Estimation of the GEV distribution from censored samples by method of partial probability weighted moments

The concept of partial probability weighted moments (PPWM), which can be used to estimate a distribution from censored samples, is introduced. Unbiased estimators of PPWM are derived. An application is made to estimating parameters and quantiles of the generalized extreme value (GEV) distribution from censored samples. Censored samples yield high quantile estimates which are almost as efficient as those obtained from uncensored samples. This could be a very useful technique for dealing with the undesirable effects of low outliers which occur in semiarid and arid zones.     

On the use of partial probability weighted moments in the analysis of hydrological extremes

The use of partial probability weighted moments (PPWM) for estimating hydrological extremes is compared to that of probability weighted moments (PWM). Firstly, estimates from at‐site data are considered. Two Monte Carlo analyses, conducted using continuous and empirical parent distributions (of peak discharge and daily rainfall annual maxima) and applying four different distributions (Gumbel, Fréchet, GEV and generalized Pareto), show that the estimates obtained from PPWMs are better than those obtained from PWMs if the parent distribution is unknown, as happens in practice. Secondly, the use of partial L‐moments (obtained from PPWMs) as diagnostic tools is considered. The theoretical partial L‐diagrams are compared with the experimental data. Five different distributions (exponential, Pareto, Gumbel, GEV and generalized Pareto) and 297 samples of peak discharge annual maxima are considered. Finally, the use of PPWMs with regional data is investigated. Three different kinds of regional analyses are considered. The first kind is the regression of quantile estimates on basin area. The study is conducted applying the GEV distribution to peak discharge annual maxima. The regressions obtained with PPWMs are slightly better than those obtained with PWMs. The second kind of regional analysis is the parametric one, of which four different models are considered. The congruence between local and regional estimates is examined, using peak discharge annual maxima. The congruence degree is sometimes higher for PPWMs, sometimes for PWMs. The third kind of regional analysis uses the index flood method. The study, conducted applying the GEV distribution to synthetic data from a lognormal joint distribution, shows that better estimates are obtained sometimes from PPWMs, sometimes from PWMs. All the results seem to indicate that using PPWMs can constitute a valid tool, provided that the influence of ouliers, of course higher with censored samples, is kept under control. Copyright © 2007 John Wiley & Sons, Ltd.     

Parameter and quantile estimation of the 2-parameter kappa distribution by maximum likelihood

Asymptotic properties of maximum likelihood parameter and quantile estimators of the 2-parameter kappa distribution are studied. Eight methods for obtaining large sample confidence intervals for the shape parameter and for quantiles of this distribution are proposed and compared by using Monte Carlo simulation. The best method is highlighted on the basis of the coverage probability of the confidence intervals that it produces for sample sizes commonly found in practice. For such sample sizes, confidence intervals for quantiles and for the shape parameter are shown to be more accurate if the quantile estimators are assumed to be log normally distributed rather than normally distributed (same for the shape parameter estimator). Also, confidence intervals based on the observed Fisher information matrix perform slightly better than those based on the expected value of this matrix. A hydrological example is provided in which the obtained theoretical results are applied.     

Statistical prediction of the next great earthquake around Tehran,Iran

The analysis of seismic activity variations with space and time is a complex problem. Several statistical methods have been adopted to study these variations. One of the tasks that has attracted the attention of the seismological and statistical community is to explain seismicity patterns by statistical models and apply the results for earthquake prediction. Here the probability distribution of recurrence times as described by Exponential, Gamma, Lognormal, Pareto, Rayleigh and Weibull probability distributions and the idea of conditional probability has been applied to predict the next great (Ms ≥ 6.0 and Ms ≥ 6.5) earthquake around Tehran (r ≤ 200 km). Conditional probability specifies the likelihood that a given earthquake will happen within a specified time. This likelihood is based on the information about past earthquake occurrences in the given region and the basic assumption that future seismic activity will follow the pattern of past activity. The rapid growth of Tehran to approximately 12 million inhabitants has resulted in a much more rapid increase in its vulnerability to natural disasters, especially earthquakes. Several earthquakes affected this region in the past, mostly on the Mosha, Taleqan, Eyvankey and Garmsar faults. The estimated recurrence times for Exponential, Gamma, Lognormal, Pareto, Rayleigh and Weibull distributions has been computed to be 66.64, 14.79, 26.88, 2.37, 67.58 and 80.47, respectively. Accordingly, one may expect that a large damaging earthquake may occur around Tehran approximately every 10 years.     

Finding the most appropriate precipitation probability distribution for stochastic weather generation and hydrological modelling in Nordic watersheds

Six precipitation probability distributions (exponential, Gamma, Weibull, skewed normal, mixed exponential and hybrid exponential/Pareto distributions) are evaluated on their ability to reproduce the statistics of the original observed time series. Each probability distribution is also indirectly assessed by looking at its ability to reproduce key hydrological variables after being used as inputs to a lumped hydrological model. Data from 24 weather stations and two watersheds (Chute‐du‐Diable and Yamaska watersheds) in the province of Quebec (Canada) were used for this assessment. Various indices or statistics, such as the mean, variance, frequency distribution and extreme values are used to quantify the performance in simulating the precipitation and discharge. Performance in reproducing key statistics of the precipitation time series is well correlated to the number of parameters of the distribution function, and the three‐parameter precipitation models outperform the other models, with the mixed exponential distribution being the best at simulating daily precipitation. The advantage of using more complex precipitation distributions is not as clear‐cut when the simulated time series are used to drive a hydrological model. Although the advantage of using functions with more parameters is not nearly as obvious, the mixed exponential distribution appears nonetheless as the best candidate for hydrological modelling. The implications of choosing a distribution function with respect to hydrological modelling and climate change impact studies are also discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Maximum product of spacings estimators for the generalized pareto and log-logistic distributions

The maximum product of spacings (MPS) method is discussed from the standpoint of information theory. MPS parameter and quantile estimates for the generalized Pareto distribution and the two parameter log-logistic distribution are compared with the maximum likelihood(ML) and probability weighted moment (PWM) estimates.     

The influence of parameter distribution uncertainty on hydrological and sediment modeling: a case study of SWAT model applied to the Daning watershed of the Three Gorges Reservoir Region, China

Parameter uncertainty involved in hydrological and sediment modeling often refers to the parameter dispersion and the sensitivity of the parameter. However, a limitation of the previous studies lies in that the assignment of range and specification of probability distribution for each parameter is usually difficult and subjective. Therefore, there is great uncertainty in the process of parameter calibration and model prediction. In this study, the impact of probability parameter distribution on hydrological and sediment modeling was evaluated using a semi-distributed model—the Soil and Water Assessment Tool (SWAT) and Monte Carlo method (MC)—in the Daning River watershed of the Three Gorges Reservoir Region (TGRA), China. The classic types of parameter distribution such as uniform, normal and logarithmic normal distribution were involved in this study. Based on results, parameter probability distribution showed a diverse degree of influence on the hydrological and sediment prediction, such as the sampling size, the width of 95% confidence interval (CI), the ranking of the parameter related to uncertainty, as well as the sensitivity of the parameter on model output. It can be further inferred that model parameters presented greater uncertainty in certain regions of the primitive parameter range and parameter samples densely obtained from these regions would lead to a wider 95 CI, resulting in a more doubtful prediction. This study suggested the value of the optimized value obtained by the parameter calibration process could may also be of vital importance in selecting the probability distribution function (PDF). Such cases, where parameter value corresponds to the watershed characteristic, can be used to provide a more credible distribution for both hydrological and sediment modeling.     

基于广义帕累托分布的地震震级分布尾部特征分析

极值理论在地震危险性分析中有着重要应用, 发震震级超过某一阈值的超出量分布可以近似为广义帕累托分布. 基于广义帕累托分布给出了若干地震活动性参数的估计公式, 包括强震震级分布、 地震复发周期和重现水平、 期望重现震级、 地震危险性概率和潜在震级上限等; 以云南地区震级资料为基础数据, 讨论了阈值选取、 模型拟合诊断和参数估计; 在此基础上计算了该地区的地震活动性参数. 结果表明, 广义帕累托分布较好地刻画了强震震级分布, 通过超阈值(POT)模型计算的复发周期与实际复发间隔统计基本一致, 高分位数估计在一定阈值范围内表现稳定, 为工程抗震中潜在震级上限的确定提供了一种途径.      

LL-moments for estimating low flow quantiles / Estimation des quantiles d'étiage grâce aux LL-moments

Abstract

A parameter estimation method is proposed for fitting probability distribution functions to low flow observations. LL-moments are variants of L-moments that are analogous to LH-moments, which were defined for the analysis of floods. LL-moments give higher weights to the small observations. Expressions are given that relate them to the probability distribution function for the case of normal, Weibull and power distributions. Sampling properties of the LL-moments and of the distribution parameters and quantiles estimated by them are found by a Monte Carlo simulation study. It is shown on an example that the low flow quantile estimates obtained by LL-moments may be significantly different from those obtained by L-moments.     

The PWM large quantile estimates of heavy tailed distributions from samples deprived of their largest element / Estimation des grands quantiles de distributions à queue à décroissance lente par la méthode des moments pondérés par les probabilités à partir d'échantillons amputés de leur plus grande valeur

Abstract

Extremes of streamflow are usually modelled using heavy tailed distributions. While scrutinising annual flow maxima or the peaks over threshold, the largest elements in a sample are often suspected to be low quality data, outliers or values corresponding to much longer return periods than the observation period. In the case of floods, since the interest is focused mainly on the estimation of the right-hand tail of a distribution function, sensitivity of large quantiles to extreme elements of a series becomes the problem of special concern. This study investigated the sensitivity problem using the log-Gumbel distribution by generating samples of different sizes and different values of the coefficient of L-variation by means of Monte Carlo experiments. Parameters of the log-Gumbel distribution were estimated by the probability weighted moments (PWM) method, both for complete samples and the samples deprived of their largest element. In the latter case Hosking's concept of the “A” type PWM with Type II censoring was employed. The largest value was censored above the random threshold T corresponding to the non-exceedence probability F _T. The effect of the F _T value on the performance of the quantile estimates was then examined. Experimental results show that omission of the largest sample element need not result in a decrease in the accuracy of large quantile estimates obtained from the log-Gumbel model by the PWM method.     

On the informative value of the largest sample element of log-Gumbel distribution

Extremes of stream flow and precipitation are commonly modeled by heavytailed distributions. While scrutinizing annual flow maxima or the peaks over threshold, the largest sample elements are quite often suspected to be low quality data, outliers or values corresponding to much longer return periods than the observation period. Since the interest is primarily in the estimation of the right tail (in the case of floods or heavy rainfalls), sensitivity of upper quantiles to largest elements of a series constitutes a problem of special concern. This study investigated the sensitivity problem using the log-Gumbel distribution by generating samples of different sizes (n) and different values of the coefficient of variation by Monte Carlo experiments. Parameters of the log-Gumbel distribution were estimated by the probability weighted moments (PWMs) method, method of moments (MOMs) and maximum likelihood method (MLM), both for complete samples and the samples deprived of their largest elements. In the latter case, the distribution censored by the non-exceedance probability threshold, F _T , was considered. Using F _T instead of the censored threshold T creates possibility of controlling estimator property. The effect of the F _T value on the performance of the quantile estimates was then examined. It is shown that right censoring of data need not reduce an accuracy of large quantile estimates if the method of PWMs or MOMs is employed. Moreover allowing bias of estimates one can get the gain in variance and in mean square error of large quantiles even if ML method is used.     

Probabilistic assessment of earthquake recurrence in the January 26, 2001 earthquake region of Gujrat, India

Kutch region of Gujrat is one of the most seismic prone regions of India. Recently, it has been rocked by a large earthquake (M _w = 7.7) on January 26, 2001. The probabilities of occurrence of large earthquake (M≥6.0 and M≥5.0) in a specified interval of time for different elapsed times have been estimated on the basis of observed time-intervals between the large earthquakes (M≥6.0 and M≥5.0) using three probabilistic models, namely, Weibull, Gamma and Lognormal. The earthquakes of magnitude ≥5.0 covering about 180 years have been used for this analysis. However, the method of maximum likelihood estimation (MLE) has been applied for computation of earthquake hazard parameters. The mean interval of occurrence of earthquakes and standard deviation are estimated as 20.18 and 8.40 years for M≥5.0 and 36.32 and 12.49 years, for M≥6.0, respectively, for this region. For the earthquakes M≥5.0, the estimated cumulative probability reaches 0.8 after about 27 years for Lognormal and Gamma models and about 28 years for Weibull model while it reaches 0.9 after about 32 years for all the models. However, for the earthquakes M≥6.0, the estimated cumulative probability reaches 0.8 after about 47 years for all the models while it reaches 0.9 after about 53, 54 and 55 years for Weibull, Gamma and Lognormal model, respectively. The conditional probability also reaches about 0.8 to 0.9 for the time period of 28 to 40 years and 50 to 60 years for M≥5.0 and M≥6.0, respectively, for all the models. The probability of occurrence of an earthquake is very high between 28 to 42 years for the magnitudes ≥5.0 and between 47 to 55 years for the magnitudes ≥6.0, respectively, past from the last earthquake (2001).     

The generalized Pareto sum

The generalized Pareto distribution has received much popularity as models for extreme events in hydrological sciences. In this note, the important problem of the sum of two independent generalized Pareto random variables is considered. Exact analytical expressions for the probability distribution of the sum are derived and a detailed application to drought data from Nebraska is provided. Copyright © 2007 John Wiley & Sons, Ltd.     

Models for extremes using the extended three-parameter Burr XII system with application to flood frequency analysis / Modèles d’extrêmes utilisant le système Burr XII étendu à trois paramètres et application à l’analyse fréquentielle des crues

Abstract

Abstract A new theoretically-based distribution in frequency analysis is proposed. The extended three-parameter Burr XII distribution includes the generalized Pareto distribution, which is used to model the exceedences over threshold; log-logistic distribution, which is also advocated in flood frequency analysis; and Weibull distribution, which is a part of the generalized extreme value distribution used to model annual maxima as special cases. The extended Burr distribution is flexible to approximate extreme value distribution. Note that both the generalized Pareto and generalized extreme value distributions are limiting results in modelling the exceedences over threshold and block extremes, respectively. From a modelling perspective, generalization might be necessary in order to obtain a better fit. The extended three-parameter Burr XII distribution is therefore a meaningful candidate distribution in the frequency analysis. Maximum likelihood estimation for this distribution is investigated in the paper. The use of the extended three-parameter Burr XII distribution is demonstrated using data from China.     

Comparative evaluation of Poisson partial duration models for mixed flood populations

Simple homogeneous formulations of two extreme value partial duration flood models are compared to more sophisticated compound formulations in terms of asymptotic performance of quantile estimates. The compound model formulations were developed to model flood series resulting from mixed climatological processes. It was found that only in the case of marked nonhomogeneity in the data samples did the compound formulation of the models offer significant advantages in terms of variance of quantile estimates. However, the estimates from the homogeneous model were significantly biased in the negative direction. This negative bias of quantile estimates from the simple model was even more pronounced when the more sophisticated Weibull model was used as the base.     

Comparative evaluation of Poisson partial duration models for mixed flood populations

Simple homogeneous formulations of two extreme value partial duration flood models are compared to more sophisticated compound formulations in terms of asymptotic performance of quantile estimates. The compound model formulations were developed to model flood series resulting from mixed climatological processes. It was found that only in the case of marked nonhomogeneity in the data samples did the compound formulation of the models offer significant advantages in terms of variance of quantile estimates. However, the estimates from the homogeneous model were significantly biased in the negative direction. This negative bias of quantile estimates from the simple model was even more pronounced when the more sophisticated Weibull model was used as the base.     

西江水文干旱历时与强度的遭遇概率分析

利用西江下游马口水文站1959 2009年月径流量数据计算径流干旱指数,经游程理论提取了水文干旱特征值.应用Copula函数分析水文干旱强度和历时之间的联合概率分布.对构建的干旱历时和强度联合分布模式进行分析,结果表明:(1)径流干旱历时和强度之间具有高关联性,秩相关系数达0.617;(2)三参数Weibull分布较好地描述了干旱历时和强度的边缘分布特征;(3)经拟合优度检验结果优选的干旱历时和强度之间的较优连接函数为Archimedean类的Gumbel-Hougaard Copula函数;(4)5～10年重现期和20年重现期的水文干旱分别达到了重旱级别和特旱级别;(5)干旱历时和强度之间的遭遇概率可为特定干旱历时与水文干旱级别或特定干旱强度与干旱历时之间的对应关系提供概率意义上的干旱特征诊断与预测.     

The distribution of daily snow water equivalent in the central Italian Alps

The statistical distribution of the daily Snow Water Equivalent (SWE) is investigated for a network of gauging stations in the Alpine part of Lombardia region, in the central Italian Alps. An event based data analysis is carried out using a 14 year long data set dating back to 1989. SWE is estimated when the new snow depth is greater than 6 cm. The SWE sample average in time is shown to be related to physiographic attributes of the gauging area, thus not being homogeneous in space. The values of SWE scaled by their average, or index value, instead show well approximated homogeneity of the second order moment, or coefficient of variation, in space. This suggests the use of a regional approach for frequency estimation of SWE. The frequency of occurrence of the normalized values of SWE is evaluated and tentatively accommodated by four probability distributions, often adopted in statistical modeling of hydrological variables. The Lognormal distribution shows the best performance. Single site distribution fitting is then carried out using the regional distribution, providing satisfactory results.     

Partial L-moments for the analysis of censored flood samples / Utilisation des L-moments partiels pour l’analyse d’échantillons tronqués de crues

Abstract

Abstract A parameter estimation method is proposed for fitting the generalized extreme value (GEV) distribution to censored flood samples. Partial L-moments (PL-moments), which are variants of L-moments and analogous to ?partial probability weighted moments?, are defined for the analysis of such flood samples. Expressions are derived to calculate PL-moments directly from uncensored annual floods, and to fit the parameters of the GEV distribution using PL-moments. Results of Monte Carlo simulation study show that sampling properties of PL-moments, with censoring flood samples of up to 30% are similar to those of simple L-moments, and also that both PL-moment and LH-moments (higher-order L-moments) have similar sampling properties. Finally, simple L-moments, LH-moments, and PL-moments are used to fit the GEV distribution to 75 annual maximum flow series of Nepalese and Irish catchments, and it is found that, in some situations, both LH- and PL-moments can produce a better fit to the larger flow values than simple L-moments.     

Estimation of confidence intervals of quantiles for the Weibull distribution

 Estimation of confidence limits and intervals for the two- and three-parameter Weibull distributions are presented based on the methods of moment (MOM), probability weighted moments (PWM), and maximum likelihood (ML). The asymptotic variances of the MOM, PWM, and ML quantile estimators are derived as a function of the sample size, return period, and parameters. Such variances can be used for estimating the confidence limits and confidence intervals of the population quantiles. Except for the two-parameter Weibull model, the formulas obtained do not have simple forms but can be evaluated numerically. Simulation experiments were performed to verify the applicability of the derived confidence intervals of quantiles. The results show that overall, the ML method for estimating the confidence limits performs better than the other two methods in terms of bias and mean square error. This is specially so for γ≥0.5 even for small sample sizes (e.g. N=10). However, the drawback of the ML method for determining the confidence limits is that it requires that the shape parameter be bigger than 2. The Weibull model based on the MOM, ML, and PWM estimation methods was applied to fit the distribution of annual 7-day low flows and 6-h maximum annual rainfall data. The results showed that the differences in the estimated quantiles based on the three methods are not large, generally are less than 10%. However, the differences between the confidence limits and confidence intervals obtained by the three estimation methods may be more significant. For instance, for the 7-day low flows the ratio between the estimated confidence interval to the estimated quantile based on ML is about 17% for T≥2 while it is about 30% for estimation based on MOM and PWM methods. In addition, the analysis of the rainfall data using the three-parameter Weibull showed that while ML parameters can be estimated, the corresponding confidence limits and intervals could not be found because the shape parameter was smaller than 2.