Reduced-bias estimator of the Conditional Tail Expectation of heavy-tailed distributions

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1 Reduced-bias estimator of the Conditional Tail Expectation of heavy-tailed distributions El Hadji Deme, Stephane Girard, Armelle Guillou To cite this version: El Hadji Deme, Stephane Girard, Armelle Guillou. Reduced-bias estimator of the Conditional Tail Expectation of heavy-tailed distributions. M. Hallin et al. Mathematical Statistics and Limit Theorems, Springer, pp.05 23, 205. <hal v2> HAL Id: hal Submitted on 23 Sep 203 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Reduced-bias estimator of the Conditional Tail Expectation of heavy-tailed distributions El hadji Deme, Stéphane Girard 2 and Armelle Guillou 3 LERSTAD, Université Gaston Berger de Saint-Louis, Sénégal 2 Team Mistis, Inria Grenoble Rhône-Alpes & Laboratoire Jean Kuntzmann 655, avenue de l Europe, Montbonnot, Saint-Ismier Cedex, France 3 Université de Strasbourg et CNRS, IRMA, UMR 750, 7 rue René Descartes, Strasbourg cedex, France Abstract Several ris measures have been proposed in the literature. In this paper, we focus on the estimation of the Conditional Tail Expectation CTE. Its asymptotic normality has been first established in the literature under the classical assumption that the second moment of the loss variable is finite, this condition being very restrictive in practical applications. Such a result has been extended by Necir et al. 200 in the case of infinite second moment. In this framewor, we propose a reduced-bias estimator of the CTE. We illustrate the efficiency of our approach on a small simulation study and a real data analysis. Keywords: Bias correction, heavy-tailed distribution, conditional tail expectation, ernel estimators. Introduction Different ris measures have been proposed in the literature and used to determine the amount of an asset to be ept in reserve in the financial framewor. We refer to Goovaerts et al. 984 for various examples and properties. One of the most popular example in hydrology or climatology is undoubtedly the return period. A frequency analysis in hydrology focuses on the estimation of quantities e.g., flows or annual rainfall corresponding to a certain return period. It is closely related to the notion of quantile which has therefore been extensively studied. For a real value random variable X with E[X] <, the quantile of order T expresses the magnitude of the event which is exceeded with a probability equal to T. T is then called the return period. In an

3 actuarial context, the Value-at Ris VaR is defined as the p quantile Qp = inf{x 0 : Fx p}, for p [0, ], with F the distribution function of the random variable X. A second important ris measure, based on the quantile notion, is the Conditional Tail Expectation CTE defined by CTE α [X] = EX X > Qα, for α 0,. The CTE satisfies all the desirable properties of a coherent ris measure see Artzner et al., 999 and it provides a more conservative measure of ris than the VaR for the same level of degree of confidence see Landsman and Valdez, For all these reasons, the CTE sometimes referred to as expected shortfall is preferable in many applications. It thus continues to receive an increased attention in the actuarial literature see for instance chapters 2 and 7 in McNeil et al., In all the sequel we assume that F is continuous, which allows us to rewrite the CTE α [X] as C α [X] = α α Qsds. Clearly, the CTE is unnown since it depends on F. Hence, it is desirable to define estimators for this quantity and to study their asymptotic properties. To this aim, suppose that we have at our disposal a sample X,..., X n of independent and identically distributed random variables from F and denote by X,n... X n,n the order statistics. The asymptotic behaviour of C αn [X] has been studied recently in Pan et al. 203 and Zhu and Li 202 when α n as n. On the contrary, in this paper we consider α fixed. A natural estimator of C α [X] can then be obtained by Ĉ n,α [X] = α α Q n sds, where Q n s is the empirical quantile function, which is equal to the ith order statistic X i,n for all s i /n, i/n], and for all i =,..., n. The asymptotic behavior of the estimator Ĉn,α[X] has been studied by Brazausas et al. 2008, when E[X 2 ] <. Unfortunately, this condition is quite restrictive. For instance, in the case of Pareto-type distributions, defined as Fx = x γ lf x where l F is a slowly varying function at infinity satisfying l F λx/l F x as x for all λ > 0, this condition of second moment implies that γ 0, /2. When γ /2,, we have E[X 2 ] = but nevertheless the CTE is well-defined and finite since E[X] <. Note that, in the case γ = /2, the finiteness of the second moment depends on the slowly varying function. This framewor will be the subject of this paper where we assume that Fx = x /γ l F x where γ > 0 is the extreme value index. We focus on the case where γ /2, and thus E[X 2 ] =, this range of values being excluded in the results of Brazausas et al The 2

4 estimation of γ has been extensively studied in the literature and the most famous estimator is the Hill 975 estimator defined as: γ H n, = j log X n j+,n log X n j,n j= for an intermediate sequence = n, i.e. a sequence such that and /n 0 as n. More generally, Csörgő et al. 985 extended the Hill estimator into a ernel class of estimators γ K n, = j= j K Z j,, + where K is a ernel integrating to one and Z j, = j log X n j+,n log X n j,n. Note that the Hill estimator corresponds to the particular case where Ku = Ku := l {0<u<}. Now remar that C α [X] can be rewritten as C α [X] = α /n α =: C α [X] +C 2 α [X]. Qsds + α /n 0 Q sds. In this spirit, Necir et al. 200 introduced the following estimator of the CTE, which taes into account different asymptotic properties of moderate and high quantiles in the case of Pareto-type distributions: C n,α [X] =: C n,α[x] + = α n j= C 2 n,α[x] j n α + j n α /n X j,n + + α γ n, H X n,n 2 where s α + is the classical notation for the positive part of s α. C The estimator n,α[x] is obtained similarly to using the well-nown properties of the empirical quantile function Q n whereas C n,α[x] is obtained using a Weissman estimator of Q: 2 Q s := X n,n n γ H n, s γh n,, s 0 see Weissman, 978. This estimator may suffer from a high bias in finite sample situations, as illustrated on Figure on a Burr distribution with extreme value index γ = 2/3. Besides, it appears that the bias heavily depends on the intermediate sequence, maing the choice of difficult in practice. The goal of this paper is twofold. First, we state an asymptotic normality result for C n,α [X] exhibiting the bias term Section 2 and thus generalizing the one of Necir et al Second, the precise nowledge of the first order of the bias allows us to propose a reduced-bias approach. The efficiency of our method is illustrated on a small simulation study and a real dataset in Section 3. All the proofs are postponed to Section 4. 3

5 CTE CTE CTE Figure : Median of C n,0.05 [X] as a function of based on 500 samples of size 500 from a Burr distribution defined as Fx = + x 3ρ 2 /ρ. From the left to the right: ρ =.5, ρ = and ρ = The horizontal line represents the true value of the CTE 0.05 [X]. 2 Main results As usual in the extreme value framewor, to prove asymptotic normality results, we need a secondorder condition on the function Ux = Q /x such as the following: Condition R U. There exist a function Ax 0 as x of constant sign for large values of x and a second order parameter ρ < 0 such that, for every x > 0, logutx logut γ log x lim t At = xρ. ρ Note that condition R U implies that A is regularly varying with index ρ see e.g. Gelu and de Haan, 987. It is satisfied for most of the classical distribution functions such as the Pareto, Burr and Fréchet ones. We start to give in Theorem, the main expansion of C n,α [X] in terms of Brownian bridges, which leads to its asymptotic normality stated in Corollary. As it exhibits some bias, we propose a reduced-bias estimator whose expansion is formulated in Theorem 2 and its asymptotic normality is given in Corollary Asymptotic results for the CTE estimator Theorem. Assume that F satisfies R U with γ /2,. Then for any sequence of integer = n satisfying, /n 0 and An/ = O as n, we have n α D= n Cn,α [X] C α [X] A ABγ, ρ +W n, +W n,2 +W n,3 + o P Un/ where ABγ, ρ := γρ ργ + ρ γ 2 4

6 and W n, := /n 0 B n sdqs /nq /n W n,2 := γ n γ B n /n W n,3 := γ n γ 2 0 s B n s/ndsks. Now, by computing the asymptotic variances of the different processes appearing in Theorem, we deduce the following corollary. Corollary. Under the assumptions of Theorem, if An/ λ R, we have n α D Cn,α [X] C α [X] N λabγ, ρ, AVγ Un/ where ABγ, ρ is as above and AVγ = γ 4 2γ γ 4. Since ρ < 0 and γ /2,, we can easily chec that ABγ, ρ is always positive and thus the sign of the function A. determines the sign of the bias of C n,α [X]. Note that the asymptotic variance AVγ does not depend on α and that this result generalizes Theorem 3. in Necir et al. 200 in case λ 0. The goal of the next section is to propose a reduced-bias estimator of C α [X]. 2.2 Reduced-bias method with the least squared approach From Theorem, it is clear that the estimator C n,α [X] exhibits a bias due to the use in its construction of the Weissman s estimator which is nown to have such a problem. To overcome this issue, we propose to use the exponential regression model introduced in Feuerverger and Hall 999 and Beirlant et al. 999 to construct a reduced-bias estimator. More precisely, using R U, Feuerverger and Hall 999 and Beirlant et al. 999, 2002 proposed the following exponential regression model for the log-spacings of order statistics: ρ j Z j, γ + An/ + ε j,, j, 3 + where ε j, are zero-centered error terms. If we ignore the term An/ in 3, we retrieve the Hill-type estimator γ n, H by taing the mean of the left-hand side of 3. By using a least-squares approach, 3 can be further exploited to propose a reduced-bias estimator of γ in which ρ is substituted by a consistent estimator ρ = ρ n, see for instance Beirlant et al., 2002 or by a canonical choice, such as ρ = see e.g. Feuerverger and Hall 999 or Beirlant et al

7 The least squares estimators of γ and An/ are then given by γ n, ρ LS = Â LS n, Z j, ρ ρ, j= Â LS 2 ρ ρ2 n, ρ = ρ 2 j + j= ρ Z j,. ρ The main asymptotic properties of γ n, LS ρ and ÂLS n, ρ as a function of Brownian bridges have been established in Deme et al. 203, Lemma 5. Note that γ n, LS ρ can be viewed as a ernel estimator where for 0 < u : γ LS n, ρ = with K ρ u = ρ/ρu ρ l {0<u<}. j K ρ + j= Z j,, K ρ u = ρ ρ Ku + ρ K ρ ρ u Now, using the second order refinements of assumption R U, we can construct the following asymptotically unbiased estimator of the quantile: Q LS, ρ s = ns/ γls n, ρ Xn,n ρ Â LS n, ρ ns/ ρ, see e.g. Matthys et al Thus, in the spirit of 2, we arrive at the following asymptotically unbiased estimator of C α [X] n C LS, ρ j j n,α [X] := α n α + n α X j,n + + j= /n α γ LS n, ρ γ LS n, Â LS n, ρ X n,n. ρ + ρ Our next goal is to establish, under suitable assumptions, the asymptotic normality of This is done in the following theorem. LS, ρ C n,α [X]. Theorem 2. Under the assumptions of Theorem, if ρ is a consistent estimator of ρ, then we have n α Un/ CLS, ρ n,α [X] C α [X] D= Wn, +W n,2 +W n,4 +W n,5 + o P where W n,, W n,2 and W n,3 are defined in Theorem, and ργ 2 n W n,4 := γ + ρ γ 2 s B n s/ndsk ρ s 0 W n,5 := γ ρ W n,3. γ + ρ 6

8 Now, by computing the asymptotic variances of the different processes appearing in Theorem 2, we deduce the following corollary. Corollary 2. Under the assumptions of Theorem, if ρ is a consistent estimator of ρ, then we have with n α CLS, ρ [X] C α [X] Un/ n,α D N 0, ÃVγ, ρ ÃVγ, ρ = γ 4 γ ρ 2 2γ γ 4 γ + ρ 2. As expected, the asymptotic bias of our new estimator of the CTE is equal to zero whereas its asymptotic variance ÃVγ, ρ is larger than the one of the original estimator AVγ exhibited in Corollary. 3 Finite sample behavior 3. A small simulation study In this section, the biased estimator C n,α [X] and the reduced-bias one LS, C n,α [X] are compared on a small simulation study. To this aim, 500 samples of size 500 are simulated from a Burr distribution defined as: Fx = + x 3 2 ρ /ρ. The associated extreme value index is γ = 2/3 and ρ is the second order parameter. Three values for α {0.05, 0.0, 0.20} are used and different values of ρ { 0.75,,.5} are considered to assess its impact. The median and median squared error MSE of these estimators are estimated over the 500 replications. The results are displayed in Figure 2 and Figure 3. It appears on Figure 2 that the closer ρ is to 0, the more important is the bias of C n,α [X] whatever the value of α is. The effect of the bias correction on the MSE is illustrated on Figure 3. We can observe that the MSE of the reduced-bias estimator C LS, n,α [X] is almost constant with respect to, especially when the bias of C n,α [X] is strong, i.e when ρ is close to Real data analysis Our real dataset concerns a Norwegian fire insurance portfolio from 972 until 992. Together with the year of occurrence, the data contain the value 000 Krone of the claims. A priority of 500 units was in force. These data were of some concern in that the number of claims had risen systematically with a maximum in 988 as illustrated in Figure 4a. We concentrate here on the year 976 where the average claim size per year reached a pea as was the case in 988. The sample size is n = 207. Figure 4b shows the histogram corresponding to this year 976. From 7

9 Figure 4c we can observe the difficulty to find a stable part in the plot of the Hill estimator γ H n, as a function of, due to the bias of this estimator. We can apply our methodology to this real dataset as the extreme value index or at least its estimator is in the interval /2, whatever the value of is. Figure 4d-f shows the biased estimator C n,α [X] dashed line and the reduced-bias LS, one C n,α [X] full line for three different values of α: 0.05, 0.0 and The reduced-bias LS, estimator C n,α [X] is almost constant for a large range of values of which maes the choice of easier in practice. 4 Proofs Let Y,..., Y n be independent and identically distributed random variables from the unit Pareto distribution G, defined as Gy = y, y. For each n, let Y,n... Y n,n be the order statistics pertaining to Y,..., Y n. Clearly X j,n D = UYj,n, j =,..., n. In order to use the results from Csörgő et al. 986, a probability space Ω, A, P is constructed carrying a sequence ξ, ξ 2,... of independent random variables uniformly distributed on 0, and a sequence of Brownian bridges B n s, 0 s, n =, 2... The resulting empirical quantile is denoted by β n t = n t V n t where V n s = ξ j,n, j n < s j n, j =,..., n and V n0 = 0. The following lemma gives an asymptotic expansion for the second random term appearing in 2. Lemma. Under the assumptions of Theorem, we have n α C2 n,α[x] C 2 D= n α [X] A ABγ, ρ +W n,2 +W n,3 + o P. Un/ Proof of Lemma. Note that C 2 n,α[x] can be rewritten as follows α C 2 n,α[x] = As a consequence, the following expansion holds: where T n, := T n,2 := T n,3 := T n,4 := /n γ n, H U Y n,n. n α C2 n,α[x] C 2 α [X] = Un/ γ n, H [ U Yn,n Un/ [ n Y n,n γ ], 4 T n,j, j= n Y n,n γ n, H H γ γ n, H γ n, γ, γ ], [ n /n Un/ αc2 α [X] Un/ γ ]. 8

10 We study each term separately. Term T n,. According to de Haan and Ferreira 2006, Theorem 2.3.9, for any δ > 0, we have U Y n,n Un/ n Y n,n where A 0 t At as t. γ n { = A 0 n Y n,n Thus, since Y n,n /n = + o P and γ H n, Term T n,2. The equality Y n,n D = ξn,n yields [ n Y n,n γ ] γ n Y ρ n,n P γ, it readily follows that ρ } γ+ρ±δ + o P n Y n,n, T n, = o P. 4 D = n γ ξ n,n = γ n ξ n,n n = γ β n n n = γ B n n + o P + O P n ν for 0 ν < /2, by Csörgő et al Thus, using again that γ H n, + o P by a Taylor expansion /2 ν + o P, n P γ, it follows that D γ n T n,2 = γ B n + o P = W n,2 + o P. 5 n Term T n,3. According to Theorem in Deme et al 203 and by the consistency in probability of γ n, H, we have T n,3 D = { A n/ n γ 2 + γ ρ = 0 s B n s } d sks + o P n ρ γ 2 A n/ +Wn,3 + o P. 6 Term T n,4. A change of variables and an integration by parts yield T n,4 = { γ x 2Unx/ } Un/ dx = Unx/ x 2 Un/ xγ dx. Thus, Theorem in de Haan and Ferreira 2006 entails that, for γ /2,, Combining 4-7, Lemma follows. T n,4 = n A 0 x γ 2 xρ dx + o ρ = n A + o. 7 γγ + ρ 9

11 Proof of Theorem. Combining Lemma with statement 4.3 in Necir et al. 200, we get n α D= n Cn,α [X] C α [X] A ABγ, ρ +W n, +W n,2 +W n,3 + o P. Un/ Theorem is thus established. Proof of Corollary. From Theorem, we only have to compute the asymptotic variance of the limiting process. The computations are tedious but quite direct. We only give below the main arguments, i.e. /n t t EWn, sdqs dqt = /n Q 2 + /n /n u u dq v dq u = /n Q 2 /n u /n /n vdq v dq u + /n Q 2 /n =: Q,n + Q 2,n + Q 3,n + Q 4,n. /n 0 t /n t sdqs dqt /n Q 2 /n /n u u vdq v dq u /n Q 2 /n /n u u /n vdq v dq u /n Q 2 /n Recall now that Q s = s γ ls with l a slowly varying function at 0. By integration by parts and using Lemma 6 in Deme et al. 203, it follows that Q,n = /n + Q2 udu 2 /nq 2 γ /n 2γ. Now remar that d u vdq v = udq u which implies that Q 2,n = vdq v /n 2 n /n Q /n 2 = o 8 this last result coming from the fact that, according to Proposition.3.6 in Bingham et al. 987: for all ε > 0, x ε lx as x 0. Thus, choosing 0 < ε < γ 2 entails 2 2 tdq t s s 0 s = s + t γ ltdt sq s s γ ls s + Cs γ ε 2 = O s +2[γ ε] = o where C is a suitable constant. Consequently, Q 2,n 0. The two others terms, Q 3,n and Q 4,n, can be treated similarly, leading to Q 3,n = Q,n γ 2γ Q 4,n = Q 2,n 0. Finally, EW 2 n, 2γ 2γ, 0

12 and direct computations now lead to EW 2 n,2 γ 2 γ 2 EWn,3 2 γ 2 γ 4 by Corollary in Deme et al. 203 EW n, W n,2 γ by 8 γ EW n, W n,3 = 0 EW n,2 W n,3 = 0. Combining all these results, Corollary follows. Proof of Theorem 2. We use the following decomposition n α CLS, ρ [X] C α [X] = Un/ n,α 7 i= S n,i where n α C S n, = n,α[x] C α [X] Un/  LS n, S n,2 = γ n, LS ρ ρ [ γ ] U Yn,n γ n, LS ρ + ρ Un/ n Y n,n  LS n, S n,3 = γ n, LS ρ ρ [ γ γ n, LS ρ + ρ n Y n,n ] S n,4 = LS γ γ n, LS ρ γ n, ρ γ S n,5 = An/ γγ + ρ γ n, LS ρ γ n, LS ρ + ρ ÂLS S n,6 = n, ρ An/ γ n, LS ρ γ n, LS ρ + ρ [ n /n S n,7 = An/ ] Un/ αc 2 α [X]. Un/ γ γ + ρ Now, we are going to study separately the terms S n,,..., S n,7. Term S n,. Statement 4.3 in Necir et al. 200 leads to S n, = W n, + o P. 9 Term S n,2. Note that S n,2 = γh n, γ LS n, ρ γ LS n,  LS n, ρ ρ + ρ T n,

13 where T n, is defined in the proof of Lemma. Thus combining Lemma 5 in Deme et al. 203 with the consistency of ρ and 4, we obtain that S n,2 = o P. 0 Term S n,3. Similarly, we observe that S n,3 = T n,2 + o P where T n,2 is defined in the proof of Lemma. Thus according to 5, we have S n,3 D = Wn,2 + o P. Term S n,4. Combining Lemma 5 in Deme et al. 203 with the consistency of γ n, LS ρ, we infer that S n,4 D = γ + ρ ργ W n,4 + o P. 2 Term S n,5. Under the assumption that An/ = O and by the consistency of ρ and γ LS n, ρ we have Term S n,6. Using Lemma 5 in Deme et al. 203, we get D γ ρ S n,6 = γγ + ρ ρ γ = γ + ρ = W n,5 + Term S n,7. Remar that n S n,5 = o P. 3 0 s B n s dsks K ρ s + o P n W n,3 γ + ρ W n,4 + o P ργ ρ γ W n,4 + o P. 4 γρ An/ S n,7 = γγ + ρ + T n,4, where T n,4 is defined in the proof of Lemma. Thus using 7 and the assumption that An/ = O, we deduce that Combining 9-5, Theorem 2 follows. S n,7 = o P. 5 Proof of Corollary 2. From Theorem 2, we only have to compute the asymptotic variance of the limiting process. As in Corollary, the computations are quite direct and the desired asymptotic 2

14 variance can be obtained by noticing that EWn,5 2 γ 2 ρ 2 γ 2 γ + ρ 2 EW n, W n,4 = 0 EW n, W n,5 = 0 EWn,4 2 = γ 4 ρ 2 γ 4 γ + ρ 2 EW n,2 W n,5 = 0 EW n,2 W n,4 = 0 ργ 3 ρ EW n,4 W n,5 = γ 3 γ + ρ 2.. Acnowledgements The first author acnowledges support from AIRES-Sud AIRES-Sud is a program from the French Ministry of Foreign and European Affairs, implemented by the "Institut de Recherche pour le Développement", IRD-DSF and from the "Ministère de la Recherche Scientifique" of Sénégal. References [] Artzner, P., Delbaen, F., Eber, J-M., Heath, D Coherent measures of ris, Mathematical Finance, 9, [2] Beirlant, J., Diercx, G., Goegebeur, M., Matthys, G Tail index estimation and an exponential regression model, Extremes, 2, [3] Beirlant, J., Diercx, G., Guillou, A., Starica, C On exponential representations of log-spacings of extreme order statistics, Extremes, 5, [4] Bingham, N.H., Goldie, C.M., Teugels, J.L Regular variation, Cambridge. [5] Brazausas, V., Jones, B., Puri, M., Zitiis, R Estimating conditional tail expectation with actuarial applications in view, Journal of Statistical Planning and Inference, 38, [6] Csörgő, M., Csörgő, S., Horváth, L., Mason, D.M Weighted empirical and quantile processes, Annals of Probability, 4, [7] Csörgő, S., Deheuvels, P., Mason, D.M Kernel estimates of the tail index of a distribution, Annals of Statistics, 3,

15 [8] de Haan, L., Ferreira, A Extreme value theory: an introduction, Springer. [9] Deme, E., Girard, S., Guillou, A Reduced-bias estimator of the Proportional Hazard Premium for heavy-tailed distributions, Insurance Mathematic & Economics, 52, [0] Feuerverger, A., Hall, P Estimating a tail exponent by modelling departure from a Pareto distribution, Annals of Statistics, 27, [] Gelu, J.L., de Haan, L Regular variation, extensions and Tauberian theorems, CWI tract 40, Center for Mathematics and Computer Science, P.O. Box 4079, 009 AB Amsterdam, The Netherlands. [2] Goovaerts, M.J., de Vlyder, F., Haezendonc, J Insurance premiums, theory and applications, North Holland, Amsterdam. [3] Hill, B. M A simple approach to inference about the tail of a distribution, Annals of Statistics, 3, [4] Landsman, Z., Valdez, E Tail conditional expectations for elliptical distributions, North American Actuarial Journal, 7, [5] Matthys, G., Delafosse, E., Guillou, A., Beirlant, J Estimating catastrophic quantile levels for heavy-tailed distributions, Insurance Mathematic & Economics, 34, [6] McNeil, A.J., Frey, R., Embrechts, P Quantitative ris management: concepts, techniques, and tools, Princeton University Press. [7] Necir, A., Rassoul, A., Zitiis, R Estimating the conditional tail expectation in the case of heavy-tailed losses, Journal of Probability and Statistics, ID , 7 pp. [8] Pan, X., Leng, X., Hu, T Second-order version of Karamata s theorem with applications, Statistics and Probability Letters, 83, [9] Weissman, I., 978. Estimation of parameters and larges quantiles based on the largest observations, Journal of American Statistical Association, 73, [20] Zhu, L., Li, H Asymptotic analysis of multivariate tail conditional expectations, North American Actuarial Journal, 6,

16 CTE CTE CTE CTE CTE CTE CTE CTE CTE Figure 2: Median of C n,α [X] dotted line and LS, C n,α [X] full line as a function of based on 500 samples of size 500 for α = 0.05 top, α = 0.0 middle and α = 0.20 bottom from a Burr distribution defined as Fx = + x 3ρ 2 /ρ. From the left to the right: ρ =.5, ρ = and ρ = The horizontal line represents the true value of the CTE α [X]. 5

17 MSE MSE MSE MSE MSE MSE MSE MSE MSE Figure 3: MSE of C n,α [X] dotted line and C LS, n,α [X] full line as a function of based on 500 samples of size 500 for α = 0.05 top, α = 0.0 middle and α = 0.20 bottom from a Burr distribution defined as Fx = + x 3ρ 2 /ρ. From the left to the right: ρ =.5, ρ = and ρ =

18 Claim size 0e+00 e+05 2e+05 3e+05 4e γ Hill Year Claim size, 976 a b c CTE CTE CTE d e f Figure 4: a Time plot for the Norwegian fire insurance data; b Histogram of the claim size for the year 976; c Hill estimator as a function of for the year 976; Biased estimator C n,α [X] LS, dotted line and reduced-bias one C n,α [X] full line as a function of for α = 0.05 d, α = 0.0 e and α = 0.20 f. 7