In many data sets, a mixture distribution formulation applies when it is known that each observation comes from one of the underlying categories. Even if there are no apparent categories, an implicit categorical structure may justify a mixture distribution. This thesis concerns the modeling of extreme values in such a setting within the peaks-over-threshold (POT) approach. Specifically, the traditional POT modeling using the generalized Pareto distribution is augmented in the sense that, in addition to threshold exceedances, data below the threshold are also modeled by means of the mixture exponential distribution. In the first part of this thesis, the conventional frequentist approach is applied for data modeling. In view of the mixture nature of the problem, the EM algorithm is employed for parameter estimation, where closed-form expressions for the iterates are obtained. A simulation study is conducted to confirm the suitability of such method, and the observation of an increase in standard error due to the variability of the threshold is addressed. The model is applied to two real data sets, and it is demonstrated how computation time can be reduced through a multi-level modeling procedure. With the fitted density, it is possible to derive many useful quantities such as return periods and levels, value-at-risk, expected tail loss and bounds for ruin probabilities. A likelihood ratio test is then used to justify model choice against the simpler model where the thin-tailed distribution is homogeneous exponential. The second part of the thesis deals with a fully Bayesian approach to the same model. It starts with the application of the Bayesian idea to a special case of the model where a closed-form posterior density is computed for the threshold parameter, which serves as an introduction. This is extended to the threshold mixture model by the use of the Metropolis-Hastings algorithm to simulate samples from a posterior distribution known up to a normalizing constant. The concept of depth functions is proposed in multidimensional inference, where a natural ordering does not exist. Such methods are then applied to real data sets. Finally, the issue of model choice is considered through the use of posterior Bayes factor, a criterion that stems from the posterior density. / published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

published_or_final_version / abstract / toc / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

The problem of efficient estimation of the parameters of the extreme value distribution has not been addressed in the literature. We obtain efficient estimators of the parameters of type I (maximum) extreme value distribution without solving the likelihood equations. This research provides for the first time simple expressions for the elements of the information matrix for type II censoring. We construct efficient estimators of the parameters using linear combinations of order statistics of a random sample drawn from the population. We derive explicit formulas for the information matrix for this problem for type II censoring and construct efficient estimators of the parameters using linear combinations of available order statistics with additional weights to the smallest and largest order statistics. We consider numerical examples to illustrate the applications of the estimators. We also perform an extensive Monte Carlo simulation study to examine the performance of the estimators for different sample sizes.

Assignment (MComm)--University of Stellenbosch, 2005. / Bibliography. Also available via the Internet.

Thesis (Ph. D.)--University of Texas at Dallas, 2006. / Includes vita. Includes bibliographical references (leaves 92-93).

In this thesis, we review the basic notions related to univariate regular variation and study some fundamental properties of regularly varying random variables. We then consider the notion of regular variation in the multivariate case. After collecting some results from multivariate regular variation for random vectors with values in $\mathbb{R}_{+}^{d}$, we discuss its properties and examine several examples of multivariate regularly varying random vectors such as independent and identically distributed random vectors, fully dependent random vectors and other models. We also present the elements of univariate and multivariate extreme value theory and emphasize the connection with multivariate regular variation. Some measures of extreme dependence such as the stable tail dependence function and the Pickands dependence function are presented. We end the study by conducting a data analysis using financial data. In the univariate case, graphical tools such as quantile-quantile plots, mean excess plots and Hill plots are used in order to determine the underlying distribution of the univariate data. In the multivariate case, non-parametric estimators of the stable tail dependence function and the Pickands dependence function are used to describe the dependence structure of the multivariate data.

The Frechet distribution was applied to a set of earthquake data in order to test its validity as a practical alternative distribution for extreme value data. It was concluded that the Frechet distribution was the best model representing that data set. Also, a Poisson model of occurrence could not be rejected for that data set. The combination of these two models resulted in a closed form unconditional extreme value distribution which was developed analytically. The appropriate statistical tests and sensitivity analyses were performed on the obtained model. / Applied Science, Faculty of / Mechanical Engineering, Department of / Graduate

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.

The crude oil, natural gas, and electricity markets are among the most widely traded and talked about commodity markets across the world. Over the past two decades each commodity has seen price volatility due to political, economic, social, and technological reasons. With that comes a significant amount of risk that both corporations and governments must account for to ensure expected cash flows and to minimize losses. This thesis analyzes the portfolio risk of the major US commodity hubs for crude oil, natural gas and electricity by applying Extreme Value Theory to historical daily price returns between 2003 and 2013. The risk measures used to analyze risk are Value-at-Risk and Expected Shortfall, with these estimated by fitting the Generalized Pareto Distribution to the data using the peak-over-threshold method. We consider both the univariate and bivariate cases in order to determine the effects that price shocks within and across commodities will have in a mixed portfolio. The results show that electricity is the most volatile, and therefore most risky, commodity of the three markets considered for both positive and negative returns. In addition, we find that the univariate and bivariate results are statistically indistinguishable, leading to the conclusion that for the three markets analyzed during this period, price shocks in one commodity does not directly impact the volatility of another commodity’s price. / Graduate

In this thesis we deal with statistical inference related to extreme value phenomena. Specifically, if X is a random vector with values in d-dimensional space, our goal is to estimate moments of ψ(X) for a suitably chosen function ψ when the magnitude of X is big. We employ the powerful tool of regular variation for random variables, random vectors and time series to formally define the limiting quantities of interests and construct the estimators. We focus on three statistical estimation problems: (i) multivariate tail estimation for regularly varying random vectors, (ii) extremogram estimation for regularly varying time series, (iii) estimation of the expected shortfall given an extreme component under a conditional extreme value model. We establish asymptotic normality of estimators for each of the estimation problems. The theoretical findings are supported by simulation studies and the estimation procedures are applied to some financial data.