This thesis considers estimation of the quantiles of the smallest extreme value distribution, sometimes referred to as the log - Weibull distribution. The estimators considered are linear combinations of two order statistics. A table of the best linear estimates (BLUE's) is presented for sample sizes two through twenty. These estimators are compared to the asymptotic estimators of Kubat and Epstein (1980).

This study attempts to quantify the shocks to a banking network and analyze the transfer of shocks through the network. We consider two sources of shocks: external shocks due to market and macroeconomic factors which impact the entire banking system, and idiosyncratic shocks due to failure of a single bank. The external shocks will be estimated by using two methods: (i) non-parametric simulation of the time series of shocks that occurred to the banking system in the past, and (ii) using the extreme value theory (EVT) to model the tail part of the shocks. The external shocks we considered in this study are due to exchange rate and treasury bill rate volatility. Also, an ARMA/GARCH model is used to extract iid residuals for this purpose. In the next step, the probability of the failure of banks in the system is studied by using Monte Carlo simulation. We calibrate the model such that the network resembles the Canadian banking system.

This study examines the frequency of extreme trading days and investment behaviour in Sweden. We show that the frequency, as well as the magnitude of extreme trading days has increased over time. We also show that the frequency of extreme trading days in a year is positively correlated to the frequency the preceding year. Furthermore, we show that aggregate cash flows into equity and bond funds are unrelated to risk measured by standard deviation of return. Our findings show that investors, individuals as well as corporations, use simple passive investment strategies and hence, do not believe in market timing or wish to risk capital on capturing far tail or black swan type returns.

This paper uses a new measure of volatility based on extreme day return occurrences and examines the relative prevailing volatility among worldwide stock markets during 1997-2009. Using several global stock market indexes of countries categorized as an emerging and developed capital markets are utilized. Additionally this study investigates well known anomalies namely Monday effect and January effect. Further using correlation analysis of co movement and extent of integration highlights the opportunities for international diversification among those markets. Evidences during this time period suggest volatility is not the only phenomena of emerging capital markets. Emerging markets offer opportunities of higher returns during volatility. Cross correlation analysis depicts markets have become more integrated during this time frame; still opportunities for higher returns prevail through global portfolio diversification.

Thesis (Ph. D.)--University of Washington, 2006. / Vita. Includes bibliographical references (leaves 87-96).

This thesis uses statistical modelling to better understand the relationship between insured losses and hazard footprint characteristics for European windstorms (extra- tropical cyclones). The footprint of a windstorm is defined as the maximum wind gust speed to occur at a set of spatial locations over the duration of the storm. A better understanding of this relationship is required because the most damaging historical windstorms have had footprints with differing characteristics. Some have a large area of relatively low wind gust speeds, while others have a smaller area of higher wind gust speeds. In addition, this insight will help to explain the surprising, sharp decline in European wind related losses in the mid 1990’s. This novel exploration is based on 5730 high resolution model generated historical footprints (1979-2012) representing the whole European domain. Functions of extreme footprint wind gust speeds, known as storm severity measures, are developed to represent footprint characteristics. Exploratory data analysis is used to compare which storm severity measures are most successful at classifying 23 extreme windstorms, known to have caused large insured losses. Summarising the footprint using these scalar severity measures, however, fails to capture different combinations of spatial scale and local intensity characteristics. To overcome this, a novel statistical model for windstorm footprints is developed, initially for pairs of locations using a bivariate Gaussian copula model; subsequently extended to represent the whole European domain using a geostatistical spatial model. Throughout, the distribution of wind gust speeds at each location is modelled using a left-truncated Generalised Extreme Value (GEV) distribution. Synthetic footprints, simulated from the geostatistical model, are then used in a sensitivity study to explore whether the local intensity or spatial dependence structure of a footprint has the most influence on insured loss. This contributes a novel example of sensitivity analysis applied to a stochastic natural hazards model. The area of the footprint exceeding 25ms−1 over land is the most successful storm severity measure at classifying extreme loss windstorms, ranking all 23 within the top 18% of events. Marginally transformed wind gust speeds are identified as being asymptotically independent and second-order stationary, allowing for the spatial dependence to be represented by a geostatistical covariance function. The geostatistical windstorm footprint model is able to quickly (∼3 seconds) simulate synthetic footprints which realistically represent joint losses throughout Europe. The sensitivity study identifies that the left-truncated GEV parameters have a greater influence on insured loss than the geostatistical spatial dependence parameters. The observed decline in wind related losses in the 1990’s can therefore be attributed to a change in the local intensity rather than the spatial structure of footprint wind gust speeds.

Magister Scientiae - MSc / The aim of this research project was to estimate parameters for the distribution of annual maximum flood levels for the Zambezi River at Katima Mulilo. The estimation of parameters was done by using the maximum likelihood method. The study aimed to explore data of the Zambezi's annual maximum flood heights at Katima Mulilo by means of fitting the Gumbel, Weibull and the generalized extreme value distributions and evaluated their goodness of fit. / South Africa

This thesis reviewed various heavy tailed distributions and Extreme Value Theory (EVT) to estimate the catastrophic losses simulated from Florida Public Hurricane Loss Projection Model (FPHLPM). We have compared risk measures such as Probable Maximum Loss (PML) and Tail Value at Risk (TVaR) of the selected distributions with empirical estimation to capture the characteristics of the loss data as well as its tail distribution. Generalized Pareto Distribution (GPD) is the main focus for modeling the tail losses in this application. We found that the hurricane loss data generated from FPHLPM were consistent with historical losses and were not as heavy as expected. The tail of the stochastic annual maximum losses can be explained by an exponential distribution. This thesis also touched on the philosophical implication of small probability, high impact events such as Black Swan and discussed the limitations of quantifying catastrophic losses for future inference using statistical methods.

This study attempts to quantify the shocks to a banking network and analyze the transfer of shocks through the network. We consider two sources of shocks: external shocks due to market and macroeconomic factors which impact the entire banking system, and idiosyncratic shocks due to failure of a single bank. The external shocks will be estimated by using two methods: (i) non-parametric simulation of the time series of shocks that occurred to the banking system in the past, and (ii) using the extreme value theory (EVT) to model the tail part of the shocks. The external shocks we considered in this study are due to exchange rate and treasury bill rate volatility. Also, an ARMA/GARCH model is used to extract iid residuals for this purpose. In the next step, the probability of the failure of banks in the system is studied by using Monte Carlo simulation. We calibrate the model such that the network resembles the Canadian banking system.

In traditional classification problems, all classes in the test set are assumed to also occur in the training set, also referred to as the closed-set assumption. However, in practice, new classes may occur in the test set, which reduces the performance of machine learning models trained under the closed-set assumption. Machine learning models should be able to accurately classify instances of classes known during training while concurrently recognizing instances of previously unseen classes (also called the open set assumption). This open set assumption is motivated by real world applications of classifiers wherein its improbable that sufficient data can be collected a priori on all possible classes to reliably train for them. For example, motivated by the DARPA WASH project at WPI, a disease classifier trained on data collected prior to the outbreak of COVID-19 might erroneously diagnose patients with the flu rather than the novel coronavirus. State-of-the-art open set methods based on the Extreme Value Theory (EVT) fail to adequately model class distributions with unequal variances. We propose the Variational Open-Set Recognition (VOSR) model that leverages all class-belongingness probabilities to reject unknown instances. To realize the VOSR model, we design a novel Multi-Modal Variational Autoencoder (MMVAE) that learns well-separated Gaussian Mixture distributions with equal variances in its latent representation. During training, VOSR maps instances of known classes to high-probability regions of class-specific components. By enforcing a large distance between these latent components during training, VOSR then assumes unknown data lies in the low-probability space between components and uses a multivariate form of Extreme Value Theory to reject unknown instances. Our VOSR framework outperforms state-of-the-art open set classification methods with a 15% F1 score increase on a variety of benchmark datasets.