with which a set of novel asymptotic confidence intervals for extreme quantiles are obtained. The ridge regression estimator for the extreme value index is combined with the proposed pivotal quantity together with the saddlepoint approximation to yield a set of confidence intervals that are accurate and narrow. The utility of these confidence intervals are illustrated by means of simulation results and applications to Belgian reinsurance data. Multivariate generalizations of sample quantiles are considered with the aim of developing multivariate risk measures, including maximum correlation risk measures and an estimator for the extreme value index. These multivariate sample quantiles are called center-outward quantiles, and are defined as an optimal transportation of the uniformly distributed points in the unit ball Sd to the observed sample points in Rd. A continuous extension of the centeroutward quantile is proposed, which yields quantile contours that are nested. Furthermore, maximum correlation risk measures for multivariate samples are presented, as well as an estimator for the extreme value index for multivariate regularly varying samples. These results are applied to Danish fire insurance data and the stock returns of Google and Apple share prices to illustrate their utility.
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