Adapting the Hill estimator to distributed inference:dealing with the bias

The distributed Hill estimator is a divide-and-conquer algorithm for estimating the extreme value index when data are stored in multiple machines. In applications, estimates based on the distributed Hill estimator can be sensitive to the choice of the number of the exceedance ratios used in each machine. Even when choosing the number at a low level, a high asymptotic bias may arise. We overcome this potential drawback by designing a bias correction procedure for the distributed Hill estimator, which adheres to the setup of distributed inference. The asymptotically unbiased distributed estimator we obtained, on the one hand, is applicable to distributed stored data, on the other hand, inherits all known advantages of bias correction methods in extreme value statistics

The Log-Logistic Weibull Distribution with Applications to Lifetime Data

In this paper, a new generalized distribution called the log-logistic Weibull (LLoGW) distribution is developed and presented. This distribution contain the log-logistic Rayleigh (LLoGR), log-logistic exponential (LLoGE) and log-logistic (LLoG) distributions as special cases. The structural properties of the distribution including the hazard function, reverse hazard function, quantile function, probability weighted moments, moments, conditional moments, mean deviations, Bonferroni and Lorenz curves, distribution of order statistics, L-moments and Renyi entropy are derived. Method of maximum likelihood is used to estimate the parameters of this new distribution. A simulation study to examine the bias, mean square error of the maximum likelihood estimators and width of the condence intervals for each parameter is presented. Finally, real data examples are presented to illustrate the usefulness and applicability of the model

On tail trend detection: modeling relative risk

The climate change dispute is about changes over time of environmental characteristics (such as rainfall). Some people say that a possible change is not so much in the mean but rather in the extreme phenomena (that is, the average rainfall may not change much but heavy storms may become more or less frequent). The paper studies changes over time in the probability that some high threshold is exceeded. The model is such that the threshold does not need to be specified, the results hold for any high threshold. For simplicity a certain linear trend is studied depending on one real parameter. Estimation and testing procedures (is there a trend?) are developed. Simulation results are presented. The method is applied to trends in heavy rainfall at 18 gauging stations across Germany and The Netherlands. A tentative conclusion is that the trend seems to depend on whether or not a station is close to the sea.Comment: 38 page

Aplicação da teoria de valores extremos para o índice pluviométrico da cidade de Juiz de Fora - MG

The analysis of events, often critical, which are at the tails of the distributions is difficult because there is little information about them, due to their rarity. The extreme value theory presents methods for dealing with these events through limit distributions, allowing inference about them. This paper provides an introduction to this theory with the use of their most widespread univariate models, the generalized extreme value distribution (GEV) and the generalized Pareto distribution (GPD). A literature review on their properties, modeling techniques, estimation and evaluation of goodness-of-fit is shown. At the end, it is used for analysis of extreme events in the daily rainfall in the city of Juiz de Fora - MG. Such events are responsible for situations of floods, landslides and burials in the city and the knowledge about the behavior of precipitation extremes must be used to minimize their impact and prevent such tragedies. For this study, we used indices of daily rainfall between January 1, 1961 and December 31, 2014, provided by the Banco de Dados Meteorológicos para Ensino e Pesquisa (BDMEP) of the Instituto Nacional de Meteorologia. Through these data, it was concluded that potential harmful rains are expected to occur once between 0.99 and 2.4 years (GEV model) and between 1.17 and 2.24 years (GPD model). These results highlight the importance of preventive actions to be performed jointly by the government and community.A análise de eventos, muitas vezes críticos, que se encontram nas caudas das distribuições é dificultada pelo fato de haver pouca informação sobre eles, devido à sua raridade. A teoria de valores extremos apresenta metodologias para lidar com estes eventos através de distribuições limite, possibilitando a inferência sobre os mesmos. Este trabalho oferece uma introdução à esta teoria com a utilização de seus modelos univariados mais difundidos, a distribuição de valor extremo generalizada (GEV) e a distribuição Pareto generalizada (GPD). É feita uma revisão bibliográfica sobre suas propriedades, técnicas de modelagem, estimação e de avaliação da qualidade do ajuste. Ao final, é utilizada para análise de eventos extremos na precipitação pluvial diária da cidade de Juiz de Fora – MG. Tais eventos são responsáveis por situações de inundações, soterramentos e desabamentos na cidade e o conhecimento sobre o comportamento dos extremos da precipitação deve ser utilizado para a minimização de seu impacto e prevenção de tais tragédias. Para este estudo, foram utilizados índices da precipitação pluvial diária entre 01 de janeiro de 1961 e 31 de dezembro de 2014, cedidos pelo Banco de Dados Meteorológicos para Ensino e Pesquisa (BDMEP) do Instituto Nacional de Meteorologia. Através destes dados, concluiu-se que chuvas com alto potencial danoso são esperadas ocorrerem uma vez entre 0,99 ano e 2,4 anos (modelo GEV) e entre 1,17 ano e 2,24 anos (modelo GPD). Estes resultados ressaltam a importância de ações preventivas que devem ser exercidas de forma conjunta pelo Poder Público e população

Existence and consistency of the maximum likelihood estimator for the extreme value index

AbstractThe paper is about the asymptotic properties of the maximum likelihood estimator for the extreme value index. Under the second order condition, Drees et al. [H. Drees, A. Ferreira, L. de Haan, On maximum likelihood estimation of the extreme value index, Ann. Appl. Probab. 14 (2004) 1179–1201] proved asymptotic normality for any solution of the likelihood equations (with shape parameter γ>−1/2) that is not too far off the real value. But they did not prove that there is a solution of the equations satisfying the restrictions.In this paper, the existence is proved, even for γ>−1. The proof just uses the domain of attraction condition (first order condition), not the second order condition. It is also proved that the estimator is consistent. When the second order condition is valid, following the current proof, the existence of a solution satisfying the restrictions in the above-cited reference is a direct consequence

Existence and consistency of the maximum likelihood estimator for the extreme value index

The paper is about the asymptotic properties of the maximum likelihood estimator for the extreme value index. Under the second order condition, Drees et al. [H. Drees, A. Ferreira, L. de Haan, On maximum likelihood estimation of the extreme value index, Ann. Appl. Probab. 14 (2004) 1179-1201] proved asymptotic normality for any solution of the likelihood equations (with shape parameter [gamma]>-1/2) that is not too far off the real value. But they did not prove that there is a solution of the equations satisfying the restrictions. In this paper, the existence is proved, even for [gamma]>-1. The proof just uses the domain of attraction condition (first order condition), not the second order condition. It is also proved that the estimator is consistent. When the second order condition is valid, following the current proof, the existence of a solution satisfying the restrictions in the above-cited reference is a direct consequence.62G05 Consistency Extreme value condition Extreme value index Maximum likelihood estimator

On Extreme Value Statistics: maximum likelihood; portfolio optimization; extremal rainfall; internet auctions

In the 18th century, statisticians sometimes worked as consultants to gamblers. In order to answer questions like "If a fair coin is flipped 100 times, what is the probability of getting 60 or more heads?", Abraham de Moivre discovered the so-called "normal curve". Independently, Pierre-Simon Laplace derived the central limit theorem, where the normal distribution acts as the limit for the distribution of the sample mean. Nowadays, statisticians sometimes work as consultants for economists, to whom the normal distribution is far from a satisfactory model. For example, one may need to model large-impact ¯nancial events in order to to answer questions like "What is the probability of getting into a crisis period similar to the credit squeeze in 2007 in the coming 10 years?". At ¯rst glance, estimating the chances of events that rarely happen or even have never happened before sounds like a "mission impossible". The development of Extreme Value Theory (EVT) shows that it is in fact possible to achieve this goal. Di®erent from the central limit theorem, Extreme Value Theory starts from the limit distribution of the sample maximum. Initiated by M. Frechet, R. Fisher and R. von Mises, the limit theory completed by B. Gnedenko, gave the fundamental assumption in EVT, the "extreme value condition". Statistically, the extreme value condition provides a semi-parametric model for the tails of distribution functions. Therefore it can be applied to evaluate the rare events. On the other hand, since the assumption is rather general and natural, the semi-parametric model can have extensive applications in numerous felds

Modelling South Africa's market risk using the APARCH model and heavy-tailed distributions.

Master of Science in Statistics. University of KwaZulu-Natal, Durban 2016.Estimating Value-at-risk (VaR) of stock returns, especially from emerging economies has recently attracted attention of both academics and risk managers. This is mainly because stock returns are relatively more volatile than its historical trend. VaR and other risk management tools, such as expected shortfall (conditional VaR) are highly dependent on an appropriate set of underlying distributional assumptions being made. Thus, identifying a distribution that best captures all aspects of financial returns is of great interest to both academics and risk managers. As a result, this study compares the relative performance of the GARCH-type model combined with heavy-tailed distribution, namely Skew Student t distribution, Pearson Type IV distribution (PIVD), Generalized Pareto distribution (GPD), Generalized Extreme Value distribution (GEVD), and stable distribution in estimating Value-at-Risk of South African all share index (ALSI) returns. Model adequacy is checked through the backtesting procedure. The Kupiec likelihood ratio test is used for backtesting. The proposed models are able to capture volatility clustering (conditional heteroskedasticity), and the asymmetric effect (leverage effect) and heavy-tailedness in the returns. The advantage of the proposed models lies in their ability to capture volatility clustering and the leverage effect on the returns, though the GARCH framework and at the same time model their heavy tailed behaviour through the heavy-tailed distribution. The main findings indicate that APARCH model combined with this heavy-tailed distribution performed well in modelling South African market’s risk at both the long and short position. It was also found that when compared in terms of their predictive ability, APARCH model combined with the PIVD, and APARCH model combined with GPD model gives a better VaR estimation for the short position while APARCH model combined with stable distribution give the better VaR estimation for long position. Thus, APARCH model combined with heavy-tailed distribution model provides a good alternative for modelling stock returns. The outcomes of this research are expected to be of salient value to financial analysts, portfolio managers, risk managers and financial market researchers, therefore giving a better understanding of the South African market