PUBLICATIONS Water Resources Research RESEARCH ARTICLE Key Points: Modified weighted function method was introduced Monte-Carlo experiment was carried out to simulate a large number of samples New method was compared to original weight function and L-M methods Correspondence to: Z. Yu, zyu@hhu.edu.cn Citation: Liang, Z., Y. Hu, B. Li, and Z. Yu (2014), A modified weighted function method for parameter estimation of Pearson type three distribution, Water Resour. Res., 50, 3216 3228, doi:10.1002/ 2013WR013653. Received 5 FEB 2013 Accepted 25 MAR 2014 Accepted article online 27 MAR 2014 Published online 10 APR 2014 A modified weighted function method for parameter estimation of Pearson type three distribution Zhongmin Liang 1,2, Yiming Hu 2, Binquan Li 1, and Zhongbo Yu 1,2,3 1 State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing, China, 2 College of Hydrology and Water Resources, Hohai University, Nanjing, China, 3 Department of Geoscience, University of Nevada Las Vegas, Las Vegas, Nevada, USA Abstract In this paper, an unconventional method called Modified Weighted Function (MWF) is presented for the conventional moment estimation of a probability distribution function. The aim of MWF is to estimate the coefficient of variation (C V ) and coefficient of skewness (C S ) from the original higher moment computations to the first-order moment calculations. The estimators for C V and C S of Pearson type three distribution function (PE3) were derived by weighting the moments of the distribution with two weight functions, which were constructed by combining two negative exponential-type functions. The selection of these weight functions was based on two considerations: (1) to relate weight functions to sample size in order to reflect the relationship between the quantity of sample information and the role of weight function and (2) to allocate more weights to data close to medium-tail positions in a sample series ranked in an ascending order. A Monte-Carlo experiment was conducted to simulate a large number of samples upon which statistical properties of MWF were investigated. For the PE3 parent distribution, results of MWF were compared to those of the original Weighted Function (WF) and Linear Moments (L-M). The results indicate that MWF was superior to WF and slightly better than L-M, in terms of statistical unbiasness and effectiveness. In addition, the robustness of MWF, WF, and L-M were compared by designing the Monte-Carlo experiment that samples are obtained from Log-Pearson type three distribution (LPE3), three parameter Log-Normal distribution (LN3), and Generalized Extreme Value distribution (GEV), respectively, but all used as samples from the PE3 distribution. The results show that in terms of statistical unbiasness, no one method possesses the absolutely overwhelming advantage among MWF, WF, and L-M, while in terms of statistical effectiveness, the MWF is superior to WF and L-M. 1. Introduction Providing the hydrological design value X p with assigned cumulative probabilities p or return period T (p 5 1/T) is often required for the hydraulic engineering design and water resources planning and management [e.g., Stedinger et al., 1992]. The accuracy on the estimation of X p directly influences the engineering investments and safety. Currently, the statistical parametric method is a dominant measure for quantiles estimation, in which the parameter estimation is a key component. In the parameter estimation, the method of moment (MOM) was once a widely used approach, but conclusions made by many researchers [e.g., Wallis et al., 1974; Kirby, 1974] showed that MOM statistics bear high bias. Another method that is known to provide asymptotically minimum variance estimates is the Maximum Likelihood (ML) estimators. Previous studies show that ML generally offers reasonable estimation with records available in hydrology and is praised especially for samples with larger sizes. Griffis and Stedinger [2007] indicated, through a Monte-Carlo analysis, that MOM with a highly informative regional skew performs as well as the ML method for the range of parameters of Log-Pearson type 3 (LP3) distribution. Greenwood et al. [1979] introduced the Probability Weighted Moment (PWM) to estimate the parameters of a probability distributions whose inverse form is explicitly defined, which was seen to compare favorably with MOM and ML [Landwehr et al., 1979]. As an extension of the traditional PWM, the self-determined PWM [Haktanir, 1997] was introduced for parameter estimation by mean of algorithms with certain mathematical manipulations for numerical convenience or special knowledge of the behavior of a sample, and later a new improvement was developed that directly implements the relevant equations without relying upon external conditions [Whalen et al., 2002], which enhances the applicability of the method. LIANG ET AL. VC 2014. American Geophysical Union. All Rights Reserved. 3216
Derived from PWM, the concept of Linear Moment (L-M) was developed by Hosking [1990] which improves the accuracy of frequency analysis. A new system of L-M ratios was proposed which is taken as an alternative for describing the shapes of probability distributions. L-M is mathematically equivalent to PWM, but is more easily interpreted and powerful in summarizing the statistical properties of samples. Recently, Hosking [2006] proved that a wide range of distributions can be characterized by their L-Ms, including those applied in the hydrological frequency analysis. Markus et al. [2007] calculated design precipitation totals for the northeastern Illinois region by using L-M and analyzed the effects of selecting different periods of the precipitation record, different regions, and different underlying distributions on the frequency and intensity of heavy rainfall events. Karvanen [2008] found that L-M compared favorably to other more complicated approaches and produced unbiased or nearly unbiased estimations. So far, L-M has been successfully applied to various problems in hydrological frequency analysis regarding the description, identification, and estimation of distributions for both at-site and regional analysis [Hosking and Wallis, 1997; Zhang and Hall, 2004; Daniele et al., 2007]. In exploring the reason why L-M (and PWM) possesses advantages in parameter estimation, Hosking [1990] and Stedinger et al. [1992] pointed out that sample estimators of L-M are linear combinations of ranked observations. As compared to the conventional MOM, they do not involve squaring and cubing samples in the moment estimations. Therefore, L-M is less subject to bias and more robust to errors of estimations of extreme flows, whereas the MOM suffers the substantial bias and less robust. Studies and applications on PWM and L-M have revealed that such a strategy of powering down the observations in sample moment estimators makes those methods perform well in frequency analysis. In the parameter estimations of PE3 distribution, the powering down effect can also be achieved to some extent in a different way. Ma [1984] expressed the coefficient of skewness (C S ) as a function of coefficient of variation (C V ) and two weighted central moments which are the first-order and second-order moments, respectively. This approach was denoted as the Weighted Function (WF) method in this paper. By introducing a normal probability density function (pdf) as a weight function in the estimation of moments of PE3, this approach makes the estimation of C S decrease from the third-order moment estimate to the second-order weighted moment estimates. Consequently, the accuracy of estimation for C S was improved. Even though the WF approach was modified by Liu [1990] through adopting two different weight functions, compared to PWM or L-M, it still needs to involve squaring samples in the moment estimations. In this paper, new functions combining two negative exponential-type functions were presented as weight functions in moment estimations, making the estimates of both C V and C S be the calculations of the firstorder sample moment for PE3 probability distribution. Therefore, a further powering down effect for parameter estimation was acquired. In the following sections, first is to introduce the methodology of Modified Weighted Function (MWF), including the original WF method, derivation of MWF, sample estimators of MWF, and determination of weight function. Following are the results of Monte-Carlo simulations demonstrating statistical properties of MWF on unbiasness, effectiveness, and robustness by comparing with WF and L-M. Finally, conclusions are presented. 2. The Methodology of Modified Weighted Function Method 2.1. The Original Weighted Function (WF) Estimator Ma [1984] defined a weight function to improve the estimation of the coefficient of the skewness (C S ) for the PE3 probability distribution, later the method was recommended as one of the national standard procedures for flood frequency analysis by the Ministry of Water Resources of China, under the name of the Weighted Function method (WF). The WF method is described in detail below with the PE3 distribution. Let X denote a random variable following a PE3 probability distribution. Then, its pdf is defined as f ðxþ5 bc CðaÞ ðx2lþc21 e 2bðx2lÞ (1) where x is a real variable with x l; l, b, and c represent the location, scale, and skewness of the distribution, respectively, and CðÞ denotes the gamma function. In addition, the relation LIANG ET AL. VC 2014. American Geophysical Union. All Rights Reserved. 3217
between distribution parameters and statistical parameter of PE3 distribution can be described as l5eðxþ2 2r C S ; b5 2 rc S ; c5 4 C S 2 (2) where EðÞ, r, and C S represent the expectation, standard variation, and coefficient of skewness of a PE3 distribution, respectively. By taking the natural logarithms on both sides of equation (1), the following expression can be derived: ln f ðxþ5ðc21þln ðx2lþ2ln CðcÞ1cln b2bðx2lþ (3) Then, taking the derivatives of both sides of equation (3), one obtains By substituting l in equation (4) with its definition in equation (2) yields f 0 ðxþ5 ðc21þfðxþ 2bf ðxþ (4) x2l f 0 ðxþ5 ðc21þfðxþ 2bfðxÞ (5) x2½eðxþ2c=bš which can be rewritten as ½x2EðXÞ1c=bŠf 0 ðxþ52f11b½x2eðxþšgf ðxþ (6) By multiplying both sides by a function uðxþ, called weight function, and taking the integral operation for both sides, one obtains ½x2EðXÞ1c=bŠuðxÞf 0 ðxþdx52 f11b½x2eðxþšguðxþfðxþdx (7) a 0 a 0 where the weight function uðxþ should satisfy two constraints: (a) uðxþ is a continuous and differentiable function; (b) Ð 11 21 uðxþdx51. Then, taking the integration by parts to the left side of equation (7), one obtains ½x2EðxÞ1c=bŠ 1 a 0 2 fðxþdf½x2eðxþ1c=bšuðxþg a 0 52 f11b½x2eðxþšguðxþfðxþdx a 0 (8) Back to the pdf of PE3 distribution, the following expression can be obtained based on the property: Thus, equation (8) can be transformed, according to equation (9), into lim f ðxþ5 lim fðxþ50 (9) x!a 0 x!1 2 fðxþuðxþdx2 fðxþðx2lþduðxþ52 f11b½x2eðxþšguðxþfðxþdx (10) a 0 a 0 a 0 which, after algebraic manipulation, can be rewritten as LIANG ET AL. VC 2014. American Geophysical Union. All Rights Reserved. 3218
l u 0 ðxþfðxþdx1b ½x2EðXÞŠuðxÞfðxÞdx5 xu 0 ðxþf ðxþdx (11) a 0 a 0 a 0 By substitution in equation (11) the quantities of l and b (except for the lower limit of integral), one obtains EðXÞ2 2r u 0 ðxþfðxþdx1 2 ½x2EðXÞŠuðxÞfðxÞdx5 xu 0 ðxþfðxþdx (12) C S a 0 rc S a 0 a 0 which, after algebraic manipulation, can be rewritten as 2 r C S 5 fð 1 a 0 ½x2EðXÞŠuðxÞfðxÞdx2r 2Ð 1 a 0 u 0 ðxþfðxþdxg Ð 1 a 0 ½x2EðxÞŠu 0 ðxþfðxþdx (13) The above development indicates that, for any weight function uðxþ, equation (13) is an identical equation, as long as the integral operators are bounded and the denominator is not equal to zero. That is to say, the solution of equation (13) has no relation with the selection of the weight function. When a data series x 1 ; x 2 ; :::; x n (in ascending order) is available, the sample estimator of C s in equation (13) is written as 2 r C S 5 fp n ½x i2xþšuðx i Þ2r 2P n u0 ðx i Þg P n ½x i2xšu 0 ðx i Þ (14) where P n ðþ represents the approximately finite sum form of the integral operator, x i is the ith ranked sample, x is the average of the sample, and n is the number of the sample. Because the value of ½x i 2xŠu 0 ðx i Þ may be positive or negative, the sum of them, i.e., the value of denominator in equation (14), is likely to approach zero. In order to avoid this case in computation and have the solution of equation (14) with satisfactory accuracy, one could define the weight function uðxþ which guarantees all ½x i 2xŠu 0 ðx i Þ to always have either a positive or a negative value. Hence, the following expression can be assumed: Solving equation (15), one obtains r 2 uðxþ52kðx2xþu0ðxþ (15) ( uðxþ5c exp 2 k ) x2x 2 2 r (16) where k is a positive constant (it is found that k 5 1 makes best results through many test cases) and C is integral constant. Substitution in the expression Ð 11 21 uðxþdx51 of the weight function defined in equation (16) yields 1 21 Then, the integral constant C can be derived with k 5 1 rffiffiffiffiffi 2p uðxþdx5cr 51 (17) k C5 p 1 r ffiffiffiffiffi (18) 2p LIANG ET AL. VC 2014. American Geophysical Union. All Rights Reserved. 3219
Consequently, it is found that the weight function is a normal pdf, i.e., uðxþ5 p 1 r ffiffiffiffiffi e 21 2½ x2x r Š 2 (19) 2p Substituting uðxþ in equation (19) into equation (14), and after some algebraic calculation, one obtains C S 54r H G (20) where H5 1 n ½x2x i Šuðx i Þ; G5 1 n ½x i 2xŠ 2 uðx i Þ (21) Obviously, H and G defined in equation (21) are functions of the first-order and the second-order sample moments weighted by uðxþ, respectively. Hence, for the sample estimation of C s, comparing with the conventional method of moments lowers the powers to samples from cubing to squaring, i.e., the powering down effect is achieved. In addition, as the weight function uðxþ is in a form of normal distribution function, it allocates more weights to data close to mean position of a ranked sample while it reduces the weights to data close to lower and upper bounds. Consequently, the accuracy of estimation of C S can be improved when equation (20) is used as a substitution to the conventional method of moments in practical applications. 2.2. The Modified Weighted Function Estimator 2.2.1. Derivation of MWF The relation between C S and C V expressed in equation (13) is a general formula, which implies that for any of a weight function /ðxþ, as long as it is continuous and differentiable in math, equation (13) is satisfied. Therefore, given two different weight functions, the joint solutions of two equations from equation (13) will provide estimators for C V and C S simultaneously. The reason that the WF method still involves squiring samples in the moment estimations is that a normaltype function was adopted as the weight function. In order to make the estimations for both C V and C S from a high-order moment computation to the first-order moment estimation only (as L-M or PWM does), a new type of weight functions is proposed in this study by subtracting two negative exponential-type functions /ðxþ5c exp 2 x 2exp 2 x m 1 k m 2 k (22) where m 1 ; m 2 ; k; c are coefficients, m 1 6¼ m 2, and x > 0. If limits /ðxþ a pdf, then c51=½ðm 1 2m 2 ÞkŠ. Without the loss of generality, it is assumed that m 1 and m 2 are positive integers, k > 0, and m 1 > m 2 hereafter. Obviously, /ðxþ is a mono-peak function and possesses various shapes with different combinations of m 1, m 2, and k. Substituting /ðxþ and its derivative into equation (8) results in C S 5 2 r U1r2 V W (23) with U5E½ðx2xÞuðxÞŠ (24) LIANG ET AL. VC 2014. American Geophysical Union. All Rights Reserved. 3220
V5 1 m 1 k E½/ðxÞŠ2 1 m 1 m 2 k 2 E exp 2 x m 2 k W52 1 m 1 k E½ðx2lÞ/ðxÞŠ1 1 m 1 m 2 k 2 E ðx2lþexp 2 x m 2 k (25) (26) Corresponding to two different sets of parameters ðm 1 ; m 2 ; kþ 1 and ðm 1 ; m 2 ; kþ 2, there are two different weight functions / 1 ðxþ and / 2 ðxþ, and then two equations from equation (23) can be obtained C s 5 2 U 1 1ðrÞ 2 V 1 (27) r W 1 C s 5 2 U 2 1ðrÞ 2 V 2 (28) r W 2 where ðu 1 ; V 1 ; W 1 Þ and ðu 2 ; V 2 ; W 2 Þ are two sets of variables corresponding to parameters of ðm 1 ; m 2 ; kþ 1 and ðm 1 ; m 2 ; kþ 2, respectively, and calculated by using equations (24) (26). Solutions of equations (27) and (28) are as following: r 2 52 U W1 2 W 2 2 U1 U 2 V W1 2 W 2 2 V1 V 2 (29) C s 5 2 U1 1r 2 V 1 r W 1 W 1 (30) It is indicated by equations (29) and (30) that both C V (5r/l) and C S are functions of weighted moments of ðx2lþ i ; i50; 1. This implies that when estimated by samples, powers to samples are not exceeded one, so that a further powering down effect in estimation is obtained in MWF. 2.2.2. Sample Estimators of MWF For the samples x 1 ; x 2 ; :::; x n (in ascending order), sample estimators for U, V, and W in the method of MWF are given as below Let ^T 1 5E½uðxÞŠ, ^T 2 5E½exp ð2 x m 2kÞŠ, then ^U5 1 n ðx i 2xÞuðx i Þ (31) ^T 2 5 1 n ^T 1 5 1 n exp uðx i Þ (32) 2 x i m 2 k V Ù 5 1 m 1 k ^T 1 2 1 m 1 m 2 k ^T 2 2 (34) ^W 1 52 1 m 1 k U1 1 1 m 1 m 2 k 2 ðx i 2lÞ exp 2 x i n m k (35) For two given /ðxþ from equation (22), U, V, and W can be estimated through equations (31 35), substituting these estimators into equations (29) and (30), the sample estimators of r (or C V ) and C S are obtained. 2.2.3. Determination of Weight Function It seems difficult to select the optimal weight function /ðxþ theoretically which gives the best estimates of C V and C s, so that two specific /ðxþ are determined under the following considerations. (33) LIANG ET AL. VC 2014. American Geophysical Union. All Rights Reserved. 3221
Establishing the relation between /ðxþ and sample size. If assuming m 1 in equation (22) equal to the sample size, i.e., m 1 5 n, and taking m 2 5 1 and m 2 5 2, respectively, equation (22) will produce the following two corresponding weight functions: / 1 ðxþ5 1 exp 2 x 2exp 2 xk1 ðn21þk 1 nk 1 / 2 ðxþ5 1 exp 2 x 2exp 2 x ðn22þk 2 nk 2 2k 2 (36) (37) It is obvious that the larger the sample size n, the flatter are u 1 ðxþ and u 2 ðxþ in shape, which manifests that more even weights are to be distributed to samples. The increase of sample size n means that there is more information from samples, i.e., the samples play a key role in estimation while the effect of weight function is decreasing. So, for this case, it is inclined to allocate even weights samples. On the other hand, for a fixed n, i.e., the given amount of samples information, the uneven weights should be assigned to different data to emphasize their distinctive roles in estimation. Equations (36) and (37) provide such two functions that satisfy the above requirements, by which it is seen that data near the mode location of u 1 ðxþ or u 2 ðxþ are allocated more weights than that of the data away from the mode location. Consequently, the weight functions play their roles in estimation under a certain amount of sample information. Allocation of weights to samples. Equations (31) (35) include the terms of moment estimations weighted by uðxþ. From the point of view of robustness of an estimator, more weights are likely given to samples close to the central location of the ranked samples series, and fewer to data away from the central location. On the other hand, overemphasizing the robustness behavior of an estimator may lose its other statistical properties. In the method of PWM, weights to samples increase with the increase of their ranks in an ascending sample series (i.e., larger weights to larger data and vice versa), which is advantageous to estimations from the point of view of unbiasedness and efficiency. Taking into account the explanation above, we formulate the shapes of / 1 ðxþ and / 2 ðxþ such that the maximum value of / 1 ðxþ appears at the mean position of the distribution, i.e., let the mode of / 1 ðxþ equal to the mean of the distribution, l. Furthermore, restricting the value of k 2 in equation (37) equals to that of k 1 in equation (36). Let u 0 1ðxÞ50, one obtains the mode of / 1 ðxþ X mod 1 5 nlnðnþk 1 n21 (38) where X mod 1 is the mode of weight function u 1 ðxþ. Let X mod 1 5l then k 1 5 n21 nlnðnþ l (39) Let u 2 0ðxÞ50 and consider k 2 5k 1, then the mode of u 2 ðxþ (denoted as X mod 2 ) is obtained X mod 2 52 11 1 12 Lnð2Þ l (40) n22 LnðnÞ For flood extreme samples with sizes n 5 20, 30, 40, and 50 which are common cases in the hydrological frequency analysis, X mod 2 5 1.62, 1.65, 1.67, and 1.68l from equation (40), respectively, which implies that weight function / 2 ðxþ allocates more weights to data larger than l in value. For the weight function / 1 ðxþ, it was assumed that its mode equals its mean, i.e., X mod 1 5l. Therefore, as a whole, through using these two weight functions / 1 ðxþ and / 2 ðxþ, more weights are assigned to data near mean-tail positions of a sample series ranked in ascending order. LIANG ET AL. VC 2014. American Geophysical Union. All Rights Reserved. 3222
3. Comparisons of MWF With WF and L-M Based on the Monte-Carlo Experiments for the PE3 Distribution Monte-Carlo simulation was carried out to demonstrate statistical properties of the MWF for the PE3 distribution, and comparison analysis were also made to the WF and L-M for quantile estimations. 3.1. The Linear Moments (L-M) Parameter Estimation for PE3 Probability Distribution Hosking [1990] defined the L-M of a random variable X to be linear combinations of PWM k 1 5b 0 k 2 52b 1 2b 0 (41) k 3 56b 2 26b 1 1b 0 where b r is the probability weighted moments [Greenwood et al., 1979]. For a sample series in ascending order, an unbiased sample estimator of b r (denoted as b r ) can be computed by using the following relationships: b 2 5 1 n b 0 5 1 n b 1 5 1 n i53 i52 x i i21 n21 x i ði21þði22þ ðn21þðn22þ x i (42) The expectation EðxÞ, standard deviation r, and coefficient of skewness C s of a PE3 probability distribution can be obtained as the following [Hosking and Wallis, 1997]: EðxÞ5k 1 r5k 2 p 1=2 t 1=2 CðtÞ=Cðt11=2Þ C s 52t 21=2 signðs 3 Þ (43) where s 3 5k 3 =k 2 and t is calculated by If 0 < absðs 3 Þ < 1=3, let z53ps 2 3 then 110:2906z t5 z10:1882z 2 10:0442z 3 (44) If 1=3 < js 3 j < 1, let z512js 3 j then t5 0:36067z20:59567z2 10:25361z 3 122:78861z12:56096z 2 20:77045z 3 (45) Equation (42) was used to estimate b m, then equation (41) was used to calculate k 1 ; k 2 ; k 3, finally equations (43) (45) were used to estimate EðxÞ; r s, and C s. 3.2. Experiments of Parameter Sets Assigned to PE3 Population Distributions In order to perform the Monte-Carlo simulation, parameter sets are assigned to a PE3 statistical model to set up the population distributions. Without the loss of generality, the mean of the population distribution can take any value, and thus in this study we fix EðxÞ51. For flood extremes, the coefficient of variation usually varies in a broad range, where a set of typical values is assigned as C v 5 0.4, 0.5, 0.6, 0.8, 1.0. The coefficient of skewness is set to the multiple of C v, i.e., C s/ C v 5 2, 3, 4. The length of the samples n equals to 20, LIANG ET AL. VC 2014. American Geophysical Union. All Rights Reserved. 3223
Table 1. Monte-Carlo Results of Quantile Estimation of All the 60 Experiments a Averaged Absolute Relative Bias (%) Averaged RRMSE (%) Method p 5 99% p 5 99.5% p 5 99.9% p 5 99.99% p 5 99% p 5 99.5% p 5 99.9% p 5 99.99% MWF 2.4 2.6 3.0 3.3 22.4 23.9 26.6 29.3 WF 3.2 3.5 3.7 4.6 23.6 25.2 29.1 33.4 L-M 2.6 2.8 3.0 3.4 23.1 24.6 27.8 31.2 a The value of averaged absolute relative bias is calculated by taking the absolute value of relative bias of each experiment first, and then averaging all the 60 experiments; averaged RRMSE is calculated in the same way; p is the cumulative probability. 30, 40, and 50. Those assignations cover most of the sample cases that may occur in the hydrological flood frequency analysis. In total, there are 60 combinations among these assignations, correspondingly forming the parameter experiments for PE3 population distributions. It should be noted that the same data sets in the Monte-Carlo experiments were used for all three methods (WF, MWF, and L-M). 3.3. Criteria for Assessing the Parameter Estimation Method Let h 0 denote a parameter or quantile of a population probability distribution, and ^h the corresponding sample estimator of h 0. Relative Bias (denoted as RB(h) or RB) and Relative Root Mean Square Error (denoted as RRMSE(h)orRRMSE) are two criteria frequently used to assess statistical properties of an estimator or parameter estimation method. RBðhÞ5 1 m X m ^h2h 0 h 0 3100% (46) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 X m ^hi 2h 0 2 RRMSEðhÞ5t 3100% (47) m where m is the number of repetition samples of Monte-Carlo experiments for each set parameter. In this study, m 5 1000 was used. According to the statistical rule, a good estimator possess the features that its RBðhÞ value is close to zero, while RRMSEðhÞ is as small as possible. These two features are also named as unbiasness and effectiveness of a statistic, respectively. h 0 3.4. Monte-Carlo Results and Analysis We investigated through Monte-Carlo experiments the advantages of MWF for quantile estimation. For flood control engineering works, the design values or quantiles with larger exceedance probabilities are required; therefore, cumulative probabilities of p 5 99, 99.5, 99.9, and 99.99% (corresponding to 100, 200, 1000, and 10,000 year return period, respectively) were considered, and the corresponding quantile was denoted as X p. Table 1 illustrates the overall performance of MWF, WF, and L-M for quantile estimations of four design probabilities of p 5 99, 99.5, 99.9, and 99.99%. It is seen from Table 1 that the averaged absolute RB of MWF by 60 experiments with the four design probabilities are in the interval [2.4%, 3.3%], while for WF it is [3.2%, 4.6%], and [2.6%, 3.4%] for L-M. It reveals from the 60 experiments that MWF and L-M are very close to each other in terms of relative biases, and all bear less bias than WF. As for the effectiveness of the estimation with the four design probabilities, it is shown that the averaged RRMSE of MWF is in the interval [22.4%, 29.3%], while for WF it is [23.6%, 33.4%], and [23.1%, 31.2%] for L-M. Obviously, the MWF is superior to WF and L-M in terms of effectiveness of quantile estimations. Table 2 presents six typical cases randomly selected among the 60 experiments, i.e., the combinations of C V 5 0.4, 0.8, C S /C V 5 2, 3, 4 with n 5 50. Similar to the overall features exhibited by the total 60 experiments, for the six typical cases the MWF offers nearly unbiased estimations for quantiles. The RB values of MWF is in the interval [26.0%, 4.8%], while WFs are [29.3%, 7.9%] and L-Ms [26.9%, 6.3%]. With regard to LIANG ET AL. VC 2014. American Geophysical Union. All Rights Reserved. 3224
Table 2. Relative Bias (RB) and Relative Root Mean Square Error (RRMSE) of Quantile Estimation for Some Specific Monte-Carlo Experiments With n 5 50, C V 5 0.4, 0.8, and Cs/ Cv 5 2, 3, 4 RB (%) RRMSE (%) C v C S /C V Method p 5 99 p 5 99.5 p 5 99.9 p 5 99.99 p 5 99 p 5 99.5 p 5 99.9 p 5 99.99 0.4 2 MWF 20.3 20.4 0.1 0.6 13.9 15.1 17.4 20.3 WF 20.6 20.9 21.1 21.6 14.6 16.2 19.4 23.3 L-M 20.5 20.7 20.8 20.9 14.6 16.0 19.0 22.5 3 MWF 0.8 0.9 1.2 1.5 11.0 12.0 14.2 16.5 WF 22.0 22.4 22.7 23.6 11.9 13.3 16.0 19.2 L-M 1.4 1.6 1.7 1.9 11.7 12.9 15.4 18.3 4 MWF 21.8 21.9 21.9 22.0 12.2 13.3 15.5 17.8 WF 23.2 23.4 23.8 24.3 13.2 14.7 17.9 21.4 L-M 2.9 2.9 3.0 3.1 12.6 13.8 16.3 19.0 0.8 2 MWF 20.1 0.1 0.2 0.5 15.7 16.5 18.1 19.7 WF 20.7 21.0 21.2 21.4 18.9 20.6 23.7 27.1 L-M 20.4 20.2 0.2 0.5 18.0 19.4 22.1 24.9 3 MWF 25.5 25.6 25.8 26.0 19.8 20.9 22.8 24.7 WF 27.7 28.1 28.8 29.3 21.5 23.0 25.6 28.3 L-M 26.7 26.8 26.8 26.9 20.3 21.6 23.8 25.9 4 MWF 3.9 4.2 4.6 4.8 25.0 26.4 28.8 31.0 WF 6.6 6.9 7.3 7.9 27.4 29.4 32.9 36.3 L-M 5.4 5.9 6.1 6.3 26.3 28.0 30.9 33.4 effectiveness, RRMSE of MWF is in the interval [11.0%, 31.0%], while that of WF is [13.3%, 36.3%], and L-M is [11.7%, 33.4%]. It implies that MWF is a better estimator for quantile estimation. It is also found from Table 2 that with the increase of design probability, values of both absolute RB and RRMSE of all the three approaches are getting larger, or the estimation precision is decreasing. 4. Robustness of MWF Compared With WF and L-M by Means of Monte-Carlo Experiments In this section, the robustness of MWF, WF, and L-M were compared by designing the Monte-Carlo experiments such that samples are randomly generated form Log-Pearson type three distribution (LPE3), three parameter Log-Normal distribution (LN3), and Generalized Extreme Value distribution (GEV), but these samples are used as from PE3 distribution; therefore, criteria denoted by equations (41 47) are adopted to assess the statistical properties of MWF, WF and L-M. Let X denote a random variable; if X follows LPE3 distribution, then Y5ln X is a PE3 distributed and its pdf is defined as f ðyþ5 bc CðaÞ ðy2lþc21 e 2bðy2lÞ (48) where y is a real variable with y l; l, b, and c represent the location, scale, and skewness of the distribution, respectively; and CðÞ denotes the gamma function. The pdf of a LN3 distribution is defined as 1 ðln ðx2bþ2ay Þ 2 fðxþ5 pffiffiffiffiffi exp ðx2bþr y 2p 2r 2 y ; x b (49) where b is location parameter and a y and r y are expectation and standard variation of variable Y5ln ðx2bþ. The pdf of a GEV distribution is defined as fðxþ5 1 x2n ð21=kþ21 x2n 21=k 11k exp 2 11k k 6¼ 0; 11k x2n a a a a > 0 (50) LIANG ET AL. VC 2014. American Geophysical Union. All Rights Reserved. 3225
Figure 1. Statistical properties of MWF, WF, and LM for quantile estimations for the cases that samples are generated from parent distributions of LPE3 but used as from PE3 distributions. (a) The comparison of three methods in terms of Absolute Relative Error (ARE) and (b) in terms of Relative Root Mean Square Error (RRMSE). fðxþ5 1 2exp a exp 2 x2n 2 x2n ; k50 (51) a a where n is the location parameter, a the scale parameter, and k the shape parameter of the distribution. To carry out the Monte-Carlo experiment, parameter sets were fixed for LPE3, LN3, and GEV. For LPE3 distribution, E(Y) 5 1.0, 1.5, C v (Y) 5 0.2, 0.3, and C s (Y)/C v (Y) 5 2, 3, 4; for LN3 distribution, b 5 1.4, 1.8, 2.2, a y 5 0.2, 0.4, and r y 5 0.6, 0.8; for GEV distribution, n 5 1.4, 1.8, 2.2, a 5 0.6, 0.8, and k 5 0.2, 0.4. The length of the samples was fixed; n 5 20, 30, 40, and 50 for each of the distribution. So, it means there are 48 sets of parameter experiments for LPE3, LN3, and GEV distributions, respectively. Samples were generated from these parent distributions, but all taken as from PE3 distributions, and then the MWF, WF, and L-M were employed, respectively, to estimate the quantiles with cumulative probabilities of p 5 99, 99.5, 99.9, and 99.99%. Figures 1 3 illustrate the results of Absolute Relative Error (ARE) and RRMSE of the three methods, among which Figure 1 is for the case that samples are from parent distribution of LPE3, Figure 2 is for LN3, and Figure 3 is for GEV. It can be found that, for the three parent distributions, MWF is better than WF and L-M in terms of RRMSE. Taking the case that parent distribution is LPE3 as an example, the value of RRMSE for MWF is in the interval [21.7, 40.6], while for WF it is [35.4, 60.9] and for L-M it is [39.5,62.1]. However, in terms of ARE, the MWF, MF, and L-M show different performances to the three parent distributions. To LPE3 parent distribution, the ARE of MWF is in the interval [6.4, 34.1], while for WF it is [5.3, 41.0] and [9.2, 43.0] for L- M. To LN3 parent distribution, the ARE of MWF is in the interval [6.1, 29.8], while for WF it is [3.5, 28.7] and [5.5, 31.0] for L-M. To GEV parent distribution, the range of ARE is [13.2, 59.8] for MWF, [5.8, 57.3] for WF, and [6.3, 67.2] for L-M. Overall, it indicates that no method among MWF, WF, and L-M shows absolutely overwhelming advantage in terms of ARE. As a whole, from the point of view of robustness, MWF is superior to WF and L-M. 5. Summary and Conclusions The main reason that PWM and L-M perform better than some other conventional parameter estimation methods such as the MOM is that only the first-order weighted sample moments need to be estimated, whereas the MOM has to deal with higher order moment estimates. The effect of powering down the data of a sample series in moment estimation makes both PWM and L-M suffer less bias and more robust in estimation. This reason, together with the pitfalls of the WF method, has provided the motivation for this paper to develop a further powering down estimation. The derivation of MWF method achieves such a goal, i.e., making the estimations of C V and C S from the original WFs squiring samples in moment computations to only the first-order moment calculations. As an example, the concept of MWF was employed for parameter estimation of the PE3 distribution. In the derivation of formulas for estimators, new forms of weighted functions were constructed by combining two LIANG ET AL. VC 2014. American Geophysical Union. All Rights Reserved. 3226
Figure 2. Statistical properties of MWF, WF, and LM for quantile estimations for the cases that samples are generated from parent distributions of LN3 but used as from PE3 distributions. (a) The comparison of three methods in terms of Absolute Relative Error (ARE) and (b) in terms of Relative Root Mean Square Error (RRMSE). negative exponential functions. The determination of weight functions are based on two considerations: (a) relating the weight functions with the sample size in order to reflect the relationship between the amount of sample information and the role of weight function and (b) allocating more weights to data close to mean and tail positions of a ranked sample in ascending order. It is shown throughout the calculations that the weight functions do not cause any problem in computing the terms of denominators in equations (29) and (30). The statistical unbiasness and effectiveness of the proposed MWF were investigated by means of Monte- Carlo experiments on the basis of sampling from PE3 parent distributions, and comparisons with WF and L- M were also presented. It manifests that for most samples the MWF method is slightly superior to L-M in terms of unbiasness or efficiency, but as a whole, these two methods have little difference, while the performance of MWF is obviously better than that of WF. It implies that this powering down of MWF reduces the bias in quantile estimation more than that of the original WF. The robustness of MWF was also studied by Monte-Carlo experiments that samples are from a non-pe3 parent distribution, but taken as from a PE3 distribution. This indicated that MWF is a relatively more robust method than WF and L-M. In conclusion, the modified weight function method is a good approach for parameter estimation of PE3 probability distribution. Even though the study was performed to the PE3 probability distribution, the thoughts of using new weight functions to achieve powering down effect on the moment calculation and improve the precision of quantile estimation can be employed to other probability distributions. However, formulas for estimators as equations (29) and (30) should be rederived. Figure 3. Statistical properties of MWF, WF, and LM for quantile estimations to the cases that samples are generated from parent distributions of GEV but used as from PE3 distributions. (a) The comparison of three methods in terms of Absolute Relative Error (ARE) and (b) in terms of Relative Root Mean Square Error (RRMSE). LIANG ET AL. VC 2014. American Geophysical Union. All Rights Reserved. 3227
Acknowledgments This study was supported by the Major Program of National Natural Science Foundation of China (51190095), the National Basic Research Program of China ( 973 Program) (2010CB951102), and by the National Natural Science Foundation of China (51079039). We are also grateful to all anonymous reviewers for their extremely helpful comments and suggestions on earlier versions of this paper. References Daniele, N., M. Borga, M. Sangati, and F. Zanon (2007), Regional frequency analysis of extreme precipitation in the eastern Italian Alps and the August 29, 2003 flash flood, J. Hydrol., 345, 149 166. Greenwood, J. A., J. M. Landwehr, N. C. Matals, and J. R. Wallis (1979), Probability weighted moments: Definition and relation to parameters of several distributions expressible in inverse form, Water Resour. Res., 15, 1049 1054. Griffis, V. W., and J. R. Stedinger (2007), Log-Pearson type 3 distribution and its application in flood frequency analysis. II: Parameter estimation methods, J. Hydrol. Eng., 12(5), 492 500. Haktanir, T. (1997), Self-determined probability-weight moments method and its application to various distributions, J. Hydrol., 194, 180 200. Hosking, J. R. M. (1990), L-moments: Analysis and estimation of distribution using linear combinations of order statistics, J. R. Stat. Soc., Ser. B, 52, 105 124. Hosking, J. R. M. (2006), On the characterization of distributions by their L-moments, J. Stat. Plann. Infer., 136(1), 193 198. Hosking, J. R. M., and J. R. Wallis (1997), Regional Frequency Analysis, Cambridge Univ. Press, Cambridge, U. K. Karvanen, J., and A. Nuutinen (2008), Characterizing the generalized lambda distribution by L-moments, Comput. Stat. Data Anal., 52(4), 1971 1983. Kirby, W. (1974), Algebraic boundedness of sample statistics, Water Resour. Res., 10, 220 222. Landwehr, J. M., N. C. Matalas, and J. R. Wallis (1979), Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantile, Water Resour. Res., 15, 1055 1064. Liu, G. W. (1990), Parameters estimation for parameter estimation of Pearson type three distribution [in Chinese with English abstract], J. China Hydrol., 4, 1 15. Ma, X. (1984), The weighted function method for calculating hydrologic frequency parameters [in Chinese with English abstract], J. China Hydrol., 21, 1 8. Markus, M., J. R. Angel, L. Yang, and M. I. Hejazi (2007), Changing estimates of design precipitation in Northeastern Illinois: Comparison between different sources and sensitivity analysis, J. Hydrol., 347(1 2), 211 222. Stedinger, J. R., R. M. Vogel, and E. Foufula-Georgiou (1992), Frequency analysis of extreme events, in Handbook of Hydrology, edited by R. Maidment, chap. 18, pp. 18.1 18.66, McGraw-Hill, New York. Wallis, J. R., N. C. Matalas, N. C. Matas, and J. R. Slack (1974), Just a moment!, Water Resour. Res., 10, 211 219. Whalen, T. M., G. T. Savage, and G. D. Jeong (2002), The method of self-determined probability weighted moments revisited, J. Hydrol., 268, 177 191. Zhang, J., and M. J. Hall (2004), Regional flood frequency analysis for the Gan-Ming River basin in China, J. Hydrol., 296, 98 117. LIANG ET AL. VC 2014. American Geophysical Union. All Rights Reserved. 3228