Journal of Statistical Planning and Inference

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1 Journal of Statistical Planning Inference Contents lists available at ScienceDirect Journal of Statistical Planning Inference journal homepage: Maximum lielihood estimation of extreme value index for irregular cases Liang Peng a,, Yongcheng Qi b,, a School of Mathematics, Georgia Institute of Technology, Atlanta, GA 333-6, USA b Department of Mathematics Statistics, University of Minnesota Duluth, SCC 4, 7 University Drive, Duluth, MN 558, USA ARTICLE INFO ABSTRACT Article history: Received February 9 Accepted 7 March 9 Available online 6 March 9 MSC: 6E 6F Keywords: Endpoint Extreme value index Generalized Pareto distribution Stable law A method in analyzing extremes is to fit a generalized Pareto distribution to the exceedances over a high threshold. By varying the threshold according to the sample size [Smith, R.L., 987. Estimating tails of probability distributions. Ann. Statist. 5, 74 7] [Drees, H., Ferreira, A., de Haan, L., 4. On maximum lielihood estimation of the extreme value index. Ann. Appl. Probab. 4, 79 ] derived the asymptotic properties of the maximum lielihood estimates MLE when the extreme value index is larger than. Recently Zhou [9. Existence consistency of the maximum lielihood estimator for the extreme value index. J. Multivariate Anal., ] showed that the MLE is consistent when the extreme value index is larger than. In this paper, we study the asymptotic distributions of MLE when the extreme value index is in between including. Particularly, we consider the MLE for the endpoint of the generalized Pareto distribution the extreme value index show that the asymptotic limit for the endpoint estimate is non-normal, which connects with the results in Woodroofe [974. Maximum lielihood estimation of translation parameter of truncated distribution II. Ann. Statist., ]. Moreover, we show that same results hold for estimating the endpoint of the underlying distribution, which generalize the results in Hall [98. On estimating the endpoint of a distribution. Ann. Statist., ] to irregular case, results in Woodroofe [974. Maximum lielihood estimation of translation parameter of truncated distribution II. Ann. Statist., ] to the case of unnown extreme value index. 9 Elsevier B.V. All rights reserved.. Introduction Let X, X, X,... be independent identically distributed i.i.d. rom variables with a distribution function d.f. F, which lies in the domain of attraction of an extreme value distribution, i.e., there exist a n > b n R such that lim P max i n X i b n x = Gx. n a n for all continuous points of G, where G is a non-degenerate distribution. Based on., one can infer the tail properties of F, hence extrapolate data into a far tail region of F; see Chapter 4 of De Haan Ferreira 6. It is nown that G in. can be written as G γ x = exp + γx /γ Corresponding author. Fax: addresses: peng@math.gatech.edu L. Peng, yqi@d.umn.edu Y. Qi. Supported by NSF Grant SES Supported by NSF Grant DMS /$ - see front matter 9 Elsevier B.V. All rights reserved. doi:.6/j.jspi.9.3.

2 336 L. Peng, Y. Qi / Journal of Statistical Planning Inference for some γ R + γx >, condition. implies that there exists a positive non-decreasing function σ such that X t lim P t θ σt > x X > t = + γx /γ for all x>+ γx >, where θ = supx : Fx < is the right endpoint of F. Therefore, given a fixed high threshold u, the conditional distribution of the exceedance above the threshold, i.e. PX u x X > u, can be approximated by the so-called generalized Pareto distribution Hx = + γx/σu /γ,. see Chapter 3. of De Haan Ferreira 6. A parametric way in analyzing extremes is to fit the generalized Pareto distribution. to the exceedances over a fixed high threshold. With this setup, it follows from Smith 985 that the rate of convergence of the maximum lielihood estimates MLE for parameters γ τ := σu/γ depends on whether γ >, γ = or < γ <. Note that the parameter τ corresponds to the endpoint of H when γ <, the setting in Smith 985 is broader than the assumption of generalized Pareto distribution. A difficulty for this parametric approach is how to quantify the influence of the fixed high threshold. An alternative way in analyzing extremes is to start from condition. directly, which is considered as a semiparametric model. Under this setup, one may choose a sequence of thresholds u = u n estimate γ σu n by maximum lielihood estimation. For example, Smith 987 derived the limit of MLE for γ, σ when γ, but did not obtain the limit for the case < γ <. Instead, Smith 987 considered a different estimator from MLE for the case < γ <. Recently, Drees et al. 4 revisited the maximum lielihood estimate of γ, σ by taing the threshold as the + -th largest order statistic, derived the limit when γ > by employing a weighted approximation of tail quantile processes. Under condition. given below, Zhou 9 showed that the MLE of γ, σ exists is consistent when γ >. Therefore, a natural question is what the limit the rate of convergence of MLE are when γ, ]. Answering this question completes the results in Smith 987 Drees et al. 4. It is nown that condition. with γ < implies that the right endpoint of F is finite i.e. θ < Fθ tx lim t Fθ t = x /γ for x >..3 Based on.3, the endpoint θ can be estimated; see Hall 98, Loh 984, Fal 995, Athreya Fuuchi 997 Hall Wang 999 for the regular case γ,, Woodroofe 974 for the irregular case γ,. Note that Woodroofe 974 assumed that γ is nown. In this paper, we focus on the irregular case as Woodroofe 974, but tae the same setup as Drees et al. 4 without assuming that γ is unnown. That is, we assume. holds for some γ, ]. Therefore, we have one endpoint for the generalized Pareto distribution. another endpoint for the underlying distribution F. From Woodroofe 974, one may conjecture that the maximum lielihood estimators for both endpoints have a faster rate of convergence compared to the MLE for γ, the corresponding limit laws for both endpoints are non-normal. In this paper, this conjecture is proved when the MLE of τ is normalized by a sequence of rom variables. It is also shown that the MLE of τ, normalized by a sequence of non-rom constants, has a normal limit with a slower convergence rate. This phenomenon can be explained by the fact that we employ a stochastic threshold in the maximum lielihood estimation procedure thus the true value of τ is essentially related to the rom threshold. When considering the maximum lielihood estimate for γ, σ, the rates of convergence for these two parameters turn out to be of the same order the limits are both normal distribution. This is different from the limit for γ, τ. At first glance, the difference is mysterious: τ is a function of γ σ studying MLE for γ, σ is equivalent to that for γ, τ. However, we should eep in mind that the case γ, ] is irregular so the invariant property of MLE does not hold. In fact, since ˆσ = ˆγˆτ, the convergence rate of ˆσ to σ is determined by that of ˆγ to γ. Based on the above study on estimating τ, we also derive the joint asymptotic limit of estimators for γ the endpoint θ in.3, which shows that the limit of the endpoint estimator for F is non-normal. Since Li Peng 8 pointed out that maximum lielihood estimation for γ θ based on the generalized Pareto distribution is the same as that based on model.3, our results extend the results in Hall 98 to irregular case generalize the results in Woodroofe 974 to the case of jointly estimating the extreme value index endpoint with an unnown extreme value index. We organize this paper as follows. The main results are given in Section. We summarize our conclusions in Section 3. All proofs are given in Appendix A.. Main results Let X n, X n,n be the order statistics of the i.i.d. rom variables X,...,X n.let = n be an intermediate sequence of integers satisfying asn.. n

3 L. Peng, Y. Qi / Journal of Statistical Planning Inference Given X n,n, the conditional lielihood function of X n,n + X n,n,...,x n,n X n,n can be approximated by Lγ, σ = hx n,n i+ X n,n ; γ, σ, i= where hx; γ, σ = / xhx is the density function of the generalized Pareto distribution Hx defined in.. It follows from Drees et al. 4 that the score equations are log X n,n i+ X n,n /τ=γ,. i= i= X n,n i+ X n,n /τ = + γ X n,n i+ X n,n /τ > for all i =,...,.4 when γ τ = σ/γ. Note that Eq..4 is not used in Drees et al. 4 because the solution of..3 automatically fulfills this requirement when γ >. But this constraint is necessary in determining the limit of MLE when < γ ;see Theorem. For deriving the asymptotic limit of the MLE, similar to Drees et al. 4, we need a second order condition: suppose there exist two real functions atat ast such that Utx Ut/at x γ /γ lim = Ψx := t At.3 x s s γ u ρ du ds.5 for some γ > someρ, where Ut denotes the left-continuous inverse of / Ft. It follows from Theorem of De Haan Ferreira 6 that σt = au t,.6 where U denotes the left-continuous inverse function of U. Throughout, we assume < γ let ˆγ, ˆτ denote the solution of γ, τ to Eqs...4. Denote the true value of γ by γ. In virtue of the rom threshold X n,n, we will normalize the MLE of τ by au X n,n. It follows from.5 that athas a positive sign for large t, hence.4 is equivalent to Put τ/au X n,n > X n,n X n,n /au X n,n : =M n..7 f n t = i= g n t = i= h n t = g n t f n t. log X n,n i+ X n,n tau +, X n,n X n,n i+ X n,n tau X n,n Put δ n = An/log γ An/. First we show that there exits τ such that τ/au X n,n lies between M n + δ n /γ + ε for some ε > as defined in the following proposition. Proposition. Under conditions.,.5, γ, ] γ An/, log An/ as n,.8

4 3364 L. Peng, Y. Qi / Journal of Statistical Planning Inference we have lim nm n + δ n > =, n.9 lim n /γ n + ε < =. lim n Ph n t < for all t [M n + δ n, /γ + ε] =,. where ε > is small enough. It follows from Proposition that, with a probability tending to one, there exits a unique solution, say ˆγ, ˆτ, which satisfies that ˆτ > X n,n X n,n + δ n, h n ˆτ/aU X n,n =,. ˆγ = log X n,n i+ X n,n / ˆτ. i= Note that there may exist a τ X n,n X n,n,x n,n X n,n + δ n such that h n τ/au X n,n =. In this case, it is obvious that this solution has the same asymptotic limit as M n. From now on, we focus on the study on ˆγ, ˆτ satisfying.. Theorem. Assume γ,. Under conditions.,.5.8 we have ˆγ γ, γ ˆτ/aU X n,n + /γ W, d V, where W V are independent rom variables, W N, γ, the distribution of V is given by PWy if y, Vy := PV y = PW y exp γ y /γ if y <, where W y is defined in Lemma. Theorem. Assume γ =. If conditions.,.8.5 hold, then we have ˆγ γ,ln γ ˆτ/aU X n,n + /γ W, d V, where W V are independent rom variables, W N, γ, V N, γ. Remar. Note that the case γ = is not studied in Woodroofe 974. Note that in Theorems, we can treat the MLE of τ as an estimate of au X n,n /γ, which is the rom endpoint of the generalized Pareto distribution H. When studying ˆσ, onemaytaean/ as the true value as in Drees et al. 4. In this case, we compare ˆτ with the non-rom endpoint an//γ. By writing ˆτ/an/ + /γ =ˆτ/aU X n,n + /γ au X n,n an/ au + γ X n,n,.3 an/ the following theorem shows that the rate of convergence is /, which is slower than γ for γ, or ln γ for γ =. Theorem 3. Let < γ. Assume conditions.,.5.8 hold. Then we have as n ˆγ γ, ˆτ/an/ + /γ W, d W, where W W are independent rom variables, W N, γ W N,. An alternative interest is to estimate σ instead of the endpoint τ. In this case, the MLE for σ is ˆσ = ˆγˆτ its asymptotic limit is given as follows. Here we compare the limits of ˆσ after normalized by both the non-rom constant an/ the rom variable au X n,n, respectively.

5 L. Peng, Y. Qi / Journal of Statistical Planning Inference Theorem 4. Assume all conditions in Theorem 3 are satisfied. Then we have ˆγ γ, ˆσ/an/, ˆσ/aU X n,n W, d γ W + W/γ, W/γ, where W W are given in Theorem 3. Next we estimate the endpoint θ of F. Basedonˆγ ˆτ, onecanestimateθ by ˆθ = X n,n + ˆτ. As pointed out in Li Peng 8,estimatorsˆγ ˆθ are the same as the maximum lielihood estimator in Hall 98. Hence the following results extend Hall 98 to irregular case. Theorem 5. Assume γ,. Under conditions.,.5.8, we have ˆγ γ, γ ˆθ θ an/ d W, V, where W, V is defined in Theorem. Theorem 6. Assume γ =. If conditions.,.8.5 hold, then we have ˆγ γ, ln γ ˆθ θ an/ where W, V is defined in Theorem. d W, V, Remar. Write γ an/ an = γ An/ n a γ n a An/. It follows from.8 Theorem.3.6 of De Haan Ferreira 6 that γ an/an/, which implies that ˆθ has the same rate of convergence as X n,n when γ,. But it is true that θ ˆθ θ X n,n + X n,n X n,n =θ X n,n, i.e., ˆθ is always closer to the true value than X n,n.whenγ =, ˆθ has a faster rate of convergence than X n,n. Remar 3. It is nown that the endpoint estimators in Deers de Haan 989 Deers et al. 989 have the rate of convergence / an/ when lim n An/,. Hence, ˆθ is better than the endpoint estimators in Deers de Haan 989 Deers et al. 989 when γ,. However, the estimators for γ in Deers de Haan 989 Deers et al. 989 have a faster rate of convergence than the maximum lielihood estimator when γ, since.8 is required in our Theorem Conclusions We generalize the study in Drees et al. 4 to the irregular case γ, ], hence the maximum lielihood estimator for a generalized Pareto distribution becomes nown for γ >. As pointed out in Li Peng 8, when γ <, models..3 are equivalent in the sense that the maximum lielihood estimators for the extreme value index the endpoint of underlying distribution are the same. Therefore, although we study model., our results generalize the study in Hall 98 from the regular case γ, to the irregular case γ, ]. Also our results extend those in Woodroofe 974 from nown γ to unnown γ. It is nown that maximum for the endpoint has a better rate than the endpoint estimators in Deers de Haan 989 Deers et al. 989 when γ <, but it has a worse rate when γ. Our results, combining with those in Drees et al. 4 show that the maximum lielihood estimator for the endpoint always achieves the best rate of convergence for γ >. Appendix A. Proofs Let Y,...,Y n be i.i.d. rom variables with distribution /x Y n, Y n,n denote the order statistics of Y,...,Y n.consider another independent sequence of i.i.d. rom variables Y,...,Y with distribution function /x.denotey, Y,

6 3366 L. Peng, Y. Qi / Journal of Statistical Planning Inference as their order statistics. It is well nown that Y n,n i+ /Y n,n d i= =Y, i+ i=. A. The following lemma comes from Lemma 5. of Draisma et al Lemma. Let f be a measurable function. Suppose there exist a real parameter α functions a t > A t such that for all x > lim t f tx f t a t A t xα α = H x = β x α+β α + β xα, α where β. Then for any ε > there exists t > such that for all t t, tx t, f tx f t xα a t α H x A t ε + x α + x α+β e ε log x. Put l x; t = log xγ γ tγ l x; t = xγ. tγ + γ In Lemmas 3 their proofs below, we use e ix to denote complex number cos x + i sin x. In the two lemmas, we only assume. One can assume that = n depends on n lie condition. but for all rom quantities of interest in these two lemmas, their distributions depend on n through n only. Lemma. Assume γ,. i For t, we have l Y j ; /γ + tγ, γ l Y j ; /γ + tγ W, d W t j= j= A. as, where W W t are independent, W N, γ the characteristic function of W t is Ee iλwt =exp exp iλ x γ iλ tγ x γ x tγ tγ dx + iλ x γ x γ tγ x dx. ii For t <, conditional on max j Y j γ /γ < t γ,a.holds with at,γ tγ Ee iλwt =exp exp iλ x γ iλ tγ x γ x tγ /γ dx + exp iλ at,γ x γ tγ x dx, where at, γ, tγ /γ is the unique solution of the following equation: x tγ φx := x γ y γ y γ tγ y dy =. A.3 + γ iii For t, we have l Y j ; /γ, γ l Y j ; /γ + tγ W, d W t j= j= A.4 as, where W W t are given in part i. iv For t <, conditional on max j Y j γ /γ < t γ,a.4holds with W t given in part ii.

7 L. Peng, Y. Qi / Journal of Statistical Planning Inference Proof. i It is easy to chec that δ t := Eexpit / l Y j ; /γ + tγ = expit / l x; /γ + t γ x dx = it / l x; /γ + t γ t l x; /γ + tγ x dx + o = it / γ log x γ t γ log x γ x dx + o = t γ + o, A.5 δ t := Eexpit γ l Y j ; /γ + tγ γ t γ γ = exp it x γ t γ exp it γ x γ t γ γ γ γ = exp it x γ t γ exp it x γ + γ dx + o γ γ = exp it x γ t γ it x γ + γ dx + o γ γ γ = exp it x γ t γ it γ x γ t γ it γ + γ γ γ + + it x γ t γ it x γ + γ dx + o = exp it / + = / = = it x γ tγ it γ x γ t γ γ it γ + γ exp it x γ tγ exp it x γ tγ exp it x γ tγ x γ tγ it it x dx + o γ exp it x + γ dx x dx γ it x + γ dx x γ x tγ dx + x dx + it x γ tγ x γ tγ / x dx + δ 3 t, t = Eexpit / l Y j ; /γ + tγ expit γ l Y j ; /γ + tγ γ γ it x γ t γ it γ x γ x dx + o it tγ x γ x γ tγ x dx + o it tγ x γ x γ tγ x dx + o A.6 = o. A.7 By A.5 A.7, E exp it / l Y j ; /γ + tγ + it γ l Y j ; /γ + tγ j= j= = + δ t + δ t + δ 3 t, t exp t γ + exp it x γ it tγ x γ x tγ dx tγ + it x γ x γ tγ x dx. A.8 Hence part i follows from A.8. ii To prove the existence uniqueness of a solution to A.3, it suffices to show that φx is strictly increasing, φ+ = φtγ /γ =. The proof is straightforward the detail is omitted.

8 3368 L. Peng, Y. Qi / Journal of Statistical Planning Inference Conditional on max j Y j γ /γ < t γ =max j Y < tγ j /γ, Y,...,Y are i.i.d with the truncated density function g t y = tγ /γ y if < y < tγ /γ, otherwise. Let Z,...,Z be iid rom variables with the density function g t y. Similar to the proof of A.5, we have δ t := Eexpit / l Z j ; /γ + t γ It is easy to chec that = t γ + o. A.9 δ t := Eexpit γ l Z j ; /γ + t γ tγ /γ = exp it γ tγ γ x γ tγ γ exp it γ tγ /γ = exp it γ tγ γ at,γ x γ tγ γ exp it γ at,γ + exp it γ tγ γ x γ tγ γ exp it γ + γ + γ + γ x tγ /γ dx x tγ /γ dx x tγ /γ dx := I + I, A. where I = tγ /γ at,γ exp it x γ x tγ dx + o, A. at,γ I = at,γ = at,γ = at,γ + at,γ = at,γ + := at,γ exp it γ tγ γ x γ tγ γ exp it γ tγ γ x γ tγ γ exp it γ tγ γ x γ tγ γ it γ tγ γ x γ tγ γ exp it γ x γ exp it x γ tγ it γ + γ exp it γ tγ γ x γ tγ γ it γ + γ x dx + o it γ tγ γ x γ tγ γ x dx it γ + γ x dx + o it x γ x tγ dx + it γ + γ x dx + o it x γ tγ at,γ it x γ x tγ dx + I 3 + I 4 + o. x dx + o it γ tγ γ x γ tγ γ it γ x γ x dx Notice that at,γ I 3 = = at,γ / = it at,γ it γ tγ γ x γ tγ γ / = it at,γ it γ x γ tγ it γ x γ it tγ x γ x γ tγ x γ x dx x dx x dx + o tγ x γ tγ x γ x dx + o

9 L. Peng, Y. Qi / Journal of Statistical Planning Inference at,γ I 4 = it γ x γ it γ + γ x dx = at,γ / = at, γ γ + γ + it x γ x dx + o + o. From the definition of at, γ, we get that I 3 + I 4 = o. Thus I = at,γ exp it x γ it tγ x γ x tγ dx + o. Furthermore, similar to the proof of part i, we have that A. δ 3 t, t = Eexpit / l Z j ; /γ + t γ expit γ l Z j ; /γ + t γ = o. A.3 By A.9 A.3, E exp it / l Z j ; /γ j= + t γ + it γ l Z j ; /γ j= + t γ = + δ t + δ t + δ 3 t, t exp at,γ t γ + exp it x γ it tγ x γ x tγ dx tγ /γ + exp it at,γ x γ x tγ dx. A.4 Hence part ii follows from A.4. Part iii Part iv can be shown in a similar way to the proofs of i ii, respectively. Hence, we complete the proof the lemma. Lemma 3. Assume γ =. i For any t we have l Y j= j ; /γ + t ln γ, l Y ln j= j ; /γ + t ln γ d N, N A.5 as, where N N are independent normal rom variables with N N, γ N Nγ t,. ii For each t <, Pmax j Y j γ /γ < t ln γ as, conditional on max j Y j γ /γ < t ln γ, A.5 holds with N N given in part i. iii For any t we have l Y j= j ; /γ, l Y ln j= j ; /γ + t ln γ d N, N A.6 as, where N N are given in i. iv For each t <, Pmax j Y j γ /γ < t ln γ as, conditional on max j Y j γ /γ < t ln γ, A.6 holds with N N given in part i. Proof. Set V,j := l Y j ; /γ + t ln I Y ln / j V,j := l Y ; /γ + t I Y ln / j ln, ln

10 337 L. Peng, Y. Qi / Journal of Statistical Planning Inference which are uniformly bounded by a sequence of constants tending to zero. Moreover, by noting γ =, one can verify that EV, = o/, EV, = γ t/ + o/, EV, V, = o/, EV, = /4 + o/, EV, = / + o/. By using Taylor's expansion we can show that E exp it V,j + it V,j exp 8 t + it γ t t. j= j= Therefore, j= V,j, j= V,j N d, N, part i is proved since P j= Y > ln / PY > j ln / = ln / as. For each given t <, we have P max j Y γ j /γ t ln γ = P max j Y j tγ /γ ln PY tγ /γ ln = Oln, yielding Pmax j Y j γ /γ < t ln γ. The rest of the proof for part ii is the same as that of part i by replacing expectations probabilities by conditional expectations conditional probabilities given on the set max j Y j γ /γ < t ln γ. Parts iii iv can be proved in a way similar to the proofs of i ii. Hence, we complete the proof of Lemma 3. Proof of Proposition. Put f n t = log Y n,n i+/y n,n γ +, tγ i= ḡ n t = Y n,n i+/y n,n γ, tγ i= ln t = Y n,n i+/y n,n γ, tγ i= M n = Y n,n/y n,n γ γ h n t = ḡ n t f n t. By.,.5 Lemma, there exist d d such that Y n,n i+ /Y n,n γ Y n,n /Y n,n γ + d An/ X n,n i+ X n,n X n,n X n,n Y n,n i+/y n,n γ Y n,n /Y n,n γ + d An/ A.7 with probability tending to one. Note that.8 implies that δ n / An/ γ δn asn, it is easy to verify that γ M n + /γ d Q, γ M n + δ n + /γ d Q, A.8 where Q is a positive rom variable with distribution exp γ x /γ, x >. Further, we can show that f n tm n + δ n, g n tm n + δ n, M n = f n t M n + δ n, ḡ n t M n + δ n, Mn + O p δ n A.9 uniformly for t. Hence, by A.9, A.8 the wea law of large numbers we have h n /γ + ε = ḡ n /γ + ε f n /γ + ε + o p p εγ E Y γ εγ Y E log γ εγ + εγ := J J. A.

11 L. Peng, Y. Qi / Journal of Statistical Planning Inference It is easy to chec that lim ε J = lim ε + γ ε ε εγ y γ εγ y γ γ = lim y γ ε y γ y γ εγ y dy = y dy A. lim ε J ε + γ = lim ε log ε = lim ε ε γ = lim ε ε γ y γ εγ d y γ εγ εγ y γ y εγ γ dy εγ y γ εγ y γ y γ dy γ = lim yγ ε y γ y γ εγ yγ dy = γ y γ y γ dy which implies = γ3 + γ, lim ε J γ 3 = ε + γ + γ 3. A. Thus,. follows from Eqs. A. to A.. Next we prove that uniformly in t [ M n +, /γ ], h n t/d p n A.3 for d n = 3+γ /4 ln 3/4. Note that d n as n. Note that h n t = t l n t + ḡ n t + t l n t + ḡ n t + t l n t + ḡ n t + t l n t + + cḡ n t f n tḡ nt f n tḡ nt f n t f n M n + ḡ n t holds with a probability tending to one, where c = / + γ + ε > for some sufficiently small ε >. Denote Z n,i t = Y n,n i+ /Y n,n γ /tγ for i.forallt [ M n +, /γ ], we have uniformly Z n,i t Z n,i /γ = Y n,n i+ /Y n,n γ. Therefore, by choosing < b < + c /γ, for all i b, where b denotes the integer part of b, wehave Z n,i t Y n,n b + /Y n,n γ p b γ < + c. Denote I t = b i= Z n,i t, I t = Z n,i t. i= b +

12 337 L. Peng, Y. Qi / Journal of Statistical Planning Inference For all t [ M n +, /γ ], we have uniformly I t I /γ = b i= Z n,i /γ = b Y n,n i+ /Y n,n γ. i= Note that / i= b + Y n,n i+/y n,n γ is bounded by a positive number. It follows from A. that if γ, γ Y n,n i+ /Y n,n γ = d γ i= Y i γ d Q, i= where Q > is a rom variable with / γ -stable distribution. Similarly, for γ =,/ i= Y n,n i+/y n,n ln converges in distribution. Therefore, for any γ, ], we have /d ns i= Y n,n i+/y n,n γ converges to infinity in probability, where d n is given in A.3. Thus, we conclude that uniformly in t [ M n +, /γ ], I t/d n p +. A.4 In a similar manner, we have uniformly in t [ M n +, /γ ] I t I M n + = Z i= b + n,i M n + b p bb γ. It is easy to see that Z n, b + M n + l n t + + cḡ n t = + c Z n,i t i= Z n,i t b + cz n,i t i= Z n,i t + + c Z n,i t i= b + = b + cz n,i t i= Z n,i t + + ci t. Notice that for any t [ M n +, /γ ], the two series + cz n,i t /Z n,i t are both positive decreasing in i when i b. By using the Chebyshev's inequality with ordering series see Jeffrey, 995, p. 8, we get that b + cz n,i t i= b b + cz n,i t Z n,i t b i= + cz n, b /γ I t. b b i= Z n,i t Since + cz n, b /γ p + cb γ >, we conclude that uniformly in t [ Mn +, /γ ] b p + cz n,i t d n i= Z n,i t +, where d n is given in A.3. Since I t is bounded in probability by a finite number, we have proved that uniformly in t [ M n +, /γ ] l n t + + cḡ n t/d n p. A.5 This proves A.3.

13 L. Peng, Y. Qi / Journal of Statistical Planning Inference Since ln /γ + ε p E lim E ε εγ Y γ εγ εγ Y γ εγ, =, ḡ n /γ p + γ, we have, uniformly in t [ /γ, /γ + ε], h n t t l n /γ + ε + ḡ n /γ + f n /γ ḡ n /γ f n /γ + ε < A.6 with a probability tending to one when ε > is small enough. Therefore,. follows from A.3, A.6,.8 A.9. Next we prove.9. By the mean-value theorem, h n M n + δ n h n /γ = h n ξ M n + δ n + /γ A.7 for some ξ between M n + δ n /γ. It follows from Lemmas 3 that h n /γ = O p / for γ, / Since / h n /γ = O p ln for γ = /. / /d n γ for γ, / /ln / /d n γ for γ = /, where d n is given in A.3, we get h n /γ d n γ p. A.8 Moreover, we have from Eqs. A.8 A.3 that h n ξ M n + δ n + /γ d n γ = h n ξ Mn + δ n + /γ d n γ which together with A.7 A.8 implies p, h n M n + δ n /d n γ + A.9 with probability tending to one. It follows from A.9 that h n M n + δ n = h n M n + δ n + O p δ n + O p δ n, which coupled with A.9 implies h n M n + δ n /d n γ p + hence.9. Therefore we complete the proof of Proposition. For proving Theorems, we introduce some notations. For any two sets A B denote the symmetric difference of A B by AΔB, i.e. AΔB = A B\A B. The following facts about the symmetric differences will be used in the proofs: D. PA n ΔB n implies PA n PB n ; D. If PA n ΔB n asn, then PA n C n PA n PC n ifonlyifpb n C n PB n PC n.

14 3374 L. Peng, Y. Qi / Journal of Statistical Planning Inference From Proposition we conclude that Pˆτ/aU X n,n > tδh n t > = o, A.3 where o is uniform for all + δ n /γ t /γ + ε, Pˆτ/aU X n,n tδh n t, M n + δ n < t = o, where o is uniform for all t + δ n /γ. Therefore, for < γ < we have P γ ˆτ/aU X n,n + /γ > tδh n /γ + t γ > = Pˆτ/aU X n,n > /γ + t γ Δh n /γ + t γ > = o A.3 for t >, P γ ˆτ/aU X n,n + /γ tδh n /γ + t γ, M n + δ n < /γ + t γ = Pˆτ/aU X n,n /γ + t γ Δh n /γ + t γ, M n + δ n < /γ + t γ = o A.3 for t <. Similarly, for γ = we have P ln γ ˆτ/aU X n,n + /γ > tδh n /γ + t ln γ > = o A.33 for t, P ln γ for t <. ˆτ au X n,n + /γ t Δh n /γ + t ln γ, M n + δ n < /γ + tln γ = o A.34 Proof of Theorem. By Lemma, A. A.9, we have for any t > lim n Ph n /γ + t γ > = lim n P +γ h n /γ + t γ > Since = lim P +γ h n /γ n + t γ > +γ = lim n P ḡ n /γ + t γ + γ > = PW t >. A.35 lim n P γ M n + /γ x = exp γ x /γ for x <, it follows from A.9 that M n = M n + δ n + O p δ n, which implies E I γ Mn + /γ < t I γ Mn + δ n + /γ < t for any fixed t <. Therefore, for any t <, Lemma implies that lim Ph n /γ n + t γ, M n + δ n < /γ + t γ = lim n P +γ hn /γ + t γ, Mn < /γ + t γ = lim P +γ hn /γ n + t γ M n < /γ + t γ P M n < /γ + t γ +γ M n < /γ + t γ = lim n P ḡ n /γ + t γ + γ P M n < /γ + t γ = PW t exp tγ /γ. A.36 Hence, it follows from A.3, A.3, A.35 A.36 that P γ ˆτ/aU X n,n + γ x Vx, A.37 which implies that ˆτ/aU X n,n + γ = o p /. A.38

15 L. Peng, Y. Qi / Journal of Statistical Planning Inference Note that g n t is monotone in t f n t = g nt /t. By the mean-value theorem there exists a variable ξ n between ˆτ/aU X n,n γ such that f n ˆτ/aU X n,n f n γ = f n ξ n ˆτ/aU X n,n + γ + maxg n ˆτ/aU X n,n, g n γ o p / = + max + ˆγ, g n γ o p /. Here we used.3 A.38. From Zhou 9, ˆγ p γ. By using A.9 Lemma i with t = we have g n γ = ḡ n γ + O pδ n p / + γ. Therefore, from. Lemma we have ˆγ γ = f n /γ γ +o p = f n /γ γ +o p d W. A.39 We still need to prove the asymptotic independence. It suffices to show that P ˆγ γ x, γ ˆτ/aU X n,n + γ t PW xpv t for x R t R. Note that the above equation is equivalent to P ˆγ γ x, γ ˆτ/aU X n,n + γ > t PW xpv > t. Here we focus on the case t >. By using A.3 it suffices to show the asymptotic independence of ˆγ γ +γ hn /γ + t γ. From Lemma i we now that f n /γ + t γ γ is negligible compared with ḡ n /γ + t γ / + γ. Hence, from A.39 A.9 we have that ˆγ γ, +γ h n /γ + t γ = f n /γ γ, +γ hn /γ + t γ + o p = f n /γ γ, +γ ḡ n /γ + t γ / + γ + o p. So the independence for the case t follows from part iii of Lemma. Similarly, we can show the independence for the case t < by using parts ii iv of Lemma. We complete the proof of Theorem. Proof of Theorem. Following the same lines of the proof of Theorem, this theorem can be proved by using Lemma 3, Eqs. A.33 A.34. Proof of Theorem 3. From Lemma 4. of Ferreira et al. 3,wehave lim t U Ut at At + γ = γ γ + ρ. A.4 Write au X n,n an/ = au X n,n U Xn,n U Un/ + γ U X γ n,n an/ + X + γ n,n Un/ an/ γ an/ := II II + II 3. A.4 By.8 A.4, we have that II p II p. A.4 It follows from Drees et al. 4 that II 3 d = γ UY n,n Un/ an/ d N, γ := γ W. A.43 Since Y n,n is independent of Y n,n i+ /Y n,n, the theorem is proved by combining.3, A.38, Theorems i= A.4 A.43.

16 3376 L. Peng, Y. Qi / Journal of Statistical Planning Inference Proof of Theorem 4. By writing ˆσ an/ = ˆγ ˆτ an/ + + ˆγ γ γ /γ ˆσ au X n,n = ˆγ ˆτ au X n,n + + ˆγ γ γ /γ, the theorem follows from Theorem 3 A.38. Proofs of Theorems 5 6. Write ˆθ θ au X n,n = UU X n,n U au X n,n γ + ˆτ au X n,n + γ. Then Theorems 5 6 follow from Theorems, A.4,.8 the fact that au X n,n /an/ p. References Athreya, K.B., Fuuchi, J.I., 997. Confidence intervals for endpoints of a c.d.f. via bootstrap. J. Statist. Plann. Inference 58, De Haan, L., Ferreira, A., 6. Extreme Value Theory, An Introduction. Springer, Berlin. Deers, A.L.M., de Haan, L., 989. On the estimation of the extreme-value index large quantile estimators. Ann. Statist. 7, Deers, A.L.M., Einmahl, J.H.J., de Haan, L., 989. A moment estimator for the index of an extreme-value distribution. Ann. Statist. 7, Draisma, G., de Haan, L., Peng, L., Pereira, T.T., 999. A bootstrap-based method to achieve optimality in estimating the extreme-value index. Extremes, Drees, H., Ferreira, A., de Haan, L., 4. On maximum lielihood estimation of the extreme value index. Ann. Appl. Probab. 4, 79. Fal, M., 995. Some best parameter estimates for distributions with finite endpoint. Statistics 7, 5 5. Ferreira, A., de Haan, L., Peng, L., 3. On optimising the estimation of high quantiles of a probability distribution. Statistics 37, Hall, P., 98. On estimating the endpoint of a distribution. Ann. Statist., Hall, P., Wang, J.Z., 999. Estimating the end-point of a probability distribution using minimum-distance methods. Bernoulli 5, Jeffrey, A., 995. Hboo of Mathematical Formulas Integrals. Academic Press, New Yor. Li, D., Peng, L., 8. Still fit generalized Pareto distributions? Technical Report. Loh, W.Y., 984. Estimating an endpoint of a distribution with resampling methods. Ann. Statist., Smith, R.L., 985. Maximum lielihood estimation in a class of nonregular cases. Biometria 7, Smith, R.L., 987. Estimating tails of probability distributions. Ann. Statist. 5, Woodroofe, M., 974. Maximum lielihood estimation of translation parameter of truncated distribution II. Ann. Statist., Zhou, C., 9. Existence consistency of the maximum lielihood estimator for the extreme value index. J. Multivariate Anal.,