Estimation of generalized Pareto distribution

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Estimatio of geeralized Pareto distributio Joa Del Castillo, Jalila Daoudi To cite this versio: Joa Del Castillo, Jalila Daoudi.

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1 Estimatio of geeralized Pareto distributio Joa Del Castillo, Jalila Daoudi To cite this versio: Joa Del Castillo, Jalila Daoudi. Estimatio of geeralized Pareto distributio. Statistics ad Probability Letters, Elsevier, 2009, 79 (5), pp.684. < /j.spl >. <hal > HAL Id: hal Submitted o 7 Aug 2010 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

Accepted Mauscript Estimatio of geeralized Pareto distributio Joa del Castillo, Jalila Daoudi PII: S0167-7152(08)00498-7 DOI: 10.

2 Accepted Mauscript Estimatio of geeralized Pareto distributio Joa del Castillo, Jalila Daoudi PII: S (08) DOI: /j.spl Referece: STAPRO 5250 To appear i: Statistics ad Probability Letters Received date: 2 October 2008 Accepted date: 20 October 2008 Please cite this article as: del Castillo, J., Daoudi, J., Estimatio of geeralized Pareto distributio. Statistics ad Probability Letters (2008), doi: /j.spl This is a PDF file of a uedited mauscript that has bee accepted for publicatio. As a service to our customers we are providig this early versio of the mauscript. The mauscript will udergo copyeditig, typesettig, ad review of the resultig proof before it is published i its fial form. Please ote that durig the productio process errors may be discovered which could affect the cotet, ad all legal disclaimers that apply to the joural pertai.

3 ESTIMATION OF GENERALIZED PARETO DISTRIBUTION JOAN DEL CASTILLO Abstract. Research partially supported by the Spaish Miisterio de Educacio y Ciecia, grat MTM : This paper provides precise argumets to explai the aomalous behavior of the likelihood surface whe samplig from the geeralized Pareto distributio for small or moderate samples. The behavior of the profile-likelihood fuctio is characterized i terms of the empirical coefficiet of variatio. A sufficiet coditio is give for global maximum of the likelihood fuctio of the Pareto distributio to be at a fiite poit. Keywords: Heavy-tailed iferece. Extreme value theory. The coefficiet of variatio. 1. Itroductio The Pareto distributio has log bee used as a model for the tails of aother log-tailed distributio, see Arold (1983). Applicatios to risk maagemet i fiace ad ecoomics are ow of icreasig importace. Sice Pickads (1975), it has bee well kow that the coditioal distributio of ay radom variable over a high threshold is approximately geeralized Pareto distributio (GPD), which icludes the Pareto distributio, the expoetial distributio ad distributios with bouded support. These distributios are closely related to the extreme value theory (Coles, 2001, ad Embrechts et al. 1997). The GPD has bee used by may authors to model excedaces i several fields such as hydrology, isurace, fiace ad evirometal sciece, see Fikestadt ad Rootzé (2003), Coles (2001) ad Embrechts et al. (1997). I geeral, GPD ca be applied to ay situatio i which the expoetial distributio might be used but i which some robustess is required agaist heavier tailed or lighter tailed alteratives, see Va Motford ad Witter (1985). The asymptotic behavior of maximum likelihood estimators was studied by Daviso (1984) ad Smith (1985). Nevertheless, there is evidece that umerical techiques for maximum likelihood estimatio do ot work well i small samples ad other estimatio methods have bee proposed, see Castillo ad Hadi (1997) ad Hoskig ad Wallis (1987). The paper provides precise argumets to explai the aomalous behavior of the likelihood surface whe samplig from the GPD distributio (Daviso ad Smith, 1990, ad Castillo ad Hadi, 1997). I Sectio 2, Theorem 1 proves that the behavior of the profile-likelihood for GPD is characterized by the sig of the empirical coefficiet of variatio of the sample. Corollary 1 proves that the maximum of the likelihood fuctio for the Pareto distributio is at a fiite poit, for samples i which the coefficiet of variatio is larger tha 1. A example i the Appedix shows that a local maximum for the likelihood fuctio of the GPD could ot exist whe the coditio is ot fulfilled. 1

4 2 JOAN DEL CASTILLO Mote Carlo simulatio i Sectio 3 raises the problem of mis-specificatio for small or moderate samples i GPD. It ca be also explaied from the differece betwee the theoretical ad the empirical coefficiets of variatio for small samples. The practical relevace of these results is also discussed. 2. Mai Results The cumulative distributio fuctio of GPD is (2.1) F (x) = 1 (1 + ξx/ψ) 1/ξ, where ψ > 0 ad ξ are scale ad shape parameters. For ξ > 0 the rage of x is x > 0 ad the GPD is just oe of several forms of the usual Pareto family of distributio ofte called the Pareto distributio. For ξ < 0 the rage of x is 0 < x < ψ/ ξ, the GPD have bouded support. The limit case ξ = 0 correspods to the expoetial distributio. A alterative parameterizatio is σ = ψ/ ξ ad ξ = s ξ, where s = sig (ξ). The, the probability desity fuctio for GPD is give by (2.2) f (x; σ, ξ) = 1 ( 1 + s x (1+ξ)/ξ, σ ξ σ) for ξ < 0, the rage of x is ow 0 < x < σ. Usig this otatio the two families of distributios correspodig to ξ > 0 ad ξ < 0 ca be studied at the same time. Give a sample {x i } of size, the log-likelihood fuctio for GPD distributio, divided by the sample size, is (2.3) l (σ, ξ) = log(s ξσ) 1 + ξ ξ log (1 + s x i /σ). If ξ < 0 it is assumed σ > M = max {x i }, otherwise the likelihood is zero. I this case the likelihood may be made arbitrarily large as σ teds to M, so the maximum likelihood estimators are take to be the values which yield a local maximum of (2.3), that ofte appears. Maximum likelihood estimatio of geeralized Pareto parameters was discussed by Daviso (1984) ad Smith (1985). I particular, for large samples, maximum likelihood estimator exist ad is asymptotically ormal ad efficiet, provided that 0.5 < ξ. The restrictio 0.5 < ξ < 0.5 is usually assumed for both practical ad theoretical reasos, sice GPD with ξ < 0.5 have fiite ed poits ad the probability desity fuctio is strictly positive at each edpoit, ad GPD with ξ > 0.5 have ifiite variace. Whe GPD is used as a alterative to the expoetial distributio, values of ξ ear 0 will be of greatest iterest, because the expoetial distributio is a GPD with ξ = 0. For moderate or small samples, aomalous behavior of the likelihood surface ca be ecoutered whe samplig from the GPD distributio (Daviso ad Smith, 1990, ad Castillo ad Hadi, 1997). This will be explaied i this paper from the coefficiet of variatio of the sample. For istace, the coefficiet of variatio for Pareto distributio (ξ < 0.5) is give by (2.4) ζ = 1/(1 2ξ) > 1,

5 ESTIMATION OF GENERALIZED PARETO DISTRIBUTION 3 but for small samples the empirical coefficiet of variatio (2.5) cv = m 2 m 2 1 /m 1, where m k = x k i / are the sample momets, may be lower tha 1. Theorem 1 ad Corollary 1 below i this Sectio provide more precise argumets. Equatig to zero the derivative of l (σ, ξ) i(2.3), with respect to ξ, we fid ˆξ = ξ (σ), where (2.6) ξ (σ) ξ (σ, s) = 1 log (1 + s x i /σ). The profile-likelihood is give by (2.7) l p (σ, s) = log[s ξ (σ) σ] ξ (σ) 1. Propositio 1. The followig limits hold: lim σ log (1 + x/σ) = x, lim σx/(σ + x) = x, lim σ2 (log (1 + x/σ) x/(σ + x)) = x 2 /2. Proof. It is a elemetary exercise i calculus, usig series expasio. Propositio 2. Let l p (σ, s) be the profile-likelihood, defied by (2.7) ad let x be the sample mea, the lim ξ (σ) = 0, lim σξ (σ) = s x. l 0 lim l p (σ, s) = log(x) 1. Proof. The first limit is straightforward. From Propositio 1 it follows: 1 lim σξ (σ) = lim σ log (1 + s x i /σ) = s x This prove the secod limit ad hece, sice s 2 = 1, the last limit follows. Remark 1. The limit of the profile-likelihood, l p (σ, s), as σ teds to ifiity, i Propositio 2, correspods to the log-likelihood of the expoetial distributio for the same sample. More precisely, the log-likelihood fuctio for the expoetial distributio, σe σx, divided by the sample size, is l (σ, 0) = log σ σ x ad the maximum likelihood estimator is σ = 1/x the, (2.8) l ( σ, 0) = l 0 = log(x) 1. Theorem 1. For the Pareto distributio (ξ > 0), if the empirical coefficiet of variatio is cv > 1 the l p (σ, 1) is a mootoous decreasig fuctio for sufficietly large σ, ad if cv < 1 it is mootoous icreasig. For the distributios with bouded support i GPD (ξ < 0), if cv > 1 the l p (σ, 1) is a mootoous icreasig fuctio for sufficietly large σ, ad if cv < 1 it is mootoous decreasig. Proof. The derivative of (2.7) with respect to σ is give by s l p (σ, s) = (ξ (σ) + σ ξ (σ) + σ ξ (σ)ξ (σ)) / (σ ξ (σ) ). ad the sig of s l p (σ, s) is the same as the sig of um (σ) = ξ (σ) + σ ξ (σ) + σ ξ (σ) ξ (σ), sice σ ξ (σ) > 0.

6 4 JOAN DEL CASTILLO Takig derivative with respect to σ i (2.6) it follows σ ξ (σ) = 1 s x i /(σ + s x i ), hece um (σ) = 1 (log (1 + s x i /σ) s x i /(σ + s x i )) ( ) ( ) 1 1 log (1 + s x i /σ) s x i /(σ + s x i ). From Propositio 1, we have lim { σ 2 um (σ) } = 1 2 x 2 i x 2 1 Fially, ote that 2 x2 i x2 > 0 is equivalet to ( m 2 m1) 2 > m 2 1 ad equivalet to cv > 1. A first cosequece of Theorem 1 is obtaied immediately. l p (σ, s) teds to l 0, from Propositio 2, ad l p (σ, s) is a mootoous fuctio for sufficietly large σ (Theorem 1) the these facts determie whether l p (σ, s) is greater or less tha l 0. If cv > 1, l p (σ, 1) is a mootoous decreasig fuctio ad l p (σ, 1) is a mootoous icreasig fuctio, for sufficietly large σ, the (2.9) l p (σ, 1) > l 0 > l p (σ, 1). I the same way, if cv < 1 the (2.10) l p (σ, 1) < l 0 < l p (σ, 1), for sufficietly large σ. Remark 2. From (2.6), as σ teds to ifiite ξ (σ) teds to zero. Hece, for ξ i a eighborhood of zero, the iequalities (2.9) ad (2.10) show that if cv > 1 the Pareto distributio is more likely tha the expoetial distributio ad if cv < 1 a bouded support distributio i GPD is more likely tha the expoetial distributio. These facts are umerically relevat for a algorithm to obtai the maximum likelihood estimator i GPD. Corollary 1. Give a sample {x i } of positive umbers with a empirical coefficiet of variatio cv > 1, the likelihood fuctio for the Pareto distributio has a global maximum at a fiite poit ad the maximum is higher tha the maximum for the likelihood fuctio of the expoetial distributio. Proof. For small values of σ we write l p (σ, 1) = (log σ + ξ (σ)) log [ξ (σ)] 1, (log σ + ξ (σ)), teds to 1 log (x i) ad log [ξ (σ)] goes to. This proves lim l p (σ, 1) =. σ 0 Sice l p (σ, 1) is a cotiuous ad mootoous decreasig fuctio for sufficietly large σ (Theorem 1) the last limit prove that a global maximum exists. Iequality (2.9) shows that it is up the maximum for the expoetial distributio.

7 ESTIMATION OF GENERALIZED PARETO DISTRIBUTION 5 Whe the coefficiet of variatio of the sample is cv < 1 there may be o maximum likelihood estimator for the Pareto distributio ad either is there a local maximum for the parameter space of the bouded support distributios i the GPD. I the Appedix we give a simple example of this situatio. If l p (σ, 1) has a local miimum, as is usual, ad cv < 1 the Theorem 1 proves that there is a local maximum for the likelihood fuctio, sice l p (σ, 1) icrease o the right side of the miimum ad is mootoous decreasig for large values of σ. 3. Discussio Aalytical maximizatio of the log-likelihood for GPD is ot possible, so umerical techiques are required takig care to avoid umerical istabilities whe ξ ear zero (Coles, 2001, pp 81). Theorem 1 clarifies the behavior of the likelihood fuctio i terms of the coefficiet of variatio of the sample. If cv > 1 the Pareto distributio is more likely tha a bouded support distributio i a eighborhood of zero, ad if cv < 1 a bouded support distributio is more likely tha a Pareto distributio. The umerical algorithms have to cosider the cv sig of the sample Corollary 1 proves that the likelihood fuctio for the Pareto distributio has a global maximum at a fiite poit for samples i which cv > 1, so it is extremely simple to fid it umerically. This is a sufficiet coditio ad we also believe, from umerical experimets, that it is ecessary, although we are ot able to prove it. Hoskig ad Wallis (1987, pp 343) say we coclude that the vast majority of failures of the algorithms are caused by the oexistece of a local maximum of the likelihood fuctio rather tha by failure of our algorithm to fid a local maximum. We agree with them. Moreover, the example give i the Appedix proves the oexistece of a local maximum for a particular sample. Now it is clear that the oexistece of a local maximum it is possible. This problem icreases whe ξ decreases, specially for the bouded support distributios i GPD, as Hoskig ad Wallis (1987) showed. Samplig from Pareto distributio i GPD shows aother problem. A simulatio experimet was ru to compute mis-specificatio for sample sizes = 15, 25, 50, 100 ad shape parameter ξ = 0.1, 0.2, 0.3, 0.4. The scale parameter σ was set to 1, sice the model (2.2) is ivariat uder scale chages. For each combiatio of values of ad ξ, 50, 000 radom samples were geerated from the Pareto distributio ad the umber of times the parameter estimates ˆξ to be positive, egative or that the algorithm does ot coverge are reported. It is oted from (2.4) that the theoretical coefficiet of variatio for Pareto distributio is ζ > 1, but for small samples the empirical coefficiet of variatio, cv, may be lower tha 1. Table 1 shows that for ξ = 0.3 (a distributio with ifiite kurtosis) ad sample size = 15, 29% of cases that lead to a wrog decisio, ˆξ < 0 (bouded support distributio), while i 4.8% cases the algorithm does ot coverge. The problem remais for larger samples. For ξ = 0.1 ad sample size = 100, 24.9% of cases have ˆξ < 0. However, if we assume Pareto distributio without cosiderig the global GPD model (with the bouded support distributios), the samples with cv > 1 lead to Pareto distributio ad samples with cv < 1 lead to the expoetial distributio, ξ = 0, i both cases the support for the distributio is (0, ). I the cotext of heavy-tailed iferece assumig that the true distributio has support i (0, ), a alterative model for samples with cv < 1 may be trucated

8 6 JOAN DEL CASTILLO ormal distributio. Castillo (1994) shows that the likelihood equatios for trucated ormal distributio have a solutio if ad oly if the empirical coefficiet of variatio is cv < 1. Pareto distributio ad trucated ormal distributio are two complemetary families of distributios the former with theoretical coefficiet of variatio ζ > 1 ad the latter with ζ < 1. I both cases the expoetial distributio is the limit distributio as ζ teds to 1. ξ ˆξ > 0 ˆξ < 0 NC ˆξ > 0 ˆξ < 0 NC ˆξ > 0 ˆξ < 0 NC ˆξ > 0 ˆξ < 0 NC Table 1. Radom samples geerated from the Pareto distributio for sample sizes = 15, 25, 50, 100 ad shape parameters ξ = 0.1, 0.2, 0.3, 0.4. The frequecy with which the parameter estimate ˆξ is positive, egative or the algorithm does ot coverge (NC), are reported. 4. Bibliography (1) Arold, B.(1983). Pareto distributios. Iteratioal Co-operative Publishig House. Fairlad, Marylad. (2) Castillo, E. ad Hadi, A. (1997). Fittig the Geeralized Pareto Distributio to Data. Joural of the America Statistical Associatio, 92, (3) Castillo, J. (1994). The Sigly Trucated Normal Distributio, a No- Steep Expoetial Family. Aals of the Istitute of Mathematical Statistics. 46, (4) Coles, S. (2001). A Itroductio to Statistical Modellig of Extreme Values. Spriger, Lodo. (5) Daviso, A (1984). Modellig Excesses Over High Thresholds, with a Applicatio, i Statistical Extremes ad Applicatios, ed. J.Tiago de Oliveira, Dordrecht: D.Reidel,pp (6) Daviso, A. ad Smith, R. (1990). Models for Exceedaces over High Thresholds. J.R. Statist. Soc. B, 52, (7) Embrechts, P. Klüppelberg, C. ad Mikosch, T. (1997). Modelig Extremal Evets for Isurace ad Fiace. Spriger, Berli. (8) Fikestadt, B.ad Rootzé, H. (edit) (2003). Extreme values i Fiace, Telecommuicatios, ad the Eviromet. Chapma & Hall. (9) Hoskig, J. ad Wallis, J. (1987). Parameter ad quatile estimatio for the geeralized Pareto distributio. Techometrics 29, (10) Pickads, J. (1975). Statistical iferece usig extreme order statistics. The Aals of Statistics 3, (11) Smith, R. (1985). Maximum likelihood estimatio i a class of oregular cases. Biometrika 72, (12) Va Motford, M. ad Witter, J. (1985). Testig Expoetiality Agaist Geeralized Pareto Distributio. Joural of Hydrology, 78,

9 ESTIMATION OF GENERALIZED PARETO DISTRIBUTION 7 5. Appedix Let us examie the sample of size two give by {x 1, x 2 } = {1, 2} for the GPD. First, we will show that the profile-likelihood l p (σ, 1), give by (2.7), is a mootoous icreasig fuctio for σ > 0 ad, hece, there is o maximum likelihood estimator for the Pareto distributio with this sample. The derivative l p (σ, 1) is give by um (σ) = 8 + 6σ σ (3 + 2σ)(log (1 + 1/σ) + log (1 + 2/σ)), divided by a positive fuctio. Hece, l p (σ, 1) ad um (σ) have equal sig ad we will see that it is positive. The followig results hold for the fuctio um (σ) ad for its derivatives: (5.1) (5.2) lim um (σ) = 4, lim um (σ) = 0, lim um (σ) = 0, um (σ) = σ + 162σ2 + 69σ 3 + 9σ 4 σ 2 (2 + 3σ + σ 2 ) 3 < 0. The, um (σ) is a mootoous decreasig fuctio ad, from the limit zero property, um (σ) > 0. The, um (σ) is mootoous icreasig ad, hece, um (σ) < 0. Therefore, um (σ) is a mootoous decreasig fuctio, hece um (σ) > 4, ad its sig is always positive, as we said. We will also show that the profile-likelihood l p (σ, 1) with the same sample, {1, 2}, is a mootoous decreasig fuctio for σ > 2 ad, hece, does ot exist a local maximum for the parameter space of the bouded support distributios i the GPD. The derivative l p (σ, 1) is give by u (σ) = 8 6σ + σ (3 2σ)(log (1 1/σ) + log (1 2/σ)), divided by a egative fuctio. We will see that the sig of u (σ) is positive for σ greater tha 2. The followig results hold for the fuctio u (σ) ad for its derivatives: (5.3) (5.4) lim u (σ) = 4, lim u (σ) = 0, lim u (σ) = 0, u (σ) = σ + 162σ2 69σ 3 + 9σ 4 σ 2 (2 3σ + σ 2 ) 3. If σ > 2, the deomiator of u (σ) is positive; the greater real root of the umerator is at σ 1 = , the u (σ) > 0, for greater values of σ. Therefore, for σ > σ 1, u (σ) is a mootoous icreasig fuctio ad, from the limit zero property, u (σ) < 0. The, u (σ) is mootoous decreasig ad, hece, u (σ) > 0, for σ > σ 1. Fially, u (σ) is a mootoous icreasig fuctio for σ > σ 1. It is also clear that lim σ 2 u (σ) =. For 2 < σ < σ 1, u (σ) has a miimum at σ 0 = ad u (σ 0 ) = > 0. The, we coclude that the sig of u (σ) is positive for σ > 2, l p (σ, 1) is a mootoous decreasig fuctio for σ > 2 ad does ot exist a local maximum for the likelihood fuctio of the GPD model with this sample. Servei d Estadística Uiversitat Autò oma de Barceloa, Jalila Daoudi, Departamet de Matemtiques, Uiversitat Autòoma de Barceloa

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