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1 Which Extreme Values are Really Extremes? Jesús Gozalo Λ Statistics ad Ecoometrics José Olmo Statistics ad Ecoometrics U. Carlos III de Madrid U. Carlos III de Madrid Jauary 2003 Abstract The aim of this paper is to give a formal defiitio ad cosistet estimates of the extremes of a populatio. This defiitio relies o a threshold value that delimits the extremes ad o the uiform covergece of the distributio of these extremes to a Pareto type distributio. The tail parameter of this Pareto type distributio is the tail idex of the data distributio. The estimator of the threshold is achored i the Kolmogorov-Smirov distace betwee cosistet estimates of those two distributios. Our estimator is cosistet ad via the costructio of cofidece itervals for the tail idex (derived from our threshold estimator) we overcome the bias problems of the usual tail idex estimators (Hill or Pickads). The paper also explores the validity of our defiitio for stadard sample sizes. For this purpose, a hypothesis test is desiged i order to reject extremes estimates that are ot really extremes. Applicatios for differet stock returs are preseted. Keywords: Bootstrap, Goodess of fit test, Hill estimator, Kolmogorov-Smirov distace, Balkema ad De-Haa, Pickads Theorem, Tail idex. JEL: C12, C13, C14, C15, G10 Λ Correspodig Address: Fiacial support DGCYT Grat (SEC ) is gratefully ackowledged. We thak participats i the Coferece o Extremal Evets i Fiace celebrated i Motreal.

2 1 Itroductio Noe doubts that Risk Maagemet is oe of the most importat iovatios of the 20th cetury. The questio oe would like to aswer is: "If thigs go wrog, how wrog ca they go?" The variace used as a risk measure is uable to aswer this questio, ad therefore alterative measures regardig possible values out of the rage of available iformatio eed to be defied. Extreme value theory (EVT) provides some tools to costruct these ew risk measures: Value at Risk (VaR), Expected Shortfall or the tail idex of a distributio. All these measures eed to start by idetifyig which values are extreme values. I practice this is doe by graphical methods like QQ-plot, Sample Mea Excess Plot or by other ad-hoc methods that impose a arbitrary threshold (5%; 10%;:::). I this paper we propose a formal way of idetifyig which extreme values are really extremes. The goal is to estimate the lower boud of these extremes for fiite samples, i.e. a threshold value. Our method is achored i three key elemets: Pickads, Balkema- De Haa theorem (BHP), a distace based o a Kolmogorov-Smirov (KS) statistic ad hypothesis testig via bootstrap methods. By Pickads, Balkema-De Haa theorem we kow that the distributio of the exceedaces of a radom variable i the limit teds to a Pareto shape distributio. Therefore, extreme values cosidered as exceedaces above certai threshold will asymptotically have this type of Pareto distributio. I order to estimate this threshold poit we propose a alterative of Pickads estimator based o miimizig a Kolmogorov-Smirov distace takig ito accout the legth of the sample tail. Oe of the cotributios of our threshold estimator is the obtetio of cofidece itervals for the tail idex capturig the tail behavior of the data distributio. Moreover, the tail idex estimators relyig o our threshold estimator are cosistet ad allow to test our defiitio of extreme values (uiform covergece of the sample distributio of extremes to apareto type distributio). The paper cocludes with some applicatios to extreme quatile estimatio for simulated kow distributios as well as for real fiacial series. For these series, extreme quatiles are the corerstoe of risk measures, as Value at Risk or Expected Shortfall. The paper is structured as follows. I sectio 2 we preset a summary of the existig methods to calculate the threshold value. Sectio 3 shows a brief review of the results from the Extreme Value Theory that we will be usig i the core of the paper. Sectio 4 is devoted to defie our cocept of extreme value ad to preset a ew estimatio method for the threshold value that is used to defie extreme observatios. Sectio 5 itroduces a bootstrap goodess of fit test to check the validity of our defiitio. The fiite sample performace of our proposed method as well as some real applicatios are show i sectio 2

3 6. The coclusios are give i sectio 7. All proofs as well as the otatio used i the paper are gathered i the Appedices. 2 Existig ad-hoc Methods for Threshold Estimatio I the existig literature, there is ot a clear defiitio of the threshold value ν that determies the extremes. There exist differet popular estimatio methods to select a threshold (^ν ) relyig o the asymptotic Pareto distributio of the exceedaces. ffl QQ-plot ffl Sample Mea Excess Plot ffl Simulatio Procedures This estimatio of the threshold has differet challeges depedig o how close ^ν is to the right ed poit. A small ^ν yields bias problems i the estimatio of the parameters of the Pareto distributio. O the other had, large ^ν implies problems of great variace due to the abscece of poits i the tail to estimate the Pareto distributio. 2.1 QQ-plot The method is based o the followig simple fact: if U (1)» U (2)» :::U () are the order statistics from i.i.d. observatios uiformly distributed o [0,1], the by symmetry E(U (i+1) U (i) ) = 1 +1 ad hece E(U (i)) = i +1. Sice U (i) should be close to its i mea, the plot of f( i ;U (+1) (+1) (i)); 1» i» g has to be liear. Suppose ow, X (1)» X (2)» :::X () are the order statistics from a i.i.d. sample of size which is suspected to come from a particular cotiuous distributio G. The plot of f( ;G(X (+1) (i)); 1» i» g should be approximately liear ad hece also the plot of fg ψ ( );X (+1) (i)); 1» i» g should be liear. i i 3

4 Figure 2.1. QQ-plots of the egative tail of Nikkei Idex returs over the period 05= =2001 with ^ν = x d0:90e ad ^ν = x d0:95e. It is ot clear from Figure 2.1 which portio of the observatios fits better to the Geeralized Pareto distributio GP D ^, with parameters estimated from the sample observatios. 2.2 Sample Mea Excess Plot Aother stadard tool for choosig suitable thresholds is the sample mea excess plot (ν; e (ν)) where e (ν) is the sample mea excess fuctio defied by e (ν) = ± (X i ν) +, ± 1 fxi>νg with x + = max(x; 0). The sample mea excess fuctio e (ν) is the empirical couterpart of the mea excess fuctio which is defied as e(ν) =E[X ν j X>ν]. If the empirical plot follows a reasoably straight lie with positive gradiet above a certai value of ν, the this is a idicatio that the exceedaces over this threshold follow a Geeralized Pareto distributio with positive tail idex (ο) parameter. This is derived from the fact that e(ν) = ff + ον 1 ο ; where ff + ον > 0adff is the stadard deviatio of the GPD (see McNeil ad Saladi, 2001). 4

5 Figure 2.2. Sample mea Excess plot for egative Nikkei Idex returs over the period 05= =2001, (left graph) ad a sample from a ormal distributio of size =1000 (right graph). Focusig o Figure 2.2 differet cadidates ca be selected for the estimated threshold value. Other methods i order to choose the threshold value ν take advatage of simulatio procedures for kow distributios. The idea is to determie a threshold ν from a sample of size ad cosider the umber of observatios over this threshold N ν. The goal is to obtai the ecessary sample size to geerate N ν exceedaces over the determied threshold ν. This sample size is employed to estimate a extreme quatile closer to the right ed poit tha the threshold ν. I this waywe ca compare this extreme estimate with the actual extreme quatile of the kow distributio ad see the reliability of the ad-hoc threshold estimate (see McNeil, 1997). 3 Extreme Value Theory Results The mathematical foudatio of EVT is the class of extreme value limit laws, first derived heuristically by Fisher ad Tippet (1928) ad later from a rigorous stadpoit by Gedeko (1943). Suppose X 1 ;:::;X are idepedet radom variables with commo distributio fuctio F (x) = P fx» xg ad let M = max(x 1 ;:::;X ). Uder some cotiuity coditios o F at its right ed poit, the maximum M properly cetered ad ormalized has a limit law H ο with ο the parameter of the limit distributio, P f M d c» xg = F (c x + d ) d! H ο (x): (1) The cotiuity o F is a sufficiet coditio but it is ot ecessary. It is oly required some smoothess ear the right ed poit. 5

6 Theorem 3.1. Let F be a distributio fuctio with right ed poit x F» 1 ad let fi 2 (0; 1). There exists a sequece (u ) satisfyig F (u )! fi if ad oly if lim x!x F F (x) F (x =1 (2) ) (see Embrechts, Klüppelberg ad Mikosch, 1997, p.117). The coditio F (u )! fi is equivalet tosay that the sample maximum has a odegeerate distributio of expoetial type P (M» u )! e fi. The asymptotic distributio of the maximum is called extreme value law. The key result of Fisher-Tippet ad Gedeko is that there are oly three fudametal types of extreme value limit laws. These are Type I: (Gumbel) Λ(x) =exp( e x ); 1 <x<1; Type II: (Fr echet) Φ ff (x) = Type III: (Weibull) Ψ ff (x) = 8 < : 8 < : 0 x» 0; exp( x ff ) x>0 1 x 0; exp( ( x) ff ) x<0 I Types II ad III ff is a positive parameter. The three types may alsobecombied ito a sigle geeralised extreme value distributio, first proposed by Vo Mises (1936), of the form H ο (x) = 8 < : e (1+οx) 1 ο ο 6= 0 e e x ο =0 with 1+οx > 0. The case ο > 0 correspods to Type II with ff = 1 ο, the case ο < 0to Type III with ff = 1=ο, ad the limit case ο! 0toType I.. (3) Figure 3.1. The desity fuctio of the extreme value limit laws. The dot lie is the Gumbel distributio. Fr echet ad Weibull distributios are plotted with ff =1. 6

7 Corollary 3.1. From expressios ( 1) ad ( 3), the followig relatioships ca be extracted depedig o the value of the parameter ο, F (c x + d ) d! (1 + οx) 1 ο if ο 6= 0ad F (c x + d ) d! e x if ο =0. These expressios ca be cosidered as the survivor fuctios of a Geeralized Pareto distributio. Moreover, the asymptotic distributio of the stadardized tail of F depeds o a parameter ο (tail idex), hece a distributio F verifyig ( 2) ca be classified accordig to this parameter. Defiitio 3.1. F belogs to the Maximum Domai of Attractio of a Extreme Value Distributio H ο, F 2 MDA(H ο ), if ad oly if there exist costats c > 0 ad d,such that c 1 (M d ) d! H ο. Notice that the commoly employed cotiuous distributio fuctios belog to the maximum domai of attractio of a extreme value limit law, F 2 MDA(H ο ). From the results of Fisher-Tippet ad Gedeko it is derived that there are oly three types of maximum domais of attractio i cotrast with the umber of domais of attractio of ff-stable processes. This maximum domai of attractio of F depeds o the sig of ο. Defiitio 3.2. A df F such that the right tail satisfies 1 F (tx) lim x!1 1 F (x) = t ff ; t>0; ff = 1 ο > 0 (4) is called regularly varyig with idex ff (F 2 RV ff ). The tail of a distributio F satisfyig ( 4) decays polyomially (F is heavy tailed). This coditio ca be rewritte as 1 F (x) =x 1 ο L(x); x!1; ο>0 (5) where L(x) is a slowly varyig fuctio L(tx) lim =1; t>0: (6) x!1 L(x) A distributio fuctio F with positive tail idex ad verifyig coditio ( 2) idicates that the sample maximum should have a o degeerate distributio of Type II. Propositio 3.1. F 2 RV ff, F 2 MDA(Φ ff ) where Φ ff is the Fréchet EVD. The ormalizig costats for this case are d = 0 ad c = F ψ (1 1 ) (see Embrechts, Klüppelberg ad Mikosch, 1997, p.132). 7

8 It is sufficiettokow the tail idex of a distributio F to kow the asymptotic distributio of the stadardized maximum. Moreover, this parameter ο provides iformatio about the behavior of the tail ad therefore the tail idex ca help to give a formal defiitio for the tail avoidig the ad-hoc selectio of arbitrary quatiles. Defiitio 3.3. The tail of a distributio is the set of extreme values; these extremes are the exceedaces over a determied threshold (ν) with ν sufficietly large. The distributio of these large observatios F ν (x) is called the coditioal excess distributio fuctio (cedf) over ν ad is defied as F ν (x) =P fx» xjx >νg; ν» x» x F ; (7) where Xisaradom variable, ν is a give threshold ad x F»1is the right edpoit of F. This distributio ca be writte i terms of F, F ν (x) = F (x) F (ν) 1 F (ν) x>ν: (8) From this expressio it is deduced that F ν (x) = F (x). The extremes of a populatio F (ν) are determied by the threshold ν ad by the tail idex ο of the distributio F. These parameters defie the asymptotic distributio of stadardized extremes. Theorem 3.2. (Balkema ad de Haa (1974), Pickads (1975)) (BHP). Let F be a distributio fuctio such that F 2 MDA(H ο ), the coditioal excess distributio fuctio F ν (x) for ν large, is where GP D ο;ff (x ν) = lim F ν (x) =GP D οff (x ν) ν!x F 8 < : 1 (1 + ο(x ν) ff ) 1 1 exp (x ν) ff ο if ο 6= 0 if ο =0 (9) is the so-called Geeralized Pareto distributio (GPD). The Geeralized Pareto is the asymptotic distributio of the extremes uder some cotiuity coditios over F. If the distributio F has a regularly varyig tail, the the distributio of the extremes as goes to ifiity ca be reduced to a Pareto distributio. Corollary 3.2. For a distributio fuctio F such that F 2 MDA(Φ ff ), the coditioal excess distributio fuctio F ν (x), for ν large, is lim F ν (x) =PD ο ( x ν!x F ν ); 8

9 where PD ο ( x ν )=1 ( x ν ) 1 ο ; x>ν is the Pareto distributio. We brig together the two approaches of the asymptotic Pareto type distributio of the tail i the otatio G, G = 8 < : GP D ο;ff (x ν); F 2 MDA(H ο ) PD ο ( x ν ); F 2 MDA(Φ ff) with = fο;ffg for the GPD case ad = fοg for the Pareto distributio. It is importat to otice that we are ot cotradictig BHP theorem. The Pareto distributio is icluded i the Geeralized Pareto family: PD ο ( x ν )=GP D ο;ff( x ν νο ). 4 Defiitio ad Estimatio of Extremes Uder defiitio ( 3.3) the tail of a distributio is the set of extreme values. We ca exted BHP theorem to give a formal defiitio of the extremes based o uiform covergece betwee the two ivolved distributio fuctios. Defiitio 4.1. Let X be aradom variable with a distributio F 2 MDA(H ο ). Let ν 2 support of F =[x 0 ;x F ] with x F»1, F ν (x) be the coditioal excess distributio fuctio ad G (x; ν) be the Pareto type distributio. The extreme values of the distributio F are defied byaparameter ν<x F such that F ν (x) coverges uiformly to G (x; ν) as ν! x F, lim ν!x F sup jf ν(x) G (x; ν)j =0 (10) with ο 2 the tail idex of the distributio F. I order to obtai the extremes of a distributio F from a sample data we eed to estimate the threshold parameter. Coditio ( 10) is ot possible to check for a give, therefore we give a characterizatio of the defiitio for sample data. Propositio 4.1. Let F (x) be a distributio fuctio verifyig coditio ( 2) ad cosider ^ν a estimator of ν. The extreme values estimates defied by^ν are extreme values if ad oly if ffl sup jf^ν p (x) G (x; ^ν )j! 0. ffl F^ν (x) =G (x; ^ν ) for almost every x 2 R ad give. 9

10 The first coditio provides the cosistecy of the estimator ^ν. This is a ecessary coditio but ot sufficiet to defie the set of extremes. We ca obtai cosistet estimators of ν that determie extremes estimates that do ot follow Pareto type distributios for agive sample size. This is the reaso to impose the secod coditio. This coditio is ot usually possible to check because the parameters of the Pareto type distributio ( ) are ukow. I cosequece, we propose a goodess of fit test to circumvet this drawback. 4.1 Coditioal Estimatio of the Parameter Set The distributio of the extremes of a distributio depeds o its tail behavior, i.e. o. This set of parameters must be estimated from the available data sample. Maximum Likelihood (ml) is the most covetioal method ad has very desirable properties; cosistecy, asymptotic efficiecy ad ormality. The estimatio of the tail parameters is coditioed o the kowledge wehave about the sample tail defied by the threshold ν. For the GPD approach, ^ (ν) =f^ο ml (ν); ^ff ml (ν)g ad for the Pareto approach, ^ (ν) =^ο ml (ν). Propositio 4.2. If F 2 MDA(Φ ff ), ^ο ml (ν) for PD ο is the Hill estimator (see Hill, 1975), 1 ^ο Hill (ν) = ( k) X i=k+1 with ν = x (k) ad x (k+1)» :::» x () the icreasig order statistics. log x (i) ν ; (11) Hill estimator is gaiig popularity i the EVT Literature because is easy to calculate ad has good asymptotic properties, but i the fiacial literature is employed eve for ot heavy tailed distributios. Therefore, cosistecy ad asymptotic ormality may ot hold ay more.there exists some cofusio about the coditios to use it. Propositio 4.3. Let ^ο ml ad ^ο Hill be the maximum likelihood estimators of the parameter ο of a Geeralized Pareto ad of a Pareto distributio respectively. These estimators are p -cosistet estimates of the tail idex of a distributio fuctio F verifyig coditio ( 2) if ο > 1 2 for ^ο ml (see Smith, 1984) ad if ο > 0 for ^ο Hill (Goldie ad Smith, 1987). The drawback of these estimators is their biases (see Guillou & Hall, 2000). This bias has two differet sources: the distributio of data is ot of a Pareto type ad the choice of the umber of order statistics used to costruct the estimator. Let us cocetrate o the Hill estimator ad assume F 2 MDA(Φ ff ). By BHP theorem the large observatios x (k 0 +1)» :::» x () greater tha ν = x (k 0 ) follow apd ο with ν sufficietly large, therefore 10

11 we elimiate the first source of bias. It is show i Hill (1975) that the radom variable V i = i [log x ( i+1) log x ( i) ] follows a expoetial distributio with mea ο. Cosider ^ν = x (k) a estimate of ν such that x (k+1)» :::» x (k 0 )» :::» x () are the exceedaces over the estimate 1. Hill estimator based o x (k) is ^ο Hill (^ν ) = 1 estimator ca be decomposed as P k i=k+1 log x (i) x (k). This ^ο Hill (^ν )= 1 k X i=k 0 +1 log x (i) ν + 1 k = k0 k ^ο Hill (ν)+ 1 k Xk 0 i=k+1 Xk 0 i=k+1 log x (i) ν log x (i) ν + log ν x (k) : + log ν x (k) = O the other had, Hill estimator based o the parameter ν ca be expressed i terms of V i, ^ο k P 0 Hill (ν) = 1 k V 0 i. This estimator is ubiased ( E[^ο Hill (ν)] = ο ), however the expected value of the Hill estimator based o the estimate ^ν is biased. This deviatio from the parameter depeds o the bias of the threshold estimator: E[^ο Hill (^ν )] = k0 k ο + 1 k Xk 0 i=k+1 Notice that the bias disappears if k = k 0. Elogx (i) Elogx (k) + k0 k logν: Therefore, the bias of the Hill estimator of a distributio F 2 MDA(Φ ff ) as goes to ifiity depeds oly o the bias of the threshold estimate. The problem is that the parameter ν = x (k 0 ) is ukow. I order to miimize bias problems, cofidece itervals are proposed as estimators of the tail idex. It is well kow that the radom variable S k = p k(^ο Hill ο) has a asymptotic N(0;ο 2 ) distributio. We costruct oparametric bootstrap cofidece itervals to approximate the exact cofidece itervals for the tail idex, ο 2 [^ο Hill (^ν ) p 1 J 1 k (F ; 1 ff k 2 ); ^ο Hill (^ν ) p 1 J 1 k (F ; ff )]; (12) k 2 with F the empirical distributio fuctio of the data, ff the sigificace level ad J k (x; F ) the approximate bootstrap distributio of S k. Note that the same procedure ca be applied to calculate cofidece itervals for the tail idex based o the maximum likelihood estimator ^ο ml (^ν ) of a Geeralized Pareto distributio. 1 Notice that if ^ν >νthere is ot a problem of bias, it is oly a matter of efficiecy of the estimator ^ν. 11

12 4.2 Estimatio Method for the Threshold Value ν Pickads (1975) proposed a method to estimate the threshold value ν based o uiform covergece (d 1 )betwee the Empirical Distributio associated to F ν ad a Geeralized Pareto distributio estimated from data GP D Pic ν : ν Pic = arg mi ν d ν 1(F ν; ;GPD Pic ν ); (13) with Pic ν the estimated parameters of the GPD. This estimator of ν is cosistet i the sese that P fsup jf ν Pic GP D Pic j >"g!0. The estimators for the parameters of the ν Pic GPD proposed by Pickads deped o the differet values of ν. Cosider ν = X ( 4i+1), i =1;:::;=4. for the tail idex ad ^ο(ν) = 1 log(2) log(x ( i+1) X ( 2i+1) ), X ( 2i+1) ν ^ff(ν) = X ( 2i+1) ν R log2 e ^οu du, 0 for the variace. This estimator for the tail idex is cosistet, but it is very sesitive to the choice of the order statistics ad it is ot efficiet (Drees, 1995). For stadard sample sizes the estimatios of the tail idex are biased ad the cofidece itervals for ο do ot give reliable iformatio. Alterative statistics have bee proposed for the tail idex to overcome these drawbacks, (see Dekker, Eimahl ad de Haa, 1989). Goldie ad Smith (1987) or Dekker ad de Haa (1993) establish the optimal umber of order statistics for differet estimators of the tail idex. O the other had, Pickads estimator for the threshold (ν Pic ) does ot take ito accout the legth of the sample tails defied by ν to compute the distaces i ( 13). As ν! x F the available samples of the tails are smaller yieldig worse estimatios of the tail parameters of the GPD. This implies worse goodess of fit of the coditioal distributios F ν to the theoretical asymptotic distributio. This is caused ot oly by the lack of fit of data to the theoretic GPD distributio but also by the estimatio mechaism of the tail parameters. Hece, ν Pic is ot ear the tail by its ow costructio. Cosequetly, the extremes estimates defied by Pickads estimator ca be very misleadig for stadard sample sizes (see Table 6.1). A atural distace to derive a good estimator to overcome Pickads drawbacks for fiite samples is a distace based o Kolmogorov-Smirov statistic. 12

13 Defiitio 4.2. (Kolmogorov-Smirov distace) ± Let F ν; (x) = 1 fν»x i»xg be the empirical distributio fuctio associated tof ν ad G be ± 1 fx i >νg apareto type distributio. The distace betwee F ν ad G is calculated by the followig KS distace d ν ks(f ν; ;G^ ν )= vu u t X 1 fxi>νg sup j P 1 fν»xi»xg P 1 fxi>νg G ^ ν (x; ν)j: (14) This statistic regards the umber of observatios of the available sample tails givig less weight to distaces of samples with less data i order to compesate the estimatio failure of the parameters of the theoretical distributio from small samples. Defiitio 4.3. Let d ν ks be the KS distace of ( 14) ad x = fx 1 :::;x g be a sample of size from a distributio F. The estimated threshold ^ν is the order statistic x (k) that makes the distace d ν ks miimum. with x (k) such that k!1, k! 0. ^ν = arg mi ν d ν ks (F ν;;g^ ν ); The latter coditios are cosequece of BHP theorem. As becomes large, -k should go to ifiity to beefit of a icreasig sample (more iformatio as icreases ad therefore smaller variace). At the same time, uless a portio of the upper tail follows exactly a Pareto type distributio we expect that k the approximatio to the theoretical distributio whe ν! x F (smaller the bias). teds to zero i order to improve as BHP theorem states Theorem 4.1. Let ^ν be the threshold estimator derived from the KS distace (d ν ks ) ad let ^ο(^ν ) be acosistet estimator of the tail idex based ox with ο 2. The, ^ν is a cosistet estimator of the threshold parameter ν i the sese that P fsup jf^ν (x) G (x; ^ν )j >"g!0, 8 ">0. The cocept of cosistecy ca be puzzlig i this cotext because the parameter ν accordig to BHP theorem must go to the right ed poit. The uiqueess of the threshold makes o sese, because as ν goes to x F the approximatio of the coditioal distributio is better. I cosequece, we prove the cosistecy of our estimator i the sese that mimics the properties of the parameter ν. However, other estimators ca mimic as well the behavior of the parameter; ν! x F, ^ο(ν)! p ο ad F ν = G (x; ν). I order to check the 13

14 performace of these other estimators we propose a hypothesis test i the ext sectio. I practice our estimator of the threshold is obtaied i the followig way, Algorithm 4.1. : 1. Fix a threshold, ν = x (k),(k = k 0 = =2) 2 2. Estimatio 3 of ^ ν = 8 < : ^ο ml (ν); ^ff ml (ν) ^ο Hill (ν) GPD approach Pareto approach 3. Compute F ν; (x) = 4. Compute G ^ ν = 8 < : ± 1 fν»x i»xg ± 1 fx i >νg GP D ^ο;^ff (x i ν). PD^ο Hill ( xi ν ) 5. Calculate the distace defied by d ν ks (F ν;;g^ ) = 6. k ++ s P 1 fxi>νg sup j GPD approach Pareto approach ± 1 fν»x i»xg ± 1 fx i >νg G ^ ν (x; ν)j Repeat the process util k = At the ed of the day, we estimate ^ν = x (^k) such that ^ν = arg mi ν d ν ks (F ν;;g^ ν ). Alterative distace measures ca be proposed for this threshold selectio. For istace the oes based o Cramér-vo Mises or Aderso-Darlig Statistics, ffl W 2 = R 1 1 (F ν;(x) G ^ ν (x)) 2 dg ^ ν (x) R ffl A 2 1 (F ν;(x) G = ^ ν (x))2 1 G ^ ν (x)(1 G (x))dg (x). ^ ν ^ ν These statistics rely o the euclidea distace. The drawback of these measures with respect to KS type statistics for threshold selectio is that these first oes are less sesitive to large deviatios from the Pareto type distributio due to isolate observatios (outlier observatios). 5 Hypothesis Testig The threshold estimate ^ν provides the lower limit of the estimatio of the extreme values i fiite samples. Our threshold estimator ^ν is such that as the sample size icreases, 2 Cosider k =1;:::; 1 is computatioally very costly. The method is implemeted takig fractios of the sample. x (k) s.t. k = Λ i ; i =50; 60; 70; 80; 90; 91;:::; The algorithm to estimate the threshold depeds o the maximum domai of attractio of the distributio F. 14

15 coditio ( 10) asymptotically holds (F^ν = G ). The key questio to aswer is whether this coditio ca be rejected or ot for the extremes estimates produced from the threshold value estimatio. I other words, are these estimates really extreme values accordig to our defiitio of extremes? The aswer boils dow to test with G = 8 < : GP D ο;ff (x ν) PD ο ( xi ν ) H 0 : F ν = G (15) GPD approach Pareto approach. The statistic proposed to test H 0 is the followig goodess of fit test T (x ; ) = p sup jf^ν ;(x) G (x; ^ν )j: (16) Although there are alteratives that are more sesitive to the deviatios from the ull distributio that occur i both tails (Modified KS tests, see Maso ad Schueemeyer, 1983) we cocetrate o the stadard KS test because our cocer is the distributio of the largest observatios exceedig the threshold value ν. The samplig distributio of this test statistic J (x; F ; ) = P ft (x ; )» xg is ot kow ad the asymptotic ull distributio J(x; F ) is parameter free (see Kolmogorov, 1933) but it is ot possible to obtai avalue of the estimator based o a sample x because the set of parameters is ukow. Therefore, the test statistic eeded to test the ull hypothesis is T (x ; ^ ), where ^ is a estimate of the true. This statistic follows asymptotically a fuctioal of a cetered gaussia process that depeds o, see Durbi (1973). The asymptotic critical values vary with H 0 ad the estimatio of this set of parameters. Bootstrap methodology ca be applied to calculate the samplig quatiles of the Bootstrap distributio J (x; ^F ; ^ Λ ) with ^ Λ the estimated set of parameters from the bootstrap sample x Λ = fx Λ 1 ;:::;xλ g ad with ^F a estimate of F. These quatiles will be close to the exact quatiles of the distributio of the statistic J (x; F ; ) if the Bootstrap is cosistet (J (x; F ; ^ ) ' J (x; ^F ; ^ Λ )) ad if ^ is a p -cosistet estimator of (J (x; F ; ) ' J (x; F ; ^ )), see Babu ad Rao (2002) for details. Propositio 5.1. Let x be a sample of size from F. Assume that ^F is a estimate of F based ox ad let J (x; F ; ^ ) be the true samplig distributio of the statistic T (x ; ^ ). If the followig two coditios hold ffl sup j ^F (x) F (x)j p! 0. ffl J (x; F ; ^ )! J(x; F ; ) with J(x; F ; ) beig a strictly icreasig cotiuous fuctio i x. 15

16 The, the Bootstrap approximatio J (x; ^F ; ^ Λ ) is cosistet (J (x; ^F ; ^ Λ ) ' J (x; F ; ^ )). 5.1 Methodology The statistic T (x ; ^ ) follows asymptotically a fuctioal of a cetered gaussia process. Therefore, i order to obtai a cosistet bootstrap approximatio (J (x; ^F ; ^ Λ )) of the true samplig distributio of T (x ; ^ ) we eed to costruct ^F verifyig uiform covergece i probability tof. Defiitio 5.1. Let ^F (x) be a mixture of F (x) for values smaller tha the estimated threshold ^ν ad of a Pareto type distributio for values above it: ^F (x) = 8 >< >: G ^ ^ν (x)+ 1 P 1 P 1 fxi»xg x» ^ν 1 fxi»^ν gg ^ ^ν (x) x>^ν : (17) It is obvious to check that ^F (x) i expressio ( 17) is a distributio fuctio. Propositio 5.2. Let x be a sample of size with distributio fuctio F (x). The, the distributio fuctio ^F is such that sup j ^F (x) F (x)j p! 0. The first task is to geerate a bootstrap sample x Λ of size from the distributio ^F. Algorithm 5.1. (Geeratig Process of Data): H Let ^ν = x (k) be the estimated threshold ad ^ be the estimated parameter space. 2. Geerate 0» j» 1 ad calculate dje 8 < : 3. x Λ x (dje) if dje»k i = z if dje >k P j 1 fxi»^ν g z = G ψ^ ^ν ( ) 4. i ++ Go to step 2 P 1 fxi>^ν g Oce a bootstrap sample is geerated it is immediate to calculate J (x; ^F ; ^ Λ ) uder Algorithm 5.2. (Bootstrap Distributio of T ): 1. l =1. 2. Geerate x Λ abootstrap sample comig from ^F. 16

17 3. Compute ^ Λ from the exceedaces of x Λ over the fixed threshold ^ν. 4. Compute T Λ l (xλ ; ^ Λ )= p 5. l =1;:::;B 6. J (x; ^F ; ^ Λ )= 1 B BP 1 ft Λ i»xg: sup j P 1 f^ν»x Λ i»xg P 1 fx Λ i >^ν g G ^ Λ^ν (x; ^ν )j: Notice that the set of parameters is cosistetly estimated by f^s 2 ; ^ο ml g for the Geeralized Pareto distributio, ad by ^ο Hill for the Pareto distributio. Both estimators of the tail idex are p -cosistet for some values of the tail idex (see propositio ( 4.3)). Therefore, the kowledge of J (x; ^F ; ^ Λ ) allows us to estimate the p-value of the test ( 16): p = P fj (x; F ; )>T (x ; )g 'P fj (x; ^F ; ^ Λ ) >T (x ; ^ )g = 1 B BP 1 ft Λ l >T g =^p: Large values of the test statistic imply rejectio of the ull hypothesis. I other words, it is rejected if ^p <fffor a give sigificace level ff. 5.2 Size of the Test Theorem 5.1. Let ^Q be a estimator of F based o a sample x of size that satisfies sup j p ^Q F (x)j! 0 wheever F 2 F H0. The, P ft (x; ^ ) >j (1 ff; ^Q ; ^ Λ )g!ff, with j (1 ff; ^Q ; ^ Λ ) the 1 ff quatile of the Bootstrap distributio J (x; ^Q ; ^ Λ ) of T (x; ^ ). The distributio fuctio ^F (x) of expressio ( 17) verifies the coditio of theorem 5.1, therefore j (1 ff; ^F ; ^ Λ ) ' j (1 ff; F ; ^ ). I cosequece, ^F is a good cadidate to estimate the size of the proposed test. Algorithm 5.3. : 1. j =1. 2. Estimate ^ν = x (k) ad G ^ ^ν by KS method from a sample x j; that follows F. (a) i =1 (b) Geerate a sample x Λ i; ο ^F from x j;. (c) Calculate T Λ i (x Λ i; ; ^ Λ ) (d) i ++. Go to step (b) while i» B: (e) Costruct J (x; ^F )= 1 B BP 3. Geerate a sample x 0 uder H Calculate T 0 (x 0 ; ^ 0 ). 1 ft Λ i»xg: 17

18 5. ^p = 1 B BX 1 ft Λ i >T 0 g : 6. Reject H 0 if ^p <ffwith ff the sigificace level. 7. ffi j = 8 < : 1 if H 0 is rejected 0 if H 0 is accepted. 8. j ++. Go to step 2 while j» m: 9. ^ff = 1 m mx ffi i, where ^ff is the estimatio of the type I error. ^ff should be close to the sigificace level ff. 5.3 Power of the Test The choice of ^Q ca brig some problems uder the alterative hypothesis (F 2 F H1 ). ^Q should satisfy three coditios uder the alterative hypothesis i order to avoid that the critical values of J (x; ^Q ; ^ Λ ) go to ifiity as icreases. ffl T (x ; ^ )!1uder F 2 F H1. ffl ^Q with F 2 F H1 such that ^Q fl F, but some F 0 uder (F H0 ). ffl The critical value should satisfy j (1 ff; ^Q ; ^ Λ ) ' j (1 ff; F 0 ; )!! j(1 ff; F 0 ) < 1. If these coditios hold, the by Slutsky's theorem, P ft (x; ^ ) >j (1 ff; ^Q ; ^ Λ )g'pft (x; ) >j (1 ff; F 0 )g!1as!1. Propositio 5.3. Let x be a sample of size from a distributio F uder the alterative hypothesis F H1 ad let T (x; ^ ) be the test statistic of ( 16) with ^ν ad G ^ ^ν estimated uder the ull hypothesis. The, T (x; ^ )!1. The problem is how to costruct ^Q such that does ot approach the distributio F, but F H0 whe the sample x comes from F H1. ^F is ot valid i this case because F 2 F H1 (x ο F H1 ). At least, a sample x o; of size uder F H0 is required to costruct ^Q, a cosistet estimate of F H0. ^Q (x) = 8 >< >: G ^ ^ν (x)+ 1 P 1 P 1 fx0;i»xg x» ^ν 1 fx0;i»^ν gg ^ ^ν (x) x>^ν (18) with ^ ^ν a cosistet estimate of uder the ull hypothesis. The algorithm to estimate the power is equivalet to the algorithm proposed for the size, but i step 3 the sample is geerated from F 2 F H1. Therefore, ^ff is a estimate of 18

19 the power of the test. The objective of this hypothesis test is to reject extremes estimates defied by ^ν which are ot really extremes. This situatio ca occur for small sample sizes where ^ν ca be ot ear the right ed poit x F defiig more extremes estimates tha there really exist. We ca also test if the extremes estimates defied by other ~ν are really extremes. 6 Simulatios ad Some Fiacial Applicatios I this sectio we preset how our estimatio ad testig methodology perform i fiite samples, with simulated data from differet distributios as well as with real data. Uder our methodology the extremes of the distributio are well estimated by the observatios exceedig a determied threshold value oce the ull hypothesis ( 15) is ot rejected. The extreme quatile estimates ad their bootstrap cofidece itervals rely upo the costructio of ^F (x). We distiguish two cases: if F has heavy tails, G is a PD ο ad a cosistet estimator is give by ^F (x) = 8 >< >: 1 P 1 1 fxi»xg x» ^ν ± 1 fx i >^νg ( x^ν ) 1 ^ο otherwise, G is a GP D ο;ff ad a cosistet estimate of F is ^F (x) = 8 >< >: 1 ± 1 ± 1 fx i >^νg By the coditioal probability theorem, x>^ν 1 fxi»xg x» ^ν (1 + (x ^ν) ^ο ^ff ) 1 ^ο x>^ν P fx» xg = P fx» νgp fx» x j X» νg + P fx >νgp fx» x j X>νg (19) with P fx» x j X» νg = 1 for x>ν. The coditioal probability P fx» x j X >νg = F ν (x) ca be well approximated by apareto type distributio G for ν large (BHP theorem). Cosider x p such that P fx» x p g =1 p, 0 <p<1 ad ^ν = x (k) estimated by our KS distace estimator. The, covertig expressio ( 19) ito its empirical couterpart ad approximatig F ν (x) by G ^ ^ν we obtai 1 p = 1 For F 2 MDA(Φ ff ), G ^ = PD^ο, X 1 fxi»^ν g + 1 X 1 fxi>^ν gg ^ ^ν : ^x p =^ν ( ± 1 fxi>^ν g p ) ^ο: (20) 19

20 For F 2 MDA(H ο ), G ^ = GP D ^ο;^ff, ^x p =^ν + ^ff^ο (( ± 1 fxi>^ν g p) ^ο 1): (21) Quatile estimatio is very importat as a risk measure i may fields. I Fiace is used as a risk idicator (Value at Risk) ad i Hydrology or Meteorology to determie security levels of raifalls or floods. Aother applicatio of ^F is to measure the ucertaity of the tail parameter estimates. There are two challeges to make iferece about these parameters. First, F ad the true samplig distributio of the statistic h (x ; ) of the extreme parameter are ot kow, ad secod, the asymptotic distributio of h depeds o uisace parameters. ^F defied from ^ν allows to geerate bootstrap samples x Λ i order to calculate the Bootstrap samplig distributio L (x; ^F ) = P (h (x Λ ; ^ )» x) of the statistic. Propositio 6.1. Let h (x ; (F )) be a statistic such that depeds o the sample x ad o the parameter (F ). Let L (x; F ) the true samplig distributio of the statistic ad L (x; ^F ) be the bootstrap approximatio. Cosider ^ ( ^F ) a estimator of (F ). The, if the Bootstrap approximatio is cosistet (L (x; F ) ' L (x; ^F )), P fl 1 ( ff 2 ; ^F )» h (x ; (F ))» L 1 (1 ff 2 ; ^F )g'1 ff: Suppose h (x ; (F )) = fl (^ ( ^F ) (F )), fl>0. The, a cofidece iterval for (F ) at sigificace levelff is I:C(ff) =[^ ( ^F ) fl L 1 (1 ff 2 ; ^F ); ^ ( ^F ) fl L 1 ( ff 2 ; ^F )]: (22) Cofidece itervals for the tail idex parameter proposed i ( 12) are calculated with this methodology but with o iformatio about the tail behavior, i.e. ^F is the empirical distributio. Oce the ull hypothesis of ( 15) is ot rejected, cofidece itervals from expressio ( 12) ca be improved approximatig F by our semi-parametric distributio ^F because we are coutig with crucial iformatio about the tail of F. 6.1 Fiite Sample Performace The scope of this sectio is to give simulated evidece about the fiite sample properties of the differet estimators of the threshold, as well as the impact of these estimators i the tail idex estimators. 20

21 p Extremes are characterized by a threshold parameter ν such that satisfies: ^ο(ν)! ο, ν! x F ad F ν = G (x; ν). Let us start with the tail idex estimator. We cosider three alterative estimators: ^οml (^ν) based o a GPD with the threshold estimated by KS distace, ^ο Hill (^ν )with^ν also estimated by KS distace ad ^ο Pic (ν Pic ) Pickads estimator with the threshold estimated by Pickads method (see Sectio 4:2). These statistics deped o the threshold, therefore the method to select ^ν is crucial to miimize possible bias effects ad to get cosistecy. We have costructed bootstrap cofidece itervals for the tail idex yielded from these three differet approaches. KS(GPD) ad KS(PD) are the methods achored i a Geeralized Pareto ad a Pareto distributio respectively. Pickads method is costructed with the estimates of the Pickads estimator obtaied from the values over the estimated threshold proposed by Pickads (1975). F ο KS (GPD) KS (PD) Pickads N(0; 1) ο =0 [ 0:41; 0:18] [0:08; 0:19] [ 0:80; 0:35] Exp(1) ο =0 [ 0:23; 1:22] [ 0:29; 0:25] [ 0:34; 0:05] t 60 ο ο 0 [ 0:39; 0:27] [0; 0:24] [ 0:6; 0:31] t 10 ο ο 0:1 [ 0:28; 0:48] [0:16; 0:30] [ 0:67; 0:09] PD 1=4;1 ο =0:25 [0:02; 0:59] [0:16; 0:37] [0:13; 0:43] PD 1=2;1 ο =0:5 [ 0:13; 1:41] [0:23; 0:81] [0:46; 0:79] Table 6.1. Cofidece itervals at ff = 0:05 for the tail idex ο yielded from the three proposed estimators, ^ο ml (^ν ), ^ο Hill (^ν ) ad ^ο Pic (ν Pic ) with ν estimated by the KS distace method ad Pickads estimator, respectively. B = 1000 bootstrap samples of size = 1000 have bee geerated from a sample of the distributio F. It ca be observed that KS(GPD) cofidece itervals always cotai the parameter, although they are loger tha the other oes. KS(PD) method outperforms the GPD method whe F has heavy tails, i other cases, this estimator ca produce biased cofidece itervals. Pickads method oly performs well for distributios with heavy tails. It is importat to otice that these bootstrap itervals rely o the empirical distributio fuctio, F. For large sample sizes it is ot relevat the bootstrap approximatio of F, however, for as the sample size decreases it is better to use ^F of expressio ( 17), because it provides us with iformatio about the tail whe there is o sufficiet available data of F. 21

22 F KS (GPD) KS (PD) F ^F F ^F N(0; 1) [ 0:48; 1:45] [ 0:67; 0:11] [ 1:38; 0:08] [0:04; 0:38] Exp(1) [ 0:35; 1:39] [ 0:48; 1:56] [0:02; 0:42] [ 2:32; 0:13] t 60 [ 1:49; 1:50] [ 0:62; 0:01] [ 0:89; 0:32] [ 0:03; 0:30] t 10 [ 0:39; 0:29] [ 0:43; 0:31] [0:20; 0:59] [ 0:25; 0:29] PD 1=4;1 [ 0:78; 0:66] [ 0:14; 0:70] [0:10; 0:42] [0:19; 0:30] PD 1=2;1 [0:06; 0:95] [0:11; 1:11] [0:18; 1:70] [0:37; 0:67] Table 6.2. Cofidece itervals at ff =0:05 for the tail idex ο yielded from ^ο ml (^ν ) ad ^ο Hill (^ν ) with ν estimated by the KS distace method. B =1000bootstrap samples of size = 250 have bee geerated from a sample of the distributio F. I the rest of the sectio we will be usig F to costruct the cofidece itervals for the tail idex, because i order to employ ^F wehave first to accept the ull hypothesis F ν = G. To check imore detail the performace of these estimators for heavy tails, i Table 6.3 we aalyze t-studet distributios with differet degrees of freedom. t 1 (ο ο 1) t 3 (ο ο 0:33) t 5 (ο ο 0:2) t 10 (ο ο 0:1) t 30 (ο ο 0) KS (GPD) [0:37; 1:11] [0:10; 1:53] [ 0:17; 0:33] [ 0:48; 0:14] [ 1:31; 0:50] KS (PD) [0:67; 1:24] [0:09; 0:42] [0:15; 0:39] [0:16; 0:30] [ 0:03; 0:24] Pickads [0:61; 1:36] [ 0:44; 0:14] [0:01; 0:90] [ 0:67; 0:09] [ 0:83; 0:36] Table 6.3. Cofidece itervals at ff = 0:05 for the tail idex ο from the three proposed estimators, ^ο ml (^ν ), ^ο Hill (^ν ) ad ^ο Pic (ν Pic ) with ν estimated by the KS distace method ad Pickads estimator, respectively. B =1000bootstrap samples of size =1000have bee geerated from a sample of the differet t-studet distributios. I practice, the problem arises whe the geeratig process of data is ukow ad there is o iformatio about the ratio of decay of the tail. The tail idex ca be estimated by both methods (KS(GPD) ad KS(PD)) ad depedig o the results we should apply a adequate estimator for the threshold parameters, ^ν GP D ;ks or ^ν ;ks PD,toachieve more accurate ad reliable estimatios of the extremes. Some fiacial idexes are cosidered i Table

23 KS (GPD) KS (PD) C.I. Pickads Dax [ 0:18; 0:89] [0:23; 0:37] [ 0:49; 0:15] Ftse [ 0:25; 0:07] [ 0:31; 0:15] [ 0:46; 0:06] Ibex [ 0:11; 0:87] [0:25; 0:47] [ 0:46; 0:04] Nikkei [ 0:11; 0:56] [0:27; 0:41] [ 0:36; 0:03] Dow-Joes [ 0:15; 1:55] [0:039; 0:53] [ 0:43; 0:03] Table 6.4. Cofidece itervals at ff =0:05 for the tail idex ο for real data over roughly the period 05= =2001. B = 1000 bootstrap samples of size = 1000 have bee geerated for the bootstrap itervals. Almost all fiacial idexes aalyzed i this Table ca be cosidered to be fat tailed ad the extremes of these distributios are well defied by ^ν yielded from the KS estimator ad the Pareto distributio (PD ο ) with ο cotaied i a precise cofidece iterval. Some doubts ca exist with respect Ftse idex. I this case we coclude that the extremes follow a GP D ο;ff. Cosider ow thesecod property ofthe threshold parameter: ν! x F. By cosistecy, the threshold estimators should go to the right ed poit as the sample size icreases. Distributio = 500 = 1000 = 1500 = 2000 =5000 N(0; 1) ^ν GP D ;ks 1:19 1:37 1:45 1:51 1:67 (0:57) (0:49) (0:47) (0:46) (0:42) ^ν Pic 0:44 0:52 0:59 0:64 0:88 (0:26) (0:29) (0:32) (0:33) (0:36) t 10 ^ν PD ;ks 2:18 2:28 2:33 2:39 2:49 (0:47) (0:43) (0:41) (0:38) (0:32) ^ν Pic 0:47 0:56 0:63 0:69 0:96 PD1 4 ;1 (0:27) (0:31) (0:34) (0:36) (0:39) ^ν PD ;ks 2:14 2:13 2:11 2:07 2:07 (0:64) (0:62) (0:62) (0:61) (0:61) ^ν Pic 1:29 1:29 1:29 1:29 1:29 (0:07) (0:07) (0:08) (0:08) (0:08) Table 6.5. Threshold estimatio with KS distace ad Pickads estimators as icreases samples of size of differet distributios are geerated. The ubiased estimated stadard deviatio from simulatios of ^ν is displayed i brackets. 23

24 As icreases, the two estimators go to the right ed poit of the distributio. Pickads estimator provides estimates far from the right ed poit ad the variace slowly icreases. This result poits out that extremes estimates produced by Pickads method may be ot very reliable. O the other had, the estimators achored i KS distace have decreasig variace ad approachtox F as!1. Notice that for PD1 4 ;1 distributio, ^ν PD ;ks estimator has a greater variace as before ad k 9 0. This is because this distributio is exactly of Pareto type but the term of the KS statistic accoutig for the sample legth of the tails produces this ucertaity i the threshold estimates from the bootstrap samples. Pickads estimator detects the shape of the distributio from the begiig. Oe of the goals of this paper is to propose a test to check if the extreme estimates yielded from a proposed threshold estimator verify the third property: F ν = G (x; ν). The rejectio of the ull hypothesis meas the extremes estimates defied by ^ν are ot really extremes. Tables 6.6 ad 6.7 show size ad power of the goodess of fit test proposed i ( 16). The proposed alteratives to measure the power of this test are costructed as deviatios from the theoretical distributio of the extremes. Table 6.6 shows the empirical rejectio rates of our test for F 2 MDA(H ο ). = 1000 Size Power (5%) 0:01 0:05 Exp(1) GP D 1=4;1 GP D 1=4;1 N(0; 1) 0:014 0:07 0:98 0:96 0:96 Exp(1) 0:014 0:04 0:5 0:72 0:75 t 60 0:02 0:05 0:97 0:95 0:96 FTSE 0:006 0: Table 6.6. B=1000 Bootstrap samples of legth = 1000 of the differet distributios with tail expoetially decayig. m=500 simulatios are geerated for the bootstrap test. Notice that the results from the expoetial distributio reflect certai lack ofpower of the test. This is because the tail of a expoetial with mea 1 is a GPD with ο =0. Thus, our proposed alteratives are very close to the ull hypothesis. Next table displays the the empirical rejectio rates of our test for F 2 MDA(Φ ff ). 24

25 = 1000 Size Power (5%) 0:01 0:05 Exp(1) PD 0:1;1 PD 0:65;1 t 10 0:012 0:038 0:79 0:74 0:97 PD 1=4;1 0:012 0:056 0:75 0:92 0:95 PD 1=2;1 0:01 0:046 0:98 0:99 0:67 Nikkei 0:014 0: Table 6.7. B=1000 Bootstrap samples of legth = 1000 of the differet heavy tailed distributios. m=500 simulatios are geerated for the bootstrap test. Aother possibility for the alterative hypothesis is to cosider more extremes tha with our defiitio of extremes, i.e. ~ν < ^ν. Let us cocetrate o distributios with heavy tails. We should test F ν = PD ο fixig the threshold ~ν i order to check ifthere are more data i the populatio that follow a Pareto distributio with tail idex ο. I additio, the opposite case ca be tested as well. Cosider a smaller set of extremes tha the oes produced with our defiitio of extremes. I this case the ull hypothesis should be accepted because F^ν = PD ο implies F ~ν = PD ο with ^ν < ~ν. Data ^ν ~ν = x (950) ~ν = x (900) ~ν = x (800) ~ν = x (700) t 10 fl 0:97 =2:27 0:19 0:01 0:00 0:00 ^s =(0:42) (0:29) (0:07) (0:00) (0:00) t 3 fl 0:97 =2:97 0:29 0:13 0:0001 0:00 ^s =(0:97) (0:33) (0:26) (0:002) (0:00) DaX x (910) =0:025 0:69 0:20 0:00 0:00 Nikkei x (920) =0:021 0:97 0:05 0:00 0:00 Table 6.8. p-values of the bootstrap hypothesis tests H 0 : F ~ν = PD ο for samples of = 1000 observatios. For the t-studet distributios m = 500 iteratios are geerated. fl p is the extreme quatile ^ν of the distributio. The ubiased estimated stadard deviatio of the p-values is displayed ibrackets. 7 Coclusio Risk ad ucertaity are ot the same thig (see Grager, 2002) ad therefore they eed to be characterized by differet measures. It is accepted that variace is well desiged to capture the latter but ot the former. To measure risk, i other words, to respod the questio if thigs go wrog how wrog they ca go? it is first ecessary to fid a aswer 25

26 to the questio which extreme values are really extremes? This is the mai goal of this paper, where followig Pickads (1975) methodology we do ot oly defie formally ad aalytically the set of extreme observatios of a give populatio, but we proposeasimple estimator of them ad costruct a test to aswer the previous questio. Idetificatio of the extreme observatios allows to estimate very accurately risk measures as Value at Risk or Expected Shortfall, as well as to make iferece o differet tail parameters of iterest. Boths issues are extesios of this paper ad costitute udergoig researchby the authors. A Appedix: Proofs Corollary 3.1: Takig logs i expressio ( 1), wehave log(1 F (c x+d )) d! logh ο (x). Therefore, log (1 F (cx+d) ) d! logh ο (x). This is equivalet to F (c x + d ) d! logh ο (x), with H ο = e (1+οx) 1 ο if ο 6= 0adHο = e e x if ο =0. We obtai F (c x + d ) d! (1 + οx) 1 ο if ο 6= 0 ad F (c x + d ) d! e x. Λ Corollary 3.2: Let F 2 MDA(Φ ff ) ad M = max(x 1 ;:::;x ). By defiitio, there exist costats, c = F ψ (1 1 ) ad d = 0 such that c 1 (M d ) d! Φ ff with Φ ff = e x ff ;x > 0, ad ff>0. By propositio ( 3.1), F 2 RV ff. Cosider ν; x 2 support(f) with x F = 1 ad x = νt with t>1. Notice that for 0 <t» 1, F ν (x) =0. Operatig i expressio ( 4), 1 F (x) 1 lim ν!1 Propositio 1 F (ν) = lim ν!1 F (x) F (ν) 1 F (ν) = lim ν!1 F ν(x) =1 ( x ν ) ff = PD ο ( x ν ): Λ 4.1: First the if part. Cosider ^ν a threshold estimator such that the values above it are extreme values. Therefore expressio ( 10) ca be writte, replacig the parameter by the estimator, as This implies lim ^ν!x F sup jf^ν (x) G (x; ^ν )j =0: P fsup jf^ν (x) G (x; ^ν )j >"g!0. I additio, if ^ν defies the set of extreme values there may exist a subset A R such that jf^ν (A ) G (A ; ^ν )j >", although from ( 10) sup x2a The, it is derived that F^ν (x) =G (x; ^ν ) 8x 2 RA. jf^ν (x) G (x; ^ν )j!0. With respect to the oly if part, this result follows from coditio ( 2). The cotiuity ear the right ed poit x F ad the cosistecy of the estimator ^ν imply that lim sup ^ν!x F jf^ν (x) G (x; ^ν )j =0. Λ 26

27 Propositio 4.2: Let x 1 ;:::;x k ο PD ο with PD ο ( x ν )=1 ( x ν ) ff ;x > ν. The desity fuctio is pd(x) =ff( x ν ) (ff+1) 1 ν. The, the likelihood fuctio is l(x 1 ;:::;x k ; ν; ff) =( ff k ν )k Π ( xi ν ) (ff+1). Let ο = 1 ff, the from the first order coditios, it is easy to obtai ^ο = 1 k Theorem kp log xi ν : Λ 4.1: Let ^ν be the threshold estimator derived from the KS distace ad let ^ο(^ν ) be a cosistet estimator of the tail idex based o x with ο 2. sup jf^ν (x) G (x; ^ν )j»sup jf^ν (x) G ^ (x; ^ν )j + sup jg (x; ^ν ) G ^ (x; ^ν )j. ^ is a cosistet estimator of, therefore, sup jg (x; ^ν ) G ^ (x; p ^ν )j! 0. s P Let X (ν) = 1 fxi>νg sup jf ν(x) G ^ ν (x; ν)j such that for values of ν sufficietly large X (ν) is a radom variable that follows a fuctioal of a cetered gaussia process depedig o the parameter (see Durbi, 1973). Cosider ow, X (^ν ) = mifx ;1 (ν 1 );:::;X ;k (ν k )g with fν 1 ;:::;ν k g greater tha a ν 0 verifyig BHP theorem ad X ;i (ν i ) radom variables. ^ν is the argumet of the miimum of this fiite set; ^ν = arg mi X (ν). The, ν P (X (^ν ) >")=P (mifx ;1 (ν 1 );:::;X ;k (ν k )g >")=P (X ;i (ν i ) >") k : As goes to ifiity k icreases as well. P (X (^ν ) >")! 0as; k!1. This expressio is equivalet to The, P f s P 1 fxi>^ν g sup I additio, P (X (ν) > ") < 1, therefore, jf^ν (x) G ^ ^ν (x; ^ν )j >"g!0. P fsup jf^ν (x) G ^ ^ν (x; ^ν )j >" Λ g!0 with 0 <" Λ <". Λ Propositio 5.1: Let x be a sample of size from F. Assume that ^F is a estimate of F based o x verifyig sup j p ^F (x) F (x)j! 0 ad let J (x; F ; ^ ) be the true samplig distributio of the statistic T (x ; ^ ). This distributio is such that J (x; F ; ^ )! J(x; F ; ) with J(x; F ; ) beig a strictly icreasig cotiuous fuctio i x. The, P ft (x; ^ )» J ψ (1 ff; F ; ^ )g!p ft (x; ^ )» J ψ (1 ff; F ; )g =1 ff. I additio, J(x; F ; ) is cotiuous ad strictly icreasig, therefore J ψ (1 ff; F ; ^ )! J ψ (1 ff; F ; ). The, as!1, J ψ (1 ff; ^F ; ^ Λ )! J ψ (1 ff; F ; ) because sup j ^F (x) F (x)j P ft (x; ^ )» J ψ (1 ff; F ; )g = 1 ff. sup p! 0. Cosequetly, P ft (x; ^ )» J ψ (1 ff; ^F ; ^ Λ )g! The, sup jj (x; F ; ^ ) J (x; ^F ; ^ Λ )j» jj (x; F ; ^ ) J(x; F ; )j + sup jj (x; ^F ; ^ Λ ) J(x; F ; )j!0. Λ Propositio 5.2: For x» ^ν, ^F (x) is the Empirical distributio fuctio. By Gliveko- Catelli theorem, sup x»^ν j ^F (x) F (x)j = sup x»^ν jf (x) F (x)j a:s:! 0. For x>^ν, uder 27