Extreme value theory (EVT) deals with the occurrence of extreme phenomena. The
tail index is a very important parameter appearing in the estimation of the probability
of rare events. Under a semiparametric framework, inference requires the choice of a
number k of upper order statistics to be considered. This is the crux of the matter
and there is no definite formula to do it, since a small k leads to high variance and
large values of k tend to increase the bias. Several methodologies have emerged in
literature, specially concerning the most popular Hill estimator (Hill, 1975). In this
work we compare through simulation well-known procedures presented in Drees and
Kaufmann (1998), Matthys and Beirlant (2000), Beirlant et al. (2002) and de Sousa
and Michailidis (2004), with a heuristic scheme considered in Frahm et al. (2005) within
the estimation of a different tail measure but with a similar context. We will see that
the new method may be an interesting alternative.Fundação para a Ciência e Tecnologia (FCT): CEMAT (UID/MULTI/04621/2013

Ferreira, Marta Susana

Rebelo, Márcio Pereira

Universidade do Minho: RepositoriUM

XII Congreso Galego de Estat´ıstica e Investigacio´n de Operacio´nsLugo, 22, 23 y 24 de octubre de 2015On the Hill estimator: a comparison of methodsMarta Ferreira1 and Ma´rcio Rebelo21Center of Mathematics of Minho University and CEMAT2Department of Mathematics and Applications of University of MinhoABSTRACTExtreme value theory (EVT) deals with the occurrence of extreme phenomena. Thetail index is a very important parameter appearing in the estimation of the probabilityof rare events. Under a semiparametric framework, inference requires the choice of anumber k of upper order statistics to be considered. This is the crux of the matterand there is no definite formula to do it, since a small k leads to high variance andlarge values of k tend to increase the bias. Several methodologies have emerged inliterature, specially concerning the most popular Hill estimator (Hill, 1975). In thiswork we compare through simulation well-known procedures presented in Drees andKaufmann (1998), Matthys and Beirlant (2000), Beirlant et al. (2002) and de Sousaand Michailidis (2004), with a heuristic scheme considered in Frahm et al. (2005) withinthe estimation of a different tail measure but with a similar context. We will see thatthe new method may be an interesting alternative.Keywords: Extreme value theory, Tail index estimation Monte-Carlo simulations1. INTRODUCTIONExtreme value statistics are being increasingly used in the environment, engineering, finance,among other sciences, given the need to account for the possible occurrence of extreme phenomenain the modeling.Let {Xn}n≥1 be a sequence of independent and identically distributed (iid) random variables(rv’s), with common distribution function (df) F . If for some sequences of real constants an > 0and bn, n ≥ 1, the limitP (max(X1, . . . , Xn) ≤ anx+ bn) −→n→∞Gξ(x)exists for some non-degenerate function Gξ with ξ real, then F is said to belong to the max-domainof attraction of Gξ, denoted F ∈ D(Gξ). This function is called generalized extreme value (GEV)and is given byGξ(x) = exp(−(1 + ξx)−1/ξ), 1 + ξx > 0,(G0(x) = exp(−e−x)). Parameter ξ is the so-called tail index and determines the shape and weightof the tail. Thus, ξ > 0 implies a heavy tail (Fre´chet max-domain of attraction) with polynomialdecay and infinite right-end-point, whenever null means an exponential tail (Gumbel max-domainof attraction) and ξ < 0 corresponds to a light tail (Weibull max-domain of attraction) with finiteright-end-point.The tail index plays a determinant role concerning the inference within rare events like, forinstance, the estimation of an unusual high quantile (e.g. the Value-at-Risk in finance) or the dualproblem of estimating the probability of exceeding a high level x, i.e., p = 1 − F (x). The Hillestimator (Hill 1975),ξ̂k,n :=1kk∑i=1logXn−i+1:nXn−k:n,with Xn:n ≥ Xn−1:n ≥ ... ≥ X1:n being the order statistics of {Xn}n≥1, is perhaps the mostapplied in literature and assumes that F belongs to max-domain of attraction of a heavy tail(ξ > 0). Observe that it depends on the k upper order statistics, where k ≡ kn →∞ and k/n→ 0,as n→∞ (i.e., k ≡ kn is an intermediate sequence), in order to achieve consistency.The choice of k is the estimation core, being somewhat ambiguous, since small values of kresults in greater variance and large values lead to an increased bias. The plot (k, ξ̂k,n), 1 ≤ k < n,usually shows a plateau region from which we may infer the values of k where ξ̂k,n approximatesthe true value (see Figure 1). This problem often accompanies extreme-values semi-parametricinference and has been much studied in the literature. See, e.g., Beirlant et al. (2004) for a survey.0 200 400 600 800 10000.51.01.52.0kFigure 1: Hill plots of 1000 realizations of a Fre´chet model with ξ = 1.For non-null ξ, Beirlant et al. (2002) stated, under little restrictive conditions,Yi := (i+ 1) logXn−i:n ξ̂i,nXn−(i+1):n ξ̂i+1,n= ξ + b(n/k)(ik)−ρ+ ǫi, i = 1, ..., k, (1)where the error term ǫi is zero-centered and b is a positive function such that b(x)→ 0, as x→∞.The so called second order tail parameter ρ is usually replaced by ρ = −1. For more details, seeBeirlant et al. 2004 and references therein. Least squares estimators of ξ and b(n/k) derived from(1) are expressed asξ˜LSk,n = Y k − b˜LSk,n/(1− ρ) and b˜LSk,n = (1−ρ)2(1−2ρ)ρ21k∑ki=1((ik)−ρ − 11−ρ)Yi. (2)One method consists in finding the k value that corresponds to the smallest estimate of theHill’s asymptotic mean squared error (AMSE)AMSE(ξ̂k,n) =ξ2k +(b(n/k)1−ρ)2. (3)More precisely, for fixed ρ = −1, compute the least squares estimates of ξ and b(n/k) by using(2). Obtain the Hill’s AMSE in (3) by replacing ξ and b(n/k) with the respective estimates. Takek̂1opt as the value of k that minimizes the obtained estimates of the AMSE and estimate ξ as ξ̂k̂1opt,n.Alternatively, the value of k for which the respective AMSE is minimal iskopt ∼ b(n/k)−2/(1−2ρ)k−2ρ/(1−2ρ)(ξ2(1−ρ)2−2ρ)1/(1−2ρ)(4)Consider again the least squares estimates for ξ and b(n/k) by using (2) and fixed ρ = −1,for k = 3, ..., n. Compute k̂opt,k according to the expression (4). Take k̂2opt = median{k̂opt,k, k =3, ..., ⌊n2 ⌋}, where ⌊x⌋ denotes the largest integer not exceeding x. Estimate ξ by ξ̂k̂2opt,n.For more details about these two methods, see Beirlant et al. 2002 Matthys and Beirlant (2000)and references therein. We denote them, respectively, AMSE and Kopt.The adaptive procedure of Drees and Kaufmann (1998), here denoted DK, is based on resultsfor a class of Generalized Pareto models, known has Hall’s class, where the maximum randomfluctuation of√i(ξ̂i,n − ξ), i = 1, ..., k − 1, with k ≡ kn an intermediate sequence, is of order√log logn. The derived stopping time criterion seeks the optimum k under which the bias startsto dominate the variance. More precisely, for ρ = −1:1. State rn = 2.5× ξ˜ × n0.25, where ξ˜ = ξ̂2√n,n.2. Compute k˜(rn) := min{k = 1, ..., n− 1 : maxi<k√i|ξ̂i,n − ξ̂k,n| > rn}. If√i|ξ̂i,n − ξ̂k,n| > rnis not satisfied for any k, impute 0.9× rn to rn and repeat step 2. Else move to step 3.3. For ε ∈ (0, 1), usually ε = 0.7, obtaink̂DK =13(2ξ˜2)1/3(k˜(rεn)(k˜(rn))ε)1/(1−ε)The procedure considered in Sousa and Michailidis (2004) looks for an estimate of the optimalvalue k based on the Hill sum plot, (k, Sk), k = 1, ..., n− 1, where Sk = kξ̂k,n. Since E(Sk) = kξ,then the sumplot is expected to be approximately linear for the values of k where ξ̂k,n ≈ ξ. Theslope of this linear part is an estimator of ξ. Thus, an algorithm was developed to establishwhere there is a breakdown of linearity. Considering the regression model y = Xξ + δ, withy = (S1, ..., Sk)′, X = [1 i]ki=1 and δ the error term, the method is based on a sequential testingprocedure discussed in McGee and Carleton (1970). This process checks the null hypothesis thata new point y0 is adjacent to the left or to the right of the set of points y = (y1, ..., yk), throughthe statisticsF = s−2((y0 − ŷ∗0)2 +k∑i=1(ŷi − ŷ∗i )2),where ∗ denotes the predictions based on k + 1 and s2 = (k − 2)−1(y′y − ξ̂X ′y). We have Fapproximately distributed by F1,k−2 and thus the null hypothesis is rejected whenever F is largerthan the (1 − α)-quantile, F1,k−2;1−α. The algorithm runs along the following steps:1. Fit a least-squares regression line to the initial k = νn upper observations, y = [yi]ki=1 (usuallyν = 0.02).2. Using the test statistic F , determine if a new point y0 = yj for j > k, belongs to the originalset of points y. Go adding points until the null hypothesis is rejected.3. Consider knew = max(0, {j : F < F1,k−2;1−α}). If knew = 0, no new points are added to y andthus move forward to step 4. Return to step 1. if knew > 0 by considering k = knew.4. Estimate ξ by considering the obtained k.Although in a different context concerning the estimation of a bivariate tail dependence mea-sure, but with a similar problematic, Frahm et al. (2005) presented a heuristic procedure aimingto find the plane area of the estimator sample path. Ferreira (2014, 2015) adapted this method tothe Hill estimator, whose algorithm is described below:1. Smooth the Hill plot (k, ξ̂k,n) by taking the means of 2b+1 successive points, ξ̂1,n, ..., ξ̂n−2b,n,where the bandwidth b = ⌊0.005× n⌋ (and thus each moving average consists in 1% of thedata).2. Define the regions pk = (ξ̂k,n, ..., ξ̂k+m−1,n), k = 1, ..., n − 2b − m + 1, with length m =⌊√n− 2b⌋. The algorithm stops at the first region satisfyingk+m−1∑i=k+1∣∣∣ξ̂i,n − ξ̂k,n∣∣∣ ≤ 2s,where s is the empirical standard-deviation of ξ̂1,n, ..., ξ̂n−2b,n.3. Consider the chosen plateau region pk∗ and estimate ξ as the mean of the values of pk∗(consider the estimate zero if no plane region fulfills the stopping condition).In the sequel it will be referred as plateau method.In this work we compare through simulation the performance of these methods within theestimation of the tail index and the exceedance probability. A simple estimator under ξ > 0 isgiven by (Dijk and de Haan, 1992)p̂k,n :=kn(xXn−k:n)−1/ξ̂.In this measure, the plateau algorithm will be applied to the respective sample path (k, p̂k,n) in asimilar manner of the variance estimation considered in Ferreira (2015). More precisely, we choosethe plane region in step 2 at the same position of the one found for the ξ estimation.2. SIMULATION STUDYWe consider 1000 independent samples of sizes n = 100, 1000, generated from the followingmodels:• Pareto(ξ) with d.f. F (x) = 1− x−1/ξ, x > 1, ξ > 0: (ξ = 1);• Cauchy with d.f. F (x) = 12 + 1pi arctanx, x real: (ξ = 1);• Burr(β,τ ,λ), with d.f. F (x) = 1 − (β/(β + xτ ))λ), x > 0, β, τ, λ > 0, ξ = 1/(τλ): we takeBurr(1,1,1) (ξ = 1)• Fre´chet(ξ) with d.f. F (x) = exp (−x−1/ξ), x > 0, ξ > 0: (ξ = 1);• Log-Gamma(τ ,λ), with d.f. F (x) = ∫ x1 λτΓ(τ) (log t)τ−1t−λ−1dt, x ≥ 0, τ, λ > 0, ξ = 1/λ: wetake Log-Gamma(2,1) (ξ = 1).The bias and root mean squared error (rmse) values obtained for the tail index are reported inTable 1. The results concerning the exceedance probabilities, p = 0.001, 0.0001, respectively in thecases n = 100, 1000, are reported in Table 2. Regarding the tail index, the AMSE method presentsthe smaller rmse, followed by the DK except in the Pareto model. Indeed, this latter methodologywas developed for the so-called Hall’s class that leaves out the simple Pareto case considered here.The plateau procedure has also a good performance, in particular, for small sample sizes. Theworst performance for n = 100 is associated with the Kopt method. The sumplot methodologywas developed under the assumption of a Pareto tail behavior and therefore it is not surprising itsbest performance within the Pareto model. However, it revealed to be the worst method in theremaining models, in particular, for n = 1000. In what concerns the exceedance probability, thebest results for n = 100 lie in the plateau and Kopt methods. In the case n = 1000, the methodshave a broadly similar performance, except the sumplot with the highest rmse.3. AcknowledgmentsThe first author research was supported by the Research Centre CEMAT through the ProjectUID/Multi/04621/2013.bias (n=100) AMSE Kopt DK Plateau SumplotBurr 0.1136 0.1798 0.1338 0.0424 0.3085Fre´chet 0.0101 0.1489 0.0827 0.0362 0.2158Cauchy 0.0070 0.1121 0.0282 -0.0306 0.2166Log-Gamma 0.1512 0.2918 0.1803 0.2242 0.4240Pareto -0.0882 0.0349 -0.1396 -0.0139 -0.0311rmse (n=100) AMSE Kopt DK Plateau SumplotBurr 0.2848 0.4546 0.2887 0.3058 0.4243Fre´chet 0.2261 0.4610 0.2371 0.2606 0.3327Cauchy 0.2953 0.5366 0.2991 0.3737 0.4118Log-Gamma 0.3003 0.5708 0.3595 0.3523 0.5045Pareto 0.1975 0.3665 0.4256 0.2010 0.1262bias (n=1000) AMSE Kopt DK Plateau SumplotBurr 0.0710 0.0475 0.0662 0.0142 0.3123Fre´chet -0.0122 0.0620 0.0577 0.0107 0.2735Cauchy 0.0199 0.0194 0.0064 -0.0147 0.1707Log-Gamma 0.1496 0.1896 0.2646 0.2025 0.5249Pareto -0.0402 0.0187 -0.0901 -0.0002 -0.0002rmse (n=1000) AMSE Kopt DK Plateau SumplotBurr 0.1319 0.1303 0.1279 0.1465 0.3372Fre´chet 0.1032 0.2446 0.1069 0.1166 0.2927Cauchy 0.1115 0.1470 0.1152 0.1765 0.2124Log-Gamma 0.1957 0.3568 0.2833 0.2380 0.5404Pareto 0.0860 0.2116 0.3648 0.0760 0.0321Table 1: Bias and rmse of the tail index estimation.REFERENCESBeirlant, J., Dierckx, G., Guillou, A. and Staˇricaˇ, C. (2002) On Exponential Representation ofLog-Spacings of Extreme Order Statistics. Extremes, 5, 157-180.Beirlant,J., Goegebeur, Y., Segers, J. and Teugels, J.L. (2004). Statistics of Extremes: Theoryand Applications. J. Wiley & Sons.de Sousa, B. and Michailidis, G. (2004) A Diagnostic Plot for Estimating the Tail Index of aDistribution. Journal of Computational and Graphical Statistics, 13(4), 1-22.Dijk, V. and de Haan, L. (1992). On the estimation of the exceedance probability of a high level.Order statistics and nonparametrics: Theory and applications. In Sen, P.K. and Salama I.A.(Editors), 79-92. Elsevier, Amsterdam.Drees, H. and Kaufmann, E. (1998) Selecting the optimal sample fraction in univariate extremevalue estimation. Stochastic Process Appl., 75, 149-172.Ferreira, M. (2014) A Heuristic Procedure to Estimate the Tail Index, Proceedings of the 14thInternational Conference in Computational Science and Its Applications - ICCSA 2014, June 30 -July 3 (2014), Guimares, Portugal, IEEE-Computer Society 4264a241, 241-245.Ferreira, M. (2015) Estimating the tail index: Another algorithmic method. ProbStat Forum, 08,45-53.Frahm, G., Junker, M. and Schmidt R. (2005) Estimating the tail-dependence coefficient: proper-ties and pitfalls. Insurance: Mathematics & Economics, 37(1), 80-100.Hill, B.M. (1975). A Simple General Approach to Inference About the Tail of a Distribution. Ann.Stat., 3, 1163-1174.Matthys, G. and Beirlant, J. (2000) Adaptive Threshold Selection in Tail Index Estimation. In:Extremes and Integrated Risk Management, ed. P. Embrechts, London: Risk Books.bias (n=100) AMSE Kopt DK Plateau SumplotBurr 0.0002 -0.0008 -0.0004 -0.0009 0.0056Fre´chet 0.0002 -0.0008 0.0001 -0.0006 0.0118Cauchy 0.0017 -0.0004 0.0009 -0.0006 0.0174Log-Gamma 0.0003 -0.0008 0.0025 -0.0004 0.0338Pareto -0.0004 -0.0007 -0.0008 -0.0004 0.0381rmse (n=100) AMSE Kopt DK Plateau SumplotBurr 0.0020 0.0009 0.0012 0.0009 0.0130Fre´chet 0.0011 0.0009 0.0015 0.0009 0.0241Cauchy 0.0046 0.0012 0.0039 0.0011 0.0335Log-Gamma 0.0011 0.0009 0.0053 0.0010 0.0878Pareto 0.0008 0.0009 0.0009 0.0008 0.0603bias (n=1000) AMSE Kopt DK Plateau SumplotBurr 0.0001 -0.0001 -0.0001 -0.0001 0.0004Fre´chet 0.0001 -0.0001 -0.0001 -0.0001 0.0012Cauchy 0.0000 -0.0001 -0.0007 -0.0001 0.0008Log-Gamma 0.0000 -0.0001 0.0003 -0.0001 0.0037Pareto -0.0001 -0.0001 -0.0001 -0.0001 0.0503rmse (n=1000) AMSE Kopt DK Plateau SumplotBurr 0.0001 0.0001 0.0001 0.0001 0.0007Fre´chet 0.0001 0.0001 0.0001 0.0001 0.0016Cauchy 0.0001 0.0001 0.0008 0.0001 0.0012Log-Gamma 0.0001 0.0001 0.0005 0.0001 0.0051Pareto 0.0001 0.0001 0.0001 0.0001 0.0729Table 2: Bias and rmse of the exceedance probability estimation, considering p = 0.001 for n = 100and p = 0.0001 for n = 1000.McGee, V.E. and Carleton, W.T. (1970) Piecewise Regression. Journal of the American StatisticalAssociation, 65, 11091124.

On the Hill estimator: a comparison of methods

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