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Journal ArticleDOI

A Comparison of Maximum Likelihood and Bayesian Estimators for the Three-Parameter Weibull Distribution

TL;DR: In this article, maximum likelihood and Bayesian estimators are developed and compared for the three-parameter Weibull distribution, and the authors conclude that there are practical advantages to the Bayesian approach.
Abstract: Maximum likelihood and Bayesian estimators are developed and compared for the three‐parameter Weibull distribution. For the data analysed in the paper, the two sets of estimators are found to be very different. The reasons for this are explored, and ways of reducing the discrepancy, including reparametrization, are investigated. Our overall conclusion is that there are practical advantages to the Bayesian approach.
Citations
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Journal ArticleDOI
TL;DR: In this article, Modelling Extremal Events for Insurance and Finance is discussed. But the authors focus on the modeling of extreme events for insurance and finance, and do not consider the effects of cyber-attacks.
Abstract: (2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.

2,729 citations

Journal ArticleDOI
TL;DR: In this article, the authors discuss the analysis of the extremes of data by modelling the sizes and occurrence of exceedances over high thresholds, and the natural distribution for such exceedances, the generalized Pareto distribution, is described and its properties elucidated.
Abstract: We discuss the analysis of the extremes of data by modelling the sizes and occurrence of exceedances over high thresholds. The natural distribution for such exceedances, the generalized Pareto distribution, is described and its properties elucidated. Estimation and model-checking procedures for univariate and regression data are developed, and the influence of and information contained in the most extreme observations in a sample are studied. Models for seasonality and serial dependence in the point process of exceedances are described. Sets of data on river flows and wave heights are discussed, and an application to the siting of nuclear installations is described

1,503 citations

Journal ArticleDOI
TL;DR: In this paper, a fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities.
Abstract: Summary. A fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities. The approach is sufficiently general to encompass some recent proposals in the literature, variously related to the skew normal distribution. The special case of skew elliptical densities is examined in detail, establishing connections with existing similar work. The final part of the paper specializes further to a form of multivariate skew t-density. Likelihood inference for this distribution is examined, and it is illustrated with numerical examples.

1,215 citations

Journal ArticleDOI
TL;DR: In this paper, a general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities.
Abstract: A fairly general procedure is studied to perturbate a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities. The approach is general enough to encompass a number of recent proposals in the literature, variously related to the skew normal distribution. The special case of skew elliptical densities is examined in detail, establishing connections with existing similar work. The final part of the paper specializes further to a form of multivariate skew $t$ density. Likelihood inference for this distribution is examined, and it is illustrated with numerical examples.

1,174 citations

Journal ArticleDOI
TL;DR: In this article, a hierarchical model for the intensity and frequency of extreme precipitation events in a region in Colorado is presented, where the authors assume that the regional extreme precipitation is driven by a latent spatial process characterized by geographical and climatological covariates.
Abstract: Quantification of precipitation extremes is important for flood planning purposes, and a common measure of extreme events is the r-year return level. We present a method for producing maps of precipitation return levels and uncertainty measures and apply it to a region in Colorado. Separate hierarchical models are constructed for the intensity and the frequency of extreme precipitation events. For intensity, we model daily precipitation above a high threshold at 56 weather stations with the generalized Pareto distribution. For frequency, we model the number of exceedances at the stations as binomial random variables. Both models assume that the regional extreme precipitation is driven by a latent spatial process characterized by geographical and climatological covariates. Effects not fully described by the covariates are captured by spatial structure in the hierarchies. Spatial methods were improved by working in a space with climatological coordinates. Inference is provided by a Markov chain Monte Carlo ...

513 citations


Cites background from "A Comparison of Maximum Likelihood ..."

  • ...Although there have been several studies using Bayesian methods in extremes (e.g., Smith and Naylor 1987; Coles and Tawn 1996a; Bottolo, Consonni, Dellaportas, and Lijoi 2003; Stephenson and Tawn 2005) only a few have built models that borrowed strength across different spatial locations....

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References
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Book
01 Jan 1958

4,153 citations

Journal ArticleDOI

2,421 citations


"A Comparison of Maximum Likelihood ..." refers background or methods in this paper

  • ...In a review in Technometrics, Lawless (1983a) claimed to have identified 80 relevant papers in that journal alone, and rightly criticised the disproportionate attention given to the distribution....

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  • ...A consequence of this is that other functionals of the distribution, such as quantiles, will be much better estimated than 0 by the method of maximum likelihood, a point already made by Lawless (1983b), Section 4....

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Journal ArticleDOI

1,803 citations


"A Comparison of Maximum Likelihood ..." refers background in this paper

  • ...1), which is the sample-minimum version of Gumbel's (1958) first asymptotic distribution....

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Journal ArticleDOI
TL;DR: In this article, the authors consider maximum likelihood estimation of the parameters of a probability density which is zero for x 2, the information matrix is finite and the classical asymptotic properties continue to hold.
Abstract: SUMMARY We consider maximum likelihood estimation of the parameters of a probability density which is zero for x 2, the information matrix is finite and the classical asymptotic properties continue to hold. For cx = 2 the maximum likelihood estimators are asymptotically efficient and normally distributed, but with a different rate of convergence. For 1 < a < 2, the maximum likelihood estimators exist in general, but are not asymptotically normal, while the question of asymptotic efficiency is still unsolved. For cx < 1, the maximum likelihood estimators may not exist at all, but alternatives are proposed. All these results are already known for the case of a single unknown location parameter 0, but are here extended to the case in which there are additional unknown parameters. The paper concludes with a discussion of the applications in extreme value theory.

826 citations

Journal ArticleDOI
TL;DR: In this paper, a general method of estimating parameters in continuous univariate distributions is proposed, which is especially suited to cases where one of the parameters is an unknown shifted origin and is shown to give consistent estimators with asymptotic efficiency equal to ML estimators when these exist.
Abstract: SUMMARY A general method of estimating parameters in continuous univariate distributions is proposed. It is especially suited to cases where one of the parameters is an unknown shifted origin. This occurs, for example, in the three-parameter lognormal, gamma and Weibull models. For such distributions it is known that maximum likelihood (ML) estimation can break down because the likelihood is unbounded and this can lead to inconsistent estimators. Properties of the proposed method are described. In particular it is shown to give consistent estimators with asymptotic efficiency equal to ML estimators when these exist. Moreover it gives consistent, asymptotically efficient estimators in situations where ML fails. Examples are given including numerical ones showing the advantages of the method.

482 citations