Probability-weighted Moments Estimators Research Articles

Bootstrapping Extreme Value Estimators

This article develops a bootstrap analogue of the well-known asymptotic expansion of the tail quantile process in extreme value theory. One application of this result is to construct confidence intervals for estimators of the extreme value index such as the Probability Weighted Moment (PWM) estimator. For the peaks-over-threshold method, we show the bootstrap consistency of the confidence intervals. By contrast, the asymptotic expansion of the quantile process of the bootstrapped block maxima does not lead to a similar consistency result for the PWM estimator using the block maxima method. For both methods, We show by simulations that the sample variance of bootstrapped estimates can be a good approximation for the asymptotic variance of the original estimator. Supplementary materials for this article are available online.

Use of Nonconventional Dispersion Measures to Improve the Efficiency of Ratio-Type Estimators of Variance in the Presence of Outliers

The use of auxiliary information in survey sampling to enhance the efficiency of the estimators of population parameters is a common phenomenon. Generally, the ratio and regression estimators are developed by using the known information on conventional parameters of the auxiliary variables, such as variance, coefficient of variation, coefficient of skewness, coefficient of kurtosis, or correlation between the study and auxiliary variable. The efficiency of these estimators is dubious in the presence of outliers in the data and a nonsymmetrical population. This study presents improved variance estimators under simple random sampling without replacement with the assumption that the information on some nonconventional dispersion measures of the auxiliary variable is readily available. These auxiliary variables can be the inter-decile range, sample inter-quartile range, probability-weighted moment estimator, Gini mean difference estimator, Downton’s estimator, median absolute deviation from the median, and so forth. The algebraic expressions for the bias and mean square error of the proposed estimators are obtained and the efficiency conditions are derived to compare with the existing estimators. The percentage relative efficiencies are used to numerically compare the results of the proposed estimators with the existing estimators by using real datasets, indicating the supremacy of the suggested estimators.

The coupling method in extreme value theory

A coupling method is developed for univariate extreme value theory, providing an alternative to the use of the tail empirical/quantile processes. We emphasize the Peak-over-Threshold approach that approximates the distribution above high threshold by the Generalized Pareto Distribution (GPD) and compare the empirical distribution of exceedances to the empirical distribution associated with the limit GPD model. Sharp bounds for their Wasserstein distance in the second order Wasserstein space are provided. As an application, we recover standard results on the asymptotic behavior of the Hill estimator, the Weissman extreme quantile estimator or the probability weighted moment estimators, shedding some new light on the theory.

Maximum likelihood estimators based on the block maxima method

The extreme value index is a fundamental parameter in univariate Extreme Value Theory (EVT). It captures the tail behavior of a distribution and is central in the extrapolation beyond observations. Among other semi-parametric methods (such as the popular Hill estimator), the Block Maxima (BM) and Peaks-Over-Threshold (POT) methods are widely used for assessing the extreme value index and related normalizing constants. We provide asymptotic theory for the maximum likelihood estimators (MLE) based on the BM method for independent and identically distributed observations in the max-domain of attraction of some extreme value distribution. Our main result is the asymptotic normality of the MLE with a non-trivial bias depending on the extreme value index and on the so-called second-order parameter. Our approach combines asymptotic expansions of the likelihood process and of the empirical quantile process of block maxima. The results permit to complete the comparison of common semi-parametric estimators in EVT (MLE and probability weighted moment estimators based on the POT or BM methods) through their asymptotic variances, biases and optimal mean square errors.

Modelling of Hydrological Drought Events in the Upper Tana Basin of Kenya

Drought is a recurring hazard which affects many parts of Kenya. In most countries in Sub-Saharan Africa, agriculture which is predominantly rain-fed is the main stay of the economies is highly prone to the impacts of drought which, whenever it occurs, leads to serious socioeconomic challenges at various levels. The study of drought duration, magnitude and severity have relevance in many areas such as waste load allocations, issuance of pollution discharge permits, location of treatment plants and sanitary landfills, determination of allowable water transfers and withdrawals both within, between and outside the affected areas and determination of minimum downstream release requirements for hydropower water supply, cooling plants and other facilities. Knowledge of the frequency distribution of the drought events is useful as it contributes to the assessment of drought risks which have implications on the long term ecological, economic and social well being of the biological and human communities that make use of water from the various streams in a basin. In this study five frequency distributions were fitted to drought duration and severity as determined from discharge data from representative river gauge stations in the upper Tana Basin of Kenya. The frequency distributions fitted to the two drought events were the Generalized Normal (GN) or 3-parameter Lognormal, Generalized Extreme Value (GEV) or the Extreme Value Type III, Generalized Pareto (GPA), Pearson Type III (P3) and Generalized Logistic (GL). The distributions of best fit for the drought events were identified using the Z value obtained from the average L-moment statistics of a particular candidate distribution and the average L-moment statistics. The Z value for each homogenous region was determined from sample estimates of Lcv , Lcs and Lck that were determined from probability weighted moment estimators and the weighted means of Lcv, Lcs and Lck using records from the river gauging stations representing each hydrologically homogenous region. A frequency distribution of best fit was selected if ׀Z Dis ׀ ≤ 1.64 and the one with the lowest ׀Z Dis ׀ value selected as the distribution

Different estimation methods for the parameters of the extended Burr XII distribution

The extended three-parameter Burr XII (EBXII) distribution has recently attracted considerable attention for modeling data from various scientific fields since it yields a wide range of skewness and kurtosis values. However, it is well known that the parameter estimates have significant effects on the success of a distribution in real-life applications. In this study, modified moment estimators (MMEs) and modified probability-weighted moments estimators (MPWMEs) are used to estimate the parameters of the EBXII distribution. These two considered estimators are also compared with the commonly used maximum-likelihood, percentiles, least-squares and weighted least-squares estimators in terms of bias and efficiency via an extensive numerical simulation. The MMEs and MPWMEs are observed to perform well in varying sample cases, and the simulation results are supported with application through a real-life data set.

Bias correction in extreme value statistics with index around zero

Applying extreme value statistics in meteorology and environmental science requires accurate estimators on extreme value indices that can be around zero. Without having prior knowledge on the sign of the extreme value indices, the probability weighted moment (PWM) estimator is a favorable candidate. As most other estimators on the extreme value index, the PWM estimator bears an asymptotic bias. In this paper, we develop a bias correction procedure for the PWM estimator. Moreover, we provide bias-corrected PWM estimators for high quantiles and, when the extreme value index is negative, the endpoint of a distribution. The choice of k, the number of high order statistics used for estimation, is crucial in applications. The asymptotically unbiased PWM estimators allows the choice of higher level k, which results in a lower asymptotic variance. Moreover, since the bias-corrected PWM estimators can be applied for a wider range of k compared to the original PWM estimator, one gets more flexibility in choosing k for finite sample applications. All advantages become apparent in simulations and an environmental application on estimating “once per 10,000 years” still water level at Hoek van Holland, The Netherlands.

Emulation of simulated earthquake catalogues

In earthquake occurrence studies, the so-called q value can be considered both as one of the parameters describing the distribution of interevent times and as an index of non-extensivity. Using simulated datasets, we compare four kinds of estimators, based on principle of maximum entropy (POME), method of moments (MOM), maximum likelihood (MLE), and probability weighted moments (PWM) of the parameters (q and τ0) of the distribution of inter-events times, assumed to be a generalized Pareto distribution (GPD), as defined by Tsallis (1988) in the frame of non-extensive statistical physics. We then propose to use the unbiased version of PWM estimators to compute the q value for the distribution of inter-event times in a realistic earthquake catalogue simulated according to the epidemic type aftershock sequence (ETAS) model. Finally, we use these findings to build a statistical emulator of the q values of ETAS model. We employ treed Gaussian processes to obtain partitions of the parameter space so that the resulting model respects sharp changes in physical behaviour. The emulator is used to understand the joint effects of input parameters on the q value, exploring the relationship between ETAS model formulation and distribution of inter-event times.

Expected probability weighted moment estimator for censored flood data

Two well-known methods for estimating statistical distributions in hydrology are the Method of Moments (MOMs) and the method of probability weighted moments (PWM). This paper is concerned with the case where a part of the sample is censored. One situation where this might occur is when systematic data (e.g. from gauges) are combined with historical data, since the latter are often only reported if they exceed a high threshold. For this problem, three previously derived estimators are the “B17B” estimator, which is a direct modification of MOM to allow for partial censoring; the “partial PWM estimator”, which similarly modifies PWM; and the “expected moments algorithm” estimator, which improves on B17B by replacing a sample adjustment of the censored-data moments with a population adjustment. The present paper proposes a similar modification to the PWM estimator, resulting in the “expected probability weighted moments (EPWM)” estimator. Simulation comparisons of these four estimators and also the maximum likelihood estimator show that the EPWM method is at least competitive with the other four and in many cases the best of the five estimators.

A new look at probability-weighted moments estimators

In this article, we propose to study, in more generality, the probability-weighted moments method used by Hosking and Wallis (1987) in the case of generalized Pareto distributions which depend on two parameters γ and σ. The objective is to extend the domain of validity: γ<1/2 required in order to obtain the asymptotic properties of their estimators. By simulations, we show the efficiency of our technique. To cite this article: J. Diebolt et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).

A comparative evaluation of the estimators of the three-parameter generalized pareto distribution

The parameters and quantiles of the three-parameter generalized Pareto distribution (GPD3) were estimated using six methods for Monte Carlo generated samples. The parameter estimators were the moment estimator and its two variants, probability-weighted moment estimator, maximum likelihood estimator, and entropy estimator. Parameters were investigated using a factorial experiment. The performance of these estimators was statistically compared, with the objective of identifying the most robust estimator from amongst them.

Bayesian approach to parameter estimation of the generalized pareto distribution

Several methods have been used for estimating the parameters of the generalized Pareto distribution (GPD), namely maximum likelihood (ML), the method of moments (MOM) and the probability-weighted moments (PWM). It is known that for these estimators to exist, certain constraints have to be imposed on the range of the shape parameter,k, of the GPD. For instance, PWM and ML estimators only exist fork>−0.5 andk≤1, respectively. Moreover, and particularly for small sample sizes, the most efficient method to apply in any practical situation highly depends on a previous knowledge of the most likely values ofk. This clearly suggests the use of Bayesian techniques as a way of using prior information onk. In the present work, we address the issue of estimating the parameters of the GPD from a Bayesian point of view. The proposed approach is compared via a simulation study with ML, PWM and also with the elemental percentile method (EPM) which was developed by Castillo and Hadi (1997). The estimation procedure is then applied to two real data sets.

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.

Applying medical survival data to estimate the three-parameter Weibull distribution by the method of probability-weighted moments

The method of probability-weighted moments is used to derive estimators of parameters and quantiles of the three-parameter Weibull distribution. The properties of these estimators are studied. The results obtained are compared with those obtained by using the method of maximum likelihood. The Weibull probability distribution has numerous applications in various areas: for example, breaking strength, life expectancy, survival analysis and animal bioassay. Because of its useful applications, its parameters need to be evaluated precisely, accurately and efficiently. There is a rich literature available on its maximum likelihood estimation method. However, there is no explicit solution for the estimates of the parameters or the best linear unbiased estimates. Further, the Weibull parameters cannot be expressed explicitly as a function of the conventional moments and iterative computational methods are needed. The maximum likelihood methodology is based on large-sample theory and the method might not work well when samples are small or moderate in size. Others have proposed a class of moments called probability-weighted moments. This class seems to be interesting as a method for estimating parameters and quantiles of distributions which can be written in inverse form. Such distributions include the Gumbel, Weibull, logistic, Tukey's symmetric lambda, Thomas Wakeby, and Mielke's kappa. It has been illustrated that rather simple expressions for the parameters can be written in inverse form in terms of probability-weighted moments (PWMs) for most of these distributions. In this paper we define the PWM estimators of the parameters for the three-parameter Weibull distribution. We investigate the properties of these estimators in a medical application setting. We also examine the added influence that censored data may have on the estimates.

Parameter and quantile estimation for the generalized extreme-value distribution: a second look

The Generalized Extreme Value distribution (GEVD) was introduced by Jenkinson (1955; Quarterly Journal of the Meteorological Society81, 158–171) as a model for extremes of natural phenomena such as sea level, rainfall, river heights and air pollutants. The GEVD is a three-parameter family with a location, scale, and shape parameter. Given a set of data, there were two traditional methods for estimating the parameters: maximum likelihood and probability-weighted moments (PWM), the latter being better (Hosking et al., 1985; Technometrics27, 251–261). Castillo and Hadi (1994; Environmetrics5, 417–432) then introduced two methods based on order statistics, LMS and MED, as improvements over PWM, claiming that the performance of the PWM worsens as the shape parameter increased. Unfortunately, their comparison of the methods was flawed as they used an inappropriate approximation to the PWM equations. Thus, PWM were inaccurately discarded. In this paper, PWM are reexamined and the correct computational procedures are discussed. We show that, even for large shape parameter, PWM still compare well to estimators based on order statistics. In fact, simulation results show that PWM estimators yield smaller bias and root mean square error (RMSE) than LMS for the estimation of the parameters and the quantiles. Furthermore, PWM yield smaller bias and RMSE than MED for the estimation of the parameters, and only slightly larger bias and RMSE for the quantiles. Based on bias and RMSE, and given the existence of asymptotic results, PWM are still a better choice. Copyright © 1999 John Wiley & Sons, Ltd.

Parameter and quantile estimation for the generalized extreme‐value distribution

AbstractThe generalized extreme‐value distribution (GEVD) was introduced by Jenkinson (1955). It is now widely used to model extremes of natural and environmental data. The GEVD has three parameters: a location parameter (−∞ &lt; Λ &lt; ∞), a scale parameter (α &gt; 0) and a shape parameter (−∞ &lt; k &lt; ∞). The traditional methods of estimation (e.g., the maximum likelihood and the moments‐based methods) have problems either because the range of the distribution depends on the parameters, or because the mean and higher moments do not exist when k ≤ − 1. The currently favoured estimators are those obtained by the method of probability‐weighted moments (PWM). The PWM estimators are good for cases where −1/2 &lt; k &lt; 1/2. Outside this range of k, the PWM estimates may not exist and if they do exist they cannot be recommended because their performance worsens as k increases. In this paper, we propose a method for estimating the parameters and quantiles of the GEVD. The estimators are well‐defined for all possible combinations of parameter and sample values. They are also easy to compute as they are based on equations which involve only one variable (rather than three). A simulation study is implemented to evaluate the performance of the proposed method and to compare it with the PWM. The simulation results seem to indicate that the proposed method is comparable to the PWM for −1/2 &lt; k &lt; 1/2 but outside this range it gives a better performance. Two real‐life environmental data sets are used to illustrate the methodology.

Further research on application of probability weighted moments in estimating parameters of the Pearson type three distribution

This paper contains further research results on the recently developed method for estimating parameters of P-III distribution by using probability weighted moments (PWM). The computing procedure can be largely simplified and the PWM method is appropriate and functional for various conditions of the skewness coefficient. Statistical experiments made based on the new procedure have shown that PWM estimators are almost unbiased and compare favorably with those by the conventional moment method.

Estimation of the Generalized Extreme-Value Distribution by the Method of Probability-Weighted Moments

We use the method of probability-weighted moments to derive estimators of the parameters and quantiles of the generalized extreme-value distribution. We investigate the properties of these estimators in large samples, via asymptotic theory, and in small and moderate samples, via computer simulation. Probability-weighted moment estimators have low variance and no severe bias, and they compare favorably with estimators obtained by the methods of maximum likelihood or sextiles. The method of probability-weighted moments also yields a convenient and powerful test of whether an extreme-value distribution is of Fisher-Tippett Type I, II, or III.

Estimation of the Generalized Extreme-Value Distribution by the Method of Probability-Weighted Moments

We use the method of probability-weighted moments to derive estimators of the parameters and quantiles of the generalized extreme-value distribution. We investigate the properties of these estimators in large samples, via asymptotic theory, and in small and moderate samples, via computer simulation. Probability-weighted moment estimators have low variance and no severe bias, and they compare favorably with estimators obtained by the methods of maximum likelihood or sextiles. The method of probability-weighted moments also yields a convenient and powerful test of whether an extreme-value distribution is of Fisher-Tippett Type I, II, or III.

Estimating a regional flood frequency distribution

When floods at different sites are assumed to arise from the same distribution except for scale (U.S. Geological Survey's Index Flood Method), one can attempt to identify the dimensionless flood‐flow frequency distribution by normalizing small samples by each sample's sample mean. Unfortunately, the resultant curve may be a poor description of the true dimensionless flood distribution. This problem can be overcome by working with the logarithms of the peak flow values and using unbiased moment or probability weighted moment estimators. However, the correlation among concurrent flood flows in a region is shown to place severe limits the accuracy with which a distribution's moments can be estimated with many such correlated records.