# L-moments and Bayesian inference for probabilistic risk assessment with scarce samples that include extremes

About: This article is published in Reliability Engineering & System Safety.The article was published on 2023-03-01. It has received None citations till now. The article focuses on the topics: Probabilistic risk assessment & Inference.

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^{1}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

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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.

1,147 citations

01 Jan 2011

TL;DR: This article reviews multivariate theory, distinguishing asymptotic independence and dependence models, followed by a description of models for spatial and spatiotemporal extreme events, and discusses inference and describe two applications.

Abstract: Statistics of extremes concerns inference for rare events. Often the events have never yet been observed, and their probabilities must therefore be estimated by extrapolation of tail models fitted to available data. Because data concerning the event of interest may be very limited, efficient methods of inference play an important role. This article reviews this domain, emphasizing current research topics. We first sketch the classical theory of extremes for maxima and threshold exceedances of stationary series. We then review multivariate theory, distinguishing asymptotic independence and dependence models, followed by a description of models for spatial and spatiotemporal extreme events. Finally, we discuss inference and describe two applications. Animations illustrate some of the main ideas.

836 citations

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TL;DR: It is shown that, for most types of radial basis functions that are considered in this paper, convergence can be achieved without further assumptions on the objective function.

Abstract: We introduce a method that aims to find the global minimum of a continuous nonconvex function on a compact subset of \dRd It is assumed that function evaluations are expensive and that no additional information is available Radial basis function interpolation is used to define a utility function The maximizer of this function is the next point where the objective function is evaluated We show that, for most types of radial basis functions that are considered in this paper, convergence can be achieved without further assumptions on the objective function Besides, it turns out that our method is closely related to a statistical global optimization method, the P-algorithm A general framework for both methods is presented Finally, a few numerical examples show that on the set of Dixon-Szego test functions our method yields favourable results in comparison to other global optimization methods

793 citations

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TL;DR: In this article, the authors presented a series of curves such that one or other of them would agree with any observational or theoretical frequency curve of positive ordinates to the following extent: (i) the areas should be equal; (ii) the mean abscissa or centroid vertical should be the same for the two curves; (iii) the standard deviation (or, what amounts to the same thing, the second moment coefficient) about this centroid horizontal vertical, and (iv) to (v) the third and fourth moment coefficients about this horizontal vertical should also be the

Abstract: In a memoir presented to the Royal Society in 1894, I dealt with skew variation in homogeneous material. The object of that memoir was to obtain a series of curves such that one or other of them would agree with any observational or theoretical frequency curve of positive ordinates to the following extent :—(i) The areas should be equal; (ii) the mean abscissa or centroid vertical should be the same for the two curves; (iii) the standard deviation (or, what amounts to the same thing, the second moment coefficient) about this centroid vertical should be the same, and (iv) to (v) the third and fourth moment coefficients should also be the same. If μs be the s th moment coefficient about the mean vertical, N the area, x ¯ be the mean abscissa, σ = √ μ 2 the standard deviation, β 1 = μ 32/ μ 23, β 4 = μ 4/ μ 22, then the equality for the two curves of N, x ¯, σ, β 1 and β 2 leads almost invariably in the case of frequency to excellency of fit. Indeed, badness of fit generally arises from either heterogeniety, or the difficulty in certain cases of accurately determining from the data provided the true values of the moment coefficients, e. g ., especially in J- and U-shaped frequency distributions, or distributions without high contact at the terminals ; here the usual method of correcting the raw moments for sub-ranges of record fails. Having found a curve which corresponded to the skew binomial in the same manner as the normal curve of errors to the symmetrical binomial with finite index, it occurred to me that a development of the process applied to the hypergeometrical series would achieve the result I was in search of, i. e ., a curve whose constants would be determined by the observational values of N, x ¯, σ, β 1 and β 2.

294 citations