CONFIDENCE REGIONS FOR HIGH QUANTILES OF A HEAVY TAILED DISTRIBUTION

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1 The Annals of Statistics 2006, Vol. 34, No. 4, DOI: 0.24/ Institute of Mathematical Statistics, 2006 CONFIDENCE REGIONS FOR HIGH QUANTILES OF A HEAVY TAILED DISTRIBUTION BY LIANG PENG AND YONGCHENG QI Georgia Institute of Technology and University of Minnesota Duluth Estimating high quantiles plays an important role in the context of ris management. This involves extrapolation of an unnown distribution function. In this paper we propose three methods, namely, the normal approximation method, the lielihood ratio method and the data tilting method, to construct confidence regions for high quantiles of a heavy tailed distribution. A simulation study prefers the data tilting method.. Introduction. In estimating high quantiles of an unnown probability distribution function, one has to infer beyond the observations. This can be done via extrapolating from intermediate quantiles when the underlying distribution has a regularly varying tail. An important application of high quantiles is to forecast rare events. Some references on this topic include [, 3, 7, 4, 23, 25]. Lie tail index estimation, only a part of the upper order statistics is involved in the estimation of high quantiles. Recently, Ferreira, de Haan and Peng [8] provided a data-driven method to choose the optimal sample fraction in terms of asymptotic mean squared errors. In this paper we are interested in obtaining confidence regions for high quantiles. More specifically, three methods will be investigated, namely, the normal approximation method, the lielihood ratio method and the data tilting method; see Section 2 for details. We demonstrate by a simulation study that the data tilting method is preferred. Our data tilting method is similar to the general data tilting method proposed by Hall and Yao [2], which is employed to tilt time series data. This general data tilting method was applied to interval estimation, robust inference and inference under constraints for linear time series. One of its advantages is that it admits a wide range of distance functions. Tilting methods to statistics have a long history; nonparametric techniques involving tilting go bac at least to wor of Grenander [0], which studies nonparametric density estimation under monotonicity constraints. To our best nowledge, not much wor has been done in applying data tilting methods or empirical lielihood methods to statistics of extremes. The empirical Received March 2004; revised September Supported in part by NSF Grant DMS and a Humboldt research fellowship. AMS 2000 subject classifications. Primary 62G32; secondary 62G02. Key words and phrases. Confidence region, data tilting, empirical lielihood method, heavy tail, high quantile. 964

2 CONFIDENCE REGIONS FOR QUANTILES 965 lielihood method, introduced in [7, 8], is a nonparametric approach for constructing confidence regions. Lie the bootstrap and the jacnife, the empirical lielihood method does not need to specify a family of distributions for the data. One of the advantages of the empirical lielihood method is that it enables the shape of a region, such as the degree of asymmetry in a confidence interval, to be determined automatically by the sample. In certain regular cases, empirical lielihood based confidence regions are Bartlett correctable; see [6, ]. For a more complete disclosure of recent references and development, we refer to the boo by Owen [9]. Recently, Lu and Peng [5] applied the empirical lielihood method to obtain confidence intervals for the tail index, and Peng [20] generalized the empirical lielihood method to the case of infinite variance. Here, we propose to employ the general data tilting method of Hall and Yao [2] to obtain confidence regions for high quantiles. We organize this paper as follows. In Section 2 three different methods for constructing confidence regions for high quantiles are introduced, and main results about asymptotic limits are also presented. In Section 3 simulation results are reported for comparisons of the performance of the three methods in terms of both coverage probability and approximate interval length, and a real data application is also included. Finally, all the proofs are given in the Appendix. 2. Methodologies and main results. Let X,...,X n be independent random variables with a common distribution function F which satisfies Fx= exx γ for x>0, where γ>0 is an unnown parameter called the tail index, and ex is a slowly varying function, that is, lim t etx/et = forallx>0. Let X n, X n,n denote the order statistics of X,...,X n. Throughout this paper we assume that p n 0, and p n 0asn. A 00 p n % quantile for the distribution F is defined as x p = F p n, where denotes the inverse function of. The main aim of this paper is to obtain confidence regions for x p. In order to introduce our methodologies, let us assume temporarily that F has the simpler form 2 Fx= cx γ for x>t. Put δ i = IX i >T. Then the lielihood function for the censored data {δ i, maxx i,t} n is Lγ, c = n γ cγ Xi δ i ct γ δ i. In the paper we actually tae T = X n,n,where = n satisfies 3 and n 0.

3 966 L. PENG AND Y. QI Then the lielihood function above becomes 4 n γ Lγ, c = cγ Xi δ i cx γ n,n δ i. Next we are ready to present our three methods for constructing confidence regions for x p. Method I: Normal approximation method. Let ˆγ n, ĉ n denote the maximum lielihood estimator of c, γ, thatis,l ˆγ n, ĉ n = max γ>0,c>0 Lγ, c. Thenitis easy to chec that ĉ n = n X ˆγ n n,n and ˆγ n = { logx n,n i+ /X n,n }. Note that ˆγ n is the well-nown Hill estimator [3]. Therefore, by 2, a natural estimator for x p is ˆx p = p n /ĉ n / ˆγ n. In order to derive the asymptotic normality of ˆx p, we need a stricter condition than : suppose there exists a function At 0 as t such that 5 Utx/Ut x /γ lim t At = x /γ xρ ρ for some ρ<0, where Ux= F x. Then At is a regularly varying function with index ρ;see[5]. Note that 5 implies. The following theorem can be derived from [8]. THEOREM. Assume 5 and 3 hold. If An/ 0, np n = O and log np n / 0 as n, then 6 ˆγ n log/np n log ˆx p x p d N0,. Hence, based on the above limit, a confidence interval with level α for x p is { / Iα n = } / ˆx p exp z α log ˆγn, ˆx p exp{ } z α log ˆγn, np n np n where z α satisfies P N0, z α = α. This confidence interval has asymptotically correct coverage probability α, thatis,px p Iα n α as n. The next theorem presents the coverage expansion for Iα n.

4 CONFIDENCE REGIONS FOR QUANTILES 967 THEOREM 2. Under the conditions of Theorem, P ˆγ n log/np n log ˆx p x p x x = 3 φx + γ 2x2 φx ρ + o log np n An/ An/ 2 xφx log uniformly for <x<, where x and φx denote the distribution function and density function of N0,, respectively. Furthermore, Px p Iα n = α z αφz α log + o log np n 2, np n An/ REMARK. Theorem 2 shows that Px p Iα n α = Olog n 2 in the case lognp n = Ologn. This means that the coverage accuracy for high quantiles is not very accurate in general. To achieve this asymptotic rate, can be of order n θ for some 0 <θ< 2ρ/ 2ρ. The unnown parameter ρ can be estimated; see, for example, [2].. np n 2 Method II: Lielihood ratio method. Define ˆγ n and ĉ n as in Method I. First set l = max log Lγ, c = log L ˆγ n, ĉ n. γ>0,c>0 Next we maximize log Lγ, c subject to γ>0, c>0, γ log x p + log pn c = 0, and denote this maximized lielihood function by l 2 x p. Note that the above equation comes from setting p n = Fx p = cxp γ. It is easy to show that where γλ= cλ = X γλ l 2 x p = log L γλ, cλ, log X n,n i+ log X n,n + λ log X n,n λ log x p, n,n λ n λ

5 968 L. PENG AND Y. QI and λ satisfies 7 8 pn γλlog x p + log = 0, cλ γ λ > 0 and λ<. Therefore, the log-lielihood ratio multiplied by minus two is lx p = 2 l 2 x p l. THEOREM 3. Suppose the conditions in Theorem hold. Then there exists a unique solution to 7 and 8, say, ˆλx p, and 9 lx p,0 d χ 2, with λ = ˆλx p,0 in the definition of l 2 x p,0, where x p,0 is the true value of x p. Therefore, based on the above limit, a confidence region with level α for x p is I l α ={x p : lx p u α }, where u α is the α-level critical point of χ 2. This confidence region has asymptotically correct coverage probability α, thatis,px p Iα l α as n. REMARK 2. The profile lielihood approach has been employed to construct confidence regions for high quantiles based on fitting a generalized Pareto distribution to exceedances over a deterministic high threshold; see [24]. The difference between our Method II and the profile lielihood method is that we tae the random high threshold into account in our censored lielihood function. Method III:Data tilting method. Here we employ a data tilting method, similar to that of Hall and Yao [2], to construct a confidence region for x p.first,forany fixed weights q = q,...,q n such that q i 0and n q i =, we solve This results in ˆγq,ĉq = arg max γ,c ˆγq= n q i log γ cγ Xi n q i δ i n q i δ i log X i log X n,n, n ĉq = X ˆγq n,n q i δ i. δ i cx γ n,n δ i.

6 CONFIDENCE REGIONS FOR QUANTILES 969 Define ρ0 ρ 0 n n nq i ρ 0, if ρ 0 0,, D ρ0 q = n n lognq i, if ρ 0 = 0, n q i lognq i, if ρ 0 =. The function D ρ0 q is a measure of distance between q and uniform distribution, that is, q i = /n. Next, we shall choose q to minimize this distance. More specifically, solve 2n Lx p = min q D ρ0 q subject to the constraints q i 0, n q i =, ˆγqlog x p X n,n = log n q i δ i p n. The constraint ˆγqlogx p /X n,n = log n q i δ i /p n is equivalent to x p = p n /ĉq / ˆγq. Here we only consider the case ρ 0 = since other cases are similar and the case ρ 0 = gives good robustness properties. Put A λ = n n e λ and A 2 λ = A λ logx p/x n,n loga λ /p n. Then, by the standard method of Lagrange multipliers, we have 0 q i = q i λ,λ 2 = n e λ, if δ i = 0, { n exp λ if δ i =, where λ and λ 2 satisfy n q i =, + λ 2 logxp /X n,n A 2 λ A λ A λ logx i /X n,n logx p /X n,n A 2 2 λ ˆγqlog x n p q i δ i = log. X n,n p n }, THEOREM 4. Suppose the conditions in Theorem hold. Then, with probability tending to one, there exists a solution to, say, ˆλ x p, ˆλ 2 x p, such that,

7 970 L. PENG AND Y. QI for λ,λ 2 = ˆλ x p, ˆλ 2 x p, log/npn log + n λ log λ 2 /4 /n log/np n log/npn n and Lx p,0 d χ 2 with λ,λ 2 = ˆλ x p,0, ˆλ 2 x p,0 in the definition of Lx p,0. Hence, based on the above limit, a confidence region with level α for x p is Iα t ={x p : Lx p u α }, where u α is the α-level critical point of χ 2. This confidence region has asymptotically correct coverage probability α, thatis,px p Iα t α as n. REMARK 3. In order to compare these three methods theoretically, it is necessary to derive corresponding coverage expansions for Iα l and I α t. This requires much wor and it will be one of our future topics. 3. Simulation study and real application. 3.. A simulation study. In order to compare the performance of confidence regions based on the normal approximation method, the lielihood ratio method and the data tilting method, we conducted a simulation study to examine coverage probabilities and approximate lengths of confidence regions. We employed the following two distributions: i the Burrα, β distribution, given by Fx= + x β α α/β α x>0;ii the Fréchetα distribution, given by Fx = exp x α x >0. Corresponding to 5, we have γ = /α, ρ = β α α and At = α t β α/α for Burrα, β, andγ = /α, ρ = and At = 2αt for Fréchetα. First, we generated 0,000 random samples of size n = 000 from the distributions Burr,.5, Burr, 2, Burr2, 3, Burr2, 4, Fréchet and Fréchet2, and then computed coverage probabilities of I0.9 n,il 0.9 and I 0.9 t for p n = 0.0 and p n = These coverage probabilities are plotted against different sample fractions = 20, 25,...,300 in Figures 4. From these figures we observe that the data tilting method is better than the other two methods in terms of coverage accuracy in general, especially for larger values of. Thus, the data tilting method is less sensitive to the bias when a large value of is employed. This may be due to the automatic choice of weights q i in the data tilting method. Although it does,

8 CONFIDENCE REGIONS FOR QUANTILES 97 FIG.. Coverage probabilities for Burr distributions with p n = 0.0. The coverage probabilities of confidence regions I0.90 n, I 0.90 l and I 0.90 t are plotted against the different sample fractions = 20, 25,...,300 for different Burr distributions. not mae much sense to compare these three methods with the empirical lielihood method for quantiles see Section 3.6 of [9], we find that the coverage probabilities based on the empirical lielihood method for quantiles are for p n = 0.0 and for p n = These coverage probabilities are not as accurate as those based on the other three methods for most sample fractions.note that the empirical lielihood method for quantiles is independent of the underlying distribution function. Second, we generated,000 random samples of size n = 000 from the distributions Burr, 2 and Fréchet, and then calculated the length of I0.9 n and the approximate lengths of I0.9 l and I 0.9 t for p n = 0.0. Let us explain how we calculate

9 972 L. PENG AND Y. QI FIG. 2. Coverage probabilities for Burr distributions with p n = The coverage probabilities of confidence regions I0.90 n, I 0.90 l and I 0.90 t are plotted against the different sample fractions = 20, 25,...,300 for different Burr distributions. the approximate length of I0.9 t. The same algorithm was employed to obtain the approximate length of I0.9 l. First, we search an x p near ˆx p such that Lx p <u 0.9. Then we both increase and decrease x p by a small step 0. until Lx p >u 0.9.The corresponding values are denoted by xp u and xl p, respectively. Thus, we approximate I0.9 t by the interval [xl p,xu p ]. These approximate lengths, xu p xl p, are plotted against the different sample fractions = 20, 30,...,300 in Figure 5. We notice that the approximate confidence interval lengths based on the data tilting method are smallest for most cases. Third, we generated a random sample of size n = 000 from the distributions Burr, 2 and Fréchet, and then computed the data tilting lielihood function

10 CONFIDENCE REGIONS FOR QUANTILES 973 FIG. 3. Coverage probabilities for Fréchet distributions with p n = 0.0. The coverage probabilities of confidence regions I0.90 n, I 0.90 l and I 0.90 t are plotted against the different sample fractions = 20, 25,...,300 for different Fréchet distributions. Lx p for p n = 0.0 and x p = x p, i, i = 0,,...,200, where x p,0 denotes the true quantile. We too = 50 and 00. Figure 6 indicates that the data tilting lielihood function is approximately convex, which suggests that I0.9 t may indeed be an interval. Unlie the empirical lielihood method for means, we were unable to prove that Iα t is an interval. FIG. 4. Coverage probabilities for Fréchet distributions with p n = The coverage probabilities of confidence regions I0.90 n, I 0.90 l and I 0.90 t are plotted against the different sample fractions = 20, 25,...,300 for different Fréchet distributions.

11 974 L. PENG AND Y. QI FIG. 5. Averages of the approximate confidence lengths with p n = 0.0. The averages of approximate lengths of confidence regions I0.90 n, I 0.90 l and I 0.90 t are plotted against the different sample fractions = 20, 30,...,300 for Burr.0, 2.0 and Fréchet distributions. In summary, our simulation study for sample size n = 000 prefers the data tilting method, which gives the best coverage accuracy in general, is less sensitive to the choice of sample fraction, and has a shorter approximate interval length in most cases. Although we do not report the simulation study for sample size n = 200, the same conclusions as above are drawn, except that Method II performs worst. FIG. 6. Data tilting lielihood function Lx p with p n = 0.0. The data tilting lielihood functions are plotted against different x p = x p, i, i = 0,,...,200, for Burr.0, 2.0 and Fréchet, where x p,0 denotes the true quantile. We too = 50 and = 00.

12 CONFIDENCE REGIONS FOR QUANTILES 975 FIG. 7. Danish fire loss data. This consists of 256 losses over one million Danish Kroner DKK from the years 980 to 990, inclusive A real application. The data set we shall analyze consists of 256 Danish fire losses over one million Danish Kroner DKK from the years 980 to 990 inclusive see Figure 7. The loss figure is a total loss for the event concerned and includes damage to buildings, damage to furnishings and personal property, as well as loss of profits. This data set was analyzed by McNeil [6]andResnic[22], where the right tail index was confirmed to be between and 2. Further, Peng [20] applied the empirical lielihood method to this data set to obtain a confidence interval for the mean. We too p n = 0.00 and plotted the confidence interval I0.90 n, and the approximate confidence intervals I0.90 l and I 0.90 t, against the different sample fraction = 60, 65,...,400 in Figure 8. We note again that the approximate interval lengths based on the data tilting method are smallest for most cases. APPENDIX A: PROOFS OF THEOREMS 2, 3 AND 4 PROOF OF THEOREM 2. Let V,V 2,...,V n be i.i.d. random variables uniformly distributed over 0, and V n, V n,2 V n,n be the order statistics of V,V 2,...,V n.definec n = n+ and d n = c n c n /n +. Then the density function of V n,n c n /d n is n!d n!n! c n + d n u n c n d n u, φ n u = if 0 <c n + d n u<, 0, otherwise.

13 976 L. PENG AND Y. QI FIG. 8. Approximate confidence intervals for Danish fire loss data. The approximate confidence intervals with level 0.90 based on the normal approximation method Method I, the lielihood ratio method Method II and the data tilting method Method III are plotted against = 60, 65,...,400. We too p n = For each c n /d n <u< c n /d n,define Y n,j u = γ { log U c n d n u V i log U c n d n u }, H n = γ ˆγ n γ, H n u = Yn,j u, Since and r n u = P j= { γ log/np n log Upn } U/ c n d n u. ˆγ n log/np n log ˆx p ˆγ n = x p log/np n log X n,n + x p ˆγ n log/np n log ˆx p = P ˆγ n = P H n x x p x/ log/np n log { x + x/ x p X n,n γ log/np n log } x p X n,n

14 CONFIDENCE REGIONS FOR QUANTILES 977 for x /4, it follows from Lemma 2.2 of [2] that 4 P ˆγ n log/np n log ˆx p x p x = P H n u x + r nu x/ φ n u du for x /4. Similar to Lemma 2.3 of [2], we can prove that P H n u x + r nu x/ { x + rn u x + = x/ rn u x + + φ x/ 3 rn u 2 } 5 x/ x + rn u γ φ x/ An/ + o + An/, ρ uniformly for x R and u /4.Since5 is equivalent to logutx logut logx/γ lim t At it follows from Potter s bounds that = xρ, ρ u r n u = log/np n + u 2 2 log/np n + u 3 3 log/np n An/ + O log/np n +, log/npn uniformly for u /4. Hence, x + rn u x/ x u = φx log/np n + φxx2 / 2 xφxu2 log 2 6 np n + o log 2 + An/ + + u 2, np n log/np n uniformly for x /4 and u /4. Thus, the theorem follows from 4 6 and the facts that u t φ n u du = O and u t φ n ui u > /4 du = o/ for any t>0. PROOF OF THEOREM 3. By the definition of γλ,wehave 7 γλ= ˆγ n λ/ ˆγ n logx p /X n,n.

15 978 L. PENG AND Y. QI Thus, equations 7 and8 are equivalent to 8 and ˆγ n logx p /X n,n λ/ ˆγ n logx p /X n,n + log n λp n = 0 λ 9 λ ˆγ n logx p /X n,n >0 and λ<, respectively. Set ˆγ n logx p /X n,n gλ = λ/ ˆγ n logx p /X n,n + log n λp n λ. Then gλ is continuous and increasing in λ under the restriction 9 since g [ˆγ n logx p /X n,n ] 2 λ = [ λ/ ˆγ n logx p /X n,n ] 2 + n n λ λ > 0. If x p /X n,n >, then 9 is equivalent to λ<min,[ˆγ n logx p /X n,n ] =: a n. Since g = log p n < 0andga n =, we conclude that there exists a unique λ<a n such that gλ = 0, that is, there exists a unique λ satisfying 8 and9. We can draw the same conclusion for the cases x p /X n,n = and x p /X n,n <. So we prove the existence and uniqueness of a solution to 7 and8. Since log ˆx p = ˆγ log p n = ˆγ log p n log n ĉ n n n ˆγ n log X n,n, we have log ˆx p /X n,n = ˆγ log p n log = ˆγ log n n n np n and ˆγ n logx p /X n,n = log ˆγ n log ˆx p. np n x p It follows from 6 that ˆx p x p p 0. Thus, ˆγ n logx p /X n,n = log 20 + op. p np n Denote the solution to 7 and8 byλ n. Thus, gλ n = 0. First we will show that 2 P λ n < [ˆγ n logx p /X n,n ] 2.

16 CONFIDENCE REGIONS FOR QUANTILES 979 This is equivalent to proving that, as n, 22 P gb n >0 and P g b n <0, where b n = /[ˆγ n logx p /X n,n ] 2. By 20 and Taylor s expansion, gb n =ˆγ n logx p /X n,n + b n log + log np n b n n ˆγ n logx p /X n,n + o p log b n =ˆγ n logx p /X n,n log + b n + op np n + b n [ˆγ n logx p /X n,n ] 2 + o p = ˆγ n log ˆx p + b n x p [ˆγ n logx p /X n,n ] 2 + o p = + o p. Similarly, g b n = + o p. This yields 22 and, hence, 2. Since 20 and2 imply λ n 23 ˆγ n logx p /X n,n p 0, using Taylor s expansion again, we have 0 = gλ n = ˆγ n log ˆx p + λ n x p [ˆγ n logx p /X n,n ] 2 + o p, that is, λ n = ˆγ n log ˆx p /x p + op [ˆγ n logx p /X n,n ] 2 = ˆγ n log ˆx p /x p 24 + op log/np n 2. Note that log Lγ, c = logcγ γ + log X n,n i+ + n log cx γ n,n = logγ γ + ˆγ n + n log cx γ n,n + logcx γ n,n logx n,n

17 980 L. PENG AND Y. QI and lx p = 2 log L γλ n, cλ n log L ˆγ n, ĉ n = 2 log γλ n γλn 2 log ˆγ n ˆγ n In view of 7 and23, we have γλ n ˆγ n = It follows from 6 and24 that λ n λ n /ˆγ n logx p /X n,n + 2n log p. 25 λ n / p 0. Hence, by 20, 6, 25 and Taylor s expansion, γλn 2 lx p = + op + O p λ 2 n ˆγ / n = λ n ˆγ n logx p /X n,n 2 + op + o p ˆγn log ˆxp /x p 2 = + op + o p ˆγ n logx p /X n,n λ n n. = ˆγn log ˆxp x 2 p + op + o p log/np n d χ 2. PROOF OF THEOREM 4. Let Z i = logx n,n i+ /X n,n for i, q i = n exp{ λ } for + i n and q i = { n exp logxp /X n,n λ + λ 2 A 2 λ A λ for i. Then is equivalent to q i = A λ Furthermore, this is equivalent to 26 q i = A λ and and A λ Z i logx p /X n,n A 2 2 λ q i Z i = A 2 λ. q i Z i q i = A 2λ A λ. }

18 CONFIDENCE REGIONS FOR QUANTILES 98 The second identity in 26 is exp{ λ 2 A λ Z i logx p /X n,n /A λ }Z i exp{ λ 2 A λ Z i logx p /X n,n /A 2 2 λ } = logx p/x n,n loga λ /p n. In order to demonstrate the existence of a solution to equation, first we will show that, with probability tending to one, there exists a continuous function λ 2 = λ 2 λ such that, for each λ, λ,λ 2 = λ,λ 2 λ is the solution to 27, and then we should prove that, for some λ, λ,λ 2 = λ,λ 2 λ is also the solution to the first identity of 26, and the solution satisfies both 2 and3. To this end, set exp{ λz i }Z i fλ= exp{ λz i }. Then it is easy to see that lim λ fλ= Z, lim λ fλ= Z and fλ is d decreasing in λ by checing that dλ log f λ < 0. Therefore, there exists a unique continuous function rx such that frx= x for any x Z,Z. From now on we restrict λ such that 2 holds, that is, log/npn A λ log/npn 28 +, n n which implies 29 and 30 3 loga λ /p n log/np n log/npn 0 logx p /X n,n log/np n log/npn logx p/x n,n loga λ /p n logx p/x n,n log/np n Set F n ={Z < logx p/x n,n log/np n logx p /X n,n log/np n +. log/npn <Z }.ThenPF n since p γ, Z p, Z p 0. By definition, fr logx p/x n,n loga λ /p n = logx p/x n,n loga λ /p n on F n.thisimpliesthat 32 λ 2 = λ 2 λ = r logxp /X n,n loga λ /p n A 2 2 λ A λ logx p /X n,n

19 982 L. PENG AND Y. QI is the unique solution to equation 27, with probability tending to one. Set R = logx p/x n,n loga λ /p n logx p/x n,n logxp /X n,n and ˆλ = r. log/np n loga λ /p n It follows from 30 that R = O p /2 log /np n /2 33 holds uniformly for λ under the restriction 2 or, equivalently, 28. Hereafter all terms O p and o p are assumed to hold uniformly for λ if λ is involved. Using 6, 33 and logx p /X n,n ˆγ n = logx p/x n,n ˆγ n + R loga λ /p n log/np n 34 = log/np n log ˆx p + R, x p we have logx p /X n,n 35 ˆγ n = O p /2. loga λ /p n On the other hand, from Taylor s expansion, we have f± /4 f0 = ± /4 f 0 + o p, wheref0 =ˆγ n,and f 0 = Z i ˆγ n = γ 2 + O p /2. For the proof for the last step of 36, see, for example, [4] or[9]. Hence, P f /4 < logx p/x n,n loga λ /p n and P f /4 > logx p/x n,n. loga λ /p n Therefore, ˆλ = r logx p/x n,n loga λ /p n satisfies Pˆλ /4, /4. Thus, from Taylor s expansion, we obtain fˆλ ˆγ n = ˆλf 0 + O p /4 = γ 2 ˆλ + O p /4, which, coupled with 34, yields 37 ˆλ = γ 2 logxp /X n,n loga λ /p n = ˆγ n + Op /4 γ 2 log/np n log ˆx p x p + Op /4.

20 CONFIDENCE REGIONS FOR QUANTILES 983 Then, by using 35 and Taylor s expansion, we have 38 Note that exp{ ˆλZ i }= + O p /2. λ 2 = λ 2 λ = ˆλA 2 2 λ A λ logx p /X n,n, where ˆλ is a function of λ as well. Plug λ 2 = λ 2 λ into q i and set hλ = q i A λ.thenhλ has the expression { n exp{ λ ˆλA 2 2 } exp λ A λ logx p /X n,n and Put It is easy to chec that logxp /X n,n } exp{ ˆλZ i } A λ. A 2 λ A λ λ = log + λ = log log/npn n log/npn n. 39 log/npn e λ = +, n log/npn, A λ = n A 2 λ = γ n log/npn log/npn + o p. The first two identities are obvious. The third one follows from the second one, equation 35 and a well-nown result for the Hill estimator, that is, ˆγ n γ d N0,γ 2. Now it follows from 39 and38 that hλ = log/npn + op. n

21 984 L. PENG AND Y. QI Similarly, hλ = log/npn + op. n Hence, with probability tending to one, there exists a λ satisfying 2 andthe first equation in 26 with λ 2 = λ 2 λ defined in 32; that is, we have shown the existence of the solution to such that 2 and3 hold. We still need to estimate ˆλ from the equation hˆλ = 0, which is equivalent to { n exp logxp /X n,n ˆλ loga ˆλ /p n logx } p/x n,n exp{ ˆλZ i } loga ˆλ /p n 2 40 = exp{ + ˆλ } n n. It follows from 35 that logx p /X n,n loga ˆλ /p n logx p/x n,n =ˆγ loga ˆλ /p n 2 n + o p. Hence, applying Taylor s expansion to both sides of 40 yields 4 + ˆλ = O p. n It is easy to show that max i n nq i =o p. Thus, n { Lx p = 2 nq i nqi 2 nq i 2 + o p } n = nq i 2 + o p n = lognq i 2 + o p = + o p n + ˆλ log nqi. It follows from 29 and35 that log logxp /X n,n nq i = + ˆλ + ˆλ loga ˆλ /p n logx p/x n,n loga ˆλ /p n Z 2 i = + ˆλ ˆλ Z i ˆγ n log/npn + O p +, log/np n

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