Keywords: L-moments, frequency analysis, goodness-of-fit test, stochastic simulation, acceptance regions, Pearson type III distribution.

Transcription

1 Establishing acceptance regions for L-moments based goodness-of-fit tests for the Pearson type III distribution Yii-Chen Wu, Jun-Jih Liou, Ke-Sheng Cheng* Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. *Corresponding author. Tel: +---1, Fax: +---, rslab@ntu.edu.tw. Abstract Goodness-of-fit tests based on the L-moment-ratios diagram for selection of appropriate distributions for hydrological variables have had many applications in recent years. For such applications, sample-size-dependent acceptance regions need to be established in order to take into account the uncertainties induced by sample L-skewness and L-kurtosis. Acceptance regions of two-parameter distributions such as the normal and Gumbel distributions have been developed. However, many hydrological variables are better characterized by three-parameter distributions such as the Pearson type III and generalized extreme value distributions. Establishing acceptance regions for these three-parameter distributions is more complicated since their L-moment-ratios diagrams plot as curves, instead of unique points for two-parameter distributions. Through stochastic simulation we established sample-size-dependent % acceptance regions for the Pearson type III distribution. The proposed approach involves two key elements the conditional distribution of population L-skewness given a sample L-skewness and the conditional distribution of sample L-kurtosis given a sample L-skewness. The established % acceptance regions of the Pearson type III distribution were further validated through two types of validity check, and were found to be applicable for goodness-of-fit tests for random samples of any sample size between 0 and 00 and coefficient of skewness not exceeding.0. Keywords: L-moments, frequency analysis, goodness-of-fit test, stochastic simulation, acceptance regions, Pearson type III distribution. Introduction Hydrological frequency analysis requires identifying adequate types of distribution for collected random samples. This is commonly done by conducting goodness-of-fit (GOF) tests. Traditionally the chi-square test and the Kolmogorov-Smirnov test are often used for selection of probability distributions for hydrological variables (Haan, 00). Another method of goodness-of-fit test is the method based on ordinary moment-ratio diagrams 1

2 (D Agostino and Stephen, 1). Moment ratios are unique properties of probability distributions and sample moment ratios of ordinary skewness and kurtosis have been used for selection of probability distribution (Kottegoda, ; D Agostino and Stephen, 1). Nevertheless, Tsukatani and Shigemitsu () indicated that ordinary moment-ratio diagrams are not always appropriate because estimation of the fourth moment lacks sufficient accuracy. Thus, they proposed using a combination of variance coefficient and skewness coefficient (see Tsukatani and Shigemitsu () for definitions) which can be derived using the first three moments of the data for GOF test of different types of Pearson distribution. Hosking and Wallis (1) demonstrated that the L-skewness and L-kurtosis are much less biased than the ordinary skewness and kurtosis. As a result, the skewness and kurtosis L-moment-ratio diagram (LMRD) has been suggested as a useful tool for discrimination between candidate distributions (Hosking, ; Hosking and Wallis, 1, Vogel and Fennesset, 1, Hosking and Wallis, 1). L-moment-ratio diagrams of different probability distributions have been derived by Hosking and Wallis (1). As shown in Figure 1, a two-parameter distribution with a location and a scale parameter (such as the normal or Gumbel distribution) plots as a single point on the LMRD, whereas a three-parameter distribution (such as the Pearson type III (PE) distribution or the generalized extreme value (GEV) distribution) with location, scale and shape parameters plots as a curve on the LMRD. However, these theoretical points or curves of various probability distributions on the LMRD are somewhat limited in practical applications since they cannot accommodate for uncertainties induced by parameter estimation using random samples. To circumvent the above problem, Liou et al. (00) established sample-size-dependent acceptance regions for L-moments-based GOF tests for the normal and Gumbel distributions using stochastic simulation. As illustrated in Figure, the acceptance regions are characterized by a set of acceptance ellipses which corresponds to various sample sizes. Empirical formulas were also developed to determine the acceptance regions for any sample size within the [0, 00] range. While the acceptance regions established for the normal and Gumbel distributions may be useful for many applications, hydrological frequency analysis often encounters variables which are better characterized by three-parameter distributions with location, scale and shape parameters. For example, the U.S. Water Resources Council recommended the log Pearson type III (LPE) distribution be used for flood flow frequency analysis (US-WRC, ). In Taiwan the Log-Pearson type III distribution (LPE) was also suggested for rainfall frequency analysis (TW-WRA, 000). Flood flow in Northern Tunisia was shown to be best represented by the GEV distribution (Abida and Elouze, 00). To our knowledge, acceptance regions for L-moments-based

3 GOF tests for three-parameter distributions such as PE and GEV have not been established. Establishing LMRD acceptance regions for three-parameter distributions is more complicated than that for normal and Gumbel distributions. For a two-parameter distribution with a location and a scale parameter, there exists a unique point characterizing its LMRD. Whereas for a three-parameter distribution with location, scale and shape parameters, the LMRD plots as a curve and each point on the curve represents a valid parameter vector. Following the same concept of establishing acceptance region for the normal distribution, it appears that the acceptance region for L-moments-based GOF test of a three-parameter distribution will vary with sample size and the L-skewness. Thus, the objectives of this study are: (1) to demonstrate how the acceptance region of L-moments- based GOF test of a three-parameter distribution can be established, and () to investigate the effect of sample size on the acceptance region using stochastic simulation. The PE distribution is chosen to exemplify the proposed method. This paper is organized as follows. Section L-moments and the L-moment ratio diagram of the PE distribution defines the L-moment-ratios, their probability-weighted-moment estimators, and the L-moment-ratio diagram of the PE distribution. Section Stochastic simulation of the Pearson type III distribution describes details of random sample generation for the PE distribution using stochastic simulation. Detailed discussions on the biasedness of the sample L-skewness and L-kurtosis are also given. Section Establishing acceptance regions for GOF test of the PE distribution explains the rationale for establishing the acceptance regions and gives a detailed account on how the % acceptance regions of GOF tests based on L-moment-ratios of the PE distribution can be established. Section Validation of the LMRD acceptance regions describes the procedures and results of two types of validity check of the sample-size-dependent % acceptance regions. An illustrative example is also given to demonstrate utilization of the sample-size-dependent acceptance regions for GOF test. L-moments and the L-moment ratio diagram of the PE distribution Since its first introduction by Hosking () applications of L-moments in the field of hydrology have been ever increasing. L-moments are defined as linear combinations of expected values of order statistics of a random variable. In terms of linear combination of order statistics, the first four L-moments ( r, r 1,,, ) can be expressed by E 1 X 1:1 1 E X : X 1: (1) ()

4 X E : 1 where X X : X E : X : X k : n 1: X : X 1: is the k-th order statistic from a random sample of size n. Similar to the ordinary moment ratios, the L-moment ratios are defined by r, r,, r Explicit expressions for the relationships between L-skewness ( ) and L-kurtosis ( ), i.e. the L-moment ratio diagram, for commonly used probability distributions have been given by Hosking () and can be expressed by the following polynomial approximations: k 0 A k k For the PE distribution, A 0 =0.1, A =0.0, A =0.1, A =-0., and A =0.1. All other A i values are zero. L-moment ratios can be estimated from random samples using the probability-weighted- moment estimator or the plotting-position estimator (Hosking and Wallis, 1; Liou et al., 00). In general, the probability-weighted-moment estimators are preferred over the plotting-position estimators since they have generally lower bias (Hosking and Wallis, 1). In this study only the probability-weighted-moment estimators were calculated and used for construction of acceptance regions of LMRD. Given a random samplex, x, 1, x n, an unbiased estimator of the probability weighted moment is given by r 1 j 1 j j r x n 1 n n r j n n b r n j r 1 : The sample L-moments ( ) r and sample L-moment ratios ( t r ) can then be calculated by 1 b 0 () b1 b 0 () b b1 b 0 () 0b 0b 1b1 b 0 () t (1) r r Stochastic simulation of the Pearson type III distribution In this study we chose the standard PE distribution (zero mean and unit standard deviation) to exemplify the proposed approach for construction of acceptance regions for () () () () ()

5 GOF tests of a three-parameter distribution based on the L-moment ratio diagram. Probability density function of the PE distribution is expressed as x (1) 1 1 x f ( x) e, x ( ) where,, and are respectively the scale, shape, and location parameters. These parameters are related to the expected value (), standard deviation (), and coefficient of skewness () of the distribution through the following equations: / (1) / (1) (1) For the PE distribution, rational-function approximations can be used to express L-moment-ratios and as functions of the shape factor (Hosking and Wallis, 1). If 1 (or equivalently ), C 1/ 0 1 A0 A 1 A A 1 1B B 1 C 1 C C ; 1D D if 1 (or equivalently ), 1E 1 E E 1F F F 1 1G G G 1. 1H 1 H H Coefficients of the above equations are shown in Table 1. Apparently, among the three distribution parameters only the shape factor affects L-moment ratiosand. From Eq. (1) we see that the shape factor is totally dependent on the coefficient of skewness. Thus, stochastic simulation of the PE distribution was conducted by setting = 0, = 1, and varying from 0.0 to at an increment of 0.0 and an additional case of = 0.001, making a total of 1 parameter settings. Such parameter settings cover a wide range of L-skewness ( ) and corresponding L-kurtosis. For each case of parameter setting (= 0, = 1, ), it corresponds to a specific combination of the scale, shape, and location parameters ( /, PE distribution are generated. /, / (1) (1) (1) (0) ) from which random samples of the

6 In order to assess the effect of sample size on acceptance regions, for each case of the 1 parameter settings, 0,000 random samples were generated with respect to each of the 1 sample sizes n = 0,,,,, 0,,,,, 0, 0, 0, 0, 0,, 00, 0 and 00. Such simulations resulted in a total of () sets of (n, γ) combinations, each having 0,000 random samples. For every (n, γ) combination, sample L-skewness (t ) and L-kurtosis (t ) were calculated for each of the 0,000 samples, using the probability- weighted-moment estimator (Eq. (1)). Figures and respectively illustrate the scattering of sample L-skewness and L-kurtosis, with sample size n = 0 and 0, using 0,000 random samples of the standard PE distributions of various coefficients of skewness. Also shown in Figures and are the % equiprobable density contours of (t, t ), assuming t and t form a bivariate normal distribution. For smaller skewness (for example, 0. 1for sample size n = 0), scattering of (t, t ) approximates a bivariate normal distribution. As the coefficient of skewness increases, approximation by the bivariate normal distribution becomes less satisfactory. In particular, for n = 0 and, the % equiprobable density contour even covers an area with no (t, t ) points. For each case of (n, γ) combination, means of the 0,000 sample estimates of L-skewness and L-kurtosis, respectively represented by (ˆ t ) and (ˆ t ), were calculated (see Table ). For a specific coefficient of skewness, biases of the probability-weighted-moment estimators (t and t ) decrease with increasing sample size, as shown in Figure. From simulated random samples, we found that biases of t and t are dependent on both the sample size n and coefficient of skewness γ(or equivalently L-skewness ) and can be respectively approximated by b(, t, bias) t, bias(, n) E( t) (ˆ t), n b (, t, bias) (1a) (1b) and t, bias b(, t, bias) (, n) E( t) (ˆ t), n b. (, t, bias) (a) (b) The regression coefficients b(γ,t,bias ) and b(γ,t,bias ) only depend on the coefficient of skewness (see Figure ) and Eqs. (1a) (b) can be used for calculating biases of t and t for any sample size n varying between 0 and 00 and coefficient of skewness γnot exceeding.0. It is noteworthy that for the PE distribution t is a biased estimator of with negative bias for all 0 < γ. In other words, t tends to underestimate although the

7 bias is generally negligible. In contrast, t is positively biased for smaller skewness (γ< 0.) and negatively biased for 0. < γ. Establishing acceptance regions for GOF test of the PE distribution Rationale for establishment of acceptance regions For the normal distribution, acceptance regions for L-moment-ratio-based GOF test can be expressed by a set of unique ellipses which almost centered on the theoretical point of (, 0. 1), and depend only on the sample size (Liou et al., 00). However, 0 LMRD of the three-parameter PE distribution plots as a curve and every point on the curve characterizes a specific set of distribution parameter. Even if the scattering of (t, t ) can be well approximated by a bivariate normal distribution, it will be necessary to establish 1 different set of ellipses for different points of, ) on the LMRD curve, as illustrated in ( 1 Figure. As a result, for a given sample pair of L-moment ratios ( t, t ), one will need to 1 identify an ellipse in order to conduct the GOF test. Unfortunately, unlike in the case of 1 normal distribution for which and 0. 1, the population L-moment-ratios (, ) for the given sample pair t, ) is never known, and thus the (, )-specific ( t acceptance ellipses approach can not be adopted for GOF test of the PE distribution. The above argument indicates that, in conducting an L-moment-ratios based GOF test for a three-parameter distribution such as PE or GEV, one needs to consider all possible points in the parameter space which may give rise to random samples with a certain sample pair of L-moment-ratios (t, t ). As illustrated in Figure, a sample pair of L-moment-ratios ( t t, t t ) may originate from PE distributions with varying mostly between a and b (or corresponding range for the coefficient of skewness γ). Such PE distributions can be identified by considering the conditional density of given t t, i.e. f ( t t ) as shown in Figure. Additionally, PE random samples with sample L-skewness t t are associated with a range of sample L-kurtosis which can be characterized by the conditional density of t given t, i.e. g t t t ). t ( 0 Given a random sample of size n, the practice of GOF test for a probability distribution with level of significance will result in a 0% chance of rejecting the null hypothesis, if the random sample is indeed from that distribution. Thus, given a large number of PE 1 random samples of size n with sample L-skewness t, the 0(1 )% t acceptance

8 region of the L-moment-ratios-based GOF test can be determined from the conditional density g t t t ) and it should result in a nearly 0% ( rejection rate. Hereafter, the 1 acceptance region for random samples of a specific sample L-skewness (for example, t ) t is referred to as the t -specific acceptance region. The same procedure can then be repeated for a range of t values, and finally an acceptance band with respect to the sample size n can be constructed by fitting a set of upper and lower bound curves to 0(1 )% upper bounds and lower bounds of the t -specific acceptance regions. Conditional density of given t As has been described in the previous section, the t -specific acceptance region can be determined from the conditional density of t givent, i.e. g ( t t). To achieve a good estimation of g t t t ), we need to ensure having a large set of random samples (with ( 1 sample L-skewness t t ) that are generated from PE distributions of all possible values of. While it is impossible to generate random samples from PE distributions of all possiblevalues, we could at least settle for a range of which accounts for a very large portion of possiblevalues. This is achieved by investigating the conditional density of given t, i.e. f ( t), using simulated random samples of various sample size n and coefficient of skewness ( ) through the following procedures. For a specific sample size (say n = 0), our simulation of PE random samples with various coefficient of skewness results in a huge data set of (t, t ) sample pairs which form the data cloud shown in Figure. For a specific sample L-skewness t, all simulated samples with sample L-skewness falling within a small range of t ( in our study) were collected. Each simulated sample within this range corresponds to a PE population with certain L-skewness. Thus a relative histogram of can be constructed and is considered as an estimate of the condition density of given t, i.e. f t t ). t ( The same procedures were applied to simulated random samples of different sample size n and different t values, yielding estimate of f ) for various sample L-skewness t. ( t Through the Kolmogorov-Smirnov GOF test we found that the conditional densities f ) are normally distributed with expected value almost equaling t and standard ( t

9 deviation ( ) decreasing with increasing sample size. The standard deviation seems to be dependent only on the sample size, and, as demonstrated by Figure, for larger sample size ( n 0 ) the dependency approaches 0. ( n). () n Equation () indicates that % of PE samples with a specific sample L-skewness t come from PE distributions with population parameterfalling within the range of ( t 1. ( n), t 1. ( )) n. Such result is important in that it affects the upper limit for establishing t -specific acceptance regions using our simulated data. For example, at sample size n = 0, equals 0.1 and % of PE samples with a specific t come from PE distributions withfalling within the range of ( t 0., t 0. ). Since our random samples were simulated for not exceeding (or equivalently0.0), the upper limit of the t -specific acceptance region for sample size n=0 is t = 0. (0.0 0.) which is approximately equivalent to γ= (or 0.). If establishment of t -specific acceptance region is desired for higher t values, it will be necessary to generate random samples of PE distributions with coefficient of skewness higher than. Determining acceptance regions The conditional density g t t ) is the basis for determining the t -specific acceptance ( region. Given a specific sample L-skewness t, all simulated samples with sample 1 L-skewness falling within the t range were collected and considered as a random 0 1 sample of g ( t t). The and 1 quantiles of this random sample were then extracted from its empirical cumulative distribution function, and they respectively represent the lower and upper bounds of the 0(1 )% acceptance region with respect to the given sample L-skewness. The acceptance regions determined in this manner not only depend on t but also sample size n. Figure demonstrates the % acceptance regions for sample size n = 0, 0, and 0. It is also worthy to note that, unlike f ) which is ( t g( t t normally distributed, normality assumption for the conditional density ) is not

10 generally valid. Figure 1 demonstrates histograms of t derived from random samples with sample size n = 0 and 0, given certain t values. For higher sample L-skewness, for example t = 0., the conditional density g t t ) appears to be positively skewed even for ( sample size n = 0. In order for the acceptance regions to be applicable for arbitrary sample size n and sample L-skewness t, the upper and lower bounds of the acceptance regions need to be expressed as functions of the sample size and sample L-skewness. We first calculate deviations of the upper and lower bounds from the theoretical ~ curve (see ( t, n U ) and ( t, n) in Figure ()) and then express these bound deviations as functions of the L sample L-skewness through the following regression models: ( t t U, n) a 0( n) a 1( n) t a ( n) t a ( n) () ( t t L, n) b 0( n) b 1( n) t b ( n) t b ( n) () Coefficients of the above equations with respect to 1 sample sizes are listed in Table. From the very high R values of the regression models, we can be sure that the dependence of bound deviations on sample L-skewnes is well characterized by the models. However, regression coefficients listed in Table are specific only to a few sample sizes we have used. In order for Equations () and () to be generally applicable the following regression model is adopted to express those coefficients ( a, ) as functions of the sample size n: w1 w n w n y 0 where n is the sample size and y represents one of the regression coefficients in Equations () and (). Regression coefficients of Equation () are listed in Table. Combination of Equations (1)-(0) and ()-() allows us to establish the % acceptance regions of PE distribution with respect to any sample size between 0 and 00 and sample L-skewness t not exceeding 0.. Validation of the LMRD acceptance regions The sample-size-dependent acceptance regions established in the previous section are further checked for their validity. Two types of validation, namely the t -specific and -specific validity checks, were conducted. The t -specific validity check began by stochastically generating 0,000 standardized PE samples (= 0, = 1), with the same sets of coefficient of skewness γand sample size n as described in the section Stochastic simulation of the Pearson type I distribution. Sample L-skewness and L-kurtosis t, ) of each random sample were calculated and ( t i bi ()

11 LMRD-based GOF test was conducted using the sample-size-dependent % acceptance regions established in the previous section. Then various sets of (t 0.00) interval for t not exceeding 0. (corresponding to γ= ) were selected. For a specific sample size n, each selected (t 0.00) interval is composed of a large number, say N, of sample (t, t ) pairs. The rejection rate, i.e. the type I error, is then calculated as ˆN r / N where N r represents the rejected number of random samples within the (t 0.00) interval. For sample-size-dependent % acceptance regions to be valid the rejection rateˆ should be very close to the level of significance ( 0.0), or equivalently the acceptance rate be very close to 0.. Results of the t -specific validity check are shown in Figure 1 for sample size n = 0, 0, 0, 0, 0, and 00. The acceptance rates all fall within a very small range of [0., 0.]. Such results strongly indicate the validity of the sample-size- dependent % acceptance regions established in this study, and that they can be applied to GOF test for PE random samples of any sample size between 0 and 00 and sample L-skewness not exceeding 0.. The above validation is done with respect to sample L-skewness t. This is normally the case since the population L-skewness is never known. However, for complete demonstration of its applicability, we also check the validity of the established acceptance regions for random samples of known population parameters (,, γ), i.e. validation with respect to population L-skewness. Again, we simulated 0,000 random samples from the PE distribution with respect to each combination of (= 1., = 1., γ), with 1 values of skewness γthe same as described in previous sections, for sample size n = 0, 0 and 0. Each combination of (,, γ) corresponds to a specific pair of (,) since () population L-skewness and L-kurtosis only depend on the coefficient of skewness γ. For every pair of (, ), GOF test using the established acceptance regions was conducted for a total of 0,000 random samples with sample size n = 0, 0 and 0. Results of such -specific validity check are shown in Figure 1. Except for few cases of sample size n = 0 and 0., acceptance rates all fall within the (0. 0.) range, further confirming the validity of the established sample-size-dependent acceptance regions. Although the PE distribution was targeted for establishment of the % acceptance regions for L-moment- 0 ratios based GOF test in this study, we believe the same treatment can be applied to 1 establishing acceptance regions of other three-parameter distributions such as the GEV distribution. We end this paper by exemplifying usage of the established acceptance regions for LMRD-based GOF test of the PE distribution. Suppose a random sample of size n = is

12 available and its sample L-skewness and L-kurtosis are calculated as t = 0. and t = 0.1, respectively. Using Eq. (), the corresponding regression coefficients for Equations () and () are calculated as a., a 0. 1, a., a and b 0 0.1, b1 0. 0, b 1., b The upper and lower bound deviations from the theoretical ~ curve are U 0. and L 0. based on Equations () and (). Corresponding to 0., we have 0. from Equation (1). Thus, the upper and lower bounds of the % acceptance region for t are 0. (0.+0.) and 0.01 ( ) respectively. The sample L-skewness t = 0.1 falls within this acceptance region and therefore the null hypothesis (the sample is from a PE distribution) is not rejected. Summary and conclusions In this paper we present a detailed account of a stochastic simulation approach for establishing % acceptance regions of L-moment-ratios-based GOF test for the PE distribution. Unlike the normal and Gumbel distributions, the LMRD of the PE distribution plots as a curve, and t -specific acceptance regions need to be established through stochastic simulation. Properties of the sample L-skewness and L-kurtosis estimated by the probability-weighted-moment estimator are discussed. The sample-size-dependent % acceptance regions of the L-moments-based GOF tests for the PE distribution are further validated through a stochastic simulation process. A few concluding remarks are drawn as follows: 1. For Pearson type III distributions of smaller coefficient of skewness, scattering of sample L-skewness and L-kurtosis can be approximated by a bivariate normal distribution. As the coefficient of skewness increases, the bivariate normal approximation becomes less satisfactory.. For the Pearson type III distribution, the asymptotic biases of the sample L-skewness and L-kurtosis of the probability-weighted-moment estimator are dependent on both the sample size and coefficient of skewness. More specifically, sample L-skewness tends to underestimate the population L-skewness, although the bias is generally negligible. In contrast, sample L-kurtosis is positively biased for smaller skewness and negatively biased for γ> 0... PE random samples of the same sample L-skewness can come from PE distributions of various population L-skewness. The likelihood of these population L-skewness can be characterized by a conditional normal density whose standard deviation is inversely 1

13 proportional to the square root of sample size.. The sample-size-dependent % acceptance regions established in this study have been thoroughly validated through a stochastic simulation process and can be applied to GOF test for random samples of any sample size between 0 and 00 and sample skewness not exceeding.. Although the PE distribution was targeted for establishment of the % acceptance regions for L-moment-ratios-based GOF test in this study, the same treatment can be applied to establishing acceptance regions of other three-parameter distributions such as the GEV distribution. Acknowledgements We gratefully acknowledge the National Science Council of Taiwan, R.O.C. for financially supporting this research. References Abida, H. and Elouze, M., 00. Probability distribution of flood flows in Tunisia. Hydrology and Earth System Sciences Discussions, 1. D Agostino, R.B. and Stephens, M.A., 1. Goodness-of-Fit Techniques. Marcel Dekker, New York. Haan, C.T., 00. Statistical Methods in Hydrology. Ames, Iowa, Iowa State Press. Hosking, J.R.M.,. L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society Series B (1), -1. Hosking, J.R.M. and Wallis, J.R., 1. Some statistics useful in regional frequency analysis. Water Resources Research (), 1-1. Hosking, J.R.M. and Wallis, J.R., 1. Regional frequency analysis: an approach based on L-moments. Cambridge University Press, Cambridge, U.K. Kottegoda, N.T.,. Stochastic water resources technology. Macmillan, London, U.K. Liou, J.J., Wu, Y.C., and Cheng, K.S., 00. Establishing acceptance regions for L-moments based goodness-of-fit tests by stochastic simulation. Journal of Hydrology,. Tsukatani, T. and Shigemitsu, K.,. Simplified Pearson distributions applied to air pollutant concentration. Atmospheric Environment 1,. U.S. Water Resources Council (US-WRS),. Guildlines for determining flood flow frequency, bulletin 1B. Office of Water Data Coordination, U.S. Geological Survey, Reston, Virginia. Vogel, R.M. and Fennesset, N.M., 1. L-moments diagrams should replace product 1

14 moment diagrams. Water Resources Research (), 1-1. Taiwan Water Resources Agency (TW-WRA), 000. Technical Code for Hydrological Design (in Chinese). Ministry of Economic Affairs, Taipei, Taiwan. 1

15 Table 1. Coefficients of the rational-function approximations of L-skewness and L-kurtosis of the Pearson type III distribution. A 0 =.01-1 C 0 = A 1 = C 1 =.0 - A =. - C =. - A = -. - C = 1. - B 1 =. -1 D 1 = B =.0-1 D =.0-1 E 1 =.0 G 1 =.1 E = 1.1 G =.1 E = G =.1 F 1 =.1 H 1 =.01 F =. H =. 1 F = 1.0 H =.1 1 1

16 Table. Sample means of t and t of the Pearson type III distributions with different values of coefficient of skewness. Sample (ˆ) t t (ˆ ) size, n 1 (τ =0.1) (τ =1) (τ =0.0) 1 (τ =0.1) (τ =1) (τ =0.0)

17 Table. Regression coefficients of the empirical relationships between the bound deviations and the sample L-skewness. Sample Coefficients for the upper bound Coefficients for the lower bound size deviations (Eq. ()) deviations (Eq. ()) n a 0 a 1 a a R b 0 b 1 b b R

18 Table. Regression coefficients of Equation (). w1 w n w n y 0 y w 0 w 1 w R a a a a b b b b

19 Figure 1. L-moment-ratio diagram of the normal, Gumbel, generalized extreme value (GEV), and Pearson type III (PT) distributions. 1

20 Normal Gumbel Figure. % acceptance regions of L-moments-based GOF test for the normal and Gumbel distributions. Acceptance ellipses correspond to various sample sizes (n = 0, 0, 0, 0, 0,, 0,, 0, 00, and 1,000). [Liou et al., 00] 1 1 0

21 Figure. Scatter plot of sample (t, t ) pairs with respect to various skewness. Each plot is constructed based on 0,000 PE random samples of size n = 0. The ellipses represent % equiprobable density contour of (t, t ), assuming t and t form a bivariate normal distribution. Red dots mark the theoretical value of (, ) of the PE distribution. 1

22 Figure. Scatter plot of sample (t, t ) pairs with respect to various skewness. Each plot is constructed based on 0,000 PE random samples of size n = 0. The ellipses represent % equiprobable density contour of (t, t ), assuming t and t form a bivariate normal distribution. Red dots mark the theoretical value of (, ) of the PE distribution.

23 Figure. Biases of sample L-skewness (t,bias ) and L-kurtosis (t, bias ) of the Pearson type III distribution vary with sample size n. See Figure for b(γ,t,bias ) and b(γ,t,bias ). 1

24 b (, t, bias ) R b(, t, bias ) R Coefficient of skewness, γ Figure. Dependence of b(γ, t,bias ) and b(γ, t,bias ) on the coefficient of skewness γ.

25 Figure. Conceptual illustration of % acceptance ellipses of three points ( i) ( i) (,, i 1,, ) on the LMRD of PE distribution. The point (t, t ) represents a sample pair of L-moment-ratios. 1

26 t Upper bound of the % acceptance region oft given t t. Conditional distribution of % t given t t. g t t t ) ( Lower bound of the % acceptance region oft given t t. ( t * *, t ) ( t, t ) f ( t t ) f ( t t * ) a b Figure. Conceptual illustration of the % acceptance region for PE random samples having sample L-skewness t. t t 0 1

27 t t 0.00 t t t Estimate of the conditional density f t ). ( Figure. Data cloud of (t, t ) using simulated PE random samples with sample size n = 0 and various coefficients of skewness. Relative histogram of corresponding to all random samples with t t is also shown in the figure.

28 n=0 n=0 0. ( n) n n Figure. Standard deviation of the conditional density f ( t) expressed as a function of sample size n.

29 n = 0 n = 0 n = 0 t U ( t, n) L ( t, n) Figure. Upper and lower bounds of the t -specific % acceptance regions with respect to sample size n = 0, 0, and 0. The solid line represents the theoretical ~ curve of the PE distribution. t 1 1

30 n 0 t n 0 t n 0 t 0. n 0 t 0. n 0 t 0. n 0 t 0. Figure 1. Empirical histograms of t for different given values of t. 0

31 t Figure 1. % acceptance regions of (t, t ) for PE random samples of size n = 0, 0, 0, 00, and 00. The theoretical curve represents the ~ relationship of the PE distribution. t % acceptance region for sample size n = 0. n = 0 n = 0 n = 0 n = 00 n = 00 Theoretical 1

32 n=0 n=0 t t n=0 n=0 t t n=0 n=00 t t Figure 1. Acceptance rates of t -specific validity check for sample-size-dependent % acceptance regions of sample L-skewness and L-kurtosis pairs.

33 n=0 n=0 n=0 Figure 1. Acceptance rates of -specific validity check for sample-size-dependent % acceptance regions of sample L-skewness and L-kurtosis pairs.