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# Doctoral Degrees (Statistics and Actuarial Science)

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### Browsing Doctoral Degrees (Statistics and Actuarial Science) by Subject "Extreme value theory"

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- ItemExtreme quantile inference(Stellenbosch : Stellenbosch University, 2020-03) Buitendag, Sven; De Wet, Tertius; Beirlant, Jan; Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
Show more 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.Show more - ItemStatistical inference for inequality measures based on semi-parametric estimators(Stellenbosch : Stellenbosch University, 2011-12) Kpanzou, Tchilabalo Abozou; De Wet, Tertius; Neethling, Ariane; Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.
Show more ENGLISH ABSTRACT: Measures of inequality, also used as measures of concentration or diversity, are very popular in economics and especially in measuring the inequality in income or wealth within a population and between populations. However, they have applications in many other fields, e.g. in ecology, linguistics, sociology, demography, epidemiology and information science. A large number of measures have been proposed to measure inequality. Examples include the Gini index, the generalized entropy, the Atkinson and the quintile share ratio measures. Inequality measures are inherently dependent on the tails of the population (underlying distribution) and therefore their estimators are typically sensitive to data from these tails (nonrobust). For example, income distributions often exhibit a long tail to the right, leading to the frequent occurrence of large values in samples. Since the usual estimators are based on the empirical distribution function, they are usually nonrobust to such large values. Furthermore, heavy-tailed distributions often occur in real life data sets, remedial action therefore needs to be taken in such cases. The remedial action can be either a trimming of the extreme data or a modification of the (traditional) estimator to make it more robust to extreme observations. In this thesis we follow the second option, modifying the traditional empirical distribution function as estimator to make it more robust. Using results from extreme value theory, we develop more reliable distribution estimators in a semi-parametric setting. These new estimators of the distribution then form the basis for more robust estimators of the measures of inequality. These estimators are developed for the four most popular classes of measures, viz. Gini, generalized entropy, Atkinson and quintile share ratio. Properties of such estimators are studied especially via simulation. Using limiting distribution theory and the bootstrap methodology, approximate confidence intervals were derived. Through the various simulation studies, the proposed estimators are compared to the standard ones in terms of mean squared error, relative impact of contamination, confidence interval length and coverage probability. In these studies the semi-parametric methods show a clear improvement over the standard ones. The theoretical properties of the quintile share ratio have not been studied much. Consequently, we also derive its influence function as well as the limiting normal distribution of its nonparametric estimator. These results have not previously been published. In order to illustrate the methods developed, we apply them to a number of real life data sets. Using such data sets, we show how the methods can be used in practice for inference. In order to choose between the candidate parametric distributions, use is made of a measure of sample representativeness from the literature. These illustrations show that the proposed methods can be used to reach satisfactory conclusions in real life problems.Show more