Maximum likelihood estimation (MLE) of the generalized extreme value distribution (GEVD) is known to sometimes over-estimate the positive value of the shape parameter for the small sample size. The maximum penalized likelihood estimation (MPLE) with Beta penalty function was proposed by some researchers to overcome this problem. But the determination of the hyperparameters (HP) in Beta penalty function is still an issue. This paper presents some data adaptive methods to select the HP of Beta penalty function in the MPLE framework. The idea is to let the data tell us what HP to use. For given data, the optimal HP is obtained from the minimum distance between the MLE and MPLE. A bootstrap-based method is also proposed. These methods are compared with existing approaches. The performance evaluation experiments for GEVD by Monte Carlo simulation show that the proposed methods work well for bias and mean squared error. The methods are applied to Blackstone river data and Korean heavy rainfall data to show better performance over MLE, the method of L-moments estimator, and existing MPLEs.
Generalized extreme value distribution (GEVD) has been used widely as a significant modelling tool to make an inference of extreme events such as heavy rainfall, wind speed, snowfall, earthquake and other related disciplines (Castillo
The two hyper-parameters (HP) should be specified in both of Martins-Stedinger and Coles-Dixon penalty (or prior) functions. They selected a specific value for the HP based on experimentation and experience. Martins and Stedinger (2000) used a Beta(6, 9) pdf. This choice of prior restricts shape parameter to a plausible range (−0.5 ≤
However, the selection of the HP are irrelevant to the present data. There might be no problem if we treat the Beta pdf as a prior probability which was specified before the current data is given. But, if we treat the Beta pdf as a penalty function, it might be desirable to select the HP using given data because the data shows what HP to use. Thus, in this paper, we propose data-adaptive methods to select the HP of a Beta pdf, based on the MLE of shape parameter
Section 2 describes the GEVD, estimations methods and penalty functions. The proposed methods are presented in Section 3. Simulation study to evaluate the performance of the proposed methods is given in Section 4. Real data examples with Blackstone river flood discharge rate and Korean heavy rainfall are provided in Section 5. Discussion and conclusion are given in Section 6.
The cumulative distribution function of the GEVD is as follows (Choi, 2015; Coles, 2001):
for 1 +
Estimates of
Regarding the range of
Under the assumption that the observations
provided that 1 +
The L-moments were introduced by Hosking (1990) as a linear combination of expectations of order statistics. The natural estimator of L-moments based on an observed sample of data is a linear combination of the ordered data values. The
The method of L-ME obtains parameter estimates by equating the first
where
These L-moments and L-ME have been used widely in many research fields including meteorology, civil engineering, and hydrology (for example, Busababodhin
For the small sample size, the MLE sometimes gives poor performance and over-estimates the large positive value of
where
Coles and Dixon (1999) proposed the following penalty function;
for non-negative values of
where
In order to select the HP (
Step 1. Compute the MLE of GEV parameters, and denote it
Step 2. Find the (
The estimator is now obtained by the MPLE with a Beta(
Figure 2 shows distributions of selected values (
Figure 3 shows probability density plot of
We will obtain the sample distribution of
After the above computation, the following measure of discrepancy is minimized with respect to (
where
As an analogy to the selection of the smoothing parameter, we consider the following prediction squared error (pse);
where
In order to evaluate the performance of the proposed method, we compared the proposed estimator to the other estimators by Monte Carlo simulation study. For the comparison of accuracy, we calculated the bias and the root mean squared error (RMSE) of
and
We have generated
Table 2 and Figure 4 show the results of the bias of
We considered the data of the annual flood discharge rates of the Blackstone River at Woonsocket, RI, USA, given in Pericchi and Rodriguez-Iturbe (1985), and Mudholkar and Hutson (1998). This data is for a period of 37 years with unit of
To judge the overall goodness-of-fit, we use the Kolmogorov-Smirnov (K-S) statistic and the
where
Table 4 provides the estimation of
where
In this section, we compared the performance of the proposed method with existing MPLEs using Korean heavy rainfall data. Annual daily maximum precipitation (unit: mm) record are considered for 75 weather stations which has at least 20 years observation (Korea Meteorological Administration, 2016). The stations with relatively large positive MLE value of
The estimates of
In this paper, we restrict the range of shape parameter to be in (−0.5, 0.5) because the variance of GEV random variable is infinite when
In Step 2 of the SHM method, one can consider refining the grid search using the increment 0.5 or 0.2. That may be able to find the HP that makes the density function very narrow and concentrate to the MLE, so that the MPLE be near the MLE. However, to our experiments showed that refining the grid did not improve the results obtained using the coarse grid as reported in this paper.
One can consider a criterion with the help of the method of L-ME. For example, the distance between population L-moments (calculated from the MPLE) and sample L-moments, i.e., |
We tried the following bootstrap based criterion, using the pse as in Efron and Tibshirani (1993);
where
A data adaptive method to select the HP of Beta pdf on the shape parameter of GEVD is presented in a MPLE framework that enables the data to tell us what HP to use. For given data, the optimal HP is obtained from the minimum distance between the MLE and MPLE. The performance evaluation experiments for GEVDs by Monte Carlo simulation show that the proposed estimators often work well. Blackstone river data and Korean heavy rainfall data are fitted to illustrate the usefulness of the proposed methods. Our recommendation is to use the SHM method among some estimations considered in this study. The details of the SHM are described in Subsection 3.1. A computer program for the proposed methods developed using R software is available upon request from the corresponding author.
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1A2B4014518). Lee’s work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1A6A3A11032852).
Description on the abbreviated names of the estimation methods considered in this paper
Methods | Description | Details |
---|---|---|
SHM | Selection of hyperparameters using the MLE | Subsection 3.1 |
BSHM | Bootstrap based selection of hyperparameters using the MLE | Subsection 3.2 |
SHPSE | Selection of hyperparameters minimizing prediction squared error | Equation (3.2) |
MPLE-P | Maximum penalized likelihood estimation using Park’s penalty | Beta(2.5, 2.5) pdf |
MPLE-MS | Maximum penalized likelihood estimation using Martins-Stedinger’s penalty | Beta(9, 6) pdf |
MPLE-CD | Maximum penalized likelihood estimation using Coles-Dixon’s penalty | Equation (2.8) |
L-ME | Method of L-moments estimation | Subsection 2.2 |
MLE | Maximum likelihood estimation | Subsection 2.1 |
pdf = probability density function.
The bias of
Methods | −0.49 | −0.4 | −0.3 | −0.2 | −0.1 | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.49 | Sum |
---|---|---|---|---|---|---|---|---|---|---|---|---|
SHM | 0.031 | −0.014 | −0.021 | −0.024 | −0.021 | − | −0.013 | −0.013 | −0.014 | − | −0.070 | |
BSHM | 0.031 | −0.019 | −0.026 | −0.027 | −0.021 | − | − | − | − | − | −0.073 | |
SHPSE | 0.041 | − | − | 0.046 | 0.080 | 0.073 | 0.039 | −0.056 | −0.148 | −0.238 | 0.740 | |
MPLE-P | 0.152 | 0.097 | 0.063 | 0.032 | −0.039 | −0.070 | −0.103 | −0.142 | −0.197 | 0.921 | ||
MPLE-MS | 0.361 | 0.293 | 0.235 | 0.174 | 0.115 | 0.058 | −0.059 | −0.119 | −0.182 | −0.248 | 1.846 | |
MPLE-CD | −0.051 | −0.051 | −0.035 | −0.03 | −0.028 | −0.036 | −0.040 | −0.054 | −0.062 | −0.073 | −0.106 | 0.567 |
L-ME | − | − | − | − | −0.022 | −0.023 | −0.039 | −0.046 | −0.064 | −0.096 | 0.316 | |
MLE | −0.051 | −0.051 | −0.034 | −0.028 | −0.021 | − | −0.012 | − |
Sum is obtained by the summation of the absolute biases. Table 1 provides descriptions on the abbreviated names of methods.
The root mean squared error of
Methods | −0.49 | −0.4 | −0.3 | −0.2 | −0.1 | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.49 | sum |
---|---|---|---|---|---|---|---|---|---|---|---|---|
SHM | 0.135 | 0.145 | 0.157 | 0.175 | 0.178 | 0.173 | 0.174 | |||||
BSHM | 0.133 | 0.149 | 0.162 | 0.179 | 0.182 | 0.181 | 0.173 | |||||
SHPSE | 0.114 | 0.117 | 0.169 | 0.215 | 0.267 | 0.295 | 0.265 | 0.225 | 0.200 | 0.222 | 0.289 | 2.377 |
MPLE-P | 0.158 | 0.122 | 0.115 | 0.132 | 0.138 | 0.158 | 0.178 | 0.227 | ||||
MPLE-MS | 0.366 | 0.300 | 0.243 | 0.185 | 0.132 | 0.192 | 0.256 | 2.057 | ||||
MPLE-CD | 0.167 | 0.155 | 0.162 | 0.151 | 0.151 | 0.162 | 0.162 | 0.165 | 0.172 | 0.161 | 0.178 | 1.786 |
L-ME | 0.159 | 0.151 | 0.150 | 0.138 | 0.140 | 0.153 | 0.162 | 0.167 | 0.185 | 0.183 | 0.211 | 1.798 |
MLE | 0.167 | 0.155 | 0.164 | 0.154 | 0.160 | 0.177 | 0.182 | 0.186 | 0.200 | 0.203 | 0.213 | 1.960 |
Table 1 provides descriptions on the abbreviated names of methods.
Result of the analysis and comparison of the estimation methods for the Blackstone river data
ASAE | CM | CIU | CIL | CI range | ||||
---|---|---|---|---|---|---|---|---|
SHM | 4432.9 | 1866.2 | 0.268 | 0.027 | 0.126 | 0.472 | 0.100 | 0.372 |
BSHM | 4474.1 | 1845.9 | 0.203 | 0.029 | 0.326 | 0.086 | ||
SHPSE | 4469.0 | 1846.8 | 0.210 | 0.029 | 0.115 | 0.368 | 0.074 | 0.294 |
MPLE-P | 4453.1 | 1851.4 | 0.232 | 0.028 | 0.118 | 0.411 | 0.084 | 0.327 |
MPLE-MS | 4467.8 | 1846.3 | 0.211 | 0.029 | 0.115 | 0.350 | 0.082 | 0.268 |
MPLE-CD | 4441.5 | 1857.2 | 0.250 | 0.027 | 0.122 | 0.481 | 0.087 | 0.394 |
L-ME | 4256.8 | 1441.3 | 0.479 | 0.616 | −0.046 | 0.662 | ||
MLE | 4430.6 | 1867.3 | 0.269 | 0.027 | 0.127 | 0.533 | 0.097 | 0.436 |
Estimates of
Estimates of shape parameter
Location | MPLE-P | MPLE-MS | SHM | ( | BSHM | ( | MLE | |
---|---|---|---|---|---|---|---|---|
Daegwallyeong | 0.221 | 0.196 | 0.285 | 0.165 | 0.283 | |||
ASAE | 0.017 | 0.017 | (4, 12) | 0.017 | (10, 14) | |||
K-S | 0.064 | 0.072 | 0.062 | 0.072 | ||||
Daejeon | 0.199 | 0.176 | 0.311 | 0.240 | 0.304 | |||
ASAE | 0.026 | 0.031 | (4, 14) | 0.027 | (6, 14) | 0.031 | ||
K-S | 0.096 | 0.098 | 0.092 | 0.084 | ||||
Pohang | 0.233 | 0.212 | 0.264 | 0.267 | ||||
ASAE | (2, 4) | 0.018 | (8, 14) | |||||
K-S | 0.073 | 0.073 | 0.073 | |||||
Gunsan | 0.228 | 0.201 | 0.286 | 0.286 | 0.286 | |||
ASAE | 0.034 | 0.034 | (4, 12) | (4, 12) | ||||
K-S | 0.108 | 0.116 | 0.116 | 0.116 | ||||
Wando | 0.232 | 0.199 | 0.312 | 0.295 | 0.313 | |||
ASAE | 0.034 | 0.034 | (4, 14) | 0.034 | (4, 12) | 0.034 | ||
K-S | 0.103 | 0.109 | 0.094 | |||||
Seogwipo | 0.206 | 0.186 | 0.250 | 0.234 | 0.250 | |||
ASAE | 0.027 | (2, 4) | 0.027 | (6, 14) | 0.027 | |||
K-S | 0.086 | 0.089 | 0.084 | |||||
Buyeo | 0.248 | 0.215 | 0.314 | 0.200 | 0.319 | |||
ASAE | 0.020 | 0.020 | (4, 14) | 0.021 | (8, 12) | 0.019 | ||
K-S | 0.087 | 0.082 | 0.095 | 0.096 | ||||
Imsil | 0.214 | 0.187 | 0.287 | 0.287 | 0.292 | |||
ASAE | 0.038 | (4, 12) | 0.038 | (4, 12) | 0.039 | |||
K-S | 0.089 | 0.092 | 0.082 | |||||
Jeongeup | 0.224 | 0.202 | 0.263 | 0.204 | 0.267 | |||
ASAE | 0.021 | 0.022 | (2, 4) | 0.023 | (8, 14) | |||
K-S | 0.083 | 0.084 | 0.084 | |||||
Haenam | 0.292 | 0.240 | 0.401 | 0.338 | 0.405 | |||
ASAE | 0.021 | 0.026 | (2, 10) | 0.017 | (4, 14) | |||
K-S | 0.068 | 0.077 | 0.061 | 0.065 | ||||
Goheung | 0.233 | 0.203 | 0.309 | 0.250 | 0.301 | |||
ASAE | 0.020 | 0.023 | (4, 14) | 0.019 | (6, 14) | 0.014 | ||
K-S | 0.081 | 0.083 | 0.080 | |||||
Yeongdeok | 0.269 | 0.222 | 0.377 | 0.411 | 0.384 | |||
ASAE | 0.024 | 0.028 | (2, 8) | 0.025 | (2, 12) | |||
K-S | 0.095 | 0.098 | 0.085 | 0.084 | ||||
No. of the best | ASAE | 4 | 4 | 8 | 1 | 6 | ||
No. of the best | K-S | 0 | 3 | 6 | 4 | 5 |
The selected values (