A data-adaptive maximum penalized likelihood estimation for the generalized extreme value distribution

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1 Communications for Statistical Alications and Methods 07, Vol. 4, No. 5, htts://doi.org/0.535/csam Print ISSN / Online ISSN A data-adative maximum enalized likelihood estimation for the generalized extreme value distribution Youngsaeng Lee a, Yonggwan Shin a, Jeong-Soo Park, a a Deartment of Statistics, Chonnam National University, Korea Abstract Maximum likelihood estimation (MLE) of the generalized extreme value distribution (GEVD) is known to sometimes over-estimate the ositive value of the shae arameter for the small samle size. The maximum enalized likelihood estimation (MPLE) with Beta enalty function was roosed by some researchers to overcome this roblem. But the determination of the hyerarameters (HP) in Beta enalty function is still an issue. This aer resents some data adative methods to select the HP of Beta enalty function in the MPLE framework. The idea is to let the data tell us what HP to use. For given data, the otimal HP is obtained from the minimum distance between the MLE and MPLE. A bootstra-based method is also roosed. These methods are comared with existing aroaches. The erformance evaluation exeriments for GEVD by Monte Carlo simulation show that the roosed methods work well for bias and mean squared error. The methods are alied to Blackstone river data and Korean heavy rainfall data to show better erformance over MLE, the method of L-moments estimator, and existing MPLEs. Keywords: annual maximum daily rainfall, Beta distribution, bootstra-based selection, L-moments estimation, enalty function, quantile estimation. Introduction Generalized extreme value distribution (GEVD) has been used widely as a significant modelling tool to make an inference of extreme events such as heavy rainfall, wind seed, snowfall, earthquake and other related discilines (Castillo et al., 005; Coles 00; Katz et al., 00; Zhu et al., 03). Several estimation methods for GEVD arameters were develoed in revious studies such as maximum likelihood estimation (MLE) or the method of L-moments estimation (L-ME). It is also found that MLE sometimes over-estimates the ositive value of shae arameter ξ (rightly heavy tail case) in a small samle size. Consequently, it causes large bias and variance of extreme uer quantiles. In order to solve this roblem, Coles and Dixon (999) roosed a maximum enalized likelihood estimation (MPLE) that gives a enalty for a large value of ξ by considering an exonential enalty function. Martins and Stedinger (000) considered a Beta robability density function (df) which can be treated as a rior for Bayesian aroach. The two hyer-arameters (HP) should be secified in both of Martins-Stedinger and Coles-Dixon enalty (or rior) functions. They selected a secific value for the HP based on exerimentation and exerience. Martins and Stedinger (000) used a Beta(6, 9) df. This choice of rior restricts shae arameter to a lausible range ( 0.5 ξ 0.5) consistent with rainfall and flood flows observed Corresonding author: Deartment of Statistics, Chonnam National University, 77 Yongbong-ro, Buk-gu, Gwangju 686, Korea. jsark@jnu.ac.kr Published 30 Setember 07 / journal homeage: htt://csam.or.kr c 07 The Korean Statistical Society, and Korean International Statistical Society. All rights reserved.

2 494 Youngsaeng Lee, Yonggwan Shin, Jeong-Soo Park worldwide (Huard et al., 00). The actual variance of the generalized extreme value (GEV) random variable is infinite when ξ > 0.5; in addition, the MLE of arameters is non-regular (asymtotic otimality is no more guaranteed) when ξ < 0.5. However, their selection of the HP in a Beta distribution, which lays a very imortant role in this framework, deends on subjective exeriences and exeriments that are not fully exlained. Thus, Park (005) introduced a Monte Carlo simulationbased systematic way to select the HP of the Beta df. His recommendation is Beta(.5,.5) df. However, the selection of the HP are irrelevant to the resent data. There might be no roblem if we treat the Beta df as a rior robability which was secified before the current data is given. But, if we treat the Beta df as a enalty function, it might be desirable to select the HP using given data because the data shows what HP to use. Thus, in this aer, we roose data-adative methods to select the HP of a Beta df, based on the MLE of shae arameter ξ. These methods are comared with the existing aroaches. The erformance evaluations for GEVD are conducted through a simulation study to illustrate that such a Beta enalty with otimally selected HP roduces desirable quantile estimates. Section describes the GEVD, estimations methods and enalty functions. The roosed methods are resented in Section 3. Simulation study to evaluate the erformance of the roosed methods is given in Section 4. Real data examles with Blackstone river flood discharge rate and Korean heavy rainfall are rovided in Section 5. Discussion and conclusion are given in Section 6.. Generalized extreme value distribution and arameter estimation The cumulative distribution function of the GEVD is as follows (Choi, 05; Coles, 00): [ ] F(x; µ, σ, ξ) = ex ξ (x µ) + ξ, if ξ 0, (.) σ for + ξ(x µ)/σ > 0, where µ, σ > 0 and ξ are location, scale and shae arameter, resectively. The case for ξ = 0 in (.) is well known as the Gumbel distribution. By inverting (.), quantiles of GEVD are given by x = µ σ ξ [ { log()} ξ ], for ξ 0. (.) Estimates of x are obtained after substituting the estimates of (µ, σ, ξ) into (.) for various -values. Regarding the range of ξ, it is reorted that ξ usually lies between 0.5 and 0.5 in hydrological ractice. In addition, the GEVD has finite variance when ξ < 0.5 and is regular when ξ > 0.5. Thus ξ is confined between 0.5 and 0.5 in Martins and Stedinger (000) and Park (005) while it is extended to near.0 in Coles and Dixon (999) (Figure )... Maximum likelihood estimation Under the assumtion that the observations x, x,..., x n are indeendent variables having the GEVD, the negative log-likelihood function of (µ, σ, ξ) is ( l(µ, σ, ξ) = n ln σ + + ) n [ ln + ξ(x ] i µ) + ξ σ i= n i= [ + ξ(x ] ξ i µ), (.3) σ rovided that + ξ((x i µ)/σ) > 0 for i =,..., n. The MLE of µ, σ and ξ can be obtained by minimizing (.3). We have to use a numerical otimization algorithm such as Newton tye otimizer

3 Penalized estimation for the generalized extreme value distribution 495 Figure : Penalty functions of Park, Martins-Stedinger, and Coles-Dixon. since no exlicit minimizer is available in minimizing (.3). In this study, R ackage ismev was used to obtain the MLE (Coles, 00)... Method of L-moments estimation The L-moments were introduced by Hosking (990) as a linear combination of exectations of order statistics. The natural estimator of L-moments based on an observed samle of data is a linear combination of the ordered data values. The r th samle L-moments (l r ) defined by Hosking (990) are unbiased estimators of the oulation L-moments. The method of L-ME obtains arameter estimates by equating the first k samle L-moments to the corresonding oulation quantities. The L-ME of GEVD are given as (Hosking et al., 985; Zhu et al., 03): ˆµ = l ˆσˆξ ˆσ = { Γ ( + ˆξ )}, (.4) l ˆξ ( ) ˆξ Γ ( ), (.5) + ˆξ ˆξ = c c, (.6) where c = /(3 + ˆτ 3 ) log()/log(3), and ˆτ 3 is the L-skewness defined as ˆτ 3 = l 3 /l from the samle. It is known that the L-ME works better than the MLE for small samle size. Moreover it is less sensitive to outlier (Hosking and Wallis, 997). These L-moments and L-ME have been used widely in many research fields including meteorology, civil engineering, and hydrology (for examle, Busababodhin et al., 06; Meshgi and Khalili, 009; Murshed et al., 04; Zhu et al., 03). We used R ackage lmom develoed by Hosking (05) to calculate the samle L-moments and the L-ME of GEVD.

4 496 Youngsaeng Lee, Yonggwan Shin, Jeong-Soo Park.3. Maximum enalized likelihood estimation For the small samle size, the MLE sometimes gives oor erformance and over-estimates the large ositive value of ξ severely. Consequently it causes large bias and variance of extreme uer quantiles. In order to solve this roblem, Coles and Dixon (999) and Martins and Stedinger (000) roosed to use enalty functions on the ositive value of ξ. The enalized negative log-likelihood to be minimized to obtain the MPLE is l en (µ, σ, ξ) = ln(l(µ, σ, ξ)) + ln((ξ)), (.7) where (ξ) is a enalty function on ξ. Coles and Dixon (999) roosed the following enalty function;, if ξ 0, { ( ) α } (ξ) = ex λ ξ, if 0 < ξ <, 0, if ξ, (.8) for non-negative values of α and λ. For the HP, Coles and Dixon (999) suggested to use the combination α = and λ = (Figure ). For this function, we call it Coles-Dixon (CD) enalty as an abbreviation. Martins and Stedinger (000) roosed the following enalty function, a Beta(α, β) df on ξ between 0.5 and 0.5: (ξ) = (0.5 + ξ)α (0.5 ξ) β, (.9) B(α, β) where B(α, β) = Γ(α)Γ(β)/Γ(α + β) is the beta function. They chose α = 9 and β = 6 based on rior hydrological information and exeriments. For this function, we call it MS enalty as an abbreviation. Park (005) recommended to use α =.5 and β =.5. Yoon et al. (00) considered a full Bayesian aroach for the selection of HP. We denote the MPLEs using CD, MS, and Park enalties as the MPLE-CD, MPLE-MS, and MPLE-P, resectively. 3. Proosed aroach 3.. Selection of HP based on distance to ξ estimator In order to select the HP (α, β) in the Beta df on ξ between 0.5 and 0.5 as in (.9), we considered the distance between the estimator (MLE for the first time) and MPLE of ξ. The roosed method is: Method SHM (selection of the HP based on the MLE): Ste. Comute the MLE of GEV arameters, and denote it ˆξ M. Ste. Find the (α, β) which minimize the distance ˆξ M ˆξ(α, β), where ˆξ(α, β) is the MPLE for given (α, β). Denote such selected HP as (α, β ). The estimator is now obtained by the MPLE with a Beta(α, β ) enalty function. In this method, we still resect the MLE from data but restrict ˆξ to be in [ 0.5, 0.5] and to follow a Beta distribution. In minimizing the distance between MLE and MPLE of ξ in Ste, a grid search was alied by

5 Penalized estimation for the generalized extreme value distribution 497 β α Figure : Distribution of selected values (α, β ) from the selection of HP based on the maximum likelihood estimator (SHM) method for ξ = 0.4 and for samle sizes n = 30. changing α and β from to 4 by increment. Then, a finer grid search is erformed around the coarsely found otimal set of (α, β ). That is, the distance comutation over the grids of (α, α 0.5, α, α + 0.5, α + ) (β, β 0.5, β, β + 0.5, β + ) are tried. We refer this method as a selection of the HP based on the MLE, or SHM in abbreviation. Figure shows distributions of selected values (α, β ) from the SHM method. It was calculated for 000 random samles generated from GEVD for ξ = 0.4 and for samle sizes n = 30. The value (, 0) was selected the most often. Figure 3 shows robability density lot of ˆξ obtained by the methods considered in this aer. It was calculated for,000 random samles generated from GEVD for ξ = 0.4 and for samle sizes n = 30. It was drawn using the function density in the R rogram in which a Gaussian kernel density estimation method is imlemented. The legends beta(.5,.5), beta(6, 9), and roosed in this figure stand for the MPLE with Park, Martins-Stedinger, and the SHM method, resectively. Figure 3 ahows that MPLE-P and MPLE-MS under-estimate severely. MLE has big variance. The roosed methods work well and has smaller variance than MLE. 3.. Bootstra-based selection of the hyerarameters We will obtain the samle distribution of ˆξ by using bootstra samles, and then find the (α, β) which minimize the distance between the distribution of Beta(α, β) and the samle distribution of ˆξ. For a given data set, we obtain B MLEs of ξ from B bootstra samles. The relative frequency is then calculated for each category. Here the number (K) of categories and the width of category are automatically selected by hist function in R software. Usually K is chosen between 0 and 5. Only the estimates ˆξ in [ 0.5, 0.5] are used when comuting relative frequencies. That means that the estimates outside [ 0.5, 0.5] are eliminated; subsequently, the total number of the remaining estimates is then used as a denominator in comuting relative frequencies. This is done to make a set of relative frequencies form a kind of robability mass function inside the interval [ 0.5, 0.5].

6 498 Youngsaeng Lee, Yonggwan Shin, Jeong-Soo Park Density of estimated xi (xi=0.4 n=30) Probability M beta(.5,.5) beta(6,9) roosed MLE M M M M Figure 3: Probability density lot of ˆξ for ξ = 0.4 and for samle sizes n = 30. The vertical line reresents for the true value of ξ. The legends Pbeta(.5,.5), Pbeta(6, 9), and roosed stand for the maximum enalized likelihood estimation with Park, and Martins-Stedinger, and the SHM (selection of hyerarameters based on the maximum likelihood estimator) method, resectively. ξ^ Method BSHM (bootstra based selection of the HP using the MLE): After the above comutation, the following measure of discreancy is minimized with resect to (α, β); Q(α, β) = K k= [ (ˆξ (k) ; α, β ) ˆ f (ˆξ (k) )], (3.) where (ˆξ (k) ; α, β) is, as in (.9), the df of Beta(α, β) at the center oint of k th category, and ˆ f (ˆξ (k) ) is the relative frequency of ˆξ at k th category. A grid search that is the same as the algorithm in the above subsection is used. Because of comutational costs, we set B = 00. We call this method a bootstra based selection of the HP using the MLE, or BSHM in short Selection of the hyerarameters minimizing rediction squared error As an analogy to the selection of the smoothing arameter, we consider the following rediction squared error (se); se(α, β) = n [ ] ˆx(i) (α, β) x (i), (3.) n i= where x (i) is the i th order statistic from the original data, and ˆx (i) (α, β) is the quantile estimate for the lotting osition i:n = (i.35)/n, as recommended by Hosking et al. (985) for the GEVD. This is obtained by lugging the MPLE with HP (α, β) into the quantile function (.). The HP which minimize se(α, β) are selected. A grid search that is the same as the algorithm in the above subsection is used with B = 00. We call this method as a selection of the HP using the se criterion, or SHPSE in abbreviation. Table rovides a descrition on the abbreviated names of the estimation methods considered in this aer.

7 Penalized estimation for the generalized extreme value distribution 499 Table : Descrition on the abbreviated names of the estimation methods considered in this aer Methods Descrition Details SHM Selection of hyerarameters using the MLE Subsection 3. BSHM Bootstra based selection of hyerarameters using the MLE Subsection 3. SHPSE Selection of hyerarameters minimizing rediction squared error Equation (3.) MPLE-P Maximum enalized likelihood estimation using Park s enalty Beta(.5,.5) df MPLE-MS Maximum enalized likelihood estimation using Martins-Stedinger s enalty Beta(9, 6) df MPLE-CD Maximum enalized likelihood estimation using Coles-Dixon s enalty Equation (.8) L-ME Method of L-moments estimation Subsection. MLE Maximum likelihood estimation Subsection. df = robability density function. Table : The bias of ξ estimators obtained by roosed and other estimation methods as the true ξ ranges from 0.49 to 0.49 with samle size of 30 Methods Sum SHM BSHM SHPSE MPLE-P MPLE-MS MPLE-CD L-ME MLE Sum is obtained by the summation of the absolute biases. Table rovides descritions on the abbreviated names of methods. 4. Monte Carlo simulation In order to evaluate the erformance of the roosed method, we comared the roosed estimator to the other estimators by Monte Carlo simulation study. For the comarison of accuracy, we calculated the bias and the root mean squared error (RMSE) of ξ estimators; Bias(ξ) = M M (ˆξ i ξ ), (4.) i= and RMSE(ξ) = M M (ˆξ i ξ ). (4.) i= We have generated M =,000 random samles from GEVD for samle sizes n = 30, 60 and for given shae arameters ξ ( 0.5, 0.5). Other arameters are fixed as location µ = 0 and scale σ =, because these arameters are location and scale equivariant. Table and Figure 4 show the results of the bias of ξ estimates (Bias(ξ)). There are negative biases both in the MLE for negative ξ and in the L-ME for ositive ξ. The MPLE-P and MPLE-MS are good only for near zero ξ, but are worst for ξ far from zero. The sizes of biases for ξ = 0.4 are larger than those for ξ = 0.. Based on the summation of the absolute biases (sum), BSHM method works the best and SHM and MLE work well. MPLE-MS is the worst in general. Table 3 and Figure 5 show the results of the RMSE of ξ estimates (RMSE(ξ)). The MLE and L-ME work similarly while those are not good for ξ far from zero. SHM and BSHM and methods work well for general ξ, secially

8 500 Youngsaeng Lee, Yonggwan Shin, Jeong-Soo Park Figure 4: The bias of ξ estimators obtained by four estimation methods (MLE, L-ME, MPLE-MS, and SHM) with samle size of 30. Descrition on the abbreviated names of methods is given in Table. Table 3: The root mean squared error of ξ estimators obtained by roosed and other estimation methods as the true ξ ranges from 0.49 to 0.49 with samle size of 30 Methods sum SHM BSHM SHPSE MPLE-P MPLE-MS MPLE-CD L-ME MLE Table rovides descritions on the abbreviated names of methods. for ξ far from zero. Note that SHPSE work badly. MPLE-MS works well only for ξ between 0.0 and 0.3, but worst for ξ far from zero. The MPLE-P works well for ξ between 0.3 and 0.0, and is the best in the sense of the summation of the RMSEs (sum). From the results based on both bias and mean squared error criterion, we would conclude that the SHM and BSHM work better than the other estimation methods. 5. Real data examles 5.. Blackstone River data We considered the data of the annual flood discharge rates of the Blackstone River at Woonsocket, RI, USA, given in Pericchi and Rodriguez-Iturbe (985), and Mudholkar and Hutson (998). This data is for a eriod of 37 years with unit of f t 3 /s. To judge the overall goodness-of-fit, we use the Kolmogorov-Smirnov (K-S) statistic and the average scaled absolute error (ASAE) (Castillo et al., 005), ASAE = n x (i) ˆx (i), (5.) n x (n) x () i=

9 Penalized estimation for the generalized extreme value distribution 50 Figure 5: The root mean squared error (RMSE) of ξ estimators obtained by four estimation methods (MLE, L-ME, MPLE-MS, and SHM) with samle size of 30. Table rovides descritions on the abbreviated names. Table 4: Result of the analysis and comarison of the estimation methods for the Blackstone river data ˆµ ˆσ ˆξ ASAE CM CIU CIL CI range SHM BSHM SHPSE MPLE-P MPLE-MS MPLE-CD L-ME MLE Estimates of µ, σ, and ξ, ASAE, CM, CIU, and CIL, and the range of CI. Table rovides descritions on the abbreviated names of methods. ASAE = average scaled absolute error; CM = Cramer-von Mises Statistic; CIU = uer bounds of 95% confidence interval of ξ; CIL = lower bounds of 95% confidence interval of ξ. where x (i) are the ascendingly ordered observations, and ˆx (i) is obtained from the quantile function (.) with the estimates lugged in the equation for the lotting osition i:n = (i 0.35)/n, as recommended by Hosking and Wallis (997) for the GEVD. Table 4 rovides the estimation of µ, σ, and ξ along with ASAE, Cramer von Mises statistic (CM), uer and lower bounds of 95% confidence interval (CIU and CIL) of ξ), and the range of confidence interval (CI range). To comute the confidence intervals of ˆξ, the rofile likelihood aroach was used for MLE based methods, while the bootstra (B =,000) method was used for L-ME. The rofile likelihood of GEV for ξ 0 is defined as (Coles, 00) L (ξ 0 ; x) = max µ,σ ξ 0 L (µ, σ, ξ 0 ; x), (5.) where L( ) is the likelihood function for given data x = x,..., x n. The L-ME works well for criteria of ASAE and CM, while BSHM works well for CM criterion and CI range. Note that the CI range of L-ME is the worst.

10 50 Youngsaeng Lee, Yonggwan Shin, Jeong-Soo Park Table 5: Estimates of shae arameter ξ, ASAE, and K-S statistic for the annual maximum of daily rainfall data in twelve Korean sites, for five estimation methods Location MPLE-P MPLE-MS SHM (α, β ) BSHM (α, β ) MLE ˆξ Daegwallyeong ASAE (4, ) 0.07 (0, 4) 0.06 n = 4 K-S ˆξ Daejeon ASAE (4, 4) 0.07 (6, 4) 0.03 n = 44 K-S ˆξ Pohang ASAE (, 4) 0.08 (8, 4) 0.07 n = 64 K-S ˆξ Gunsan ASAE (4, ) (4, ) n = 45 K-S ˆξ Wando ASAE (4, 4) (4, ) n = 4 K-S ˆξ Seogwio ASAE (, 4) 0.07 (6, 4) 0.07 n = 5 K-S ˆξ Buyeo ASAE (4, 4) 0.0 (8, ) 0.09 n = 4 K-S ˆξ Imsil ASAE (4, ) (4, ) n = 4 K-S ˆξ Jeongeu ASAE (, 4) 0.03 (8, 4) 0.09 n = 43 K-S ˆξ Haenam ASAE (, 0) 0.07 (4, 4) 0.0 n = 4 K-S ˆξ Goheung ASAE (4, 4) 0.09 (6, 4) 0.04 n = 4 K-S ˆξ Yeongdeok ASAE (, 8) 0.05 (, ) 0.03 n = 40 K-S No. of the best ASAE No. of the best K-S The selected values (α, β ) from SHM and BSHM are rovided. Table rovides descritions on the abbreviated names of methods. ASAE = average scaled absolute error; K-S = Kolmogorov-Smirnov. 5.. Korean heavy rainfall data In this section, we comared the erformance of the roosed method with existing MPLEs using Korean heavy rainfall data. Annual daily maximum reciitation (unit: mm) record are considered for 75 weather stations which has at least 0 years observation (Korea Meteorological Administration, 06). The stations with relatively large ositive MLE value of ˆξ (> 0.5) are selected. Table 5 shows the stations. The estimates of ξ, ASAE, and K-S statistic for various methods are given in Table 5 for the stations. The selected values (α, β ) from SHM and BSHM are rovided. Based on the number of the best at the bottom of Table 5, the SHM method works best comared to other methods.

11 Penalized estimation for the generalized extreme value distribution Discussion In this aer, we restrict the range of shae arameter to be in ( 0.5, 0.5) because the variance of GEV random variable is infinite when ξ > 0.5 and the MLE of arameters is non-regular (asymtotic otimality is no more guaranteed) when ξ < 0.5. Therefore, our concern was concentrated on Beta df only. One can release this restriction to (.0,.0). For this case, one can use the Coles-Dixon (CD) enalty function (.8). Based on our exerimental exerience on the CD enalty function, the HP α =, λ = work well. If a data-adative selection of HP (α, λ) in CD enalty function is recommended, one can consider a criterion like a se(α, λ) similar as in (3.) to choose the best α and λ from the data. In Ste of the SHM method, one can consider refining the grid search using the increment 0.5 or 0.. That may be able to find the HP that makes the density function very narrow and concentrate to the MLE, so that the MPLE be near the MLE. However, to our exeriments showed that refining the grid did not imrove the results obtained using the coarse grid as reorted in this aer. One can consider a criterion with the hel of the method of L-ME. For examle, the distance between oulation L-moments (calculated from the MPLE) and samle L-moments, i.e., τ 3 t 3 can be used to select the HP, where τ 3 and t 3 are oulation L-skewness and samle L-skewness, resectively. The HP minimizing the above distance is then selected. We exect this MPLE, guided by L-ME, to work well because L-ME works better than the MLE for small samle size. It is a toic for future research. We tried the following bootstra based criterion, using the se as in Efron and Tibshirani (993); se (α, β) = n n [ ] ˆx (i) (α, β) x (i), (6.) i= where ˆx (i) (u, v) is the same as the above but is obtained from the MPLE for a bootstra samle x i, i =,,..., n. Averaging this quantity se (u, v) over B bootstra samles rovides an estimate of the exected rediction squared error. We denote this average by se(u, v). The HP minimizing se(u, v) can be selected; however, the disadvantage of this bootstra aroach was not reorted it in this aer because it a comutational cost, and it did not work well in our brief simulation study. 7. Conclusion A data adative method to select the HP of Beta df on the shae arameter of GEVD is resented in a MPLE framework that enables the data to tell us what HP to use. For given data, the otimal HP is obtained from the minimum distance between the MLE and MPLE. The erformance evaluation exeriments for GEVDs by Monte Carlo simulation show that the roosed estimators often work well. Blackstone river data and Korean heavy rainfall data are fitted to illustrate the usefulness of the roosed methods. Our recommendation is to use the SHM method among some estimations considered in this study. The details of the SHM are described in Subsection 3.. A comuter rogram for the roosed methods develoed using R software is available uon request from the corresonding author. Acknowledgments This work was suorted by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 06RAB40458). Lee s work was suorted by Basic Science

12 504 Youngsaeng Lee, Yonggwan Shin, Jeong-Soo Park Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (07RA6A3A0385). References Busababodhin P, Seo YA, Park JS, and Kumhon B (06). LH-moment estimation of Wakeby distribution with hydrological alications, Stochastic Environmental Research and Risk Assessment, 30, Castillo E, Hadi AS, Balakrishnan N, and Sarabia JM (005). Extreme Value and Related Models with Alications in Engineering and Science, Wiley-Interscience, New Jersey. Choi H (05). A note on the deendence conditions for stationary normal sequences, Communications for Statistical Alications and Methods,, Coles S (00). An Introduction to Statistical Modeling of Extreme Values, Sringer, New York. Coles SG and Dixon MJ (999). Likelihood-based inference for extreme value models, Extremes,, 5 3. Efron B and Tibshirani RJ (993). An Introduction to the Bootstra, Chaman & Hall/CRC, Boca Raton. Hosking JRM (990). L-moments: analysis and estimation of distributions using linear combinations of order statistics, Journal of the Royal Statistical Society Series B (Methodological), 5, Hosking JRM (05). Package lmom, version.5, Retrieved June 9, 06, from: htts://cran.rroject.org/web/ackages/lmom/lmom.df Hosking JRM and Wallis JR (997). Regional Frequency Analysis: An Aroach based on L-Moments, Cambridge University Press, Cambridge. Hosking JRM, Wallis JR, and Wood EF (985). Estimation of the generalized extreme-value distribution by the method of robability weighted moments, Technometrics, 7, 5 6. Huard D, Mailhot A, and Duchesne S (00). Bayesian estimation of intensity-duration-frequency curves and of the return eriod associated to a given rainfall event, Stochastic Environmental Research and Risk Assessment, 4, Katz RW, Parlange MB, and Naveau P (00). Statistics of extremes in hydrology, Advances in Water Resources, 5, Korea Meteorological Administration (06). Annual daily maximum reciitation record for 75 weather stations, Retrieved May, 07, from: htt:// Martins ES and Stedinger JR (000). Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data, Water Resources Research, 36, Meshgi A and Khalili D (009). Comrehensive evaluation of regional flood frequency analysis by L- and LH-moments. I. A re-visit to regional homogeneity, Stochastic Environmental Research and Risk Assessment, 3, Mudholkar GS and Hutson AD (998). LQ-moments: analogs of L-moments, Journal of Statistical Planning and Inference, 7, Murshed MS, Seo YA, and Park JS (04). LH-moment estimation of a four arameter kaa distribution with hydrologic alications, Stochastic Environmental Research and Risk Assessment, 8, Park JS (005). A simulation-based hyerarameter selection for quantile estimation of the generalized extreme value distribution, Mathematics and Comuters in Simulation, 70, Pericchi LR and Rodriguez-Iturbe I (985). On the statistical analysis of floods. In Atkinson AC,

13 Penalized estimation for the generalized extreme value distribution 505 Fienberg SE (Eds), A Celebration of Statistics (. 5 54), Sringer, New York. Yoon S, Cho W, Heo JH, and Kim CE (00). A full Bayesian aroach to generalized maximum likelihood estimation of generalized extreme value distribution, Stochastic Environmental Research and Risk Assessment, 4, Zhu J, Forsee W, Schumer R, and Gautam M (03). Future rojections and uncertainty assessment of extreme rainfall intensity in the United States from an ensemble of climate models, Climatic Change, 8, Received June, 07; Revised July 30, 07; Acceted August, 07