Dissertation submitted for Masters of Science degree in Mathematical Statistics in the FacultyofScience, SchoolofStatisticsandActuarialScience, University of the Witwatersrand Johannesburg, January 2018 / This dissertation focuses on demonstrating the use of extreme value theory in modelling temperature in South Africa. The purpose of modelling temperature is to investigate the frequency of occurrences of extremely low and extremely high temperatures and how they influence the demand of electricity over time. The data comprise a time series of average hourly temperatures that are collected by the South African Weather Service over the period 2000−2010 and supplied by Eskom. The generalized extreme value distribution (GEVD) for r largest order statistics is fitted to the average maximum daily temperature (non-winter season) using the maximum likelihood estimation method and used to estimate extreme high temperatures which result in high demand of electricity due to use of cooling systems. The estimation of the shape parameter reveals evidence that the Weibull family of distributions is an appropriate fit to the data. A frequency analysis of extreme temperatures is carried out and the results show that most of the extreme temperatures are experienced during the months January, February, November and December of each year. The generalized Pareto distribution (GPD) is firstly used for modelling the average minimum daily temperatures for the period January 2000 to August 2010. A penalized regression cubic smoothing spline is used as a time varying threshold. We then extract excessesabovethecubicregressionsmoothingsplineandfitanon-parametricmixturemodel to get a sufficiently high threshold. The data exhibit evidence of short-range dependence and high seasonality which lead to the declustering of the excesses above the threshold and fit the GPD to cluster maxima. The estimate of the shape parameter shows that the Weibullfamilyofdistributionsisappropriateinmodellingtheuppertailofthedistribution. The stationary GPD and the piecewise linear regression models are used in modelling the influence of temperature above the reference point of 22◦C on the demand of electricity. The stationary and non-stationary point process models are fitted and used in determining the frequency of occurrence of extremely high temperatures. The orthogonal and the reparameterizationapproachesofdeterminingthefrequencyandintensityofextremeshave i been used to establish that, extremely hot days occur in frequencies of 21 and 16 days per annum, respectively. For the fact that temperature is established as a major driver of electricity demand, this dissertation is relevant to the system operators, planners and decision makers in Eskom and most of the utility and engineering companies. Our results are furtherusefultoEskomsinceitisduringthenon-winterperiodthattheyplanformaintenance of their power plants. Modelling temperature is important for the South African economy since electricity sector is considered as one of the most weather sensitive sectors of the economy. Over and above, the modelling approaches that are presented in this dissertation are relevant for modelling heat waves which impose several impacts on energy, economy and health of our citizens. / XL2018

In this paper, we study the properties of k-omnisequences of length n, defined to be strings of length n that contain all strings of smaller length k embedded as (not necessarily contiguous) subsequences. We start by proving an elementary result that relates our problem to the classical coupon collector problem. After a short survey of relevant results in coupon collection, we focus our attention on the number M of strings (or words) of length k that are not found as subsequences of an n string, showing that there is a gap between the probability threshold for the emergence of an omnisequence and the zero-infinity threshold for E(M).

In this paper, we study the properties of k-omnisequences of length n, defined to be strings of length n that contain all strings of smaller length k embedded as (not necessarily contiguous) subsequences. We start by proving an elementary result that relates our problem to the classical coupon collector problem. After a short survey of relevant results in coupon collection, we focus our attention on the number M of strings (or words) of length k that are not found as subsequences of an n string, showing that there is a gap between the probability threshold for the emergence of an omnisequence and the zero-infinity threshold for E(M).

The problems encountered when empirical fit is used as the sole criterion for choosing a distribution to represent annual flood data are discussed. Some theoretical direction is needed for this choice. Extreme value theory is established as a viable tool for analyzing annual flood data. Extreme value distributions have been used in previous analyses of flood data. How�ver, no systematic investigation of the theory has previously been applied. Properties of the extreme value distributions are examined. The most appropriate distribution for flood data has not previously been fit to such data. The fit of the chosen extreme value distribution compares favorably with that of the Pearson and log Pearson Type III distributions.

This dissertation implements an optimal trading strategy based on the machine learning method and extreme value theory (EVT) to obtain an excess return on investments in the capital market. The trading strategy outperforms the benchmark S&P 500 index with higher returns and lower volatility through effective market timing. In addition, this dissertation starts by modeling the market tail risk using the EVT and reinforcement learning methods, distinguishing from the traditional value at risk method. In this dissertation, I used EVT to extract the characteristics of the tail risk, which are inputs for reinforcement learning. This process is proved to be effective in market timing, and the trading strategy could avoid market crash and achieve a long-term excess return. In sum, this study has several contributions. First, this study takes a new method to analyze stock price (in this dissertation, I use the S&P 500 index as a stock). I combined the EVT and reinforcement learning to study the price tail risk and predict stock crash efficiently, which is a new method for tail risk research. Thus, I can predict the stock crash or provide the probability of risk, and then, the trading strategy can be built. The second contribution is that this dissertation provides a dynamic market timing trading strategy, which can significantly outperform the market index with a lower volatility and a higher Sharpe ratio. Moreover, the dynamic trading process can provide investors an intuitive sense on the stock market and help in decision-making. Third, the success of the strategy shows that the combination of EVT and reinforcement learning can predict the stock crash very well, which is a great improvement on the extreme event study and deserves further study. / Business Administration/Finance

Classical models for multivariate or spatial extremes are mainly based upon the asymptotically justified max-stable or generalized Pareto processes. These models are suitable when asymptotic dependence is present. However, recent environmental data applications suggest that asymptotic independence is equally important. Therefore, development of flexible subasymptotic models is in pressing need. This dissertation consists of four major contributions to subasymptotic modeling of multivariate and spatial extremes. Firstly, the dissertation proposes a new spatial copula model for extremes based on the multivariate generalized hyperbolic distribution. The extremal dependence of this distribution is revisited and a corrected theoretical description is provided. Secondly, the dissertation thoroughly investigates the extremal dependence of stochastic processes driven by exponential-tailed Lévy noise. It shows that the discrete approximation models, which are linear transformations of a random vector with independent components, bridge asymptotic independence and asymptotic dependence in a novel way, whilst the exact stochastic processes exhibit only asymptotic independence. Thirdly, the dissertation explores two different notions of optimal prediction for extremes, and compares the classical linear kriging predictor and the conditional mean predictor for certain non-Gaussian models. Finally, the dissertation proposes a multivariate skew-elliptical link model for correlated highly-imbalanced (extreme) binary responses, and shows that the regression coefficients have a closed-form unified skew-elliptical posterior with an elliptical prior.

The work reported in this thesis is in the area of computational modeling of reliability of engineering structures. The emphasis of the study is on developing methods that are suitable for analysis of large-scale structures such as aircraft structure components. This class of problems continues to offer challenges to an analyst with the most difficult aspect of the analysis being the treatment of nonlinearity in the structural behavior, non-Gaussian nature of uncertainties and quantification of low levels of probability of failure (of the order of 10-5 or less), requiring significant computational effort. The present study covers static/ dynamic behavior, Gaussian/ non-Gaussian models of uncertainties, and (or) linear/ nonlinear structures. The novel elements in the study consist of two components: • application of modeling tools that already exists in the area of design and analysis of computer experiments, and . • application of data based extreme value analysis procedures that are available in the statistics literature. The first component of the work provides opportunity to combine space filling sampling strategies (which have promise for reducing variance of estimation) with kriging based modeling in reliability studies-an opportunity that has not been explored in the existing literature. The second component of the work exploits the virtues of limiting behavior of extremes of sequence of random variables with Monte Carlo simulations of structural response-a strategy for reliability modeling that has not been explored in the existing literature. The hope here is that failure events with probabilities of the order of 10-5 or less could be investigated with relatively less number of Monte Carlo runs. The study also brings out the issues related to combining the above sources of existing knowledge with finite element modeling of engineering structures, thereby leading to newer tools for structural reliability analysis. The thesis is organized into four chapters. The first chapter provides a review of literature that covers methods of reliability analysis and also the background literature on design and analysis of computer experiments and extreme value analysis. The problem of reliability analysis of randomly parametered, linear (or) nonlinear structures subjected to static and (or) dynamic loads is considered in Chapter 2. A deterministic finite element model for the structure to analyze sample realization of the structure is assumed to be available. The reliability analysis is carried out within the framework of response surface methods, which involves the construction of surrogate models for performance functions to be employed in reliability calculations. These surrogate models serve as models of models, and hence termed as meta-models, for structural behavior in the neighborhood of design point. This construction, in the present study, has involved combining space filling optimal Latin hypercube sampling and kriging models. Illustrative examples on numerical prediction of reliability of a ten-bay truss and a W-seal in an aircraft structure are presented. Limited Monte Carlo simulations are used to validate the approximate procedures developed. The reliability of nonlinear vibrating systems under stochastic excitations is investigated in Chapter 3 using a two-stage Monte Carlo simulation strategy. Systems subjected to Gaussian random excitation are considered for the study. It is assumed that the probability distribution of the maximum response in the steady state belongs to the basin of attraction of one of the classical asymptotic extreme value distributions. The first stage of the solution strategy consists of an objective selection of the form of the extreme value distribution based on hypothesis tests, and the next involves the estimation of parameters of the relevant extreme value distribution. Both these steps are implemented using data from limited Monte Carlo simulations of the system response. The proposed procedure is illustrated with examples of linear/nonlinear single-degree and multi-degree of freedom systems driven by random excitations. The predictions from the proposed method are compared with results from large-scale Monte Carlo simulations and also with classical analytical results, when available, from theory of out-crossing statistics. The method is further extended to cover reliability analysis of nonlinear dynamical systems with randomly varying system parameters. Here the methods of meta-modeling developed in Chapter 2 are extended to develop response surface models for parameters of underlying extreme value distributions. Numerical examples presented cover a host of low-dimensional dynamical systems and also the analysis of a wind turbine structure subjected to turbulent wind loads and undergoing large amplitude oscillations. A summary of contributions made along with a few suggestions for further research is presented in Chapter 4.

In this thesis we will elaborate on multivariate extreme value modelling, re- lated practical and theoretical aspects. We will mainly focus on the dependence models, the extreme value copulas in particular. Extreme value copulas effec- tively unify the univariate extreme value theory and the copula framework itself in a single view. We familiarize ourselves with both of them in the first two chapters. Those chapters present generalized extreme value distribution, gen- eralized Pareto distribution and Archimedean copulas, that are suitable for the multivariate maxima and the threshold exceedances description. These two top- ics will be addressed in the third chapter in detail. Taking into consideration rather practical focus of this thesis, we examine the methods of data analysis extensively. Furthermore, we will employ these methods in a comprehensive case study, that will aim to reveal the importance of extreme value theory application in the Catastrophe Insurance. 1

In the present thesis we will firstly familiarize ourselves with the term of operational risk, it's definition presented in the directives Basel II and Solvency II, and afterwards with the methods of calculation Capital Requirements for Operational Risk, set by these directives. In the second part of the thesis we will concentrate on the methods of modelling operational loss data. We will introduce the Extreme Value Theory which describes possible approaches to modelling data with significant values that occur infrequently; the typical characteristic of operational risk data. We will mainly focus on the model for threshold exceedances which utilizes Generalized Pareto Distribution to model the distribution of those excesses. The teoretical knowledge of this theory and the appropriate modelling will be applied on simulated loss data. Finally we will test the ability of presented methods to model loss data distributions.

Neste trabalho é introduzido um novo modelo estatístico para modelar dados limitados no intervalo continuo (0;1). O modelo proposto é construído sob uma transformação de variáveis, onde a variável transformada é resultado da combinação de uma variável com distribuição normal padrão e a função de distribuição acumulada da distribuição valor extremo generalizado. Para o novo modelo são estudadas suas propriedades estruturais. A nova família é estendida para modelos de regressão, onde o modelo é reparametrizado na mediana da variável resposta e este conjuntamente com o parâmetro de dispersão são relacionados com covariáveis através de uma função de ligação. Procedimentos inferênciais são desenvolvidos desde uma perspectiva clássica e bayesiana. A inferência clássica baseia-se na teoria de máxima verossimilhança e a inferência bayesiana no método de Monte Carlo via cadeias de Markov. Além disso estudos de simulação foram realizados para avaliar o desempenho das estimativas clássicas e bayesianas dos parâmetros do modelo. Finalmente um conjunto de dados de câncer colorretal é considerado para mostrar a aplicabilidade do modelo. / In this research a new statistical model is introduced to model data restricted in the continuous interval (0;1). The proposed model is constructed under a transformation of variables, in which the transformed variable is the result of the combination of a variable with standard normal distribution and the cumulative distribution function of the generalized extreme value distribution. For the new model its structural properties are studied. The new family is extended to regression models, in which the model is reparametrized in the median of the response variable and together with the dispersion parameter are related to covariables through a link function. Inferential procedures are developed from a classical and Bayesian perspective. The classical inference is based on the theory of maximum likelihood, and the Bayesian inference is based on the Markov chain Monte Carlo method. In addition, simulation studies were performed to evaluate the performance of the classical and Bayesian estimates of the model parameters. Finally a set of colorectal cancer data is considered to show the applicability of the model