A robust estimator for the tail index of Pareto-type distributions

Vandewalle, Björn; Beirlant, Jan; Christmann, Andreas; Hubert, Mia

doi:10.1016/j.csda.2007.01.003

Cited by 73 publications

(60 citation statements)

References 17 publications

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“…Following Scott (2004), Vandewalle et al (2007) make use of the fact that in derivation of (22) it is assumed that only f is a real density function, but not necessarily the model f θ . Hence, also an incomplete mixture model wf θ can be considered…”

Section: Partial Density Component Estimatormentioning

confidence: 99%

“…This relative computational simplicity of the PITSE can be considered as an argument in its favour, especially if the results of our comparison would suggest that it delivers a satisfactory degree of protection against data contamination and model deviation. Vandewalle et al (2007) introduced a robust estimator for the tail index of Paretotype distributions based on the so-called partial density component estimation, which extends the integrated squared error approach (Scott 2001(Scott , 2004. 7 In general, the approach of Vandewalle et al (2007) uses a minimum distance criterion based on integrated squared error as a measure of discrepancy between the estimated density function and the true but unknown density.…”

Section: Probability Integral Transform Statisticmentioning

confidence: 99%

“…Vandewalle et al (2007) introduced a robust estimator for the tail index of Paretotype distributions based on the so-called partial density component estimation, which extends the integrated squared error approach (Scott 2001(Scott , 2004. 7 In general, the approach of Vandewalle et al (2007) uses a minimum distance criterion based on integrated squared error as a measure of discrepancy between the estimated density function and the true but unknown density. More specifically, they use the approach of Scott (2001Scott ( , 2004, who considered estimation of mixture models by this method.…”

Section: Probability Integral Transform Statisticmentioning

confidence: 99%

“…We investigate the properties of the estimators by Monte Carlo simulations under various data contaminations and model deviations, which produce outliers that can be found in real data sets. In particular, the paper compares the optimal bias-robust estimator (OBRE) (Hampel et al 1986;Victoria-Feser and Ronchetti 1994), the weighted maximum likelihood estimator (WMLE) (Dupuis and Morgenthaler 2002;Dupuis and Victoria-Feser 2006), the generalized median estimator (GME) Serfling 2000, 2001a), the partial density component estimator (PDCE) (Vandewalle et al 2007) and the probability integral transform statistic estimator (PITSE) (Finkelstein et al 2006). 4 The OBRE, WMLE and PDCE have been applied in robust modelling of income distribution Victoria-Feser 2007, 2008;Alfons et al 2013).…”

Section: Introductionmentioning

confidence: 99%

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Robust estimation of the Pareto tail index: a Monte Carlo analysis

Brzeziński

2015

Empir Econ

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The Pareto distribution is often used in many areas of economics to model the right tail of heavy-tailed distributions. However, the standard method of estimating the shape parameter (the Pareto tail index) of this distribution-the maximum likelihood estimator (MLE), also known as the Hill estimator-is non-robust, in the sense that it is very sensitive to extreme observations, data contamination or model deviation. In recent years, a number of robust estimators for the Pareto tail index have been proposed, which correct the deficiency of the MLE. However, little is known about the performance of these estimators in small-sample setting, which often occurs in practice. This paper investigates the small-sample properties of the most popular robust estimators for the Pareto tail index, including the optimal B-robust estimator (Victoria-Feser and Ronchetti in Can J Stat 22:247-258, 1994), the weighted maximum likelihood estimator (Dupuis and Victoria-Feser in Can J Stat 34:639-658, 2006), the generalized median estimator (Brazauskas and Serfling in Extremes 3:231-249, 2001), the partial density component estimator (Vandewalle et al. in Comput Stat Data Anal 51:6252-6268, 2007), and the probability integral transform statistic estimator (PITSE) (Finkelstein et al. in N Am Actuar J 10:1-10, 2006). Monte Carlo simulations show that the PITSE offers the desired compromise between ease of use and power to protect against outliers in the small-sample setting.

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Section: Partial Density Component Estimatormentioning

confidence: 99%

Section: Probability Integral Transform Statisticmentioning

confidence: 99%

Section: Probability Integral Transform Statisticmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Robust estimation of the Pareto tail index: a Monte Carlo analysis

Brzeziński

2015

Empir Econ

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“…However, the influence function of Hill estimator is slowly increasing but unbounded, thus, the Hill procedure is not robust. Further approaches of robustifying the original Hill estimator were given in Beran and Schell (2012) and Vandewalle et al (2007). In Fabián (2001) a new score method of score moment estimators has been proposed.…”

Section: Introductionmentioning

confidence: 99%

Weak properties and robustness of t-Hill estimators

et al. 2016

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We describe a novel method of heavy tails estimation based on transformed score (t-score). Based on a new score moment method we derive the t-Hill estimator, which estimates the extreme value index of a distribution function with regularly varying tail. t-Hill estimator is distribution sensitive, thus it differs in e.g. Pareto and log-gamma case. Here, we study both forms of the estimator, i.e. t-Hill and t-lgHill. For both estimators we prove weak consistency in moving average settings as well as the asymptotic normality of t-lgHill estimator in iid setting. In cases of contamination with heavier tails than the tail of original sample, t-Hill outperforms several robust tail estimators, especially in small samples. A simulation study emphasizes the fact that the level of contamination is playing a crucial role. The larger the contamination, the better are the t-score moment estimates. The reason for this is the bounded t-score of heavy-tailed distributions (and, consequently, bounded influence functions of the estimators). We illustrate the developed methodology on a small sample data set of stake measurements from Guanaco glacier in Chile.

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Threshold detection for the generalized Pareto distribution: Review of representative methods and application to the NOAA NCDC daily rainfall database

Langousis

Mamalakis

Puliga

et al. 2016

Water Resources Research

115

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In extreme excess modeling, one fits a generalized Pareto (GP) distribution to rainfall excesses above a properly selected threshold u. The latter is generally determined using various approaches, such as nonparametric methods that are intended to locate the changing point between extreme and nonextreme regions of the data, graphical methods where one studies the dependence of GP‐related metrics on the threshold level u, and Goodness‐of‐Fit (GoF) metrics that, for a certain level of significance, locate the lowest threshold u that a GP distribution model is applicable. Here we review representative methods for GP threshold detection, discuss fundamental differences in their theoretical bases, and apply them to 1714 overcentennial daily rainfall records from the NOAA‐NCDC database. We find that nonparametric methods are generally not reliable, while methods that are based on GP asymptotic properties lead to unrealistically high threshold and shape parameter estimates. The latter is justified by theoretical arguments, and it is especially the case in rainfall applications, where the shape parameter of the GP distribution is low; i.e., on the order of 0.1–0.2. Better performance is demonstrated by graphical methods and GoF metrics that rely on preasymptotic properties of the GP distribution. For daily rainfall, we find that GP threshold estimates range between 2 and 12 mm/d with a mean value of 6.5 mm/d, while the existence of quantization in the empirical records, as well as variations in their size, constitute the two most important factors that may significantly affect the accuracy of the obtained results.

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A robust estimator for the tail index of Pareto-type distributions

Cited by 73 publications

References 17 publications

Robust estimation of the Pareto tail index: a Monte Carlo analysis

Robust estimation of the Pareto tail index: a Monte Carlo analysis

Weak properties and robustness of t-Hill estimators

Threshold detection for the generalized Pareto distribution: Review of representative methods and application to the NOAA NCDC daily rainfall database

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