# Comparative Study on the Selection Criteria for Fitting Flood Frequency Distribution Models with Emphasis on Upper-Tail Behavior

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Typical Probability Distributions

#### 2.2. Model Selection Methods

- (1)
- The log-likelihood function value for each probability model was computed according to Table 1. Where parameters P (scale, location, shape) are the parameter values that maximize the log-likelihood function. The estimation method for parameter P of flood frequency probability models is the maximum likelihood, which was used to compute the log-likelihood function for each probability model.
- (2)
- The values of AIC, BIC, AICc can be computed according to Table 1 on the basis of the value of log-likelihood function and the number of parameters.

#### 2.3. Parameter Estimation

#### 2.4. Rigorous Program to Select the Optimal Distribution by Hypothesis Tests and Information-Based Criteria

- (1)
- The candidate distributions are ordered from most to least favourite with AIC, BIC, AICc criteria. If the first distribution with the highest number of occurrences was selected respectively by AIC, BIC, AICc, then it is selected as the first optimal distribution of the information criteria.
- (2)
- After selecting the first optimal distribution, it is removed from the candidate distributions. Repeat step (1) to find the best distribution from the remaining distributions as the second optimal distribution.
- (3)
- In step (1), if two or more distributions have the same number of times appearing at the first position, then they will be sorted by the total number of occurrences in the preferred distribution (two or more distributions) selected respectively by AIC, BIC, AICc; the distribution with more occurrences is preferred.

#### 2.5. Composite Criterion for Model Selection with Focus on the High Flow Part

- (1)
- Choose a distribution from which the simulated data are generated. The Kappa and Wakeby distributions are widely recommendable choices [12]. Hosking (1997) used the four-parameter Kappa distribution as the overall simulation in regional flood frequency analysis and obtained reliable simulation results. The same distribution was used for the simulations in this study [39].
- (2)
- The four-parameter Kappa distribution, as the parent distribution, was estimated by L-moments of samples for the observed flood flow to determine parameter values. The synthetic samples, with the same length of the observations, were randomly simulated from the fitted four-parameter Kappa distribution. The detailed steps are described below:First, the first four order linear moments are obtained based on the observed sequence. Then, based on the linear moment of the observed data, the L-moments method is used to estimate the parameters of the Kappa distribution. Finally, a random sample is generated using the Kappa distribution with the estimated parameter values. The length of the random sample is the same as the length of the observed sequence.
- (3)
- The simulated samples were fitted by eight distributions as recommended before. All eight probability distributions were then used to estimate the design floods with return periods T=5, 10, 20, 30, 50, 70, 90, 100 and 200 years.
- (4)
- Repeat steps (2) and (3) for a given number of times (denoted by N
_{sim}), and save the calculated results. N_{sim}= 500 in this study. - (5)
- The relative error of the design value (RE) for each simulation was calculated by$$RE=\frac{{\stackrel{\wedge}{X}}_{i,T}-{X}_{T}}{{X}_{T}}$$
- (6)
- The root-mean-square error(RMSE) was calculated as the quantile corresponding to the assigned return periods, T = 5, 10, 20, 30, 50, 70, 90, 100 and 200 years.$$RMSE(T)=\sqrt{\frac{1}{{N}_{sim}}{\displaystyle \sum _{i=1}^{{N}_{sim}}{\left(\frac{{\widehat{X}}_{i,T}-{X}_{T}}{{X}_{T}}\right)}^{2}}}$$
_{sim}is the number of Monte Carlo simulations; other notations are the same as in Equation (1). - (7)
- The arithmetic mean $\overline{RMSE}$ of the RMSE was calculated for the return period T for a given distribution.
- (8)
- The $\overline{RMSE}$ and Box plots of REs are the composite criteria used for assessing the degree of the goodness-of-fit at the high flow part. The smaller $\overline{RMSE}$ value means a better fitting.

#### 2.6. Verify the Performance of the Five Selection Criteria by Using a Composite Criterion

- (1)
- The optimal (ranked as the top two) distributions selected by hypothesis tests and information-based criteria are listed first.
- (2)
- Test the performance of distribution selected by hypothesis tests and information-based criteria on the large floods with a long return period by a composite criterion.
- (3)
- Based on the test results by the composite criterion, compare the estimation error of distribution selected by hypothesis tests and information-based criteria for large floods. If the estimation error is small, this criterion which selected the distribution is better for high flow part (Shown as Box plots of RMSE and RE).

#### 2.7. Change Point of Flood Series Detection

## 3. Study Area and Data

^{2}. It is located at a temperate climate zone with high humidity and relatively stable temperature. Kingston station, located at the lower reach of Thames River, is used in the study. The skewness coefficient Cs of the flood series at Kingston station is large with the value of 1.181, which implies a steep upper tail of the optimal frequency distribution.

^{2}, is used in the study. The small Cs value of 0.280 for the flood series at Lafayette station indicates that the upper tail of the optimal frequency distributions is gentle at this station.

^{2}) located at the lower reach of the Beijiang River, is used in this study. The small Cs value of 0.230 for the flood series at Shijiao station shows a gentle upper tail frequency distribution at this station.

^{2}, is selected as a case study in this paper. For Lutaizi Station, the large Cs value of 1.198 infers a steep upper tail frequency distribution.

## 4. Results and Discussions

#### 4.1. Optimal Frequency Distribution for Different Model Selection Methods

#### 4.2. Composite Criterion for Model Selection

#### 4.3. Comparison on Hypothesis Tests and Information-Based Criteria for Upper Tail

#### 4.3.1. Characteristics of Statistical Hypothesis Test

#### (1) Kolmogorov–Smirnov (KS)

#### (2) Anderson–Darling Criterion (AD)

#### (3) Characteristics Summary

#### 4.3.2. Characteristics of Information-Based Criteria

#### (1) AIC, AICc Criteria

#### (2) BIC Criterion

#### (3) Characteristics Summary

## 5. Conclusions

- (1)
- There are different selections of frequency distributions in the four rivers by using hypothesis tests and information-based criteria approaches. Hypothesis tests are more likely to choose complex, parametric models, and information-based criteria prefer to choose simple, effective models. Different selection criteria have no particular tendency toward the tail of the distribution.
- (2)
- The information-based criteria perform better than hypothesis test methods most of the time when focusing on the goodness of predictions of the extreme upper tails of PDs. The distributions selected by information-based criteria are more likely to be close to true values than the distributions selected by hypothesis test methods in the upper tail of the frequency curve.
- (3)
- The composite criterion not only can select the optimal distribution, but also can evaluate the error of the estimated value. In order to decide on a particular distribution to fit the high flow, it would be better to use the composite criterion.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Hussain, Z.; Pasha, G.R. Regional flood frequency analysis of the seven sites of Punjab, Pakistan, using L-Moments. Water Resour. Manag.
**2009**, 23, 1917–1933. [Google Scholar] [CrossRef] - Chérif, R.; Bargaoui, Z. Regionalisation of Maximum Annual Runoff Using Hierarchical and Trellis Methods with Topographic Information. Water Resour Manag.
**2013**, 27, 2947–2963. [Google Scholar] [CrossRef] - Faulkner, D.; Keef, C.; Martin, J. Setting design inflows to hydrodynamic flood models using a dependence model. Hydrol. Res.
**2012**, 43, 663–674. [Google Scholar] [CrossRef] - Xia, J. Identification of a constrained nonlinear hydrological system described by Volterra Functional Series. Water Resour. Res.
**1991**, 27, 2415–2420. [Google Scholar] - Laio, F.; Baldassarre, D.G.; Montanari, A. Model selection techniques for the frequency analysis of hydrological extremes. Water Resour. Res.
**2009**, 45. [Google Scholar] [CrossRef] - Gomes, O.; Combes, C.; Dussauchoy, A. Parameter estimation of the generalized gamma distribution. Math. Comput. Simul.
**2008**, 79, 955–963. [Google Scholar] [CrossRef] - Cicioni, G.; Guiliano, G.; Spaziani, F.M. Best fitting of probability functions to a set of data for flood studies. In Proceedings of the 2nd International Symposium on Hydrology of Floods and Droughts, Fort Collins, CO, USA, 11–13 September 1972; Water Resource Publication: Fort Collins, CO, USA, 1973; pp. 304–314. [Google Scholar]
- Akaike, H. Information theory and an extension of the maximum likelihood principle. Proceeding of the International Symposium on Information Theory, Budapest, Hungary, 2–8 September 1971; Peter, B.N., Csaki, F., Eds.; Akademiai Kiado: Budapest, Hungary, 1973; pp. 267–281. [Google Scholar]
- Hosking, J.R.M.; Wallis, J.R. The Value of Historical Data in Flood Frequency Analysis. Water Resour. Res.
**1986**, 22, 1606–1612. [Google Scholar] [CrossRef] - Haktanir, T.; Horlacher, H.B. Evaluation of various distributions for flood frequency analysis. Hydrol. Sci. J.
**1993**, 38, 15–32. [Google Scholar] [CrossRef] - Haddad, K.; Rahman, A.; Stedinger, J.J.R. Regional flood frequency analysis using Bayesian generalized least squares: A comparison between quantile and parameter regression techniques. Hydrol. Process.
**2012**, 26, 1008–1021. [Google Scholar] [CrossRef] - Baldassarre, G.D.; Laio, F.; Montanaric, A. Design flood estimation using model selection criteria. Phys. Chem. Earth
**2009**, 34, 606–611. [Google Scholar] [CrossRef] - Calenda, G.; Mancini, C.P.; Volpi, E. Selection of the probabilistic model of extreme floods: The case of the River Tiber in Rome. J. Hydrol.
**2009**, 371, 1–11. [Google Scholar] [CrossRef] - Burnham, K.P.; Anderson, D.R. Model Selection and Multimodal Inference: A Practical Information-Theoretic Information-Theoretic Approach, 2nd ed.; Springer-Verlag: New York, NY, USA, 2002. [Google Scholar]
- Önöz, B.; Bayazit, M. Best-fit distributions of largest available flood samples. J. Hydrol.
**1995**, 167, 195–208. [Google Scholar] [CrossRef] - House, P.K.; Baker, V.R. Paleohydrology of flash floods in small desert watersheds in western Arizona. Water Resour. Res.
**2001**, 37, 1825–1839. [Google Scholar] [CrossRef] - Hosking, J.R.M.; Wallis, J.R. Paleoflood Hydrology and Flood Frequency Analysis. Water Resour. Res.
**1986**, 22, 543–550. [Google Scholar] [CrossRef] - Luo, P.P.; He, B.; Takara, K.; APIP; Nover, D.; Kobayashi, K.; Yamashiki, Y. Paleo-hydrology and Paleo-flow Reconstruction in the Yodo River Basin. Annu. Disas. Prev. Res. Inst. Kyoto Univ.
**2011**, 54, 119–128. [Google Scholar] - Begueria, S.; Angulo-Martinez, M.; Vicente-Serrano, S.M.; Lopez-Moreno, J.I.; El-Kenawy, A. Assessing trends in extreme precipitation events intensity and magnitude using non-stationary peaks-over-threshold analysis: A case study in northeast Spain from 1930 to 2006. Int. J. Climatol.
**2011**, 31, 2102–2114. [Google Scholar] [CrossRef] - Magilligan, F.J.; Nislow, K.H. Changes in hydrologic regimes by dams. Geomorphology
**2005**, 71, 61–78. [Google Scholar] [CrossRef] - Salas, J.D.; Obeysekera, J. Revisiting the concepts of return period and risk for nonstationary hydrologic extreme events. J. Hydrol. Eng.
**2014**, 19, 554–568. [Google Scholar] [CrossRef] - Milly, P.C.D.; Betancourt, J.B.; Falkenmark, M.; Hirsch, R.M.; Kundzewicz, Z.W.; Lettenmaier, D.P.; Stouffer, R.J. Stationarity is dead: Whither water management? Science
**2008**, 319, 2. [Google Scholar] [CrossRef] [PubMed] - Khaliq, M.N.; Ouarda, T.B.M.J.; Ondo, J.C.; Gachon, P.; Bobée, B. Frequency analysis of a sequence of dependent and/or non-stationary hydro-meteorological observations: A review. J. Hydrol.
**2006**, 329, 534–552. [Google Scholar] [CrossRef] - Chen, X.H.; Zhang, L.J.; Xu, C.-Y.; Zhang, J.M.; Ye, C.Q. Hydrological design of non-stationary flood extremes and durations in Wujiang river, South China: Changing properties, causes and impacts. Math. Probl. Eng.
**2013**, 2013. [Google Scholar] [CrossRef] - Yan, L.; Xiong, L.H.; Liu, D.D.; Hu, T.S.; Xu, C.-Y. Frequency analysis of nonstationary annual maximum flood series using the time-varying two-component mixture distributions. Hydrol. Process.
**2017**, 31, 69–89. [Google Scholar] [CrossRef] - Rao, A.R.; Hamed, K.H. Flood Frequency Analysis; CRC Press: Boca Raton, FL, USA, 2000; pp. 10–12. [Google Scholar]
- Reiss, R.-D.; Thomas, M. Statistical Analysis of Extreme Values: With Applications to Insurance, Finance, Hydrology and Other Fields; Birkäuser: Basel, Switzerland, 2001; pp. 49–50. [Google Scholar]
- El Adlouni, S.; Bobée, B.; Ouarda, T.B.M.J. On the tails of extreme event distributions in hydrology. J. Hydrol.
**2008**, 355, 16–33. [Google Scholar] [CrossRef] - Reis, D.D.S.; Stedinger, J.J.R. Bayesian MCMC flood frequency analysis with historical information. J. Hydrol.
**2005**, 313, 97–116. [Google Scholar] [CrossRef] - Huang, W.R.; Xu, S.D.; Nnaji, S. Evaluation of GEV model for frequency analysis of annual maximum water levels in the coast of United States. Ocean Eng.
**2008**, 35, 1132–1147. [Google Scholar] [CrossRef] - Martins, E.S.; Stedinger, J.R. Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data. Water Resour. Res.
**2000**, 36, 737–744. [Google Scholar] [CrossRef] - Hirose, H. Maximum likelihood estimation in the 3-parameter Weibull distribution: A look through the generalized extreme-value. IEEE Trans. Dielectr. Electr. Insul.
**1996**, 3, 43–55. [Google Scholar] [CrossRef] - Otten, A.; Montfort, V. Maximum-likelihood estimation of the general extreme-value distribution parameters. J. Hydrol.
**1980**, 47, 187–192. [Google Scholar] [CrossRef] - Yevjevich, V. Probability and Statistics in Hydrology; Water Resources Publications: Fort Collins, CO, USA, 1972; pp. 214–232. [Google Scholar]
- Ben-Zvi, A. Rainfall intensity–duration–frequency relationships derived from large partial duration series. J. Hydrol.
**2009**, 367, 104–114. [Google Scholar] [CrossRef] - Laio, F. Cramer-von Mises and Anderson-Darling goodness of fit tests for extreme value distributions with unknown parameters. Water Resour. Res.
**2004**, 40, W09308. [Google Scholar] [CrossRef] - Burnham, K.P.; Anderson, D.R. Multimodel Inference: Understanding AIC and BIC in model selection. Sociol. Methods Res.
**2004**, 33, 261–304. [Google Scholar] [CrossRef] - Moriasi, D.N.; Arnold, J.G.; Van Liew, M.W.; Bingner, R.L.; Harmel, R.D.; Veith, T.L. Model evaluation guidelines for systematic quantification of accuracy in watershed simulations. Trans. Asabe,
**2007**, 50, 885–900. [Google Scholar] [CrossRef] - Hosking, J.R.M.; Wallis, J.R. Regional Frequency Analysis: An Approach Based on L-moments; Cambridge University Press: Cambridge, UK, 1997; pp. 19–20. [Google Scholar]
- Xie, P.; Lei, H.F.; Chen, G.C.; Li, J. A Spatial and Temporal Variation Analysis Method of Watershed Rainfall Based on Hurst Coefficient. J. China hydrol.
**2008**, 28, 6–10. [Google Scholar] - Wallis, J.R.; Matalas, N.C. Small sample properties of H and K—Estimations of the Hurst coefficient h. Water Resour. Res.
**1970**, 6, 1583–1594. [Google Scholar] [CrossRef]

**Figure 3.**A comparison of the eight typical frequency distributions for four rivers with parameters estimated by MLE. (

**a**) Thames River; (

**b**) Wabash River; (

**c**) Beijiang River and (

**d**) Huai River.

**Figure 4.**Box plots of the relative errors (REs) of the Kingston at Thames River for sample series length 127, with Kappa as the parent probability distribution (PD).

**Figure 5.**Box plots of the relative errors (REs) of the Lafayette at Wabash River for sample series length 85, with Kappa as the parent PD.

**Figure 6.**Box plots of the relative errors (REs) of the Shijiao at Beijiang River for sample series length 53, with Kappa as the parent PD.

**Figure 7.**Box plots of the relative errors (REs) of the Lutaizi at Huai River for sample series length 48, with Kappa as the parent PD.

Goodness-of-Fit Test (GOFT) | Statistic Value | Description | Characteristic |
---|---|---|---|

KS | ${D}_{n}=\underset{1\le i\le n}{\mathrm{max}}\left[\frac{i}{N}-F({x}_{(i)}),F({x}_{(i)})-\frac{i-1}{N}\right]$ [34] | ${x}_{(i)}$ is a plot on the Empirical frequency curve and F^{−1}(p) is the Inverse function of cumulative distribution function F(x) for probability P_{(i).} N is the size of samples. | KS test measures the greatest discrepancy between the observed and hypothesized distributions. |

AD | ${A}^{2}=-N-\frac{1}{N}{\displaystyle \sum _{i=1}^{N}\left(2i-1\right)\left[\mathrm{ln}F\left({x}_{(1)}\right)+\mathrm{ln}\left\{1-F\left({x}_{(N-i+1)}\right)\right\}\right]}$ [35] | AD uses the sum of the squared differences between the empirical and theoretical distributions with weights to emphasize discrepancies in the tails. AD Statistic has shown good capabilities for a small sample size and heavy tailed distributions [ 15,36]. | |

AIC | $AIC=-2\mathrm{ln}\left[L(D|\stackrel{\u2322}{\theta})\right]+2m$ [8] | $L(D|\stackrel{\u2322}{\theta})$ is the likelihood function of a certain distribution with parameter set $\stackrel{\u2322}{\theta}$ and data array D. m is the number of parameters P and n is the size of the sample. | The log-likelihood maximised function value is used to select the model and penalize heavier for the number of estimated parameters P. In some situations where the sample size n is small with respect to the number of estimated parameters P, the AIC may perform inadequately [11]; a second-order variant of AIC, called AICc, should be used. |

BIC | $BIC=-2\mathrm{ln}\left[L(D|\stackrel{\u2322}{\theta})\right]+\mathrm{ln}(n)m$ [37] | Similar to the AIC, but developed in a Bayesian framework. BIC penalizes heavier than AIC for number of estimated parameters P and small sample sizes [11]. | |

AICc | $AICc=-2\mathrm{ln}\left[L(D|\stackrel{\u2322}{\theta})\right]+2m(\frac{n}{n-m-1})$ [14] | The AICc penalizes heavier than AIC for number of estimated parameters P and can be adopted when n/P <40 to reduce bias [13]. |

Basin and Station Name | Country | Area (Km^{2}) | Terrain | Climate Zone | Data Length | Cs of the Flood Series |
---|---|---|---|---|---|---|

Kingston at Thames | UK | 9948 | Plain | Temperate | 1883–2009 | 1.181 |

Lafayette at Wabash | USA | 18,821 | Alluvial Plain | Humid Continental Climate | 1907–1991 | 0.280 |

Shijiao at Beijiang | China | 38,363 | Hill | Subtropical Monsoon | 1956–2008 | 0.230 |

Lutaizi at Huai | China | 91,620 | Hill | Warm Temperate and Half Wet Monsoon Climate | 1951–1998 | 1.198 |

Study Area | Significance | Persistency | Trend | Jump |
---|---|---|---|---|

t | Spearman | Hurst Coefficient | ||

Kingston at Thames | Stats | 0.057 | 1.446 | 0.568 |

Critical Value (5%) | 1.979 | 1.96 | 0.628 | |

Accept or Not | yes | yes | yes | |

Lafayette at Wabash | Stats | 1.286 | −0.453 | 0.491 |

Critical Value (5%) | 1.989 | 1.96 | 0.323 | |

Accept or Not | yes | yes | yes | |

Shijiao at Beijiang | Stats | −0.953 | 0.927 | 0.500 |

Critical Value (5%) | 2.008 | 1.96 | 0.674 | |

Accept or Not | yes | yes | yes | |

Lutaizi at Huai | Stats | −0.534 | −0.925 | 0.435 |

Critical Value (5%) | 2.014 | 1.96 | 0.255 | |

Accept or Not | yes | yes | yes |

Study Area | PDs | Parameters (MLE) | ||
---|---|---|---|---|

Scale | Shape | Location | ||

Kingston at Thames | P3 | 0.027 | 8.51 | 15.31 |

GLO | 56.67 | −0.16 | 310.86 | |

GEV | 89.059 | 0.036 | 278.97 | |

Weibull | 282.29 | 2.36 | 76.12 | |

Gumbel | 88.47 | —— | 277.22 | |

LN3 | 0.26 | 5.94 | −69.57 | |

LN2 | 0.33 | 5.73 | —— | |

LP3 | 39.104 | 171.62 | 1.34 | |

Lafayette at Wabash | P3 | 0.011 | 32.36 | −1644.73 |

GLO | 299.93 | −0.065 | 1365.28 | |

GEV | 506.81 | 0.203 | 1185.48 | |

Weibull | 1433.11 | 2.56 | 117.46 | |

Gumbel | 490.76 | —— | 1133.93 | |

LN3 | 0.12 | 8.41 | −3140.96 | |

LN2 | 0.45 | 7.15 | —— | |

LP3 | 28.22 | 171.62 | 1.082 | |

Shijiao at Beijiang | P3 | 0.0019 | 36.96 | −9811.59 |

GLO | 1819.32 | −0.068 | 9436.47 | |

GEV | 3054.67 | 0.22 | 8325.95 | |

Weibull | 8802.03 | 2.64 | 1782.32 | |

Gumbel | 2912.39 | —— | 8000.08 | |

LN3 | 0.11 | 10.26 | −19384.8 | |

LN2 | 0.37 | 9.11 | —— | |

LP3 | 34.13 | 171.36 | 4.087 | |

Lutaizi at Huai | P3 | 0.00055 | 1.89 | 566.84 |

GLO | 1264.17 | −0.36 | 3480.44 | |

GEV | 1671.23 | −0.12 | 2861.17 | |

Weibull | 3676.23 | 1.39 | 672.028 | |

Gumbel | 1750.16 | —— | 2942.83 | |

LN3 | 0.43 | 8.47 | −1281.45 | |

LN2 | 0.62 | 8.109 | —— | |

LP3 | 20.55 | 166.68 | 0.00001 |

**Table 5.**A comparison of the test statistic values of the eight typical frequency distributions for hypothesis tests and information-based criteria.

Study Area | Frequency Distributions | KS | AD | AIC | BIC | AICc |
---|---|---|---|---|---|---|

Kingston at Thames | P3 | 0.064 | 0.502 | 1542.419 | 1550.951 | 1542.614 |

GLO | 0.053 | 0.292 | 1538.728 | 1547.260 | 1538.923 | |

GEV | 0.054 | 0.355 | 1540.212 | 1548.745 | 1540.407 | |

Weibull | 0.089 | 1.366 | 1552.091 | 1560.623 | 1552.286 | |

Gumbel | 0.055 | 0.384 | 1538.708 | 1544.396 | 1538.804 | |

LN3 | 0.057 | 0.389 | 1540.801 | 1549.334 | 1540.996 | |

LN2 | 0.056 | 0.396 | 1540.227 | 1545.916 | 1540.324 | |

LP3 | 0.072 | 0.532 | 1544.700 | 1553.300 | 1544.900 | |

Lafayette at Wabash | P3 | 0.060 | 0.443 | 1313.154 | 1320.482 | 1313.450 |

GLO | 0.070 | 0.397 | 1314.564 | 1321.892 | 1314.860 | |

GEV | 0.063 | 0.455 | 1312.927 | 1320.255 | 1313.224 | |

Weibull | 0.073 | 0.563 | 1312.474 | 1319.802 | 1312.770 | |

Gumbel | 0.086 | 1.040 | 1317.343 | 1322.228 | 1317.489 | |

LN3 | 0.060 | 0.437 | 1313.190 | 1320.518 | 1313.487 | |

LN2 | 0.110 | 1.849 | 1324.446 | 1329.332 | 1324.593 | |

LP3 | 0.114 | 2.174 | 1331.104 | 1338.432 | 1331.401 | |

Shijiao at Beijiang | P3 | 0.106 | 0.436 | 1013.854 | 1019.764 | 1014.343 |

GLO | 0.122 | 0.573 | 1012.876 | 1018.787 | 1013.366 | |

GEV | 0.098 | 0.424 | 1010.321 | 1016.232 | 1010.811 | |

Weibull | 0.096 | 0.416 | 1010.571 | 1016.481 | 1011.060 | |

Gumbel | 0.109 | 0.644 | 1012.438 | 1016.379 | 1012.678 | |

LN3 | 0.106 | 0.439 | 1010.753 | 1016.664 | 1011.243 | |

LN2 | 0.114 | 0.709 | 1014.045 | 1017.986 | 1014.285 | |

LP3 | 0.119 | 0.867 | 1018.547 | 1024.458 | 1019.037 | |

Lutaizi at Huai | P3 | 0.077 | 0.181 | 873.463 | 879.077 | 874.008 |

GLO | 0.069 | 0.290 | 876.908 | 882.521 | 877.453 | |

GEV | 0.085 | 0.255 | 875.763 | 881.377 | 876.308 | |

Weibull | 0.070 | 0.188 | 872.889 | 878.503 | 873.435 | |

Gumbel | 0.096 | 0.370 | 874.770 | 878.513 | 875.037 | |

LN3 | 0.088 | 0.237 | 875.352 | 880.966 | 875.898 | |

LN2 | 0.080 | 0.233 | 873.109 | 876.852 | 873.376 | |

LP3 | 0.077 | 0.289 | 875.939 | 881.553 | 876.485 |

River | PD | T = 5 | T = 10 | T = 20 | T = 30 | T = 50 | T = 70 | T = 90 | T = 100 | T = 200 | $\overline{\mathit{R}\mathit{M}\mathit{S}\mathit{E}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

Thames | GUM | 0.039 | 0.044 | 0.049 | 0.047 | 0.050 | 0.048 | 0.052 | 0.052 | 0.059 | 0.049 |

GLO | 0.032 | 0.042 | 0.056 | 0.066 | 0.083 | 0.094 | 0.111 | 0.114 | 0.145 | 0.083 | |

GEV | 0.038 | 0.045 | 0.055 | 0.063 | 0.076 | 0.088 | 0.089 | 0.093 | 0.126 | 0.075 | |

Wabash | LN3 | 0.043 | 0.050 | 0.062 | 0.067 | 0.079 | 0.092 | 0.086 | 0.079 | 0.095 | 0.073 |

P3 | 0.049 | 0.066 | 0.070 | 0.081 | 0.081 | 0.088 | 0.086 | 0.095 | 0.101 | 0.080 | |

WEI | 0.043 | 0.046 | 0.052 | 0.058 | 0.064 | 0.075 | 0.075 | 0.076 | 0.088 | 0.064 | |

GEV | 0.043 | 0.048 | 0.058 | 0.064 | 0.077 | 0.077 | 0.085 | 0.090 | 0.107 | 0.072 | |

GLO | 0.041 | 0.045 | 0.059 | 0.068 | 0.092 | 0.110 | 0.129 | 0.136 | 0.176 | 0.095 | |

Beijiang | WEI | 0.046 | 0.051 | 0.061 | 0.055 | 0.057 | 0.060 | 0.063 | 0.063 | 0.064 | 0.058 |

GEV | 0.048 | 0.052 | 0.059 | 0.058 | 0.069 | 0.071 | 0.078 | 0.085 | 0.096 | 0.068 | |

P3 | 0.055 | 0.061 | 0.085 | 0.078 | 0.080 | 0.089 | 0.089 | 0.093 | 0.114 | 0.083 | |

Huai | WEI | 0.099 | 0.103 | 0.114 | 0.118 | 0.126 | 0.133 | 0.145 | 0.146 | 0.146 | 0.126 |

P3 | 0.103 | 0.109 | 0.136 | 0.161 | 0.169 | 0.171 | 0.190 | 0.188 | 0.216 | 0.160 | |

LN2 | 0.097 | 0.107 | 0.138 | 0.166 | 0.214 | 0.245 | 0.273 | 0.294 | 0.369 | 0.211 | |

GLO | 0.101 | 0.132 | 0.234 | 0.331 | 0.443 | 0.468 | 0.508 | 0.500 | 0.551 | 0.363 |

River | Hypothesis Tests | Information-Based Criteria | Composite Criterion | |||||||
---|---|---|---|---|---|---|---|---|---|---|

KS | AD | KS, AD | Average Number of Parameters | AIC | BIC | AICc | AIC, BIC, AICc | Average Number of Parameters | ||

Kingston at Thames | GLO, GEV, GUM | GLO, GEV, GUM | Glo, Gev (Heavy or Mixed) | 3 | GUM, GLO, GEV | GUM, LN2, GLO | GUM, GLO, LN2 | Gum, Glo (Mixed or Heavy) | 2.5 | Gum (Mixed) |

Lafayette at Wabash | LN3, P3, GEV | GLO, LN3, P3 | Ln3, P3 (Mixed or Light) | 3 | WEI, GEV, P3 | WEI, GEV, P3 | WEI, GEV, P3 | Wei, Gev (Light or Mixed) | 3 | Wei (Light) |

Shijiao at Beijiang | WEI, GEV, P3 | WEI, GEV, P3 | Wei, Gev (Light or Mixed) | 3 | GEV, WEI, P3 | GEV, GUM, WEI | GEV, WEI, LN3 | Gev, Wei (Mixed or Light) | 3 | Wei (Light) |

Lutaizi at Huai | GLO, WEI, P3 | P3, WEI, LN2 | P3, Wei (Light) | 3 | WEI, LN2, P3 | LN2, WEI, GUM | LN2, WEI, P3 | Ln2, Wei (Mixed or Light) | 2.5 | Wei (Light) |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, X.; Shao, Q.; Xu, C.-Y.; Zhang, J.; Zhang, L.; Ye, C.
Comparative Study on the Selection Criteria for Fitting Flood Frequency Distribution Models with Emphasis on Upper-Tail Behavior. *Water* **2017**, *9*, 320.
https://doi.org/10.3390/w9050320

**AMA Style**

Chen X, Shao Q, Xu C-Y, Zhang J, Zhang L, Ye C.
Comparative Study on the Selection Criteria for Fitting Flood Frequency Distribution Models with Emphasis on Upper-Tail Behavior. *Water*. 2017; 9(5):320.
https://doi.org/10.3390/w9050320

**Chicago/Turabian Style**

Chen, Xiaohong, Quanxi Shao, Chong-Yu Xu, Jiaming Zhang, Lijuan Zhang, and Changqing Ye.
2017. "Comparative Study on the Selection Criteria for Fitting Flood Frequency Distribution Models with Emphasis on Upper-Tail Behavior" *Water* 9, no. 5: 320.
https://doi.org/10.3390/w9050320