Parametric method of flood frequency analysis involves fitting of a probability distribution to observed flood data. When record length at a given site is relatively shorter and hard to apply the asymptotic theory, an alternative distribution to the generalized extreme value (GEV) distribution is often used. In this study, we consider the beta-
Estimations of the upper extreme quantiles corresponding to low probabilities of exceedance are needed in risk analysis, extreme value analysis, and reliability engineering (Castillo
The BPD and beta-
One goal of the present study is to illustrate the feasibility of the BPD in dealing with hydrologic events. Two real applications using the annual maximum stream flow data of Colorado and rainfall data from cloud seeding experiments in Southern Florida are illustrated. The comparison with other models such as the beta-
The remainder of the paper is organized as follows. Section 2 defines the BPD and presents its properties including moments. Section 3 describes the estimation methods. The simulation study is presented in Section 4. Section 5 illustrates the effectiveness of the distribution using real data examples. The discussion and conclusion are in Section 6.
The BPD was given as a special case of the generalized second kind beta distribution (Mielke and Johnson, 1974). The cumulative distribution function (cdf) and the probability density function (pdf) of the BPD are
respectively, where
Figure 1 illustrates the shapes of the beta-
Let
where
The parameter estimates obtained from L-moments are sometimes more accurate in small samples than using MME or MLE (Hosking, 1990). As a result, L-ME has received considerable attention for analyzing skewed data or extreme data, such as extreme rainfall, flood frequency, and extreme wind speed data. See Hosking (1990) for a definition of L-moments as well of its properties and applications. Let
for
where
where Γ(·) is a gamma function, for
A convenient way of representing the L-moments of different distributions is the L-moments ratio diagram, exemplified by Figure 2. This diagram shows the L-moments on a graph whose axes are L-skewness and L-kurtosis. A two-parameter distribution with a location and a scale parameter plots as a single point, while a three-parameter distribution with location, scale and shape parameters plots a line. Distributions with more than one shape parameters generally cover two-dimensional areas on the graph (Hosking and Wallis, 1997). The shaded area of Figure 2 represents the L-skewness and L-kurtosis values attained from the BPD for all possible combinations of
The L-moment ratio diagram is used as a tool for aiding in the identification of a suitable frequency distribution to model the available samples. It typically plots the sample L-kurtosis against the sample L-skewness values and compare these to equivalent theoretical relationships derived for a range of candidate distributions. The closeness of the sample values to the theoretical lines or areas can then be used as a selection criterion for the most appropriate type of distribution (Peel
The higher-order L-moments, known as LH-moments (Wang, 1997), provide another way in which to estimate the parameters. This method has been used by Murshed
The MME can be obtained by equating the first three population moments of the BPD with the corresponding sample moments. Let
The MME of
Let
For a given sample (
for
We conducted Monte-Carlo simulations to investigate the performance of the three estimation methods (MME, L-ME, MLE) for high-quantile estimation. We generated random samples from the BPD by using the inverse transformation method. Sample sizes are
To examine the performance of the estimation methods, we used two measures: the relative bias (Rbias) and relative root mean squared error (RRMSE). These are
where
where
Figure 3 displays the upper-quantile Rbias and RRMSE for the samples of size
Figure 4 presents the boxplots of the ASAE, which were computed from 2,000 Monte-Carlo trials for
This example is based on time series data that consists of annual maximum stream flow amounts in cubic feet per second. Data were obtained from the U.S. Geological Survey (USGS) station 06714310 (Sand Creek tributary at Denver, Colorado). It covers 22 years from May 1 to August 31 between 1971 and 1992; however, the sample size is 21 years because the records for 1989 are missing. The data is accessed through the USGS website, nwis.waterdata.usgs.gov (U.S. Geological Survey, 2010).
The MLE and L-ME of the respective parameters for some distributions (i.e., GEV, Gumbel, beta-
Figure 5 shows the estimated return levels (in cubic feet per second), using the L-ME method, with a 95% confidence interval. Here, the
The
This example consists of 26 rainfall amounts (units in acre-feet) from seeded clouds of experiments conducted in Southern Florida (Simpson, 1972). It is based on radar-evaluated rainfall data from 52 Southern Florida cumulus clouds. Mielke and Johnson (1974) used this data to evaluate the performance of beta-
Based on the K-S and A-D test criteria, the BPD is better in most cases (Table 2). In addition, the results from the AIC and BIC in Table 3 suggest the better fitting ability of the BPD to the data compared with the GEV, beta-
In this study, we examined the beta-
The presented simulation study suggests that the BPD via the L-ME is satisfactory.
The L-moment ratio diagram implies that the BPD has wider applications than the GLO, LN3, PE3, and GEV distributions.
Our findings suggest that the BPD can be a useful alternative to the GEV, GLO, PE3, and LN3 distributions for modeling (extreme) hydrologic events.
The BPD has no location parameters; rather, it has two shape parameters (
The authors gratefully acknowledge Professor Daniel Wilks at Cornell University for supplying the Fortran code for the MLE calculation. They also thank Tae Kyoung Kim for his help in computation. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1A2B4014518). Seo’s work was supported by the “Research and Development of KMA Weather Climate, and Earth System Services” of the National Institute of Meteorological Sciences (NIMS) of the Korea Meteorological Administration (KMA). Lee’s work was also supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1A6A3A11032852).
Parameter estimates of the Gumbel, GEV, beta-
Distribution | Parameter | Stream flow data | Rainfall data | ||
---|---|---|---|---|---|
MLE | L-ME | MLE | L-ME | ||
GEV | 121.50 | 128.70 | 113.80 | 135.70 | |
84.10 | 98.40 | 145.80 | 184.90 | ||
0.46 | −0.24 | 0.91 | −0.53 | ||
Gumbel | 145.40 | 140.90 | 211.50 | 200.60 | |
109.40 | 130.00 | 309.60 | 418.20 | ||
beta- | 242.00 | 259.40 | 973.20 | 1290.60 | |
1.71 | 1.83 | 3.10 | 3.80 | ||
1.93 | 1.87 | 0.88 | 0.82 | ||
beta- | 109.59 | 266.16 | 382.19 | 767.66 | |
α̂ | 1.69 | 0.39 | 0.53 | 0.23 | |
1.89 | 2.89 | 1.51 | 2.02 |
GEV = generalized extreme value; L-ME = L-moments estimation; MLE = maximum likelihood estimation.
K-S and A–D goodness-of-fit test statistics with p-values in brackets, using the Gumbel, GEV, beta-P, and beta-
Estimation | Criterion | Distribution | Stream flow data | Rainfall data |
---|---|---|---|---|
MLE | K-S | GEV | 0.134 (0.809) | 0.111 (0.905) |
Gumbel | 0.178 (0.467) | 0.221 (0.159) | ||
beta- | ||||
beta- | 0.171 (0.516) | 0.089 (0.985) | ||
A-D | GEV | 0.504 (0.741) | 0.319 (0.923) | |
Gumbel | 0.525 (0.719) | 2.042 (0.088) | ||
beta- | ||||
beta- | 0.479 (0.765) | 0.233 (0.979) | ||
L-ME | K-S | GEV | 0.134 (0.800) | |
Gumbel | 0.143 (0.733) | 0.247 (0.083) | ||
beta- | 0.116 (0.870) | |||
beta- | 0.176 (0.481) | 0.147 (0.631) | ||
A-D | GEV | 0.380 (0.867) | 0.326 (0.917) | |
Gumbel | 0.513 (0.731) | 2.167 (0.075) | ||
beta- | 0.637 (0.611) | 0.669 (0.583) |
K-S = Kolmogorov-Smirnov; A-D = Anderson-Darling; GEV = generalized extreme value; MLE = maximum likelihood estimation; L-ME = L-moments estimation.
AIC and BIC values for the Gumbel, GEV, beta-
Data | Distribution | ln | AIC | BIC | |
---|---|---|---|---|---|
Stream flow | 3 | − | |||
Gumbel | 2 | −133.14 | 270.28 | 272.37 | |
− | |||||
beta- | 3 | −131.61 | 269.22 | 272.35 | |
Rainfall | GEV | 3 | −183.35 | 372.70 | 370.95 |
Gumbel | 2 | −194.48 | 392.96 | 391.79 | |
− | |||||
beta- | 3 | −181.69 | 369.38 | 373.15 |
AIC = Akaike information criterion; BIC = Bayesian information criterion; GEV = generalized extreme value.