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Improved estimation procedures for a positive extreme value index
Thesis (PhD (Statistics))--University of Stellenbosch, 2010.
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| Format: | Thesis |
| Language: | English |
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Stellenbosch : University of Stellenbosch
2010
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| _version_ | 1867614107594653696 |
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| access_status_str | Open Access |
| author | Berning, Thomas Louw |
| author2 | De Wet, Tertius |
| author_browse | Berning, Thomas Louw De Wet, Tertius |
| author_facet | De Wet, Tertius Berning, Thomas Louw |
| author_sort | Berning, Thomas Louw |
| collection | Thesis |
| dc_rights_str_mv | University of Stellenbosch |
| description | Thesis (PhD (Statistics))--University of Stellenbosch, 2010. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/5260 |
| institution | Stellenbosch University (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:46:46.943Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2010 |
| publishDateRange | 2010 |
| publishDateSort | 2010 |
| publisher | Stellenbosch : University of Stellenbosch |
| publisherStr | Stellenbosch : University of Stellenbosch |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/5260 Improved estimation procedures for a positive extreme value index Berning, Thomas Louw De Wet, Tertius University of Stellenbosch. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science. Extreme value index (EVI) Extreme value theory (EVT) Dissertations -- Statistics and actuarial science Theses -- Statistics and actuarial science Thesis (PhD (Statistics))--University of Stellenbosch, 2010. ENGLISH ABSTRACT: In extreme value theory (EVT) the emphasis is on extreme (very small or very large) observations. The crucial parameter when making inferences about extreme quantiles, is called the extreme value index (EVI). This thesis concentrates on only the right tail of the underlying distribution (extremely large observations), and specifically situations where the EVI is assumed to be positive. A positive EVI indicates that the underlying distribution of the data has a heavy right tail, as is the case with, for example, insurance claims data. There are numerous areas of application of EVT, since there are a vast number of situations in which one would be interested in predicting extreme events accurately. Accurate prediction requires accurate estimation of the EVI, which has received ample attention in the literature from a theoretical as well as practical point of view. Countless estimators of the EVI exist in the literature, but the practitioner has little information on how these estimators compare. An extensive simulation study was designed and conducted to compare the performance of a wide range of estimators, over a wide range of sample sizes and distributions. A new procedure for the estimation of a positive EVI was developed, based on fitting the perturbed Pareto distribution (PPD) to observations above a threshold, using Bayesian methodology. Attention was also given to the development of a threshold selection technique. One of the major contributions of this thesis is a measure which quantifies the stability (or rather instability) of estimates across a range of thresholds. This measure can be used to objectively obtain the range of thresholds over which the estimates are most stable. It is this measure which is used for the purpose of threshold selection for the proposed PPD estimator. A case study of five insurance claims data sets illustrates how data sets can be analyzed in practice. It is shown to what extent discretion can/should be applied, as well as how different estimators can be used in a complementary fashion to give more insight into the nature of the data and the extreme tail of the underlying distribution. The analysis is carried out from the point of raw data, to the construction of tables which can be used directly to gauge the risk of the insurance portfolio over a given time frame. AFRIKAANSE OPSOMMING: Die veld van ekstreemwaardeteorie (EVT) is bemoeid met ekstreme (baie klein of baie groot) waarnemings. Die parameter wat deurslaggewend is wanneer inferensies aangaande ekstreme kwantiele ter sprake is, is die sogenaamde ekstreemwaarde-indeks (EVI). Hierdie verhandeling konsentreer op slegs die regterstert van die onderliggende verdeling (baie groot waarnemings), en meer spesifiek, op situasies waar aanvaar word dat die EVI positief is. ’n Positiewe EVI dui aan dat die onderliggende verdeling ’n swaar regterstert het, wat byvoorbeeld die geval is by versekeringseis data. Daar is verskeie velde waar EVT toegepas word, aangesien daar ’n groot aantal situasies is waarin mens sou belangstel om ekstreme gebeurtenisse akkuraat te voorspel. Akkurate voorspelling vereis die akkurate beraming van die EVI, wat reeds ruim aandag in die literatuur geniet het, uit beide teoretiese en praktiese oogpunte. ’n Groot aantal beramers van die EVI bestaan in die literatuur, maar enige persoon wat die toepassing van EVT in die praktyk beoog, het min inligting oor hoe hierdie beramers met mekaar vergelyk. ’n Uitgebreide simulasiestudie is ontwerp en uitgevoer om die akkuraatheid van beraming van ’n groot verskeidenheid van beramers in die literatuur te vergelyk. Die studie sluit ’n groot verskeidenheid van steekproefgroottes en onderliggende verdelings in. ’n Nuwe prosedure vir die beraming van ’n positiewe EVI is ontwikkel, gebaseer op die passing van die gesteurde Pareto verdeling (PPD) aan waarnemings wat ’n gegewe drempel oorskrei, deur van Bayes tegnieke gebruik te maak. Aandag is ook geskenk aan die ontwikkeling van ’n drempelseleksiemetode. Een van die hoofbydraes van hierdie verhandeling is ’n maatstaf wat die stabiliteit (of eerder onstabiliteit) van beramings oor verskeie drempels kwantifiseer. Hierdie maatstaf bied ’n objektiewe manier om ’n gebied (versameling van drempelwaardes) te verkry waaroor die beramings die stabielste is. Dit is hierdie maatstaf wat gebruik word om drempelseleksie te doen in die geval van die PPD beramer. ’n Gevallestudie van vyf stelle data van versekeringseise demonstreer hoe data in die praktyk geanaliseer kan word. Daar word getoon tot watter mate diskresie toegepas kan/moet word, asook hoe verskillende beramers op ’n komplementêre wyse ingespan kan word om meer insig te verkry met betrekking tot die aard van die data en die stert van die onderliggende verdeling. Die analise word uitgevoer vanaf die punt waar slegs rou data beskikbaar is, tot op die punt waar tabelle saamgestel is wat direk gebruik kan word om die risiko van die versekeringsportefeulje te bepaal oor ’n gegewe periode. Doctoral 2010-11-23T13:38:40Z 2010-12-15T10:27:14Z 2010-11-23T13:38:40Z 2010-12-15T10:27:14Z 2010-12 Thesis http://hdl.handle.net/10019.1/5260 en University of Stellenbosch 240 p. : ill. application/pdf Stellenbosch : University of Stellenbosch |
| spellingShingle | Extreme value index (EVI) Extreme value theory (EVT) Dissertations -- Statistics and actuarial science Theses -- Statistics and actuarial science Berning, Thomas Louw Improved estimation procedures for a positive extreme value index |
| title | Improved estimation procedures for a positive extreme value index |
| title_full | Improved estimation procedures for a positive extreme value index |
| title_fullStr | Improved estimation procedures for a positive extreme value index |
| title_full_unstemmed | Improved estimation procedures for a positive extreme value index |
| title_short | Improved estimation procedures for a positive extreme value index |
| title_sort | improved estimation procedures for a positive extreme value index |
| topic | Extreme value index (EVI) Extreme value theory (EVT) Dissertations -- Statistics and actuarial science Theses -- Statistics and actuarial science |
| url | http://hdl.handle.net/10019.1/5260 |
| work_keys_str_mv | AT berningthomaslouw improvedestimationproceduresforapositiveextremevalueindex |