Uncertainty of quantile estimators using the population index flood method

Transcription

1 WATER RESOURCES RESEARCH, VOL. 39, NO. 8, 1206, doi: /2002wr001594, 2003 Uncertainty of quantile estimators using the population index flood method O. G. B. Sveinsson 1 Department of Civil Engineering, Colorado State University, Fort Collins, Colorado, USA J. D. Salas Department of Civil Engineering, Colorado State University, Fort Collins, Colorado, USA D. C. Boes Department of Statistics, Colorado State University, Fort Collins, Colorado, USA Received 17 July 2002; revised 7 January 2003; accepted 31 March 2003; published 8 August [1] The population index flood (PIF) method is an analytical model that has been recently suggested for regional frequency analysis. In this paper, explicit equations based on Fisher s information are derived for estimating the standard error of at-site quantile estimators for two regional PIF methods utilizing the generalized extreme value distribution with maximum likelihood estimation. These explicit equations are used to calculate the asymptotic gain in using regional frequency analysis as opposed to single site frequency analysis. Simulation experiments for different sized regions and different values of the shape parameter show that the suggested methods for estimating the standard error of at-site quantile estimators give values close to the actual or true values. In addition, similar simulation experiments are also used to test the accuracy of a newly suggested procedure for estimating the standard errors of at-site quantile estimators for the Hosking and Walls regional index flood method. The results of the simulations indicate that these estimated standard errors can in some cases give unreliable results. In general, this study shows that the PIF models are a useful addition to existing regional frequency analysis models. Their analytic structure, which is not present in other regional models, has important theoretical and practical implications. INDEX TERMS: 1821 Hydrology: Floods; 1854 Hydrology: Precipitation (3354); 1860 Hydrology: Runoff and streamflow; 1894 Hydrology: Instruments and techniques Citation: Sveinsson, O. G. B., J. D. Salas, and D. C. Boes, Uncertainty of quantile estimators using the population index flood method, Water Resour. Res., 39(8), 1206, doi: /2002wr001594, Introduction [2] Regional frequency analysis of hydrologic data has become a standard practice for improving the estimation of event quantiles at sites with short records and estimating quantiles at ungauged sites. Various regional procedures have been proposed in literature [e.g., Kuczera, 1982; Cunnane, 1988; Gabriele and Arnell, 1991; Gupta et al., 1994; Stedinger and Lu, 1995]. Among the various methods a widely used regional analysis procedure is the so-called index flood method introduced by Dalrymple [1960]. In the index flood method a statistically homogeneous region is defined as that region where the underlying data (at each site) have identical frequency distribution apart from a scale factor (the index flood). While Dalrymple [1960] assumed a Gumbel world and estimated the index flood by the 2.33 year event from the empirical frequency curve at each site, the standard today is to estimate the index flood at each site 1 Now at International Research Institute for Climate Prediction, Columbia University, New York, USA. Copyright 2003 by the American Geophysical Union /03/2002WR SWC 2-1 by the corresponding at-site sample mean [e.g., Natural Environmental Research Council (NERC ), 1975; Wallis, 1980; Hosking et al., 1985a; Hosking and Wallis, 1997; Rao and Hamed, 2000]. The introduction of probability weighted moments (PWMs) by Greenwood et al. [1979] and Landwehr et al. [1979] led to simple and efficient procedures for estimating parameters of many probabilistic models and sparked a new interest in the index flood method. Subsequently, Hosking [1986, 1990] introduced L moments as a linear combination of PWMs. [3] A complete index flood procedure based on L moments has been introduced by Hosking and Wallis [1993, 1997] and a particular attention has been paid to the generalized extreme value (GEV) distribution as the underlying regional distribution [e.g., Hosking et al., 1985a, 1985b; Lettenmaier et al., 1987; Chowdhury et al., 1991; Lu and Stedinger, 1992a, 1992b; Sveinsson et al., 2001]. In addition the effects of heterogeneous regions have been widely studied [e.g., Lettenmaier et al., 1987; Hosking and Wallis, 1988; Stedinger and Lu, 1995; Hosking and Wallis, 1997]. Furthermore, a detailed literature review on the various aspects related to the index flood method has been made by Stedinger and Lu [1995]. Hosking and Wallis

2 SWC 2-2 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF [1988] concluded that regional frequency analysis is more accurate than at-site analysis even when both heterogeneity and inter-site dependence are present and the form of the underlying regional distribution is misspecified. In the subsequent text we refer as the Hosking and Wallis scheme (HW scheme) as that where the index flood approach and the L moments and L moment ratios are utilized as the basis of the regional analysis methodology. [4] Some authors have expressed reservations about the physical basis for the index flood method [e.g., Gupta and Dawdy, 1994; Gupta et al., 1994; Gupta and Dawdy, 1995; Robinson and Sivapalan, 1997; Blöschl and Sivapalan, 1997]. Estimating the index flood by the at-site sample mean assumes that that the moment ratios (coefficient of variation, skewness, etc.) and the L moment ratios are the same for all sites in the statistically homogeneous region. This is equivalent to assuming a simple scaling property of probability distributions across sites [Gupta et al., 1994]. Gupta and Dawdy [1994, 1995] argue for multiscaling theories, and Robinson and Sivapalan [1997] and Blöschl and Sivapalan [1997] suggest that the coefficient of variation should not be treated as constant but vary as a function of the catchment area. Nevertheless, the standard before applying the index flood method is to test if any site within the region is statistically different from the other sites [e.g., Chowdhury et al., 1991; Hosking and Wallis, 1993, 1997; Sveinsson et al., 2002b]. Other authors have expressed concerns about estimation of the index flood by the at-site sample mean [e.g., Stedinger, 1983; Sveinsson et al., 2001, 2002b] because the data after being indexed by the sample mean become correlated and in addition are bounded from above by the sample size if the underlying distribution is assumed positive [Sveinsson et al., 2001]. Despite these limitations, simulation experiments based on the GEV [Sveinsson et al., 2001] comparing the HW scheme against an alternative method dubbed as the population index flood (PIF) method, suggest that the HW scheme is quite robust, with the PIF method performing better in some cases and the HW scheme in others. The study presented in this paper is an extension of the study reported by Sveinsson et al. [2001]. [5] The population index flood (PIF) method has recently been suggested as an alternative to the traditional index flood procedures for regional frequency analyzes of extreme hydrologic events [Sveinsson et al., 2001]. In the PIF method the index flood (or the indexing function) at each site is taken to be a function of certain unknown at-site population quantities (e.g., population mean) and as a result, the homogeneity of the region is embedded in the structure of the parameter space of the underlying distribution model. More precisely, depending on the regional distribution model and the type of the indexing function, some of the distribution parameters are site-specific, while other parameters are common for all sites within the statistically homogeneous region. Because of the analytical framework underlying the PIF regional method, the method of maximum likelihood can be used for parameter estimation and in addition, when regularity conditions are satisfied, the variance-covariance matrix of the maximum likelihood estimators can be used to estimate the standard error of estimated quantiles. Some of the concepts underlying the PIF method have been discussed in other studies [e.g., Boes et al., 1989; Heo et al., 2001a, 2001b] assuming the Weibull model and by Sveinsson et al. [2002a] for the upper order statistics from a Pareto model. In this paper, newer and more complete formulations are presented. [6] Asymptotic and sample variances of quantile estimators are estimated for the PIF method based on the GEV distribution with maximum likelihood estimation. This is done using a formula of the Cramer Rao lower bound (CRLB) for the variance of unbiased estimators and the estimated asymptotic and sample variance-covariance information matrix of the maximum likelihood GEV parameter estimators. The estimated asymptotic and sample variances of the quantile estimators are compared using simulation experiments for different sized regions and two types of indexing functions: (1) sample data at each site are indexed by dividing them by the at-site population mean; and (2) sample data at each site are indexed by standardizing them using at-site population statistics. In addition, the referred simulation experiments are applied to test the accuracy of methods and procedures suggested by De Michele and Rosso [2001] for estimating the standard error of at-site quantile estimators in the well-known Hosking and Wallis regional estimation scheme [Hosking and Wallis, 1997]. Lastly, the proposed PIF regional method is compared with the referred HW scheme using extreme precipitation data from northeastern Colorado. 2. Basic Notation and Formulation [7] As a way of introducing the reader with the notation and concepts used herein, the generalized extreme value (GEV) distribution for a single site is first defined with parameters estimated based on maximum likelihood. Furthermore, methods for estimating the uncertainty of quantile estimators are introduced. All methods are presented in such a way to be easily extended to the multisite regional case. Algorithms for estimation of GEV parameters by maximum likelihood are presented by Prescott and Walden [1980] and Hosking [1985] for the single site case, and by Prescott and Walden [1983] for the single site case with censored data GEV Distribution and Estimation of Parameters by Maximum Likelihood [8] A random variable X is GEV distributed with parameters a (location), b (scale), and k (shape) if its probability density function (pdf) is given by f X ðþ¼ x 1 b 1 k b ðx aþ 1 k 1 ( exp 1 k ) 1=k b ðx aþ where 1 < a < 1, b > 0, and 1 < k < 1. The range of X is: a + b/k < x < 1 for k < 0, and 1 < x < a + b/k for k > 0. The mean and the variance are E[X] =a + b [1 (1 + k)]/k for k > 1, and Var(X)=b 2 [ (1 + 2k) 2 (1 + k)]/ k 2 for k > 1/2, respectively. For k = 0 the GEV becomes the Gumbel distribution. [9] The log likelihood function of a random sample X 1,..., X n of size n from the GEV distribution is given by ln LðQ; xþ ¼ Xn ln f X ðx i Þ where x =[x 1,..., x n ] is the observed sample vector and Q =[a, b, k] 2 is the parameter vector of f X (x), in which ð1þ ð2þ

3 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF SWC 2-3 is the corresponding parameter space as specified above. The maximum likelihood estimates (ML estimates) are found by taking the gradient of the log likelihood, setting it equal to zero, and solving for the parameters, that is r ln LðQ; xþ ¼ 0 ð3þ where r =[D 1, D 2, D 3 ] and D k is the partial derivative with respect to the kth element in Q, for example D There is no explicit solution for Q in equation (3), thus an optimization algorithm such as the Newton-Raphson iteration can be used to obtain a numerical solution for the ML estimates ^Q ¼ ^a; ^b; ^k. The Newton-Raphson iteration is utilized herein, where ^Q T iþ1 ¼ ^Q T i h þ SI x; ^Q i 1½r i ln Lð^Qi ; xþš T ð4þ represents the new parameter estimates at iteration i + 1,the superscript T denotes transpose, and SIðx; ^QÞ is Fisher s sample information matrix. The latter is equal to the negative Hessian matrix of the log likelihood 2 3 SI x; ^Q D 11 ln L D 21 ln L D 31 ln L ¼ 4D 21 ln L D 22 ln L D 32 ln L 5 D 31 ln L D 32 ln L D 33 ln L with D ij = D i D j. Furthermore, ½SIðx; QÞŠ 1 is the sample variance-covariance matrix of ^Q. Throughout this paper the iterative procedure is repeated until the relative change in all parameters is less than 0.01%, that is maxjð^q iþ1 ^Q i Þ=^Q iþ1 j < 0:0001: 2.2. Uncertainty of GEV Quantile Estimators [10] The qth quantile of the GEV distribution of equation (1) is Q¼^Q ð5þ xðqþ ¼ a þ b ½1 ln q k ð Þk Š ð6þ Under regularity conditions [Mood et al., 1974] the Cramer- Rao lower bound (CRLB) is a lower bound for the variance of unbiased estimators, and the CRLB is also the asymptotic variance (AVar) of maximum likelihood estimators. The CRLB for the variance of unbiased estimators of x(q) is CRLBðxðqÞÞ ¼ rxðqþ½eiðqþš 1 ½rXðqÞŠ T ð7þ where EI(Q) is Fisher s expected information matrix given by EIðQÞ ¼ E½SIðX; QÞŠ ð8þ Furthermore [EI(Q)] 1 is the asymptotic variance-covariance matrix of the ML-estimator of Q. In the case of the GEV distribution the regularity conditions are satisfied if the diagonal elements of EI(Q) exist and are positive, which is the case when k < 1/2. [11] Given the population parameters Q, the theoretical CRLB for unbiased quantile estimators is given by equation (7) and it is the AVar of the ML-estimator of x(q). In most practical cases the population parameters Q are unknown; in which case equation (7), evaluated at Q ¼ ^Q, gives an estimate of the AVar of the ML-estimator ^x ðqþ. Another alternative is to use the sample information matrix SIðx; ^QÞ instead of EIð^QÞ, in which case equation (7) would give a type of a sample asymptotic variance, here dubbed SVar, of the ML-estimator ^x ðþ. q Prescott and Walden [1980, 1983] compared the simulated variance of estimators of the shape parameter k, Varð^k Þ, with the CRLB(k), AVarð^k Þ, and SVarð^k Þ for various values of k and a sample size n = 100. Their results showed that SVarð^k Þ was the closest to Varð^k Þ, and that CRLB(k) and AVarð^k Þ tended to underestimate Varð^k Þ. 3. Uncertainty Based on the Population Index Flood Method [12] The population index flood (PIF) method is an analytical model for regional frequency analysis. A detailed description of the PIF method and models for many commonly used two- and three-parameter distributions is given by Sveinsson et al. [2001]. Instead of using a sample property as the index flood (as is commonly done in the traditional index flood approach) in the PIF method the index flood is taken to be a function of unknown population parameters at each site, and the homogeneity of the region under consideration is embedded in the structure of the parameter space of the underlying distribution model. In the work of Sveinsson et al. [2001] two types of indexing functions are considered and x j ðqþ ; j ¼ 1;...; m ð9þ m j x j ðqþ m j ; j ¼ 1;...; m ð10þ s j where m is the number of independent sites in the region, x j (q) istheqth population quantile at site j, and m j and s j are respectively the population mean and standard deviation at site j. We will refer to indexing based on equation (9) as PIF 1, and indexing based on equation (10) as PIF 2. A statistically homogeneous region is defined as a region where either equation (9) or equation (10) (depending on the method) does not depend on j. That is, for both the PIF 1 and the PIF 2 methods a statistically homogeneous region implies that the skewness, kurtosis, and all higher order moment ratios are the same for all sites in the region. In addition, for the PIF 1 method the coefficient of variation (s j /m j ) is also the same for all sites within the region. Hence PIF 1 is more restrictive, i.e., has fewer overall parameters, than PIF 2. That is PIF 2 can always be used instead of PIF 1 but not vice versa. For a statistically homogeneous region of m sites we will denote the sample at site j by X j1,..., X jnj for j =1,..., m, where n j is the sample size at site j PIF by Indexing by the Population Mean: PIF 1 [13] For a statistically homogeneous region of m sites, Sveinsson et al. [2001] demonstrate that equation (9) independent of j implies for the GEV distribution in equation (1) that the ratio a j /b j = g and the shape parameter k must be the same for all sites. Thus the parameter space has dimension (m + 2), where either a j or b j is estimated for each site j in the region, and g and k are estimated commonly for all sites in the region. By the invariance property of ML-estimators the two cases above are equiv-

4 SWC 2-4 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF alent. For the case where the location parameter a j is 1 estimated at each site j in the region, then for q j = a j the log likelihood is ln LðQ; x 1 ;...; x m Þ ¼ Xm j¼1 þ Xnj ( n j ln q j þ ln g ) 1 k 1 ln z z 1=k ð11þ where Q =[q 1,..., q m, g, k], x j =[x j1,..., x jnj ] is the sample vector for site j, and z =1 gk(q j x 1). The first and the second partial derivatives of the log likelihood in equation (11), needed for the Newton-Raphson procedure in equation (4), along with the first partial derivatives of the qth GEV quantile in equation (6), needed for the evaluation of the CRLB in equation (7), are given in Appendix A. [14] The elements of Fisher s expected information matrix, EI(Q), are found by taking the expected value of the sample information matrix, whose elements are given in equation (A2) in Appendix A. When k < 1/2 regularity conditions are satisfied and the nonzero elements of EI(Q) are given by " # E ¼ n j f1 þ hd½hd ðnþ 2 2 j q 2 j k2 n j E j q j gk 2 f1 þ hhd ½ ðnþ ðd þ 1Þ ðhþšg E ¼ n j q j k 3 fk ½ hd 0 ðhþ 0 ð1þš þ hhd ½ ðnþ ðhd þ 1Þ ðhþþ1šg 2 ¼ n T g 2 k 2 f1 þ hh ½ ðnþ 2 ðhþšg E ¼ n gk 3 fk ½ h 0 ðhþ 0 ð1þš þh½h ðnþ ðh þ 1Þ ðhþþ1šg 2 ¼ n T 1 þ 2k h 0 k 4 ½ ðhþ 0 ð1þ 1Š h i þ k 2 p 2 =6 þ ð1 þ 0 ð1þþ 2 þh 2 ½ ðnþ 2 ðhþš ð12þ where h =(1 k), n =(1 2k), d =(gk + 1), and n T = n n m. In addition 0 (y) is the first derivative of the gamma function with argument y, and 0 (1) is the negative Euler s constant. For example, referring to equation 7, for a region of two sites the CRLB of the qth quantile estimator at site 2, x 2 (q), is given by CRLBðx ðqþþ D ¼ 2 x D 3 x 2 ðqþ5 D 4 x 2 ðqþ D 2 x D 3 x 2 ðqþ5 D 4 x 2 ðqþ where Q =[q 1, q 2, g, k]. T 2 3 D 11 ln L 0 D 31 ln L D 41 ln L 0 D 22 ln L D 32 ln L D 42 ln L 6 7 4D 31 ln L D 32 ln L D 33 ln L D 43 ln L5 D 41 ln L D 42 ln L D 43 ln L D 44 ln L PIF by Standardizing Using Population Parameters: PIF 2 [15] For a statistically homogeneous region of m sites Sveinsson et al. [2001] demonstrate that equation (10) independent of j implies for the GEV distribution in equation (1) that the shape parameter k must be the same for all sites. Thus the parameter space has dimension (2m + 1), where a j and b j are estimated for each site j in the region, and k is estimated commonly for all sites. For q j = b 1 j the log likelihood is given by ln LðQ; x 1 ;...; x m Þ¼ Xm j¼1 ( ) n j ln q j þ Xnj 1 k 1 ln z z 1=k ð13þ where Q =[a 1,..., a m, q 1,..., q m, k], and z =1 q j k(x a j ). The first and the second partial derivatives of the log likelihood in equation (11) along with the first partial derivatives of the qth GEV quantile in equation (6) are given in Appendix B. [16] For k < 1/2 the nonzero elements of Fisher s expected information matrix, EI, ofq are given by " # E ¼ n j q 2 j h2 2 j j " j # E 2 j ¼ n jh k ¼ n jq j h k 2 n j q 2 j k2 f ðhþ h ðnþg fh½ ðhþ ðnþš k 0 ðhþg f1 þ hh ½ ðnþ 2 ðhþšg E ¼ n j q j k 3 f1 þ kh 0 ½ ðhþ 0 ð1þ 1Š þ hh ½ ðnþ ðh þ 1Þ ðhþšg 2 ¼ n T ½ ðhþ 0 ð1þ 1Š 1 þ 2k h 0 k 4 h i þ k 2 p 2 =6 þ ð1 þ 0 ð1þþ 2 þ h 2 ½ ðnþ 2 ðhþš where as before h =(1 k), n =(1 2k), and n T = n n m. For example, referring to equation 7, for a region of two sites the CRLB of the qth quantile estimator at site 2, x 2 (q), is given by 2 3T 0 D 2 x 2 ðqþ CRLBðx 2 ðqþþ ¼ D 4 x 2 ðqþ5 D 5 x 2 ðqþ D 11 ln L 0 D 31 ln L 0 D 51 ln L 0 D 22 ln L 0 D 42 ln L D 52 ln L D 31 ln L 0 D 33 ln L 0 D 53 ln L D 42 ln L 0 D 44 ln L D 45 ln L 5 D 51 ln L D 52 ln L D 53 ln L D 54 ln L D 55 ln L D 2 x 2 ðqþ D 4 x 2 ðqþ5 D 5 x 2 ðqþ where Q =[a 1, a 2, q 1, q 2, k]. ð14þ

5 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF SWC Analytical Comparison of Uncertainty of Quantile Estimators [17] A measure of efficiency of single site analysis relative to regional analysis for estimation of the qth quantile at site j is given by the ratio Reff ¼ CLRB x jðqþ regional ð15þ CLRB x j ðqþ singlesite Reff depends on the sample sizes through n T /n j for both the PIF 1 and the PIF 2 models. Furthermore for the PIF 1 method, Reff depends on k and the ratio g = a j /b j, but it does not depend on the individual values of a j and b j.on the other hand, for the PIF 2 method, Reff depends only on k. For the case when all sites within the region have the same sample size, i.e., n T /n j = m, then for both PIF 1 and PIF 2 methods Reff does not depend on j and we refer to Reffðm 1 : m 2 Þ; m 2 m 1 as the efficiency of regional analysis of m 2 sites relative to regional analysis of m 1 sites for estimation of the qth quantile at any site j in the region. [18] Typical ranges of regional parameters for hydrologic data are g 2 (1, 4) and k 2 ( 0.3, 0.1). In Figure 1 the relationship of Reff(6:1) versus q is plotted for typical values of g and k for both the PIF 1 and the PIF 2 models. The effect of g on Reff for the PIF 1 model is clearly seen in Figure 1, while g does not have any effect on Reff for the PIF 2 model. Furthermore, from Figure 1 the advantages of using regional analysis over single site analysis for estimation of lower and upper extreme quantiles are clear. In addition, the gain of using regional analysis based on the PIF 1 model is generally greater than that based on the PIF 2 model. Recall though that the PIF 1 model may not be suitable in some situations where the PIF 2 model is applicable, while the PIF 2 model is applicable in all situations where the PIF 1 model can be used. An interesting property for the PIF 2 method is observed in Figure 1, where for every k there are two quantiles where there is no gain in using regional analysis over single site analysis. Boes et al. [1989] showed that for the Weibull model there is no asymptotic gain in using regional analysis over single site analysis for the quantile. Figure 2 shows the relationship Reff(m:1) versus q for various values of m, k = 0.1, and g = 2 for both the PIF 1 and the PIF 2 models. Focusing on the tails, notice how close the curves based on 6, 12, and 36 sites are, suggesting that in practice a bigger region may not necessarily be better than a smaller region. In addition, in real world situations heterogeneity usually increases with the size of the region. Note that Stedinger and Lu [1995, Figure 6] concluded that the number of sites for estimation of the 100-year quantile estimator is not a significant factor if m 10 and n j = 20. Furthermore, their results for m = 5 do not appear to be that different from those for m = 10. Also, Hosking and Wallis [1988] concluded that adding more sites to a region becomes quickly profitless if heterogeneity is present. 5. Comparison of Uncertainty of Quantile Estimators from Simulation Experiments [19] We performed simulation experiments to compare several ways of estimating the uncertainty of quantile estimators. In the work of Sveinsson et al. [2001] a statistical homogeneous region of three sites was simulated and the bias and the mean squared error (MSE) of GEV quantile estimators was investigated using the PIF 1 GEV model with parameters estimated using maximum likelihood and probability weighted moments methods. The region was simulated for various sample sizes, g = 2, and k = 0.1. These values of g and k appear typical for extreme annual precipitation of short duration in eastern Colorado. Since regions suitable for the PIF 1 model are also suitable for the PIF 2 model, we will use the same region as in the work by Sveinsson et al. [2001], except that in our case the shape parameter k will vary. Thus the population parameters are [a 1,b 1,k 1 ] =[2, 1,k] for site 1, and [a 2, b 2, k 2 ]= [4, 2, k], and [a 1, b 1, k 3 ]=[8,4,k] forsites2and3, respectively. That is in distribution X d 2 2X 1 and X d 3 4X 1 where X j is the random variable at site j. To cover the practical range of k for hydrologic data, our simulation experiments will be made for k 2 { 0.3, 0.2, 0.1, 0.1}. The probability for any of the three sites to have negative flows is , , , and.0020, for k = 0.3, k = 0.2, k = 0.1, and k = 0.1, respectively. Thus these parameters can be considered realistic for hydrologic data, except perhaps for k = 0.1 where there is 1 in 500 chance of getting negative values. For these parameter sets the result of our simulation are the same or similar for individual sites, thus we will in most cases (unless otherwise indicated) show results based on estimation of quantiles at site 1 only. In the simulation the MSEð^x ðþþis q compared with CRLB(x(q)), AVarð^x ðþþ, q and SVarð^x ðqþþ, where the MSE is defined as MSE ^x ðqþ ¼ Var ^x h ðqþ þ E ^x i 2 ðqþ xðqþ ð16þ where E½^x ðqþš xðþis q the bias of ^x ðþ. q Furthermore, the relatives bias of quantile estimators is defined by h E ^x i ðqþ RBIAS ¼ xðqþ xðqþ ð17þ where RBIAS > 0 indicates overestimation, while RBIAS < 0 indicates underestimation. [20] Simulation results based on the PIF 1 model and k 2 { 0.3, 0.2, 0.1, 0.1} are shown in Figures 3, 4, and 5 for the 0.95, 0.99, and quantiles, respectively. Simulation results based on the PIF 2 model for the 0.95, 0.99, quantiles are shown in Figures 6, 7, and 8 for k = 0.1, k = 0.1, and k = 0.2, respectively. All simulation results in Figures 3 8 are based on 10,000 generated samples for each case shown. Furthermore, note that in Figures 3 8 the same scale is used horizontally (for each k), displaying the relative gain resulting from increasing the size of the region, where for m = 12 four sets of random samples from each site are used (in the work by Sveinsson [2002], results for m = 6 are also shown). Overall, for both PIF 1 and PIF 2 models the results for the AVar and SVar are similar, and both represent the MSE closely. That is, either one can be used for estimating the variance of quantile estimators. In more detail, for m = 3, SVar appears closest to MSE for

6 SWC 2-6 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF Figure 1. Asymptotic efficiency of estimating the qth quantile at a single site using single site analysis relative to regional analysis of six sites, where the sample size n j is the same at all sites and g = a j /b j and k are the same for all sites. k = 0.1, and AVar appears closest to MSE for k 2 { 0.3, 0.2, 0.1}. For m = 6 and m = 12 (only cases m = 3 and m = 12 are shown in Figures 3 8) SVar appears closest to MSE for all k s, where for say n j > 40 SVar and AVar are almost identical. When comparing the results across the two different PIF 1 and PIF 2 models, then for this scenario (underlying model is PIF 1) it should be expected that for the same n j the AVar, SVar, and MSE are a little higher for the PIF 2 model than the PIF 1 model, since more parameters need to be estimated in the PIF 2 model. This is reflected in Figures 3 8 (note n j starts at 25 for the PIF 1 model and at 30 for the PIF 2 model). 6. Assessing the Variance of Quantile Estimators in the Hosking and Wallis Regional Estimation Scheme [21] The Hosking and Wallis regional estimation scheme [Hosking and Wallis, 1997], dubbed here as the HW scheme, is a widely used scheme for regional Figure 2. Asymptotic efficiency of estimating the qth quantile at a single site using single site analysis relative to regional analysis of m sites, where the sample size n j is the same at all sites and g = a j /b j and k are the same for all sites.

7 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF SWC 2-7 Figure 3. Simulation results based on the PIF 1 model and k 2 { 0.3, 0.2, 0.1, 0.1} for the quantile x(0.95) at site 1. The results are shown for a region with 3 sites and 12 sites. frequency analysis of extreme precipitation. In a recent study [De Michele and Rosso, 2001], approximate equations for estimating the variance of at-site quantile estimators in the HW scheme have been suggested, where the GEV is assumed as the underlying regional distribution. In this section, the accuracy of these procedures is tested using similar simulation experiments as in section 5.

8 SWC 2-8 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF Figure 4. Simulation results based on the PIF 1 model and k 2 { 0.3, 0.2, 0.1, 0.1} for the quantile x(0.99) at site 1. The results are shown for a region with 3 sites and 12 sites.

9 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF SWC 2-9 Figure 5. Simulation results based on the PIF 1 model and k 2 { 0.3, 0.2, 0.1, 0.1} for the quantile x(0.998) at site 1. The results are shown for a region with 3 sites and 12 sites.

10 SWC 2-10 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF Figure 6. Simulation results based on the PIF 2 model with k = 0.1 for the quantiles x(0.95), x(0.99), and x(0.998) at site 1. The results are shown for a region with 3 sites and 12 sites. [22] In the HW scheme the index flood at site j is estimated by the at-site sample mean, ^m j ¼ X j, so that the at-site sample observations are indexed by dividing them by the at-site sample mean, and a single regional growth curve is estimated for the whole region. Once the regional growth curve has been estimated, the qth quantile at site j, ^x j ðqþ, is simply ^xj ðqþ ¼ X j ^x R ðqþ; j ¼ 1;...; m ð18þ where ^x R ðþis q the regional qth quantile. The practice of estimating the index flood by sample statistics has been questioned by Sveinsson et al. [2001, 2002b], where for example certain analytical limitations are pointed out. [23] The parameters of the regional growth curve in the HW scheme are estimated in terms of PWMs [Greenwood et al., 1979], where the rth PMW that is commonly used in estimation of upper extreme events is b r ¼ E XFX r ðþ x ; r ¼ 0; 1;... ð19þ where F X (x) is the CDF of the random variable X. L moments (Hosking, 1990) are linear combinations of the PWMs with the (r + 1)th L moment defined as l rþ1 ¼ Xr k¼0 ð 1Þ r k r r þ k k k b k ; r ¼ 0; 1;... ð20þ

11 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF SWC 2-11 Figure 7. Simulation results based on the PIF 2 model with k = 0.1 for the quantiles x(0.95), x(0.99), and x(0.998) at site 1. The results are shown for a region with 3 sites and 12 sites. The regional L moments (l r R ) are estimated as ^l R r ¼ 1 n T X m j¼1 ðþ j n j^l r ; r ¼ 1; 2;... ð21þ where P ^l ðþ r j is the rth sample L moment at site j and n T = j=1 m n j. Because the data at each site are scaled by the at-site sample mean then ^l ðþ j 1 = 1 for all sites and consequently ^l R 1 = 1. Then the regional L moment ratios (t R r ) are estimated from ^t R 2 ¼ ^l R 2 ; ^t ^l R R r ¼ ^l R r ; r ¼ 3; 4;... ð22þ 1 ^l R 2 For further explanation on estimation of the regional GEV distribution using regional L moments the interested reader is referred to Hosking and Wallis [1997] or Sveinsson et al. [2001]. Note that Sveinsson et al. [2001] showed that the bias of the 0.95, 0.99, and GEV quantile estimators were significantly reduced when the regional L moment ratios were estimated from equation (22) as opposed to being estimated in terms of weighted averages of at-site L moment ratios ð^t R r ¼ ð 1=n TÞ P m j¼1 n j^t ðþ r j Þ as by Hosking and Wallis [1997]. [24] De Michele and Rosso [2001] suggest estimating the variance of at-site quantile estimators in the HW

12 SWC 2-12 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF Figure 8. Simulation results based on the PIF 2 model with k = 0.2 for the quantiles x(0.95), x(0.99), and x(0.998) at site 1. The results are shown for a region with 3 sites and 12 sites. scheme, with the GEV as the underlying regional distribution, from Var ^x j ðqþ ¼ X 2 j Var ^x R ðqþþ^x 2 R ðqþvar X j þ Var ^x R ðqþvar X j where ð23þ Var ^x R ðqþ ¼ b2 expf lnð ln qþexp½ 1:823k 0:165Šg ð24þ n T is a fitted formula to tabulated values based on simulations by Lu and Stedinger [1992b]. Note that equation (23) assumes that X j and ^x R ðþare q independent which generally is not the case [Stedinger and Lu, 1995]. In this paper the estimated variance based on equation (23) will be dubbed as DRVar HW to indicate that the referred De Michele-Rosso s equation has been used in conjunction with the HW scheme as above noted. [25] The simulation experiment in section 5 is repeated here to test the accuracy of estimating the variance of at-site quantile estimators in the HW scheme using equations (23) and (24). The results of the simulation experiments are shown in Figures 9, 10 and 11 for the 0.95, 0.99, and quantiles, respectively. As for the PIF methods, the MSE should be considered as the true or actual error variance of the quantile estimators in the HW scheme. In addition, the CRLB of equation (7) is included from Figures 3 5 for comparison with the PIF 1 method. For small values of m

13 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF SWC 2-13 Figure 9. Simulation results based on the HW scheme and k 2 { 0.3, 0.2, 0.1, 0.1} for the quantile x(0.95) at site 1. The approximate variance (DRVar HW ) is estimated from equations (23) and (24). The results are shown for a region of with 3 sites and 12 sites.

14 SWC 2-14 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF Figure 10. Simulation results based on the HW scheme and k 2 { 0.3, 02, 0.1, 0.1} for the quantile x(0.99) at site 1. The approximate variance (DRVar HW ) is estimated from equations (23) and (24). The results are shown for a region with 3 sites and 12 sites.

15 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF SWC 2-15 Figure 11. Simulation results based on the HW scheme and k 2 { 0.3, 0.2, 0.1, 0.1} for the quantile x(0.998) at site 1. The approximate variance (DRVar HW ) is estimated from equations (23) and (24). The results are shown for a region with 3 sites and 12 sites.

16 SWC 2-16 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF Figure 12. study. The location of the 12 sites used in the case (i.e., m =3)DRVar HW tends to overestimate the MSE, in fact for values of k = 0.2 and 0.3 and high quantiles (say q 0.99) the DRVar HW overestimates the MSE by several orders of magnitude and in some cases (e.g., n j < 50) DRVar HW may be unrealistically high. However, for large m(m = 12) DRVar HW is close to the MSE except perhaps for k = 0.3 and q = Thus as a general recommendation, the DRVar HW of Equations (23) and (24) should be used cautiously for estimating the approximate variance of quantile estimators in the HW scheme, especially for small regions with large negative values of k. 7. Case Study [26] We analyzed the 3-hr annual maximum precipitation for the sites included in subregion SE utilized by Sveinsson et al. [2002b]. The region consists of 12 sites located in the northeastern plains of Colorado with sample sizes ranging from 17 years to 50 years, and elevations ranging from 1133 m to 1635 m above sea level. The total sample size for the region is 465 years. The sites and some of their statistical characteristics are shown in Figure 12 and Table 1. Each site in the region passed Wilk s multivariate outlier test [Caroni and Prescott, 1992; Hosking and Wallis, 1993; Sveinsson et al., 2002b] or discordancy test as referred to by Hosking and Wallis [1993, 1997]. The test was applied on the vector u = ½^t ðþ j 2 ; ^t ðþ j 3 ; ^t ðþ j 4 Š for each site j at the 10% significance level. In addition, the referred region was tested for homogeneity and passed the so-called X-10 regional homogeneity test of Lu and Stedinger [1992a]. Furthermore, a heterogeneity test applied to t 2, t 3, and t 4 [Hosking and Wallis, 1997] gave values of the heterogeneity measure (H) equal to 0.12, 0.86, and 0.69, respectively (i.e., values smaller than 1). Thus the region was classified as an acceptable homogeneous region. The PIF 1 and PIF 2 regional models and the HW scheme are used for estimation of the growth curves at the sites within the region. [27] Empirical and estimated growth curves with approximate 95% confidence bounds for the even numbered sites based on the HW scheme and the regional PIF 1 and the PIF 2 models are shown in Figure 13 for the six even numbered sites. The shape parameter of the estimated growth curves is k = for the HW scheme, k = for the PIF 1 method, and k = for the PIF 2 method. Note that when different models are compared using empirical data, it can be difficult to judge the performance of the different models, since the empirical data will never reflect the structure of any one model. Regardless, one may state that overall the results based on the HW scheme and the PIF 1 model appear similar (apart from the estimated confidence intervals), and perhaps it can be argued that overall the PIF 2 model gives the best fit to the data. For the odd numbered sites, which are not shown, all models fitted the data quite well [Sveinsson, 2002]. On the other hand, the poorest fit to the empirical data appears to be for sites 4 and 6, where all models seem to overestimate the empirical growth curve for site 4, and underestimate the empirical growth curve for site 6. This is an interesting observation, since the observed coefficient of variation and the observed skewness for site 4 are the lowest among all sites in the region (refer to Table 1), while for site 6 they are the highest. Recall that for the PIF 1 and the HW scheme, the coefficient of variation, skewness, and all higher order moment ratios are assumed to be the same for all sites within the homogeneous region, while for the PIF 2 the skewness and all higher order moment ratios are assumed to be the same for all sites within the region. Furthermore, only the PIF 2 model appears to fit site 12 well. The skewness for site 12 is typical for the sites in the region, while the coefficient of variation is the second highest of all sites (Table 1). This, might explain why the PIF 2 model appears to perform better than the other models for site Summary and Conclusions [28] Formulas for the asymptotic and sample variances of maximum likelihood quantile estimators at each site within a statistically homogeneous region were derived for the regional PIF models and the three-parameter GEV distribution assuming independence in space. The formulas were based on the Cramer Rao lower bound (CRLB) for the variance of unbiased estimators and the observed and expected Fisher s information matrix of the maximum likelihood estimators of the parameters of the GEV distribution. The CRLB was used to calculate the theoretical gain of regionalization, where for extreme upper or lower quantiles there is always a significant gain in using regional analysis over single site analysis. [29] In addition, for the PIF 1 model there is always gain in regionalization, while for the PIF 2 model there are always two quantiles where there is no gain in regionalization. Furthermore, in examining the gain that may be obtained by regionalization for regions where the number of sites varies from 3 to 36, this study suggests that after a Table 1. Sample Characteristics of the Annual Maximum 3-Hour Duration Precipitation Data for the Sites in the Case Study Station Name Site Number Sample Size, years Mean, cm Coefficient of Variation Skewness Coefficient Bonny Dam 2 NE Arapahoe Paoli Eckley Joes 2 SE Eads Seibert Akron 4 E Hugo 1 NW Ordway 21 N Kutch 6 SSE New Raymer

17 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF SWC 2-17 Figure 13. Estimated growth curve and approximate 95% confidence limits for the even numbered sites based on the HW scheme, PIF 1 model, and PIF 2 model.

18 SWC 2-18 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF Figure 13. (continued) significant gain using a few sites further gains with increasing number of sites often become marginal. This result has been obtained under a true homogeneous region and under assumed spatial independence. The effects of an imperfect homogeneous region and spatial dependence will likely reduce even further the gain that may be obtained by regionalization [e.g., Lettenmaier et al., 1987; Hosking and Wallis, 1988; Stedinger and Lu, 1995; Hosking and Wallis, 1997]. That is, in practice a bigger region may not necessarily be better than a smaller one [e.g., Hosking and Wallis, 1988; Stedinger and Lu, 1995]. [30] Simulation experiments for different sized regions and different values of the GEV shape parameter were used to test the derived formulas for estimating the variance of maximum likelihood quantile estimators of the PIF methods. The formulas based on Fisher s expected information matrix (AVar) and Fisher s observed information matrix (SVar) generally give similar results and represented the simulated MSE reasonably well. AVar appears more accurate for a small region (3 sites) while SVar appears more accurate for larger regions with small sample sizes at each site. For larger sample sizes, say n j > 50, AVar and SVar were not significantly different from each other. Similar simulation experiments were also used to test the accuracy of the procedure suggested by De Michele and Rosso [2001] for estimating the variance of at-site quantile estimators for the Hosking and Wallis regional estimation scheme (HW scheme) utilizing the generalized extreme value distribution. The results of the simulations indicate that these estimated variances are fairly accurate for k = 0.1 and k = 0.1, but they can in some cases be very unreliable, especially when k 0.2, and should be used with caution. [31] A region of 12 sites in the northeastern plains of Colorado was chosen to illustrate and compare the applicability of the two PIF models, PIF 1 and PIF 2, and the Hosking and Wallis regional estimation scheme. The growth curves and confidence bounds based on the PIF 1 model and the HW scheme were similar, while overall the more flexible PIF 2 model seemed to give the best fit to the empirical growth curves. [32] As a general conclusion, it appears that the new analytic PIF models can be quite useful additions to existing models for use in regional frequency analysis. These PIF models provide a mathematical framework for regional frequency analysis that enables one to specify conditions for homogeneous regions. They are amenable for parameter estimation using various methods and offer the possibility of using ML-estimation and the Fisher information for estimation of the standard error of quantile estimators. Appendix A: PIF 1 Derivatives [33] The partial derivatives of the log likelihood for the PIF 1 model in equation (11) with respect to the parameters are given ln ln ¼ n j g Xnj x ; j ¼ 1;...; m q j ( ) ¼ Xm Xnj n j ðq j x 1Þ g ¼ 1 k j¼1 X m j¼1 X nj 1 k g q j x 1 ða1þ where = z 1 (1 k z 1/k ) and =(z 1/k 1) ln z, with z =1 gk (q j x 1). So the elements of the Fisher sample information matrix, SI, of the ML-estimators are given by 2 j ¼ 0; ¼ Xnj j ¼ n j q 2 þ g j j i 6¼ j ¼ g ( 2 ¼ Xm n j g 2 þ q j x ¼ 2 j ¼ 1 k j¼1 X m X nj j¼1 X nj ðq j g q j x 1 ¼ gð1 k 1 k Þx z 2 k þ z 1=k ða2þ ða3þ

19 SVEINSSON ET AL.: UNCERTAINTY OF QUANTILE ESTIMATORS USING PIF SWC ¼ g q ¼ q jx 1 ð1 kþz 2 k þ ¼ 1 z 1 k 2 z1=k ln z 1 þ z 1 1 z 1=k 1 þ 1 k ln z ða4þ ða5þ 1 k 2 z1=k ln 2 z ða6þ In addition, under this parameterization the elements of the gradient of the GEV qth quantile in equation (6) for site j, j =1,..., m, j ðqþ ¼ 1 h j q 2 gk þ 1 ln q j gk ð j ¼ 1 h i q j g 2 1 ln q k ð j ¼ 1 h q j gk 2 1 ð ln q Þk þk lnð ln qþð ln q Þ k i ða7þ Appendix B: PIF 2 Derivatives [34] In the same way as for the PIF 1 model in Appendix A, the first partial derivatives of the log likelihood of the PIF 2 model in equation (13) are given ln ln j ¼ q j X nj ¼ n j q j ln ¼ 1 X m k j¼1 ; X nj j ¼ 1;...; m x a j ; j ¼ 1;...; m 1 k q j x a j ðb1þ where = z 1 (1 k z 1/k ) and =(z 1/k 1) ln z as before, but with z =1 q j k (x a j ). The elements of the Fisher sample information matrix, SI, of Q are then given 2 j ¼ q j X j ¼ 0; i 6¼ j ¼ 2 j ¼ q j þ q j ¼ n j q 2 þ Xnj j ¼ 0; i 6¼ j ¼ 2 ¼ 1 X m @ x a x a X nj 1 k 1 q j x a ðb2þ where the partial derivatives of and in equation (B2) are given @a j ¼ q j ð1 k ¼ x ¼ 1 ¼ q j x a j k 2 z1=k k þ z 1=k Þz 2 ð1 k z 1 1 z 1=k Þz 2 k þ z 1=k ln z 1 1 þ 1 k ln z þ q j ðb3þ ðb4þ ðb5þ 1 k 2 z1=k ln 2 z ðb6þ Under this parameterization the gradient of the GEV qth quantile in equation (6) has the following elements for site j, j =1,..., j ¼ 1 q j k j ðqþ ¼ j ðqþ ¼ 1 j q 2 1 ln q j k ð h 1 ð ln q Þk þ k lnð ln qþð ln q Þk i Þ k i ðb7þ [35] Acknowledgments. Support from the Colorado Agricultural Experiment Station project on Predictability of Extreme Hydrologic Events Related to Colorado s Agriculture and National Science Foundation grant CMS on Uncertainty and Risk Analysis Under Extreme Hydrologic Events are gratefully acknowledged. In addition we would like to thank anonymous reviewer number one for useful comments that enhanced the manuscript. References Blöschl, G., and M. Sivapalan, Process controls on regional flood frquency: Coefficient of variation and basin scale, Water Resour. Res., 33(12), , Boes, D. C., J. Heo, and J. D. Salas, Regional flood quantile estimation for a Weibull model, Water Resour. Res., 25(5), , Caroni, C., and P. Prescott, Sequential application of Wilks s multivariate outlier test, Appl. Stat., 41(2), , Chowdhury, J. U., J. R. Stedinger, and L.-H. Lu, Goodness-of-fit tests for regional generalized extreme value flood distributions, Water Resour. Res., 27(7), , Cunnane, C., Methods and merits of regional flood frequency analysis, J. Hydrol., 100(1/3), , Dalrymple, T., Flood frequency analysis, U.S. Geol. Surv. Water Supply Pap., 1543-A, De Michele, C., and R. Rosso, Uncertainty assessment of regionalized flood frequency estimates, J. Hydrol. Eng., 6(6), , Gabriele, S., and N. Arnell, A hierarchical approach to regional flood frequency analysis, Water Resour. Res., 27(6), , Greenwood, J. A., J. M. Landwehr, N. C. Matalas, and J. R. Wallis, Probability weighted moments: Definition and relation to parameters of several distributions expressable in inverse form, Water Resour. Res., 15(5), , Gupta, V. K., and D. R. Dawdy, Regional analysis of flood peaks: Multiscaling theory and its physical basis, in Advances in Distributed Hydrol-

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B. Sveinsson, International Research Institute for Climate Prediction, Columbia University, Monell Building 133, 61 Route 9W, Palisades, NY 10964, USA. (oli@iri.columbia.edu)