Journal ArticleDOI

# Probability Weighted Moments: Definition and Relation to Parameters of Several Distributions Expressable in Inverse Form

01 Oct 1979-Water Resources Research (John Wiley & Sons, Ltd)-Vol. 15, Iss: 5, pp 1049-1054
TL;DR: In this article, Probability weighted moments are introduced and shown to be potentially useful in expressing the parameters of these distributions, such as Tukey's lambda, which may present problems in deriving their parameters by more conventional means.
Abstract: Distributions whose inverse forms are explicitly defined, such as Tukey's lambda, may present problems in deriving their parameters by more conventional means. Probability weighted moments are introduced and shown to be potentially useful in expressing the parameters of these distributions.
##### Citations
More filters
Journal ArticleDOI
TL;DR: The authors define L-moments as the expectations of certain linear combinations of order statistics, which can be defined for any random variable whose mean exists and form the basis of a general theory which covers the summarization and description of theoretical probability distributions.
Abstract: L-moments are expectations of certain linear combinations of order statistics. They can be defined for any random variable whose mean exists and form the basis of a general theory which covers the summarization and description of theoretical probability distributions, the summarization and description of observed data samples, estimation of parameters and quantiles of probability distributions, and hypothesis tests for probability distributions. The theory involves such established procedures as the use of order statistics and Gini's mean difference statistic, and gives rise to some promising innovations such as the measures of skewness and kurtosis and new methods of parameter estimation

2,668 citations

### Cites background from "Probability Weighted Moments: Defin..."

• ...Probability Weighted Moments Greenwood et al. (1979) defined probability weighted moments (PWMs) to be the quantities...

[...]

Journal ArticleDOI
Jonathan R. M. Hosking
TL;DR: In this paper, the authors use the method of probability-weighted moments to derive estimators of the parameters and quantiles of the generalized extreme-value distribution, and investigate the properties of these estimators in large samples via asymptotic theory, and in small and moderate samples, via computer simulation.
Abstract: 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.

1,275 citations

### Cites background from "Probability Weighted Moments: Defin..."

• ...Greenwood et al. (1979) generally favored the latter approach, but here we will consider the moments fl, = M t,,,o = ECX{F(X))‘I (I = 0, L2, ....

[...]

• ...Probability-weighted moments, a generalization of the usual moments of a probability distribution, were introduced by Greenwood et al. (1979)....

[...]

Journal ArticleDOI
TL;DR: In this paper, the authors show that unless the sample size is 500 or more, estimators derived by either the method of moments or probability-weighted moments are more reliable.
Abstract: The generalized Pareto distribution is a two-parameter distribution that contains uniform, exponential, and Pareto distributions as special cases. It has applications in a number of fields, including reliability studies and the analysis of environmental extreme events. Maximum likelihood estimation of the generalized Pareto distribution has previously been considered in the literature, but we show, using computer simulation, that, unless the sample size is 500 or more, estimators derived by the method of moments or the method of probability-weighted moments are more reliable. We also use computer simulation to assess the accuracy of confidence intervals for the parameters and quantiles of the generalized Pareto distribution.

1,233 citations

### Cites background or methods from "Probability Weighted Moments: Defin..."

• ...Greenwood et al. (1979) exhibited several distributions for which the relationship between the parameters of the distribution and the PWM's M, r, s is simpler than the relationship between the parameters and the conventional moments Mp, o....

[...]

• ...Property 3 of Section 1 implies a close connection between generalized Pareto and GEV distributions with equal values for their shape parameters, and, as Hosking, Wallis, and Wood (1985) remarked, applications of the GEV distributions, particularly in hydrology, usually involve the case - 2 < k < 1....

[...]

• ...Greenwood et al. (1979) exhibited several distributions for which the relationship between the parameters of the distribution and the PWM's M, r, s is simpler than the relationship between the parameters and the conventional moments Mp, o. Hosking et al. (1985) showed that efficient estimators of parameters and quantiles of the GEV distribution can be obtained using PWM's....

[...]

• ...Greenwood et al. (1979) exhibited several distributions for which the relationship between the parameters of the distribution and the PWM’s MI,.,. is simpler than the relationship between the parameters and the conventional moments M ~, 0,0, Hosking et al. (1985) showed that efficient estimators of…...

[...]

• ...Greenwood et al. (1979) exhibited several distributions for which the relationship between the parameters of the distribution and the PWM's M, r, s is simpler than the relationship between the parameters and the conventional moments Mp, o. Hosking et al. (1985) showed that efficient estimators of parameters and quantiles of the GEV distribution can be obtained using PWM's. Hosking (1986) gave a general exposition of the theory of PWM's....

[...]

Journal ArticleDOI
TL;DR: In this article, the authors describe three statistics useful in regional frequency analysis: a discordancy measure, for identifying unusual sites in a region, a heterogeneity measure, assessing whether a proposed region is homogeneous, and a goodness-of-fit measure, which assesses whether a candidate distribution provides an adequate fit to the data.
Abstract: Regional frequency analysis uses data from a number of measuring sites. A “region” is a group of sites each of which is assumed to have data drawn from the same frequency distribution. The analysis involves the assignment of sites to regions, testing whether the proposed regions are indeed homogeneous, and choice of suitable distributions to fit to each region's data. This paper describes three statistics useful in regional frequency analysis: a discordancy measure, for identifying unusual sites in a region; a heterogeneity measure, for assessing whether a proposed region is homogeneous; and a goodness-of-fit measure, for assessing whether a candidate distribution provides an adequate fit to the data. Tests based on the statistics provide objective backing for the decisions involved in regional frequency analysis. The statistics are based on the L moments [Hosking, 1990] of the at-site data.

865 citations

Journal ArticleDOI
TL;DR: In this paper, a new family of generalized distributions for double-bounded random processes with hydrological applications is described, including Kw-normal, Kw-Weibull and Kw-Gamma distributions.
Abstract: Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 462 (1980), pp. 79–88] introduced a distribution for double-bounded random processes with hydrological applications. For the first time, based on this distribution, we describe a new family of generalized distributions (denoted with the prefix ‘Kw’) to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distributions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments (PWMs) of the parent distribution. We also obtain the ordinary moments of order statistics as functions of PWMs of the baseline distribution. We use the method of maximum likelihood to fit the distributions in the new class and illustrate the potentiality of the new model with a...

742 citations

### Cites methods from "Probability Weighted Moments: Defin..."

• ...For example, for the Gumbel and Weibull distributions [15], we have...

[...]

##### References
More filters
Journal ArticleDOI
TL;DR: For a long time I have thought I was a statistician, interested in inferences from the particular to the general as mentioned in this paper. But as I have watched mathematical statistics evolve, I have had cause to wonder and to doubt.
Abstract: For a long time I have thought I was a statistician, interested in inferences from the particular to the general. But as I have watched mathematical statistics evolve, I have had cause to wonder and to doubt. And when I have pondered about why such techniques as the spectrum analysis of time series have proved so useful, it has become clear that their “dealing with fluctuations” aspects are, in many circumstances, of lesser importance than the aspects that would already have been required to deal effectively with the simpler case of very extensive data, where fluctuations would no longer be a problem. All in all, I have come to feel that my central interest is in data analysis, which I take to include, among other things: procedures for analyzing data, techniques for interpreting the results of such procedures, ways of planning the gathering of data to make its analysis easier, more precise or more accurate, and all the machinery and results of (mathematical) statistics which apply to analyzing data.

1,569 citations

Journal ArticleDOI
TL;DR: A method for generating values of continuous symmetric random variables that is relatively fast, requires essentially no computer memory, and is easy to use is developed.
Abstract: A method for generating values of continuous symmetric random variables that is relatively fast, requires essentially no computer memory, and is easy to use is developed. The method, which uses a uniform zero-one random number source, is based on the inverse function of the lambda distribution of Tukey. Since it approximates many of the continuous theoretical distributions and empirical distributions frequently used in simulations, the method should be useful to simulation practitioners.

430 citations

Journal ArticleDOI
TL;DR: The Wakeby distribution as discussed by the authors is defined to overcome this deficiency, and its analytical form proves easy to use in many applications, and it has been shown that a significant number of flood records are inadequately fit by the three-parameter log normal.
Abstract: Traditional distributions, such as the log normal distribution, inadequately model flood flows for certain records. The Wakeby distribution is defined to overcome this deficiency, and its analytical form proves easy to use in many applications. Tests show that a significant number of flood records are inadequately fit by the three-parameter log normal. The Wakeby demonstrates the ‘separation effect’ that other distributions fail to show. Most other distributions can be mimicked by the Wakeby but not always vice versa. Thus choosing to use the Wakeby does not limit the shape of the resulting curve to as great a degree as choosing a conventional distribution.

185 citations

Journal ArticleDOI
TL;DR: In this article, a family of random variables defined by the transformation Z = [Uλ - (1 - U)γ]/λ where U is uniformly distributed on [0, 1] is described with emphasis on properties of the sample range.
Abstract: Tukey introduced a family of random variables defined by the transformation Z = [Uλ - (1 - U)γ]/λ where U is uniformly distributed on [0, 1]. Some of its properties are described with emphasis on properties of the sample range. The rectangular and logistic distributions are members of this family and distributions corresponding to certain values of λ give useful approximations to the normal and t distributions. Closed form expressions are given for the expectation and coefficient of variation of the range and numerical values are computed for n = 2(1)6(2)12, 15, 20 for several values of λ. It is observed that Plackett's upper bound on the expectation of the range for samples of size n is attained for a λ distribution with λ = n − 1.

134 citations

Journal ArticleDOI
TL;DR: The relationship between the mean and the standard deviation of regional estimates of skewness for historical flood sequences is not compatible with the relations derived from several well-known distribution functions as discussed by the authors.
Abstract: The relationship between the mean and the standard deviation of regional estimates of skewness for historical flood sequences is not compatible with the relations derived from several well-known distribution functions.

127 citations