Academic literature on the topic 'R-largest order statistics'

Extreme Value Theory (EVT) is an important tool to predict efficient gains and losses. Its main areas of analyses are economic and environmental. Initially, for that form of event, it was developed the use of patterns of parametric distribution such as Normal and Gamma. However, economic and environmental data presents, in most cases, a heavy-tailed distribution, in contrast to those distributions. Thus, it was faced a great difficult to frame extreme events. Furthermore, it was almost impossible to use conventional models, making predictions about non-observed events, which exceed the maximum of observations. In some situations EVT is used to analyse only the maximum of some dataset, which provide few observations, and in those cases it is more effective to use the r largest-order statistics. This paper aims to propose Bayesian estimators' for parameters of the r largest-order statistics. During the research, it was used Monte Carlo simulation to analyze the data, and it was observed some properties of those estimators, such as mean, variance, bias and Root Mean Square Error (RMSE). The estimation of the parameters provided inference for its parameters and return levels. This paper also shows a procedure to the choice of the r-optimal to the r largest-order statistics, based on the Bayesian approach applying Markov chains Monte Carlo (MCMC). Simulation results reveal that the Bayesian approach has a similar performance to the Maximum Likelihood Estimation, and the applications were developed using the Bayesian approach and showed a gain in accurary compared with otherestimators.

The objective of this study is to compute 100-yr return value estimates of significant wave height using a new hindcast developed by the Norwegian Meteorological Institute. This regional hindcast covers the northeast Atlantic and spans the period 1958–2009. The return value estimates are based upon three different stationary models commonly applied in extreme value statistics: the generalized extreme value (GEV) distribution, the joint GEV distribution for the r largest-order statistic (rLOS), and the generalized Pareto (GP) distribution. Here, the qualitative differences between the models and their corresponding confidence intervals are investigated.

General relativity (GR) has been well tested up to solar system scales, but it is much less certain that standard gravity remains an accurate description on the largest, that is cosmological, scales. Many extensions to GR have been studied that are not yet ruled out by the data, including by that of the recent direct gravitational wave detections. Degeneracies among the standard model (ΛCDM) and modified gravity (MG) models, as well as among different MG parameters, must be addressed in order to best exploit information from current and future surveys and to unveil the nature of dark energy. We propose various higher-order statistics in the weak-lensing signal as a new set of observables able to break degeneracies between massive neutrinos and MG parameters. We have tested our methodology on so-called f(R) models, which constitute a class of viable models that can explain the accelerated universal expansion by a modification of the fundamental gravitational interaction. We have explored a range of these models that still fit current observations at the background and linear level, and we show using numerical simulations that certain models which include massive neutrinos are able to mimic ΛCDM in terms of the 3D power spectrum of matter density fluctuations. We find that depending on the redshift and angular scale of observation, non-Gaussian information accessed by higher-order weak-lensing statistics can be used to break the degeneracy between f(R) models and ΛCDM. In particular, peak counts computed in aperture mass maps outperform third- and fourth-order moments.

The design wind pressures specified in the 2005 National Building Code of Canada (NBCC) have been derived from the Gumbel distribution fitted to annual maximum wind speed data collected up to early 1990s. The statistical estimates of the annual maxima method are affected by a relatively large sampling variability, since the method considers a fairly small subset of available wind speed records. Advanced statistical methods have emerged in recent years with the purpose of reducing both sampling and model uncertainties associated with extreme quantile estimates. The two most notable methods are the peaks-over-threshold (POT) and annually r largest order statistics (r-LOS), which extend the data set by including additional maxima observed in wind speed time series. The objective of the paper is to explore the use of advanced extreme value theory for updating the design wind speed estimates specified in the Canadian building design code. The paper re-examines the NBCC method for design wind speed estimation and presents the analysis of the latest Canadian wind speed data by POT, r-LOS, and annual maxima methods. The paper concludes that r-LOS method is an effective alternative for the estimation of extreme design wind speed.Key words: wind speed, extreme value theory, order statistics, return period, maximum likelihood method, peaks-over-threshold method, generalized extreme value distribution, Gumbel distribution, generalized Pareto distribution.

Eventos extremos são raros, mas quando ocorrem têm um enorme impacto social e uma atenção mediática considerável. São exemplo os recordes no mundo do desporto – raros de acontecerem, mas quando ocorrem não apenas são divulgados nos mais variados meios de comunicação social, como são motivo de modificação da metodologia de treino e do comportamento do atleta. Como prever esta ocorrência? Qual a probabilidade de ocorrência? Qual a magnitude da ocorrência? Quanto tempo de espera? A teoria de valores extremos, baseada no teorema dos tipos extremais de Fisher-Tippett- Gnedenko, proporciona um rigoroso quadro de análise dos valores extremos, estimando a probabilidade de ocorrência de eventos que estão para além da amostra disponível. Assim, sob a questão “qual o limite da performance humana?”, este trabalho no domínio da Estatística de Extremos tem aplicações ao desporto de alto rendimento, particularmente às especialidades de lançamentos e saltos do atletismo. Foram utilizadas as metodologias: (i) r-maiores observações, (ii) excessos acima de um limiar, e (iii) máximos de blocos não-estacionários. Decidida a distribuição limite do máximo associado ao evento em estudo e o modelo que melhor se adequa aos dados disponíveis, a estimação pontual dos parâmetros extremais foi realizada por máxima verosimilhança e a estimação intervalar pelo método delta e pela função profile log-likelihood. Em cada método foram testados diversos modelos. Recorremos a técnicas gráficas, estabilidade dos erros-padrão, intervalos de confiança, testes de hipóteses e algumas métricas de erro, para verificação do ajustamento dos modelos aos dados disponíveis. Estamos particularmente interessados na estimação de quantis extremais, probabilidades de excedência, limite superior do suporte, níveis de retorno e período de retorno. Os resultados mostram que nos lançamentos do martelo, disco e dardo feminino existe uma forte probabilidade de se conseguir um novo recorde do mundo e que nos lançamentos masculinos tal probabilidade é reduzida. Com exceção do triplo-salto, nas restantes especialidades de saltos, o período de retorno (i.e., número de máximos individuais) até à ocorrência de um novo recorde do mundo é menor nas mulheres do que nos homens; ABSTRACT: Statistics of Extremes: limits of human performance - study with throwers and jumpers in athletics Extreme events are rare, but when they do occur, they have an enormous social impact and they receive a considerable media attention. Such is the case of world records in sport – they rarely happen, but when they do, not only are they disclosed by all media, but they also cause changes in the training methodology and in the athlete's behaviour. How to predict this occurrence? What is the probability of occurrence? What is the magnitude of the occurrence? How long is the wait? The extreme value theory, based on the Fisher-Tippett-Gnedenko theorem, provides a rigorous framework for analysing extreme values, estimating the probability of the occurrence of events that are beyond the sample. Thus, research within the area of Extreme Statistics provide information to answer the question “what is the limit of human performance?” in the framework of high-performance sport, particularly in the case of the specialties of throwing and jumping in athletics. The methodologies used were: (i) r-largest order statistics, (ii) peaks over threshold, and (iii) non-stationary annual maximum. Once decided the limit distribution of the maximum associated with the event under study and the model that best fits the available data, the point estimation of the extremal parameters were performed by maximum likelihood estimation, with Nelder- Mead or BFGS optimization and the interval estimation using the delta method and the profile loglikelihood function. In each method, several models were tested. We used graphical techniques, stability of standard errors, confidence intervals, hypothesis tests and some error metrics, to verify if the models fit the available data. We were particularly interested in the estimation of extreme quantiles, exceedance probability, right endpoint, return levels and return period. The results suggest that in hammer, discus and javelin throwing there are a strong probability of a new world record will be achieved. In the case of male, throwing events, the forecast of a new world record being achieved with reduced probability. With exception of triple-jump, in all other jumping specialities, the return period (i.e., number of individual maximums) until the occurrence of a new world record is shorter in women than in men.

MSc (Statistics)
Department of Statistics
Haulage trucks and cross border traders cross through Beitbridge border post from landlocked countries such as Zimbabwe and Zambia for the sake of trading. Because of global warming, South Africa has lately been experiencing extreme weather patterns in the form of very high temperatures and heavy rainfall. Evidently, in 2013 tra c could not cross the Limpopo River because water was owing above the bridge. For planning, its important to predict the likelihood of such events occurring in future. Extreme value models o er one way in which this can be achieved. This study identi es suitable distributions to model the annual maximum heights of Limpopo river at Beitbridge border post. Maximum likelihood method and the Bayesian approach are used for parameter estimation. The r -largest order statistics was also used in this dissertation. For goodness of t, the probability and quantile- quantile plots are used. Finally return levels are calculated from these distributions. The dissertation has revealed that the 100 year return level is 6.759 metres using the maximum likelihood and Bayesian approaches to estimate parameters. Empirical results show that the Fr echet class of distributions ts well the ood heights data at Beitbridge border post. The dissertation contributes positively by informing stakeholders about the socio- economic impacts that are brought by extreme flood heights for Limpopo river at Beitbridge border post

This paper provides an overview of different methods of extrapolating environmental data to low probability levels based on the extreme value theory. It discusses the Annual Maxima method and the Peak Over Threshold method, using unified terminology and notation. Furthermore, it describes a method based on the r largest order statistics that has the advantage of providing more accurate parameters and quantile estimates than the Annual Maxima method. Several examples illustrate the methodology and reveal strengths and weaknesses of the various approaches.

Data on Health Literacy in the population of Latvia is limited. The aim of the study was to determine the Health Literacy impacting factors of inhabitants of Vidzeme Statistical region in Latvia (LV008). Respondent survey (n = 383), using a paper-and-pencil self-administered approach and telephone interviews, was conducted based the European Health Literacy Survey Questionnaire (HLS-EU-Q47). In order to ensure internal consistency and reliability, the authors used Cronbach’s α test (α = 0.965). The confirmatory factor analysis (CFA) allowed to determine that factor results differentiate between genders and there is a strong positive correlation (r = 0.945), that impacts results. Factors Access, Appraise and Apply explained each 30 % of the variance, and factors Understand explained 31 %. HL index division by gender indicated that 47.4 % of female respondents and 46.6 % of male respondents have “limited health literacy” (“inadequate” + “problematic”). The largest age group among respondents are 18-39-year old where there is lower level of education and lower income. However, cases have been observed when respondents even with higher education have “limited health literacy,” which indicates towards a need for further HL research in Latvia, because compared to HL of other member states, LV008 HL index is by 38.9 % larger than the EU average (47.6 %).