On an improvement of Hill and some other estimators

Paulauskas, Vygantas; Vaičiulis, Marijus

doi:10.1007/s10986-013-9212-x

Cited by 35 publications

(43 citation statements)

References 22 publications

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“…where < 0 is a second-order shape parameter, which measures the rate of convergence and |A(t)| must then be a regular variation function with index . To obtain additional information on the asymptotic bias of the EVI estimators, we shall further assume a third-order condition, ruling now the rate of convergence in (9), and which guarantees that…”

Section: Second-and Third-order Conditions For a Heavy Right Tail Modelmentioning

confidence: 99%

“…In this paper, we shall consider the following generalizations of the classical Hill estimator, defined as

{\overset{ξ}{true}}_{n}^{scriptP false(ω false)} false(k false) : = \frac{ω}{k} true \sum_{i = 1}^{k} {((), \frac{i}{k})}^{ω - 1} U_{i}, 1 \leq k < n

and

{\overset{ξ}{true}}_{n}^{scriptL false(ω false)} false(k false) : = \frac{1}{k normalΓ false(ω false)} true \sum_{i = 1}^{k} {((), - \ln ((), \frac{i}{k}))}^{ω - 1} U_{i}, 1 \leq k < n

where ω >0 is a tuning parameter. Different classes of generalized Hill estimators can be found in previous studies . The classes of EVI‐estimators in and belong to the class of kernel EVI‐estimators studied in Csörgő et al and Goegebeur et al,

{\overset{ξ}{true}}_{n}^{scriptK} false(k false) = \frac{1}{k} true \sum_{i = 1}^{n} scriptK ((), \frac{i}{k}) U_{i}, 1 \leq k < n,

where

scriptK

is a kernel function integrating to one.…”

Section: Introductionmentioning

confidence: 99%

“…Different classes of generalized Hill estimators can be found in previous studies. [7][8][9][10][11][12] The classes of EVI-estimators in (5) and (6) belong to the class of kernel EVI-estimators studied in Csörgő et al 13 and Goegebeur et al, 14̂…”

Section: Introductionmentioning

confidence: 99%

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Reduced‐bias kernel estimators of a positive extreme value index

Caeiro

Henriques‐Rodrigues

2019

Math Methods in App Sciences

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In this paper, we deal with the semi‐parametric estimation of the extreme value index, an important parameter in extreme value analysis. It is well known that many classic estimators, such as the Hill estimator, reveal a strong bias. This problem motivated the study of two classes of kernel estimators. Those classes generalize the classical Hill estimator and have a tuning parameter that enables us to modify the asymptotic mean squared error and eventually to improve their efficiency. Since the improvement in efficiency is not very expressive, we also study new reduced bias estimators based on the two classes of kernel statistics. Under suitable conditions, we prove their asymptotic normality. Moreover, an asymptotic comparison, at optimal levels, shows that the new classes of reduced bias estimators are more efficient than other reduced bias estimator from the literature. An illustration of the finite sample behaviour of the kernel reduced‐bias estimators is also provided through the analysis of a data set in the field of insurance.

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Section: Second-and Third-order Conditions For a Heavy Right Tail Modelmentioning

confidence: 99%

“…In this paper, we shall consider the following generalizations of the classical Hill estimator, defined as

{\overset{ξ}{true}}_{n}^{scriptP false(ω false)} false(k false) : = \frac{ω}{k} true \sum_{i = 1}^{k} {((), \frac{i}{k})}^{ω - 1} U_{i}, 1 \leq k < n

and

{\overset{ξ}{true}}_{n}^{scriptL false(ω false)} false(k false) : = \frac{1}{k normalΓ false(ω false)} true \sum_{i = 1}^{k} {((), - \ln ((), \frac{i}{k}))}^{ω - 1} U_{i}, 1 \leq k < n

{\overset{ξ}{true}}_{n}^{scriptK} false(k false) = \frac{1}{k} true \sum_{i = 1}^{n} scriptK ((), \frac{i}{k}) U_{i}, 1 \leq k < n,

where

scriptK

is a kernel function integrating to one.…”

Section: Introductionmentioning

confidence: 99%

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Reduced‐bias kernel estimators of a positive extreme value index

Caeiro

Henriques‐Rodrigues

2019

Math Methods in App Sciences

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“…See also, Paulaskas and Vaičiulis (2013). For a MOP location-invariant EVI-estimation, see Gomes et al (2016b).…”

Section: Asymptotic Comparison Of Optimal Mop Evi-estimatorsmentioning

confidence: 99%

Mean-of-order p reduced-bias extreme value index estimation under a third-order framework

et al. 2016

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Reduced-bias versions of a very simple generalization of the 'classical' Hill estimator of a positive extreme value index (EVI) are put forward. The Hill estimator can be regarded as the logarithm of the mean-of-order-0 of a certain set of statistics. Instead of such a geometric mean, it is sensible to consider the mean-oforder-p (MOP) of those statistics, with p real. Under a third-order framework, the asymptotic behaviour of the MOP, optimal MOP and associated reduced-bias classes of EVI-estimators is derived. Information on the dominant non-null asymptotic bias is also provided so that we can deal with an asymptotic comparison at optimal levels of some of those classes. Large-scale Monte-Carlo simulation experiments are undertaken to provide finite sample comparisons.

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“…The Hill estimator procedure with the score moment estimator has been investigated in Stehlík et al (2012) for optimal testing for normality against Pareto tail. Recently, nice generalizations of tHill have been published, see Brilhante et al (2013), Paulauskas and Vaiciulis (2013) and Beran et al (2014). Resnick and Starica (1993) generalize the Hill estimator for more general settings with possibly dependent data.…”

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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On an improvement of Hill and some other estimators

Cited by 35 publications

References 22 publications

Reduced‐bias kernel estimators of a positive extreme value index

Reduced‐bias kernel estimators of a positive extreme value index

Mean-of-order p reduced-bias extreme value index estimation under a third-order framework

Weak properties and robustness of t-Hill estimators

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