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Distribution theory and inference for bivariate extremes
Thesis (MCom)--Stellenbosch University, 2024.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2024
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| _version_ | 1867613745292771328 |
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| access_status_str | Open Access |
| author | Van Tonder, Jana |
| author2 | Steyn, Matthys Lucas |
| author_browse | Steyn, Matthys Lucas Van Tonder, Jana |
| author_facet | Steyn, Matthys Lucas Van Tonder, Jana |
| author_sort | Van Tonder, Jana |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MCom)--Stellenbosch University, 2024. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/130600 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:41:01.634Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2024 |
| publishDateRange | 2024 |
| publishDateSort | 2024 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/130600 Distribution theory and inference for bivariate extremes Van Tonder, Jana Steyn, Matthys Lucas De Wet, Tertius Stellenbosch University. Faculty of Economic and Management Sciences. Dept. of Statistics and Actuarial Science. Multivariate analysis Statistics -- Data processing Extreme value theory UCTD Thesis (MCom)--Stellenbosch University, 2024. ENGLISH SUMMARY: Various scenarios exist where the interest is in the modelling and prediction of rare or extreme events. Extreme value theory is an important branch of statistics, where limit theory is used to analyse extremes and to estimate the tail of the underlying distribution. Extreme value theory is the most developed for the univariate case, i.e. modelling the extremes of only a single variable. In many scenarios, however, more than one variable has an effect on the probability of occurrence of extreme events. In such cases, multivariate extreme value theory will play a valuable role in the modelling procedure by taking into account the joint effect of multivariate extremes. In this thesis, the focus will be on bivariate extreme value theory, i.e., multivariate extreme value theory restricted to two dimensions. Two approaches will be considered: (1) componentwise maxima and (2) a pair of random variables above a large threshold vector. A mathematical derivation of the limiting distribution of normalised componentwise maxima, called the bivariate extreme value distribution, will be given. For the threshold exceedance approach, it will be shown how the underlying distribution can be approximated by the bivariate extreme value distribution at transformed points. Unfortunately, no parametric form exists for the bivariate extreme value distribution. However, the distribution can be expressed in terms of the two marginal distributions and a dependence function. The latter is important in characterising the dependence structure of the distribution. Various characterisations are proposed in the literature. A few popular dependence functions will be discussed. It will also be shown how they are related through appropriate transformations. Since dependence plays an important role in bivariate extreme value theory, different measures of extremal dependence will be examined. For an independent and identically distributed random bivariate sample with asymptotic dependence between the two variables, it will be shown how the limit theory, based on the bivariate extreme value distribution, can be applied and how inference can be performed. Different ways of estimating the dependence structure of the bivariate extreme value distribution will be described, which include parametric and non-parametric techniques. When data exhibit asymptotic independence, the bivariate extreme value distribution is not suitable to use in the modelling procedure. Therefore, other models will be explored which better describe the tail of an asymptotically independent distribution. For illustration, the above-mentioned methods will be applied to two South African bivariate environmental datasets. For further interpretation and visualisation, graphs of the estimated distributions and quantile curves will also be given. Finally, it will be demonstrated that an asymptotic dependent model can lead to an overestimation of the joint exceedance probability when working in the tail of an asymptotic independent distribution, which agrees with the findings in the literature. AFRIKAANSE OPSOMMING: Daar bestaan verskeie scenario’s waar belangstelling in die modellering en voorspelling van seldsame of ekstreem gebeurtenisse is. Ekstreemwaardeteorie is ’n belangrike tak van statistiek, waar limietteorie gebruik word om ekstreme te analiseer en om die stert van die onderliggende verdeling te beraam. Ekstreemwaardeteorie is die mees ontwikkeld vir die eenveranderlike geval, m.a.w. modellering van die esktreme van slegs ’n enkele veranderlike. In baie scenario’s het meer as een veranderlike egter ’n effek op die waarskynlikheid van die voorkoms van ekstreem gebeurtenisse. In sulke gevalle sal meerveranderlike ekstreemwaardeteorie ’n nuttige rol speel in die modelleringsprosedure deur die gesamentlike effek van meerveranderlike ekstreme in ag te neem. In hierdie tesis sal die fokus op tweeveranderlike ekstreemwaardeteorie wees, m.a.w. meerveranderlike ekstreemwaardeteorie beperk tot 2-dimensies. Twee benaderings sal oorweeg word: (1) komponentgewyse maksima en (2) twee stogastiese veranderlikes bo ’n groot drempelvektor. ’n Wiskundige afleiding van die limietverdeling van genormaliseerde komponentgewyse maksima, genoem die tweeveranderlike ekstreemwaardeverdeling, sal gegee word. Vir die drempeloorskrydingsbenadering gaan dit aangetoon word hoe die onderliggende verdeling deur die tweeveranderlike ekstreemwaardeverdeling by getransformeerde punte benader kan word. Ongelukkig bestaan daar geen parametriese vorm vir die tweeveranderlike ekstreemwaardeverdeling nie. Nogtans kan die verdeling in terme van twee marginale verdelings en ’n afhanklikheidsfunksie uitgedruk word. Laasgenoemde is belangrik in die karakterisering van die verdeling se afhanklikheidstruktuur. Verskeie karakteriserings bestaan in die literatuur. ’n Aantal populere karakteriserings sal bespreek word. Dit sal ook aangetoon word hoe hulle verwant is deur gepaste transformasies. Aangesien afhanklikheid ’n belangrike rol in tweeveranderlike ekstreemwaardeteorie speel, gaan verskillende maatstawwe van ekstreme afhanklikheid ondersoek word. Vir ’n onafhanklike en identiese verdeelde stogastiese tweeveranderlike steekproef met asimptoties afhanklikheid tussen die twee veranderlikes, gaan dit aangetoon word hoe die limietteorie, gebaseer op die tweeveranderlike ekstreemwaardeverdeling, toegepas kan word en hoe inferensie uitgevoer kan word. Verskillende maniere om die afhanklikheidstruktuur te beraam gaan beskryf word, wat parametriese en nie-paramatriese tegnieke insluit. Wanneer data asimptotiese onafhanklikheid vertoon, is die tweeveranderlike ekstreemwaardeverdeling nie toepaslik vir die modelleringsproses nie. Om hierdie rede, gaan ander modelle ondersoek word wat die stert van ’n asimptotiese onafhanklike verdeling beter beskryf. Vir illustrasie, gaan die bogenoemde metodes op twee Suid-Afrikaanse tweeveranderlike omgewingsdatastelle toegepas word. Vir verdere interpretasie en visualisering, gaan grafieke van die beraamde verdelings en kwantielkurwes ook gegee word. Laastens, gaan dit gedemonstreer word dat ’n asimptotiese afhanklike model tot ’n oorberaming van die gesamentlike waarskynlikheid kan lei wanneer daar in die stert van ’n asimptotiese onafhanklike verdeling gewerk word, wat ooreenstem met die bevindinge in die literatuur. Masters 2024-02-02T06:09:50Z 2024-04-26T23:30:51Z 2024-02-02T06:09:50Z 2024-04-26T23:30:51Z 2024-03 Thesis https://scholar.sun.ac.za/handle/10019.1/130600 en_ZA Stellenbosch University xvi, 156 pages : illustrations, includes annexures application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Multivariate analysis Statistics -- Data processing Extreme value theory UCTD Van Tonder, Jana Distribution theory and inference for bivariate extremes |
| title | Distribution theory and inference for bivariate extremes |
| title_full | Distribution theory and inference for bivariate extremes |
| title_fullStr | Distribution theory and inference for bivariate extremes |
| title_full_unstemmed | Distribution theory and inference for bivariate extremes |
| title_short | Distribution theory and inference for bivariate extremes |
| title_sort | distribution theory and inference for bivariate extremes |
| topic | Multivariate analysis Statistics -- Data processing Extreme value theory UCTD |
| url | https://scholar.sun.ac.za/handle/10019.1/130600 |
| work_keys_str_mv | AT vantonderjana distributiontheoryandinferenceforbivariateextremes |