A Bias-reduced Estimator for the Mean of a Heavy-tailed Distribution with an Infinite Second Moment

We use bias-reduced estimators of high quantiles, of heavy-tailed distributions, to introduce a new estimator of the mean in the case of infinite second moment. The asymptotic normality of the proposed estimator is established and checked, in a simulation study, by four of the most popular goodness-of-fit tests for different sample sizes. Moreover, we compare, in terms of bias and mean squared error, our estimator with Peng's estimator (Peng, 2001) and we evaluate the accuracy of some resulting confidence intervals.Comment: Submitte

An automatic procedure for the estimation of the tail index

Extreme Value Theory is increasingly used in the modelling of financial time series. The non-normality of stock returns leads to the search for alternative distributions that allows skewness and leptokurtic behavior. One of the most used distributions is the Pareto Distribution because it allows non-normal behaviour, which requires the estimation of a tail index. This paper provides a new method for estimating the tail index. We propose an automatic procedure based on the computation of successive normality tests over the whole of the distribution in order to estimate a Gaussian Distribution for the central returns and two Pareto distributions for the tails. We find that the method proposed is an automatic procedure that can be computed without need of an external agent to take the decision, so it is clearly objective.Tail Index; Hill estimator; Normality Test

Kernel alternatives to aproximate operational severity distribution: an empirical application

The estimation of severity loss distribution is one the main topic in operational risk estimation. Numerous parametric estimations have been suggested although very few work for both high frequency small losses and low frequency big losses. In this paper several estimation are explored. The good performance of the double transformation kernel estimation in the context of operational risk severity is worthy of a special mention. This method is based on the work of Bolancé and Guillén (2009), it was initially proposed in the context of the cost of claims insurance, and it means an advance in operational risk research

Modelling temperature in South Africa using extreme value theory

Dissertation submitted for Masters of Science degree in Mathematical Statistics in the FacultyofScience, SchoolofStatisticsandActuarialScience, University of the Witwatersrand Johannesburg, January 2018This 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.XL201

Extreme Value Theory for Tail-Related Risk Measures

Many fields of modern science and engineering have to deal with events which are rare but have significant consequences. Extreme value theory is considered to provide the basis for the statistical modeling of such extremes. The potential of extreme value theory applied to financial problems has only been recognized recently. This paper aims at introducing the fundamentals of extreme value theory as well as practical aspects for estimating and assessing statistical models for tail-related risk measures.Extreme Value Theory; Generalized Pareto Distribution, Generalized Extreme Value Distribution; Quantile Estimation, Risk Measures; Maximum Likelihood Estimation; Profile Likelihood Confidence Intervals.

Estimating the tail index: another algorithmic method

The tail index is a determinant parameter within extreme value theory. Under a semiparametric approach, one has often to choose the number of the largest order statistics to include in estimates. This is a hard task since it is not possible to know for sure where the tail of data really begins. This crucial topic has been largely addressed in literature and several methods were developed. In this paper we analyze, through simulation, a heuristic method and compare it with two very popular methodologies. It will be seen that the new method can be a good alternative.Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the ”Fundação para a Ciência e a Tecnologia”, through the Project PEstOE/ MAT/UI0013/2014

An Application of Extreme Value Theory for Measuring Financial Risk

Assessing the probability of rare and extreme events is an important issue in the risk management of financial portfolios. Extreme value theory provides the solid fundamentals needed for the statistical modelling of such events and the computation of extreme risk measures. The focus of the paper is on the use of extreme value theory to compute tail risk measures and the related confidence intervals, applying it to several major stock market indice