Bootstrapping Extreme Value Estimators

This article develops a bootstrap analogue of the well-known asymptotic expansion of the tail quantile process in extreme value theory. One application of this result is to construct confidence intervals for estimators of the extreme value index such as the Probability Weighted Moment (PWM) estimator. For the peaks-over-threshold method, we show the bootstrap consistency of the confidence intervals. By contrast, the asymptotic expansion of the quantile process of the bootstrapped block maxima does not lead to a similar consistency result for the PWM estimator using the block maxima method. For both methods, We show by simulations that the sample variance of bootstrapped estimates can be a good approximation for the asymptotic variance of the original estimator. Supplementary materials for this article are available online.


Introduction
Statistical inference based on extreme value theory uses only large observations in a sample.The large observations are selected by either the peaks-over-threshold (POT) method or by the block maxima method.Asymptotic theories for most estimators based on these two methods are established using the tail quantile process of the peaks-over-threshold or the quantile process of block maxima.However, the asymptotic variance obtained in such asymptotic theories, though mostly explicit, can be intricate.One example is the much used probability weighted moment (PWM) estimator in the block maxima method; see Ferreira and de Haan (2015).We derive a bootstrap version of the fundamental expansions for the (tail) quantile process in both the POT and block maxima methods.Consequently, for any statistical estimator in extreme value theory whose asymptotic property can be established via the tail quantile process in the POT method, the bootstrap mimics faithfully the original asymptotic behavior of the estimator.
Although the bootstrap is a widely used method for obtaining the distribution of a statistical estimator, it does not work automatically for estimators based on extreme value theory.A somewhat trivial example is that the bootstrapped full sample maxima lies below the actual full sample maxima.More formally, Bickel and Freedman (1981) shows the nonconsistency of the bootstrap method, when using the sample maxima as an estimator of the right endpoint of a distribution.For the definition of the consistency of the bootstrap method, see Van der Vaart (1998, sec. 23.2).Broadly speaking, proving consistency of the bootstrap in extreme value theory is a difficult issue because even if observations are drawn from a distribution that satisfies the extreme value conditions, the corresponding empirical distribution function does not satisfy the extreme value CONTACT Chen Zhou zhou@ese.eur.nlEconometric Institute, Erasmus University Rotterdam, 3000DR Rotterdam, The Netherlands.Supplementary materials for this article are available online.Please go to www.tandfonline.com/r/JASA.conditions.Consequently, it is not obvious that the bootstrap can be used to obtain the distribution of statistical estimators in extreme value theory.Therefore, our somewhat surprising result provides confidence for using the bootstrap in extreme value theory.
The first question regarding the bootstrap in extreme value theory is what to bootstrap.For the POT method, one can bootstrap just the selected peaks or one can bootstrap the original sample and reconstruct the peaks.We choose the latter for better expected performance because the former seems not to randomize sufficiently the observations.In the block maxima method, one can bootstrap just the block maxima of the original sample or one can bootstrap the original sample and reconstruct the block maxima.We again choose the latter for a similar reason.
For the POT method we wish to construct a bootstrap analogue of the following fundamental expansion of the tail quantile process (due to Drees 1998).Let X1 , X2 , . . .be a sequence of iid random variables with a common distribution function F. Denote the order statistics from a sample of n observations as X1,n ≤ • • • ≤ Xn,n .Suppose that the sample maximum Xn:n , properly normalized, converges to one of the extreme value distributions G γ (x) = exp{−(1 + γ x) −1/γ }.Note that G γ is a Fréchet distribution for γ positive, a negative Weibull distribution for γ negative and a Gumbel distribution for γ = 0.For any intermediate sequence k = k(n) satisfying k → ∞, k/n → 0 as n → ∞, the process Xn−[ks],n 0≤s≤1 is called the tail quantile process.
The asymptotic expansion of the tail quantile process relies on the second-order condition of extreme value theory (see, Appendix B, de Haan and Ferreira 2006) as follows.Define U(t) = F ← 1 − 1 t for t > 0 where ← indicates the left-continuous inverse function.Assume that there exist ρ ≤ 0, a positive function a and an eventually positive or negative function A with lim t→∞ A(t) = 0 such that for x > 0 where Here γ and ρ are called the extreme value index and the second order index, respectively, while a and A are called the first and second order auxiliary functions, respectively.This second order condition implies the mentioned convergence of the sample maximum.
Obviously, there is some freedom in choosing the normalizing functions U, a, and A in relation (1.1).A useful tool in applying (extended) regular variation is a sharp uniform inequality due to Drees (1998).This inequality can only be obtained for a more restricted choice of the normalizing functions, hence, we have to use those (see, de Haan and Ferreira 2006, Theorem B.3.10).This inequality is needed for obtaining Proposition 1.1, which is a version of Theorem 2.1 in Drees (1998); see also Corollary 2.4.5 in de Haan and Ferreira (2006). 1   Proposition 1.1.Assume the second order condition (1.1) holds.
There exist an appropriate version of the auxiliary functions a and A, denoted as a 0 and A 0 , a sequence of Brownian motions W 1 , W 2 , . . .such that, for any ε > 0, as n → ∞, holds uniformly for x ∈ (0, 2], where The first goal of this article is to establish a parallel result for the tail quantile process based on the bootstrap observations.Take a bootstrap sample, that is, conditional on the observations, draw iid random variables X * 1 , X * 2 , . . ., X * n from the empirical distribution function F n of X1 , X2 , . . ., Xn .After ordering the bootstrapped observations as X * 1,n ≤ • • • ≤ X * n,n , we consider the bootstrap tail quantile process X * n− [ks],n 0≤s≤1 for an intermediate sequence k and will prove a result analogue to (1.2); see Section 2. The bootstrap analogue that we develop has a similar structure except that in the expansion we have two independent 1 Notice that Corollary 2.4.5 in de Haan and Ferreira (2006)  Brownian motion terms, one due to the randomness of the original sample and the other one due to the bootstrap randomness.
The same holds-mutatis mutandis-for the fundamental expansion of the quantile process of the block maxima.The original result in Ferreira and de Haan (2015) is as follows.Recall that X1 , X2 , . . .are a sequence of iid random variables with a common distribution function F. Define the block maxima of the sample X1 , X2 , . . ., Xn as relies on a different second order condition, this time regarding the function V: with replacing U by V in the second order condition (1.1), we assume that a similar limit relation holds with different first order and second order auxiliary functions ã and Ã and a different second order index ρ.The asymptotic expansion is given in the following Proposition; see Theorem 2.1 in Ferreira and de Haan (2015).
Proposition 1.2.Assume the second order condition (1.1) holds for the function where Ã is the second order auxiliary function for the function V.Then, there exist an appropriate version of the auxiliary functions ã and Ã, denoted as ã0 and Ã0 , and an appropriate sequence of Brownian bridges Next, recall that conditional on the original observations, the bootstrapped observations are iid random variables X * 1 , X * 2 , . . ., X * n drawn from the empirical distribution function F n of X1 , X2 , . . ., Xn .Then, we construct the bootstrapped block maxima by This setup shows that we bootstrap the original sample and reconstruct the bootstrapped block maxima from the bootstrapped sample instead of bootstrapping the block maxima from the original sample.
The second goal of the article is to prove for the bootstrap block maxima a result similar to (1.3) for the original sample.This will be done in Section 3. The bootstrap analogue of this result that we develop has a similar structure except that in the expansion we have a Brownian bridge term due to the randomness of the bootstrap and a Brownian motion term due to the randomness of the original sample.
The bootstrap expansions that we prove lead easily (by integrating the various terms) to the asymptotic distribution of extreme value estimators.In Section 4, a few examples are given for applying our bootstrap expansions in the POT and the block maxima methods to estimate the distribution of estimators for the extreme value index.For the POT method, we use the PWM estimator (Hosking and Wallis 1987) as an example.For the block maxima method, we use the PWM estimator in Hosking, Wallis, and Wood (1985) as an example.Theoretically, we focus on the bootstrap consistency of the confidence intervals for these estimators.Besides, we show by simulations that the sample variance of bootstrapped estimates can be a good approximation for the asymptotic variance of the original estimator.
The proof of our main results uses a simple representation of the bootstrap sample.For the POT method the representation for the bootstrap tail quantile process is as follows ,n the tail quantile process for the standard Pareto distribution.The two processes are independent.Then we combine the expansions of both processes.The combined expansion requires that Dn (s) is in the correct range of the process Xn−[kx],n .The proof for the block maxima method is considerably more complicated.It is based on a similar representation for the bootstrap quantile process of the block maxima; see Equation (3.4).
In Section 2 the fundamental expansion for the POT method is given along with an outline of the proof.The rest of the proof can be found in the Appendix A.1, supplementary materials.Section 3 handles the block maxima method in a similar way with leaving detailed proofs to the Appendix A.2, supplementary materials.Section 4 provides a few examples for applications.We show that the asymptotic expansion of the tail quantile process of the bootstrapped sample leads to a consistency result for the bootstrap PWM estimator using the POT method.However, the asymptotic expansion of the quantile process of the bootstrapped block maxima does not lead to a similar consistency result for the PWM estimator using the block maxima method.Nevertheless, the asymptotic expansions of the bootstrapped PWM estimators using both methods suggest to use the sample variance of bootstrapped estimates as an approximation for the asymptotic variance of the original PWM estimator.Section 5 provides simulation studies illustrating the theoretical findings.
We prove the following bootstrap analogue of the tail quantile process result in (1.2).A paper related to this result is Litvinova and Mervyn (2018).
Recall the functions a 0 and A 0 and the sequence of Brownian motions {W n (s)} as in Proposition 1.1.Then there exists a sequence of and of holds uniformly for all s ∈ (0, 1], where Remark 1.The Brownian motions {W n } stem from the randomness of the original sample whereas the Brownian motions W * n stem from the randomness of the bootstrap procedure. When considering a limited region s ∈ [1/(k + 1), 1], we have the following corollary, which shows the validity of the bootstrap for the tail quantile process in the following sense.Proposition 1.1 shows that the tail quantile process, after suitable standardization possesses weighted convergence toward a stochastic process characterized by x −γ −1 W n (x) and a bias term.The Corollary below shows that the bootstrapped tail quantile process, after centered by the original tail quantile process, possesses weighted convergence toward a similar stochastic process x −γ −1 W * n (x) , conditional on the observations, whereas the bias term is canceled out.Therefore, bootstrap consistency for the tail quantiles holds with further assuming Corollary 2.2.Under the conditions in Theorem 2.1, as n → ∞, The processes W * n (s) are the Brownian motions in Theorem 2.1, independent of the observations.To prove the theorem, we need three auxiliary results.First, recall Proposition 1.1, which gives the expansion of the tail quantile process of the original observations Xj n j=1 .Second, Lemma 1.3 relates the tail quantile process of the bootstrapped observations to the tail quantile process of the original observations.Finally, Lemma 2.3 guarantees that we can use the asymptotic expansion of the latter process to obtain the expansion of the former process.
We introduce the notation Then, as n → ∞, uniformly for all s ∈ [1/(k + 1), 1], Dn (s) P s and Pr(sup The first step in the proof of Theorem 2.1 is the substitution of x in (1.2) with Dn (s), which is made available by Lemma 2.3.We obtain that, under the conditions of the Theorem, as n → ∞, √ k For the proof of Theorem 2.1, the next step is to expand the four terms involving Dn (s) on the left hand side of (2.3) can be handled by taking ξ = −γ in the following lemma.
Lemma 2.4.Assume that k → ∞ and k/n → 0 as n → ∞.There exists a sequence of Brownian motions The series of Brownian motions W * n is independent of Xj n j=1 .
The three terms on the right hand side of (2.3) are handled by the following lemmas.

The Block Maxima Method
In this section we present a bootstrap analogue of Equation (1.3).
Theorem 3.1.Suppose the second order condition (1.1) holds for the function holds uniformly for all s ∈ [1/(k + 1), k/(k + 1)], where The general idea behind the proof of the main theorem is similar to the proof of Theorem 2.1, but the steps taken are more complicated.First, we establish an extended version for the asymptotic expansion of the tail quantile process of the original observations (see Proposition 3.2).Second, we relate the quantile process of the bootstrapped block maxima X * i k i=1 to the tail quantile process of the original observations Xj n j=1 (see Lemmas 3.3 and 3.4).Finally, we can use the asymptotic expansion of the latter process to obtain the expansion of the former process.We first present the three auxiliary results, and then show the steps toward proving the main theorem at the end of this Section.
Proposition 3.2.Assume that the second order condition (1.1) holds.Assume that an intermediate sequence k Recall the appropriate function a 0 defined in Proposition 1.1 and the function b0 defined in Theorem 3.1.Then there exist a sequence of Brownian motions Remark 5.This proposition is a version of Proposition 1.1 with a different range of x: On the left corner, it does not reach 0, which allows for a deterministic shift function b0 .On the right corner, it is extended to a high level tending to infinity.The extension is allowed by requiring a slightly stronger condition on k, which is useful in the proof of Theorem 3.1.
Recall that F n is the empirical distribution function of Xj .Then defines a sample of bootstrapped block maxima with block size m.With this notation, we have that where The representation in (3.4) suggests that the expansion for the process can be obtained by sub- The following lemma guarantees that such a substitution is allowed.
Recall the notation P defined in (2.2).
Lemma 3.4.Assume that m/ log k → ∞ as n → ∞.Then, as n → ∞, uniformly for all s ∈ [1/(k + 1), k/(k + 1)], For the proof of Theorem 3.1 we need some auxiliary results.Lemma 3.4 imply that under the conditions of Theorem 3.1, with probability tending to 1, uniformly for all ]. Thus, we can apply Proposition 3.2 with replacing x by D n (s).Together with Lemma 3.3, we get that, as n → ∞, uniformly for all s ∈ [1/(k+1), k/(k+1)].Next, we approximate the various terms in (3.5) under the same conditions as in Theorem 3.1.The term (D n (s)) −γ −1 γ on the left hand side of (3.5) can be handled by taking ξ = −γ in the following lemma.
Next, the two terms on the right hand side of (3.5) are handled by the following lemmas.

Applications
In this section, we apply our main results, Theorem 2.1 for the POT method and Theorem 3.1 for the block maxima method to obtain the asymptotic behavior of the bootstrap version of estimators in extreme value theory.As in the case of the original estimator, the asymptotic behavior of the bootstrap estimator follows directly from the expansion of the bootstrap (tail) quantile process (Theorems 2.1 and 3.1).For the POT method, we use the PWM estimator (Hosking and Wallis 1987) as an example and show that the bootstrap is consistent for the PWM estimator.Here consistency refers to the consistency of the bootstrap defined in Bickel and Freedman (1981).For the block maxima method, we use the PWM estimator in Hosking, Wallis, and Wood (1985) as an example.We suggest to use the sample variance of bootstrapped estimates as an approximation for the asymptotic variance of the original PWM estimator based on the asymptotic expansion of the bootstrapped PWM estimator.The POT and block maxima methods are handled in two separate sections.

The POT Method
We start with the PWM estimator for the extreme value index using the POT method; see Hosking and Wallis (1987).Let X1 , X2 , . . ., Xn be a sequence of iid random variables with common distribution function F satisfying the second-order condition (1.1).Denote X1,n ≤ • • • ≤ Xn,n as the order statistics from a sample of n observations.Define the PWM estimator in the POT method as γPOT := where the probability weighted moments I q are given by The asymptotic behavior of the PWM estimator using the POT method is as follows; see for example, eq.(3.4) in Cai, de Haan, and Zhou (2013) with q = 2, r = 1.Proposition 4.1.Assume that γ < 1/2 and √ kA(n/k) = O(1) as n → ∞.With the same Brownian motions W n defined in Proposition 1.1, we have that as n → ∞, for q = 1, 2, where the functional L(•) is defined by and the asymptotic bias term is bn We remark that the proof of the asymptotic expansion of I q is based on taking an integral over a weighted version of the tail quantile process in Proposition 1.1 on the interval s ∈ (0, 1].The operators Lq and consequently L are linear functionals on continuous and suitably integrable functions.
From this proposition, one may calculate the asymptotic variance of the PWM estimator as follows, see Theorem 3.6.1 in de Haan and Ferreira (2006).Practically, one may use the estimated γ to obtain a consistent estimate of the asymptotic variance.Therefore, using the bootstrap is not a necessary step for obtaining the variance of the estimator.Nevertheless, we show that using bootstrap can achieve the consistency of the estimator.We consider the bootstrap sample X * 1 , X * 2 , . . ., X * n and construct probability weighted moments based on the bootstrap sample, denoted as I * q for q = 1, 2. Then we construct the PWM estimator based on the bootstrap sample, denoted as γ * POT .Notice that the expansion in Theorem 2.1 has a structure very similar to the one in Proposition 1.1, except an extra term.Similar to the proof of Proposition 4.1, with taking an integral over the asymptotic expansion of the bootstrap tail quantile process on the interval s ∈ (0, 1], we can obtain the asymptotic expansion of I * q and consequently the following Proposition. Proposition 4.2.Assume that γ < 1/2 and √ kA(n/k) = O(1) as n → ∞.With the same Brownian motions W n and W * n defined in Theorem 2.1, we have that as n → ∞, for q = 1, 2, where A 0 is the same as in Proposition 1.1, the constant b (q) n (γ , ρ) and the operator Lq are defined in Proposition 4.1.Again, with applying a bivariate version of Cramér's delta method, we have that as where L( .The relation (4.1) serves as an important step in proving the consistency of the bootstrap procedure as in the following Theorem.The proof is postponed to Appendix A.3, supplementary materials.
Theorem 4.3.Assume that γ < 1/2 and √ kA(n/k) = o(1) as n → ∞.The PWM estimator using the POT method is consistent: as n → ∞, The asymptotic expansion (4.1) motivates the following procedure to approximate the asymptotic variance of γPOT : by bootstrapping d times to obtain estimators γ * l , l = 1, 2, . . ., d, the sample variance of these bootstrapped estimates can be an approximation for the variance of 1 , which equals the asymptotic variance of the original estimator γPOT .Note that we do not claim that this approximation is a consistent estimator for the asymptotic variance.
We remark that Theorem 4.3 requires the condition √ kA(n/k) = o(1) as n → ∞ which assumes away the bias in the original estimator.This is only necessary for obtaining the consistency result.Nevertheless the aforementioned procedure for obtaining the asymptotic variance is also valid if the bias is present: that is, √ kA(n/k) = O(1) as n → ∞.In this case, the Equation (4.1) is still valid.Therefore, the sample variance of γ * l , is still a good approximate of var( L(W n ))/k.An alternative way to approximate the asymptotic variance involves the original estimator γPOT .Theorem 4.3 implies that the following statistic is also an approximation for the variance of 1 , which equals to the asymptotic variance of the original estimator.Again, we do not claim that this estimator for the asymptotic variance is consistent either.
For any other estimator for the extreme value index using the POT method, as long as its asymptotic behavior can be developed as a linear functional of the tail quantile process in Proposition 1.1, similar results as in Proposition 4.2 and Theorem 4.3 can be established.In other words, the bootstrap procedure is consistent if the asymptotic bias is zero.Further, the sample variance of the bootstrapped estimates can approximate the variance of the original estimator.Examples of such estimators are the Pickands' estimator (Pickands 1975), the maximum likelihood estimator (Smith 1987) and the negative Hill estimator (Falk 1995).

The Block Maxima Method
Analogously to the POT method, we investigate the bootstrap PWM estimator for the extreme value index using the block maxima method, which differs from the PWM estimator using the POT method.The estimator was introduced in Hosking, Wallis, and Wood (1985) with its asymptotic normality proved in Ferreira and de Haan (2015).
Let X1 , X2 , . . ., Xn be a sequence of iid random variables with common distribution function F satisfying the second order condition (1.1).Define the block maxima of the sample as be the order statistics of the block maxima X 1 , X 2 , . . ., X k .The PWM estimator using the block maxima method, denoted as γBM , is the solution of the equation where the probability weighted moments β q are given by Ferreira and de Haan (2015) shows the asymptotic behavior of the PWM estimator using the block maxima method.We cite this result in a simpler form where the sequence k = k(n) is chosen such that no asymptotic bias appears.Ferreira and de Haan (2015)).Assume the conditions in Theorem 3.1 for the function V instead of U. Further assume γ < 1/2.There exist suitable series of constant a m and b m and a series of standard Brownian bridge B 1 , B 2 , . . .such that as n → ∞, for q = 0, 1, 2

Proposition 4.4 (A simpler version of Theorem 2.3 in
where L q (z) = (q + 1) 1 0 s q z(s)ds, for all z ∈ C(0, 1) satisfying By applying a bivariate version of Cramér's delta method to 3β 2 − β 0 and 2β 1 − β 0 , we obtain that as n → ∞, where . We remark that the proof of the asymptotic expansion of β q is based on taking an integral over a weighted version of the quantile process in Proposition 1.2 on the interval s ∈ (0, 1].However, different from the POT method, Proposition 1.2 provides a uniform expansion on the interval s ∈ [1/(k + 1), k/(k + 1)] only.Consequently, the proof of Proposition 4.4 is more involved by handling the two corners close to 0 and 1.Again, the operators L q and consequently L are linear functionals on continuous and suitably integrable functions.We consider the bootstrap sample X * 1 , X * 2 , . . ., X * n , reconstruct the block maxima.Then we construct the probability weight moments β * q based on the bootstrapped block maxima, from which we can obtain the bootstrap PWM estimator denoted as γ * BM .Based on the asymptotic expansion of the bootstrap quantile process in Theorem 3.1, similar to the proof of Proposition 4.4, we get the following result.The proof is postponed to Appendix A.3, supplementary materials.To estab-lish this Proposition, we need the second order condition for U instead of V since we have to deal with the tail quantile process Xn−[ks]:n whose asymptotic behavior depends on U.This is somewhat inconsistent with the classical block maxima approach as in Proposition 4.4  We remark that a consistency result for the bootstrap PWM estimator using the block maxima method, analog to Theorem 4.3, is out of reach.Nevertheless, we may still use the bootstrap procedure to approximate the asymptotic variance by considering the sample variance of the bootstrapped estimates.Similar to the POT method, if an estimator of the extreme value index can be expanded as a linear functional of the quantile process of the block maxima as in in Proposition 3.2, then the sample variance of bootstrapped estimates can be a good approximation for the asymptotic variance of the original estimator.An example of such an estimator is the maximum likelihood estimator using the block maxima method as analyzed in Dombry and Ferreira (2019).

Simulations
In this section, we perform three simulation studies to show the usefulness of the results obtained in Section 4 in practice.Throughout the simulation study, we consider the PWM estimators using the POT and block maxima methods.We simulate observations from three different distributions: the Pareto distribution Hall and Welsh (1984 All three distributions possess the same extreme value index γ = 0.2.Notice that the second order condition (1.1) holds for the Fréchet distribution and the HW distribution with ρ = −1 and ρ = −1/2, respectively, while the Pareto distribution can be viewed as a limit case with ρ = −∞.We consider typical sample size used in application n = 2000.To make the choice of k comparable across the POT and block maxima methods, we first fix the potential block size m at 5, 6, . . ., 100 and then calculate the corresponding k as k = [n/m].
First, we show that the bootstrap approximation for the standard deviation is close to its theoretical counterpart, both for the POT and block maxima methods.Figure 1 shows the standard deviation estimated (or calculated) in three different ways.The solid horizontal line indicates the theoretical value of the asymptotic standard deviation from the corresponding asymptotic theory with plugging in γ = 0.2.Since the asymptotic theory may not be valid for finite sample size, we further obtain the standard deviation of the estimators by conducting a pre-simulation of 1000 independent samples.For each k we calculate the sample standard deviation across the 1000 estimates of the extreme value index, scaled by the corresponding √ k such that they are comparable across different levels of k.The results are plotted as the dotted line.Finally, the dash line shows the bootstrap standard deviation averaged across 100 samples.For each simulated sample and each k, we obtain the bootstrap standard deviation based on b = 100 bootstrapped samples, again scaled by the corresponding √ k.For each distribution, Figure 1 shows the results for the POT and block maxima methods in the left and right column, respec- tively.For the POT method, the bootstrap standard deviation is around the value obtained via pre-simulations, which can deviate from the theoretical value when k is low.A potential explanation is that with relatively low k, the asymptotic distribution is not a good approximation for the actual distribution of the estimator.For the block maxima method, the standard deviation obtained from pre-simulations is in general lower than the theoretical value.The bootstrap standard deviation is again closer to that obtained from pre-simulations, than the theoretical value.
Second, we compare the accuracy of the bootstrap standard deviation to a practically feasible "plug-in" estimator across 100 samples.For each simulated sample and each k, we obtain the standard deviation (scaled by the corresponding √ k) by two ways: (a) based on b = 100 bootstrap samples, (b) based on plugging in the estimated γ to the asymptotic variance following the asymptotic theory. 2 After estimating the standard deviation, we calculate the mean squared error (MSE) with respect to the theoretical value of the asymptotic standard deviation (with γ = 0.2) across 100 samples.
Figure 2 shows the results for the POT (left) and block maxima (right) methods, where the solid (dash) line shows the MSE when using the bootstrap (plug-in) standard deviation.We observe that the bootstrap standard deviation shows a comparable performance as the plug-in estimation for the 2 Since the formula for the asymptotic variance based on the block maxima method involves numerical integral, computational complexity arises for γ < −0.2 or γ > 0.4.As a consequence, whenever the estimated γ is below −0.2 (above 0.4), we simply plug-in γ = −0.2(γ = 0.4) to the corresponding asymptotic variance.Given that the true extreme value index is γ = 0.2, such a "truncation" approach may artificially reduce the MSE for the plug-in estimator.
Fréchet distribution and the HW distribution when using the block maxima method.However, for the POT method, the plugin estimation performs better.Notice that the formula of the asymptotic variance is particularly complicated for the block maxima method, whereas that for the POT method is rather straightforward.We conclude that the bootstrap standard deviation is useful when it is needed the most.Third, we compare the number of coverages by the confidence intervals.For each simulated sample and each k, we construct the 95% confidence interval around the point estimate obtained from the PWM estimator in three ways: (a) by plugging in the point estimate of γ to the asymptotic distribution derived from the asymptotic theory (solid line); (b) by using the bootstrap standard deviation as the standard deviation in the normal distribution (dash line); (c) by taking the 2.5% and 97.5% empirical quantiles among the bootstrap estimates (dotted line).To mitigate the estimation uncertainty in empirical quantiles, for (b) and (c) we increase the number of bootstraps to b = 200.In addition, to obtain a fair view on the number of coverages, we increase the number of samples to 1000 in this study.Across the 1000 samples, we evaluate how many times the constructed confidence intervals cover the true value γ = 0.2.The expected level is thus 950.
For each distribution, Figure 3 shows the results for the POT (left) and block maxima (right) methods.For the POT method, the confidence intervals using the bootstrap standard deviation performs best among the three approaches.For the block maxima method, except the HW distribution, the three types of confidence intervals perform similarly, all close to the expected level.Lastly, for the HW distribution and the block maxima method, all three types of confidence intervals have an unsatisfactory performance for large k.This is potentially due to the asymptotic bias of the estimators, since the HW distribution possesses the highest second order index among the three distributions: ρ = −1/2.
To obtain further insight on the coverage of the confidence intervals, we plot the averaged length of the intervals across the 1000 samples in Figure 4.For both the POT and block maxima methods, across all simulated distributions, the length of the bootstrap confidence interval is very close to that of the confidence intervals using bootstrap standard deviation.This is potentially due to the fact that bootstrapped estimates are close to normally distributed.For the POT method, the plug-in confidence intervals are narrower than those obtained using bootstrap.Together with their overall good performance and the fact that the plug-in standard deviation is easy to compute, the plug-in method is preferred.By contrast, for the block maxima method, the plug-in confidence intervals are wider than those obtained using bootstrap.Recall that they achieve similar coverage performance.The confidence intervals based on bootstrap, either by using empirical quantiles from bootstrapped estimates, or by using the bootstrap standard deviation, are preferred in practice.
is subject to a technical error in the random shift function b 0 n k , which is fixed in this Proposition (personal communication with the authors).
Lemma 1.3.Let F n be the empirical distribution function of Xj n j=1 .Let Y * 1 , . . ., Y * n be iid random variables following the standard Pareto distribution 1−1/x, for x > 1[kx],n is the tail quantile process of the original sample and Dn (s) = n kY * n−[ks],n with Y * n−[ks] Recall the function a 0 and the sequence of Brownian motions {W n (s)} in Proposition 1.1.Then there exist a sequence of Brownian bridges B * k (s) independent of Xj n j=1 and of {W n n ≤ X2,n ≤ • • • ≤ Xn,n are the order statistics of the original observations X1 , X2 , . . ., Xn .Denote (x) := exp(−1/x) for x > 0, the distribution function of the standard Fréchet distribution.We have the following representation result.Lemma 3.3.Let Z * 1 , • • • , Z * k be iid random variables with distribution function , independent of Xj n j=1

Figure
Figure Standard deviations: the left (right) column shows the result using the POT (block maxima) method.

Figure 2 .
Figure 2. MSEs for estimating the standard deviation: the left (right) column shows the result using the POT (block maxima) method.
Repeat the bootstrap procedure for d times and obtain estimators γ * l , l = 1, 2, . . ., d.When considering γ * l s) 1+γ , γ cancels out.Consequently, the sample variance of γ * l , as an approximation of the variance of 1 log s) 1+γ , γ , which equals to the asymptotic variance of the original estimator γBM .Again, no consistency is claimed.

Figure 3 .
Figure 3. Coverages of the confidence intervals: the left (right) column shows the result using the POT (block maxima) method.

Figure 4 .
Figure 4. Lengths of the confidence intervals: the left (right) column shows the result using the POT (block maxima) method.
Remark 2. The Brownian motions {W n } stem from the randomness of the original sample whereas the Brownian bridges B * Remark 4. For the block maxima method, a result similar to Corollary 2.2 cannot be obtained from Theorem 3.1 due to the following reason.Comparing the expansion in (3.1) with that for the original sample in (1.3), there are at least two key differences.First, on the left-hand side, the shift and scale functions are different.Second, on the right-hand side, the Brownian bridges B * k and {B k } stem from different sources of randomness.
and A 0 is the same as in Proposition 1.1.It then follows from a bivariate version of Cramér's delta method that as n