Extreme quantile inference

dc.contributor.advisorDe Wet, Tertiusen_ZA
dc.contributor.advisorBeirlant, Janen_ZA
dc.contributor.authorBuitendag, Svenen_ZA
dc.contributor.otherStellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science.en_ZA
dc.descriptionThesis (PhD)--Stellenbosch University, 2020.en_ZA
dc.description.abstractENGLISH 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.en_ZA
dc.description.abstractAFRIKAANSE OPSOMMING : ‘n Nuwe benadering om ekstreem kwantiel inferensie uit te voer, word voorgestel deur rif regressie en die saalpunt benadering toe te pas in ekstreemwaarde teorie. Om dit te bewerkstellig, word rif regressie toegepas op die log verskille van die grootste steelproef kwantiele om ‘n sydigheid-verminderde beramer vir die ekstreemwaarde indeks te verkry. Hierdie parameter in ekstreemwaarde teorie speel ‘n sentrale rol in die beraming van ekstreme kwantiele. Die nut van die rif regressie beramers vir die ekstreemwaarde indeks word geïllustreer deur middel van simulasie resultate en toepassings op daaglikse windspoed data. ‘n Spilpunt grootheid word voorgestel waarmee nuwe asimptotiese vertrouensintervalle vir ekstreme kwantiele verkry kan word. Die rif regressie beramer vir die ekstreemwaarde indeks tesame met die voorgestelde spilpunt grootheid word saam met die saalpunt benadering gebruik om ‘n versameling vertrouensintervalle te verkry wat akkuraat en nou is. The nut van hierdie vertrouensintervalle word geïllustreer deur simulasie resultate en toepassings op Belgiese herversekering data. Meerveranderlike veralgemenings van steekproef kwantiele word ondersoek met die doel om meerveranderlike risikomaatstawwe te ontwikkel, insluitend maksimum korrelasie risikomaatstawwe en ‘n beramer vir die ekstreemwaarde indeks. Hierdie meerveranderlike steekproef kwantiele word sentrum-uitwaartse kwantiele genoem. Hulle word gedefinïeer as ‘n optimale transportasie van punte in die eenheid bal na die waargenome punte in . ‘n Kontinue uitbreiding van die sentrum-uitwaartse kwantiel word voorgestel wat ingenesde kwantiele lewer. Ook word maksimum korrelasie risikomaatstawwe vir meerveranderlike steekproewe voorgestel asook ‘n beramer vir die ekstreemwaarde indeks vir meerveranderlike reëlmatig variërende steekproewe. Ter illustrasie van die nut van hierdie resultate, word dit toegepas op Deense brand versekering data en die aandeel opbrengste verkry op Google en Apple aandeelpryse.af_ZA
dc.format.extentvi, 109 pages ; illustrations, includes annexure
dc.publisherStellenbosch : Stellenbosch Universityen_ZA
dc.subjectExtreme value analysisen_ZA
dc.subjectQuantile regressionen_ZA
dc.subjectMathematical statisticsen_ZA
dc.subjectExtreme value theoryen_ZA
dc.subjectMultivariate risken_ZA
dc.subjectMultivariate analysisen_ZA
dc.titleExtreme quantile inferenceen_ZA
dc.rights.holderStellenbosch Universityen_ZA

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