Hill Estimator Research Articles

A new extreme quantile estimator for heavy-tailed distributions

The classical estimation method for extreme quantiles of heavy-tailed distributions was presented by Weissman (J. Amer. Statist. Assoc. 73 (1978) 812–815) and makes use of the Hill estimator (Ann. Statist. 3 (1975) 1163–1174) for the positive extreme value index. This index estimator can be interpreted as an estimator of the slope in the Pareto quantile plot in case one considers regression lines passing through a fixed anchor point. In this Note we propose a new extreme quantile estimator based on an unconstrained least squares estimator of the index, introduced by Kratz and Resnick (Comm. Statist. Stochastic Models 12 (1996) 699–724) and Schultze and Steinebach (Statist. Decisions 14 (1996) 353–372) and we study its asymptotic behavior. To cite this article: A. Fils, A. Guillou, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

A Hill Type Estimator of the Weibull Tail-Coefficient

We present a new estimator of the Weibull tail-coefficient. The Weibull tail-coefficient is defined as the regular variation coefficient of the inverse cumulative hazard function. Our estimator is based on the log-spacings of the upper order statistics. Therefore, it is very similar to the Hill estimator for the extreme value index. We prove the weak consistency and the asymptotic normality of our estimator. Its asymptotic as well as its finite sample performances are compared to classical ones.

Value at Risk: Concept and It's Implementation for Indian Banking System

Value-at-Risk (VaR) has been widely promoted by the Bank for International Settlement (BIS) as well as central banks of all countries as a way of monitoring and managing market risk and as a basis for setting regulatory minimum capital standards. The revised Basle Accord, implemented in January 1998, makes it mandatory for banks to use VaR as a basis for determining the amount of regulatory capital adequate for covering market risk. For market participants like Banks and Primary Dealers (PDs) in the Indian financial sector, it has become imperative to use VaR methods to calculate the regulatory capital charge required. The RBI has issued guidelines for PDs. We have adopted three categories of VaR methods, viz., Variance-Covariance (Normal) methods including Risk-Metric, Historical Simulation (HS) and Tail-Index Based approach. The Zero-Coupon Yield Curve (ZCYC) compiled by National Stock Exchange (NSE) has been used to price the bonds as well as portfolios. Estimated VaRs are validated by carrying out back testing based on last one year's data. Empirical results show that normal methods, in particularly the Risk-Metric approach, underestimate VaR numbers substantially resulting to too many failures in backtesting. Historical simulation provides more accurate VaR estimates, and indicates capital charge higher than those obtained through normal methods. The tail-index (Hill's estimator) based method also perform quite well though at times it provides too conservative VaR estimates - higher than all other competing VaR models considered here and occasions of VaR violation in backtesting are less than expected number for each bond/portfolio.

Kernel-type estimators for the extreme value index

A large part of the theory of extreme value index estimation is developed for positive extreme value indices. The best-known estimator of a positive extreme value index is probably the Hill estimator. This estimator belongs to the category of moment estimators, but can also be interpreted as a quasi-maximum likelihood estimator. It has been generalized to a kernel-type estimator, but this kernel-type estimator can, similarly to the Hill estimator, only be used for the estimation of positive extreme value indices. In the present paper, we introduce kernel-type estimators which can be used for estimating the extreme value index over the whole (positive and negative) range. We present a number of results on their distributional behavior and compare their performance with the performance of other estimators, such as moment-type estimators for the whole range and the quasi-maximum likelihood estimator, based on the generalized Pareto distribution. We also discuss an automatic bandwidth selection method and introduce a kernel estimator for a second-order parameter, controlling the speed of convergence.

Censoring estimators of a positive tail index

In this paper, and in the context of regularly varying tails, we analyse some variants of a maximum likelihood estimator of a positive tail index γ, under a type II censoring scheme. These estimators are compared with the Hill estimator, for a Fréchet model and by means of a Monte Carlo simulation. Asymptotic normality of the estimators is derived, and a robustness simulation study of the estimators is undertaken.

Bias reduction and explicit semi-parametric estimation of the tail index

In this paper, and in a context of regularly varying tails, we analyse particular but interesting cases of the maximum likelihood and least squares estimators proposed by Feuerverger and Hall (Ann. Statist. 27 (1999) 760). All these estimators are alternatives to a well-known estimator of the tail index, the Hill estimator (Ann. Statist. 3 (1997) 1163), and jointly with the generalized jackknife estimators in Gomes et al. (Extremes 2 (2000) 207, Portug. Math. 59 (2002) 393) have essentially in mind a reduction in bias, preferably without increasing mean squared error, leading to semi-parametric estimators of the tail index with a better performance than the classical estimators, provided we may use extreme-value data relatively deep into the sample.

Subsampling the distribution of diverging statistics with applications to finance

In this paper we propose a subsampling estimator for the distribution of statistics diverging at either known or unknown rates when the underlying time series is strictly stationary and strong mixing. Based on our results we provide a detailed discussion of how to estimate extreme order statistics with dependent data and present two applications to assessing financial market risk. Our method performs well in estimating Value at Risk and provides a superior alternative to Hill's estimator in operationalizing Safety First portfolio selection.

TTT 타점법을 이용한 웹서버 파일 분포의 후미성 분석

본 논문에서는 TTT 타점법을 이용하여 웹 서버가 서비스하는 파일의 크기에 대한 통계적 분포는 꼬리부분이 두꺼운 분포라는 것을 판단하는 방법을 제시한다. TTT 타점법은 신뢰성 공학에서 사용되는 방법으로써 TTT 통계량 타점결과의 직선성으로 지수분포 여부를 판단하는 방법이다. 본 연구에서 제안하는 방법을 모의실험과 실제 운영중인 웹서버의 자료를 사용하여 실험한 결과, 기존의 방법인 Hill 추정법과 LLCD 타점법에 비하여 후미성을 정확하게 판단하고 있으며, 판단의 효율성 면에서도 그들보다 우수하다는 것을 확인하였다. 특히 제안하는 방법은 기존의 방법이 웹서버의 파일 분포판정이나 통계학에서의 파레토 분포 판정시 나타날 수 있는 판정의 오류 가능성을 개선할 수 있다는 점도 확인하였다. In this paper, we propose a method of analysis to show the heavy-tailed statistical distribution of file sizes in web servers, using TTT plot technique. TTT plot technique, a well-known method in the area of reliability engineering, determines that a distribution of samples fellows a heavy tailed one when their TTT statistical plots are lied on a straight line. We performed an intensive simulation using data gathered from real web servers. The simulation indicates that the proposed method is superior to Hill estimation technique or LLCD plot method in efficiency of data analysis. Moreover, the proposed method eliminates the possible decision error, which Pareto distribution or traditional method might cause.

Hill's estimator for the tail index of an ARMA model

This paper investigates Hill's estimator for the tail index of an ARMA model with i.i.d. residuals. Based on the estimated residuals, it is shown that Hill's estimator is asymptotically normal. This method can achieve a smaller asymptotic variance than applying Hill's estimator to the original data. These results are the same as those in Resnick and Starica (Commun. Statist.—Stochastic Models 13 (4) (1997) 703) for an AR model. However, Resnick and Starica (Commun. Statist.—Stochastic Models 13 (4) (1997) 703) imposed one more condition on the choice of sample fraction than the i.i.d. case. This condition is removed in this paper so that data-driven methods for choosing optimal sample fraction based on i.i.d. data can be applied to our case. As an auxiliary theorem, we establish the weak convergence of the tail empirical process of the estimated residuals, which may be of independent interest.

Research on the multifractal characteristics of the temporal-spatial distribution of earthquakes over New Zealand area

Using the Hill estimator, general multifractal characteristics of events in the New Zealand area have been discussed. Results show that the spatial distribution of shallow events has apparent clustering characteristics, independent of the threshold magnitude; but for deep events these characteristics are not clear. While the time interval distribution has obvious clustering characteristics both for deep and shallow events, although with a different scaling range, the Hill estimates tend to indicate that the time interval distribution has a unifractal rather than a multifractal nature. All above reveal that the seismicity nature for shallow and deep events is apparently different.

How Can Non-invariant Statistics Work in Our Benefit in the Semi-parametric Estimation of Parameters of Rare Events

In this article, and in a context of regularly varying tails, we suggest a simple generalization of the classical Hill estimator of a positive tail index γ. Such an estimator is merely the Hill estimator associated to artificially shifted data. The shift a imposed to the data is the tuning parameter of the statistical procedure of estimation. Such a tuning parameter enables us, in a great diversity of situations, to reduce the main component of the bias of Hill's estimator, getting thus estimates with stable sample paths around the target value γ, and with small mean squared error.

Maximum likelihood revisited under a semi-parametric context - estimation of the tail index

In this paper, and in a context of regularly varying tails, we study computationally the classical Maximum Likelihood (ML) estimator based on the Paretian behaviour of the excesses over a high threshold, denoted PML-estimator, a type II Censoring estimator based specifically on a Fréchet parent, denoted CENS-estimator, and two ML estimators based on the scaled log-spacings, and denoted SLS-estimators. These estimators are considered under a semi-parametric set-up, and compared with the classical Hill estimator and a Generalized Jackknife (GJ) estimator, which has essentially in mind a reduction of the bias of Hill's estimator.

A New Estimator for a Tail Index

We investigate properties of a new estimator for a tail index introduced by Davydov and co-workers. The main advantage of this estimator is the simplicity of the statistic used for the estimator. We provide results of simulation by comparing plots of our's and Hill's estimators.

A class of asymptotically unbiased semi-parametric estimators of the tail index

In this paper we consider a class of consistent semi-parametric estimators of a positive tail index γ, parameterized in atuning orcontrol parameter α. Such a control parameter enables us to have access, for any available sample, to an estimator of γ with a null dominant component of asymptotic bias, and with a reasonably flatMean Squared Error pattern, as a functional ofk, the number of top order statistics considered. Moreover, we are able to achieve a high efficiency relatively to the classical Hill estimator, provided we may have access to a larger number of top order statistics than the number needed for optimal estimation through the Hill estimator.

Generalized least-squares estimators for the thickness of heavy tails

For a probability distribution with power law tails, a log–log transformation makes the tails of the empirical distribution function resemble a straight line, leading to a least-squares estimate of the tail thickness. Taking into account the mean and covariance structure of the extreme order statistics leads to improved tail estimators, and a surprising connection with Hill's estimator.

Abelian and Tauberian Theorems on the Bias of the Hill Estimator

The bias of Hill's estimator for the positive extreme value index of a distribution is investigated in relation to the convergence rate in the regular variation property of the tail function of the common distribution of the sample and the corresponding tail quantile function. Based on the theory of generalized regular variation, natural second‐order conditions are proposed which both imply and are implied by convergence of the expectation of Hill's estimator to the extreme value index at certain rates. A comparison with second‐order conditions encountered in the literature is made.

On Adaptive Tail Index Estimation for Financial Return Models

Estimation of the tail index of stationary, fat-tailed return distributions is non-trivial since the well-known Hill estimator is optimal only under iid draws from an exact Pareto model. We provide a small sample simulation study of recently suggested adaptive estimators under ARCH-type dependence. The Hill estimator's performance is found to be dominated by a ratio estimator. Dependence increases estimation error which can remain substantial even in larger data sets. As small sample bias is related to the magnitude of the tail index, recent standard applications may have overestimated (underestimated) the risk of assets with low (high) degrees of fat-tailedness.

Likelihood Based Confidence Intervals for the Tail Index

For the estimation of the tail index of a heavy tailed distribution, one of the well-known estimators is the Hill estimator (Hill, 1975). One obvious way to construct a confidence interval for the tail index is via the normal approximation of the Hill estimator. In this paper we apply both the empirical likelihood method and the parametric likelihood method to obtaining confidence intervals for the tail index. Our limited simulation study indicates that the normal approximation method is worse than the other two methods in terms of coverage probability, and the empirical likelihood method and the parametric likelihood method are comparable.

On exponential representations of log-spacings of extreme order statistics

In Beirlant et al. (1999) and Feuerverger and Hall (1999) an exponential regression model (ERM) was introduced on the basis of scaled log-spacings between subsequent extreme order statistics from a Pareto-type distribution. This lead to the construction of new bias-corrected estimators for the tail index. In this note, under quite general conditions, asymptotic justification for this regression model is given as well as for resulting tail index estimators. Also, we discuss diagnostic methods for adaptive selection of the threshold when using the Hill (1975) estimator which follow from the ERM approach. We show how the diagnostic presented in Guillou and Hall (2001) is linked to the ERM, while a new proposal is suggested. We also provide some small sample comparisons with other existing methods.

Strong Domain of Attraction of Extreme Generalized Order Statistics

It is a well-known result in extreme value theory that the von Mises conditions imply the strong convergence of extreme order statistics. We extend this result to extreme generalized order statistics. A characterization of strong domains of attraction of joint distributions of a fixed number of extreme generalized order statistics by means of the corresponding result for generalized maxima is given. In particular, we determine the asymptotic joint distribution of (upper and lower) extreme generalized order statistics. Finally, we show that the Hill estimator based on extreme generalized order statistics is asymptotic normal.