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Plotting position formula for general extreme value distribution

- , 2016,
Abstract: 【目的】研究并建立广义极值分布无偏经验概率计算公式,为广义极值分布经验概率的计算提供支持。【方法】应用次序统计量原理,推导广义极值分布无偏经验概率计算公式,采用统计试验方法将推导的公式与现有的经验频率公式进行对比,对其进行检验,最后以陕北地区12个水文测站的年最大洪峰流量系列为例进行模型应用。【结果】推导出了便于工程设计应用的广义极值分布经验概率计算公式GEVQG。统计试验和实例应用表明:推导的公式GEVQG和现有经验频率公式Cunnane公式的相对误差和偏差均较小,并且对研究区的拟合效果良好。【结论】推导的计算公式GEVQG和Cunnane公式均可以作为广义极值分布经验概率计算的优选公式,为工程水文经验概率计算提供了新的选择。
【Objective】This study developed the unbiased plotting position formula for general extreme value (GEV) distribution.【Method】Based on the theory of order statistics,this paper developed an unbiased plotting position formula for the GEV distribution.The statistical testing was used to compare the developed formula and other existing formulas.At last,the parameters of annual maximum floods peak flow series of twelve stations in Northern Shaanxi were estimated.【Result】This paper developed a practical and convenient plotting position formula for the GEV distribution.The case study and statistical testing showed that the developed formula and Cunnane formula had small error and bias as well as good fitness in the study area.【Conclusion】The developed formula and Cunnane formula can be used as better methods for the GEV distribution,which provides new choice for empirical probability calculation in hydrology engineering

WANG Shui-yong,REN Ai,

腐蚀科学与防护技术 , 2008,
Abstract: Corrosion depth data of partial backwash pipes of a sea water system in nuclear power plant was treated statistically using the extreme value probability distribution i.e.the so called Gumbel distribution.And the maximum corrosion depth 10.8±1.3 mm of the whole backwash pipes had been predicted with the return period which equals to 149.The corrosion depth data were subjected to Gumbel distribution treatment and the parameters of its probability function were estimated by MVLUE method.The Gumbel probability distribution was checked by Kolmogorov-Smirnov test and proved to be accurate.As a result,the maximum depth predicted by it was confident.
Estimadores de Momentos de Probabilidad Pesada para la Distribución General de Valores Extremos para Máximos Probability Weighted Moments Estimators for the Parameters of the General Extreme Value Distribution  [cached]
José A. Raynal
Información Tecnológica , 2005,
Abstract: Se propone un método alternativo para la estimación de parámetros de la función de distribución de probabilidad general de valores extremos (GVE), por medio del método de momentos de probabilidad pesada, para el análisis de gastos máximos anuales. El método propuesto presenta una mayor flexibilidad de modelación que el existente y se ha podido aplicar a una gran cantidad de datos de gastos máximos anuales sin ninguna dificultad observada. El artículo contiene ejemplos numéricos de estimación de parámetros usando las dos metodologías citadas. Los resultados producidos por ambos métodos son prácticamente iguales, tanto en la evaluación de los parámetros de la distribución GVE, como en la obtención de los eventos de dise o. En cuanto a las características estadísticas, el método propuesto es muy superior al existente, como ha sido demostrado por medio de experimentos de muestreo distribucional An alternative method for estimating the parameters of the general extreme value (GEV) probability distribution function, using the method of probability weighted moments, is compared with existing methods, to analyze maximum annual floods. The proposed method shows better modeling flexibility than the existing methods, and has been applied to a large number of maximum annual floods samples with no observed difficulty. The paper contains numerical examples for the estimation of parameters using the methods cited. The results produced by both methods are practically the same, both in the parameter estimation phase for the GEV distribution, as well as in the evaluation of the design events. Regarding statistical characteristics, the proposed methodology is much better than the existing one, as it has been shown through distribution sampling experiments.
Extreme Value Theory and Value at Risk
Viviana Fernandez
Revista de Análisis Económico (RAE) , 2003,
Abstract: Value at Risk (VaR) is a measure of the maximum potential change in value of a portfolio of financial assets with a given probability over a given time horizon. VaR became a key measure of market risk since the Basle Committee stated that banks should be able to cover losses on their trading portfolios over a ten-day horizon, 99 percent of the time. A common practice is to compute VaR by assuming that changes in value of the portfolio are normally distributed, conditional on past in-formation. However, assets returns usually come from fat-tailed distri-butions. Therefore, computing VaR under the assumption of conditional normality can be an important source of error. We illustrate this point with Chilean and U.S. returns series by resorting to extreme value theory (EVT) and GARCH-type models. In addition, we show that dynamic estimation of empirical quantiles can also give more accurate VaR estimates than quantiles of a standard normal.
The Extreme Value Machine  [PDF]
Ethan M. Rudd,Lalit P. Jain,Walter J. Scheirer,Terrance E. Boult
Computer Science , 2015,
Abstract: This paper provides a novel characterization of the max-margin distribution in input space. The use of the statistical Extreme Value Theory (EVT) is introduced for modeling margin distances, allowing us to derive a scalable non-linear model called the Extreme Value Machine (EVM). Without the need for a kernel, the EVM leverages a margin model to estimate the probability of sample inclusion in each class. The EVM selects a near-optimal subset of the training vectors to optimize the gain in terms of points covered versus parameters used. We show that the problem reduces to the NP-hard Set Cover problem which has a provable polynomial time approximation. The resulting machine has comparable closed set accuracy (i.e., when all testing classes are known at training time) to optimized RBF SVMs and exhibits far superior performance in open set recognition (i.e., when unknown classes exist at testing time). In open set recognition performance, the EVM is more accurate and more scalable than the state of the art.
Condensation and Extreme Value Statistics  [PDF]
Martin R. Evans,Satya N. Majumdar
Statistics , 2008, DOI: 10.1088/1742-5468/2008/05/P05004
Abstract: We study the factorised steady state of a general class of mass transport models in which mass, a conserved quantity, is transferred stochastically between sites. Condensation in such models is exhibited when above a critical mass density the marginal distribution for the mass at a single site develops a bump, $p_{\rm cond}(m)$, at large mass $m$. This bump corresponds to a condensate site carrying a finite fraction of the mass in the system. Here, we study the condensation transition from a different aspect, that of extreme value statistics. We consider the cumulative distribution of the largest mass in the system and compute its asymptotic behaviour. We show 3 distinct behaviours: at subcritical densities the distribution is Gumbel; at the critical density the distribution is Fr\'echet, and above the critical density a different distribution emerges. We relate $p_{\rm cond}(m)$ to the probability density of the largest mass in the system.
Improving extreme value statistics  [PDF]
Ashivni Shekhawat
Statistics , 2014, DOI: 10.1103/PhysRevE.90.052148
Abstract: The rate of uniform convergence in extreme value statistics is non-universal and can be arbitrarily slow. Further, the relative error can be unbounded in the tail of the approximation, leading to difficulty in extrapolating the extreme value fit beyond the available data. We show that by using simple nonlinear transformations the extreme value approximation can be rendered rapidly convergent in the bulk, and asymptotic in the tail, thus fixing both issues. The transformations are often parameterized by just one parameter which can be estimated numerically. The classical extreme value method is shown to be a special case of the proposed method. We demonstrate that vastly improved results can be obtained with almost no extra cost.
Extreme-Value Copulas  [PDF]
Gordon Gudendorf,Johan Segers
Statistics , 2009,
Abstract: Being the limits of copulas of componentwise maxima in independent random samples, extreme-value copulas can be considered to provide appropriate models for the dependence structure between rare events. Extreme-value copulas not only arise naturally in the domain of extreme-value theory, they can also be a convenient choice to model general positive dependence structures. The aim of this survey is to present the reader with the state-of-the-art in dependence modeling via extreme-value copulas. Both probabilistic and statistical issues are reviewed, in a nonparametric as well as a parametric context.
Extreme Value Laws for Superstatistics  [PDF]
Pau Rabassa,Christian Beck
Physics , 2014, DOI: 10.3390/e16105523
Abstract: We study the extreme value distribution of stochastic processes modeled by superstatistics. Classical extreme value theory asserts that (under mild asymptotic independence assumptions) only three possible limit distributions are possible, namely: Gumbel, Fr\'echet and Weibull distribution. On the other hand, superstatistics contains three important universality classes, namely $\chi^2$-superstatistics, inverse $\chi^2$-superstatistics, and lognormal superstatistics, all maximizing different effective entropy measures. We investigate how the three classes of extreme value theory are related to the three classes of superstatistics. We show that for any superstatistical process whose local equilibrium distribution does not live on a finite support, the Weibull distribution cannot occur. Under the above mild asymptotic independence assumptions, we also show that $\chi^2$-superstatistics generally leads an extreme value statistics described by a Fr\'echet distribution, whereas inverse $\chi^2$-superstatistics, as well as lognormal superstatistics, lead to an extreme value statistics associated with the Gumbel distribution.
Methodology for assessing probability of extreme hydrologic events coincidence  [PDF]
Prohaska Stevan,Blagojevi? Borislava,Ili? Aleksandra
Facta Universitatis Series : Architecture and Civil Engineering , 2010, DOI: 10.2298/fuace1002187p
Abstract: The aim of the presented research is improvement of methodology for probability calculation of coinciding occurrence of historic floods and droughts in the same year. The original procedure was developed in order to determine the occurrence probability of such an extreme historic event. There are two phases in calculation procedure for assessment of both extreme drought and flood occurrence probability in the same year. In the first phase outliers are detected as indicators of extreme events, their return periods are calculated and series' statistics adjusted. In the second phase conditional probabilities are calculated: empirical points are plotted, and both extreme drought and flood occurrence probability in the same year is assessed based on the plot. Outlier detection is performed for the territory of Serbia. Results are shown as maps of regions (basins) prone to floods, hydrologic drought, or both. Step-by-step numeric example is given for assessing conditional probability of occurrence of flood and drought for GS Raska on the river Raska. Results of assessment of conditional probability in two more cases are given for combination of extreme flood and 30 day minimum flow.
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