広島工業大学紀要研究編
第 45 巻（2011）pp.281-289

論文

A Comparison of Different Procedures Relating to Parameter Estimationin Extreme Value Type I Distribution Through Simulation

Satyabrata PAL*, Sanpei KAGEYAMA** and Subhabaha PAL***(Received Aug. 31, 2010)

Abstract
The extreme value (EV) distribution is widely used for fitting and analyzing a long range ofdata emanated from different real-life situations. Annual maximum water-flow data collectedfrom a river over a considerable period of time (say, about 30 years) are found to follow the EV
distribution very satisfactorily. Also the distribution of annual maximum rainfall data in a region closely follows the EV distribution. Thus these distributions have important applicationsin the works related to the flood frequency estimation/estimation of return period flood. The
estimation of parameters from these types of data (data following the EV type I with two parameters, EV I(2), distribution) has important use in the field of hydrology to understand thesituation more deeply. Here an investigation is made through simulation for several pairs ofvalues of a and b (parameters of the EV I(2) distribution) using different sample sizes in order
to compare the accuracy of maximum likelihood method as well as method of moments in estimating the original parameters (the pairs of values of a and b for which the simulation is done).
A module is developed in SAS/IML which deals with a random number generation in the EV
I(2) distribution and the subsequent estimation of parameters with method of moments andmaximum likelihood method. It is found that when the parameter value of a is less than orequal to 0.5, the method of moments produce more accurate estimators of the parameters thanthe corresponding maximum likelihood estimators for large samples (250 or more). An error
distribution investigation with respect to the parameters is also made with CAPABILITY andQQPLOT procedures in SAS/QC. It is found that when the sample size is small (10, 100, etc.)
the error distribution follows near normality but with the increase in sample sizes (200, 300,
etc.) the error distribution gradually deviates much away from normality. Some procedures in
SAS/BASE and SAS/STAT are used in the calculation leading to the final results.
Key Words: EV I(2) distribution, Moment estimator, MLE, Simulation, Normality.

1. Introduction
The generalized extreme value (GEV) distribution is a flexible three-parameter model that combines the
Gumbel, Fréchet and Weibull maximum extreme value distributions, which has a probability density function (pdf) as＊＊＊

Swami Vivekananda Institute of Management and Computer Science, Karbala, Sonarpur, Kolkata 700103, West Bengal, India＊＊＊

Department of Environmental Design, Hiroshima Institute of Technology, Hiroshima 731-5193, Japan＊＊＊

Department of e-Learning, Sikkim Manipal University, Bangalore, Karnataka, India― 281 ―

Satyabrata PAL, Sanpei KAGEYAMA and Subhabaha PAL

where z = (x–μ)/σ, and k, σ, μ are the shape, scale, location parameters respectively (cf. Gumbel, 1958). The
scale parameter σ must be positive, the shape and location parameters can take on any real value. The
range of definition of the GEV distribution depends on k and 1 + kz > 0. In what follows, σ is replaced by a,and μ is replaced by b. Thus a pdf of the GEV distribution is of the form.

The range of the variable x depends on the sign of the parameter k. The GEV distribution function is of theform, a>0, b>0.

When k = 0, we have a pdf of the GEV distribution given by,,which is here called a pdf of the extreme value type I (with two parameters) distribution, denoted by EV
I(2) distribution. The distribution function of the EV I(2) distribution is shown by.

The random samples are generated through simulation from the EV I(2) distribution considering thevalues of the parameters, a and b, lying in different ranges. To achieve the above, random numbers fromthe EV I(2) distribution are drawn (in case of sample sizes of 10, 50, 100, 150, 200, 250, and 300 when simulated samples are created 200 times against each size) for the above parameters values of which lie in different ranges. Based on the above sets of samples, the estimators obtained by the method of moments andby the method of maximum likelihood are found out and subsequently, the differences of the estimated values of the parameters from the original values of parameters are calculated. Finally, appropriate conclusion in regard to the accuracy of the different parameter estimation procedures is presented with reference
to an important feature revealed when the parameters lie in the range of specific values. Table 1 presents
a complete list of parameter values and sample sizes used for the simulation purpose.

2. Materials and methods
2.1 Parameter estimation by method of moments
The first moment of the EV I(2) distribution is given by μ1' = b+0.5772157a (cf. Kite, 1977), where the value
0.5772157 is an approximation of Euler’s constant. The second central moment is given by.

The parameter estimates are obtained by replacing μ1′ and μ2 by their corresponding sample estimates
m1′ and m2. Thus the expressions of the above estimators reduce respectively to,― 282 ―(1)

A Comparison of Diﬀerent Procedures Relating to Parameter Estimation in Extreme Value Type I Distribution Through Simulation
Table 1: Detailed list of parameters and sample sizes for simulation
Study
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

Parameters
a
b
2
5
0.5
10
2
0.5
0.5
5
0.4
0.5
5
0.8
0.9
2
1
2
0.7
2
0.8
2
0.6
2
0.55
2
0.53
2
0.52
2
0.51
2
0.505
2
0.5
2
0.495
2
0.49
2
0.48
2
0.45
2
0.4
2
0.4
10
0.35
2

10
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200

50
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200

100
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200

Sample sizes
150
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200

250
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200

300
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200

Number of repetitions of
the simulation with fixedparameters (i.e., fixed values of a and b) and with a
specified sample size (i.e.,
generating a fixed numberof random samples fromthe EV1(2) distribution
with specified values of theparameters repeatedly).

(2)

The estimates of the parameters obtained by the method of moments from the simulated random
samples generated from the EV I(2) distribution are calculated using (1) and (2). Random samples are generated by an SAS/IML module and the method of moments estimates are also estimated by the same module in SAS/IML.
2.2 Parameter estimation by maximum likelihood method
The likelihood function for a sample of size n from an EV I(2) distribution is given bywhich leads to the following.

(3)

Differentiating (3) with respect to a and b and then equating to zero, we get the following equations,,

which ultimately reduce to the following equation.

(4)

Equation (4) on a cannot be solved analytically and thus it is to be solved iteratively. An initial value of ais required and âMOM is taken as the initial value of a. The value of a is updated by the following formula(5)

where G′(a) is the derivative of G(a) in Equation (4). The iteration in Equation (5) is repeated until G(a) is― 283 ―

Satyabrata PAL, Sanpei KAGEYAMA and Subhabaha PAL

sufficiently close to zero. Hence âML is calculated after several iterations making G(a) sufficiently close to
zero. After âML is obtained, b̂ML is calculated from the following equation.

In fact, âML and b̂ML are also calculated by the same SAS/IML module.
Briefly, following the above procedure the random samples are generated from the EV I(2) distribution at first and from the samples the estimates âMOM, b̂MOM, âML and b̂ML subsequently, defining the modules
in SAS/IML through coding, are obtained.
The following estimates are generated as outcomes:âMOM(k) = method of moments estimate of a at the k-th sample drawing, k = 1 (1) 200,b̂MOM(k) = method of moments estimate of b at the k-th sample drawing, k = 1 (1) 200,âML(k) = maximum likelihood estimate of a at the k-th sample drawing, k = 1 (1) 200,b̂ML(k) = maximum likelihood estimate of b at the k-th sample drawing, k = 1 (1) 200,

and based on the above estimates, mean squared errors (MSE’s) in case of the method of moments estimates and of the maximum likelihood estimates of a and b are obtained respectively as
,
,

,
.

Again, these MSE’s are calculated with the help of SAS/BASE and SAS/STAT procedures. The final conclusion is achieved on examining the MSE values obtained by adopting the two estimation procedures on
the above samples generated through simulation. The error distributions are generated in case of bothmethods with the help of CAPABILITY and QQPLOT procedures in SAS/QC.

3. Results and discussion
Tables 2 to 7 contain the results which reveal distinctive features in respect of the objective (comparison of
the two methods) of the investigation. These tables are the selected results of the 24 studies. In the study
mentioned in Table 1, the initial values were taken around a = 1 and after observing the outcomes, valuesof a were chosen in the order a = 0.9, a = 0.7, a = 0.6, a = 0.5 and a = 0.4, etc. A striking feature (contrast)
is noticed at this stage, which is mentioned below.
In the range a ≤ 0.5, the method of moments estimator is found to be a better estimator than thecompeting maximum likelihood estimator with the increase in sample size (as found in Table 6, the contrast exists for a = 0.5, as in case of any value of a < 0.5). The same feature does not exist for a > 0.5. The
above conclusion is evident on examining Table 2 (a > 0.5), Table 3 (a < 0.5), Table 4 (a > 0.5), Table 5 (a <0.5), Table 6 (a = 0.5) and Table 7 (a > 0.5), that is, a ≤ 0.5 for Tables 3, 5 and 6, while a > 0.5 for Tables 2,
4 and 7.

― 284 ―A Comparison of Diﬀerent Procedures Relating to Parameter Estimation in Extreme Value Type I Distribution Through Simulation
Table 2a=2

b=5

n = 200

Samplesize

MSE(a)MOM

MSE(b)MOM

100.292506

0.386844

0.163807

50

0.086835

0.083953

0.053938

100

0.063344

0.051452

150

0.031392

0.036124200

0.030698250

0.03047300

0.027172

Number of times the methodof moments estimate of a isnear to the original parameters

Number of times the maximum likelihood estimate of ais near to the original parameters0.386032

71

129

0.08236774

1260.051601

0.037089

60

140

0.028461

0.02994464

1340.020222

0.029706

0.014581

60

140

0.020716

0.029624

0.010578

62

138

0.010962

0.024307

0.011545

51

149

Number of times the methodof moments estimate of a isnear to the original parameters

Number of times the maximum likelihood estimate of ais near to the original parameters

MSE(a)ML

MSE(b)ML

Table 3a = 0.4

b = 0.5

n = 200

Samplesize

MSE(a)MOM

MSE(b)MOM

100.054437

0.043247

0.055646

0.046346

88

112

50

0.043496

0.039187

0.045742

0.041645

97

103

100

0.03358

0.0331278

0.034612

0.03459

104

96

150

0.02438

0.025821

0.025513

0.026921

126

74

200

0.014015

0.013446

0.015472

0.015041

144

56

250

0.009543

0.008347

0.011534

0.008399

160

40

300

0.002375

0.000153

0.003222

0.000275

171

29

Number of times the methodof moments estimate of a isnear to the original parameters

Number of times the maximum likelihood estimate of ais near to the original parameters

MSE(a)ML

MSE(b)ML

Table 4
a = 0.52b=2

n = 200

Samplesize

MSE(a)MOM

MSE(b)MOM

100.143213

0.321942

0.113142

0.304219

93

107

50

0.104962

0.303216

0.094312

0.294219

92

108

100

0.094312

0.294216

0.082142

0.264219

94

106

150

0.073242

0.243216

0.064232

0.234162

90

110

200

0.064312

0.201312

0.059422

0.19321984

116

MSE(a)ML

MSE(b)ML250

0.043212

0.184190.043012

0.174236

80

120

300

0.032114

0.104862

0.032104

0.093612

75

125

Number of times the methodof moments estimate of a isnear to the original parameters

Number of times the maximum likelihood estimate of ais near to the original parameters

65

135

Table 5a = 0.45

b=2

Samplesize

MSE(a)MOM

MSE(b)MOM

100.316432

0.543162

n = 200
MSE(a)ML

0.308269

MSE(b)ML

0.531682500.246819

0.396219

0.231942

0.364198

80

120

100

0.144312

0.261483

0.141362

0.251362

89

111

150

0.093168

0.134231

0.094368

0.121368

100

100

200

0.054263

0.032162

0.064193

0.031492

115

85

250

0.014986

0.014269

0.019269

0.011268

132

68

300

0.009216

0.009421

0.009643

0.009341

147

53― 285 ―

Satyabrata PAL, Sanpei KAGEYAMA and Subhabaha PAL
Table 6
a = 0.5b=2

n = 200

Samplesize

MSE(a)MOM

100.243210.543216

0.04325150

0.132196

0.489326

0.035192100

0.094312

0.396214

0.024196150

0.023196

0.294361

0.233421

200

0.019432

0.194321

250

0.014163

0.101642

300

0.009421

0.094321

MSE(b)MOM

Number of times the methodof moments estimate of a isnear to the original parameters

Number of times the maximum likelihood estimate of ais near to the original parameters0.544216

84

116

0.49623192

108

0.398242

96

104

0.316431

101

99

0.219632

0.194319107

93

0.141796

0.101649

110

900.095219

0.094412

117

83

Number of times the methodof moments estimate of a isnear to the original parameters

Number of times the maximum likelihood estimate of ais near to the original parameters

MSE(a)ML

MSE(b)ML

Table 7
a = 0.7b=2

n = 200

Samplesize

MSE(a)MOM

MSE(b)MOM

100.094212

0.364921

0.0942090.364212

78

128

50

0.061212

0.271316

0.061021

0.263214

73

127

100

0.054121

0.143213

0.049131

0.123121

70

130

150

0.048316

0.084131

0.043212

0.071231

68

132

200

0.0321293

0.074212

0.029834

0.070012

67

133

250

0.021941

0.0694212

0.014131

0.068421

63

137

300

0.012131

0.059421

0.009832

0.05212959

141

MSE(a)ML

MSE(b)ML

4. Error distribution
Samples generated for one set of parameter values (i.e., a = 0.5 and b = 10), using different sample sizes,
are used for the examination of error distribution. Here our objective is to investigate the underlying pattern (i.e., whether the error distribution deviates from normality or not) of the error distribution in the cases of the four estimated parameters (for a and b using the methods of maximum likelihood and of moments)
with the help of the Q-Q plot as well as with CAPABILITY procedure in SAS/QC. In what follows, we present the results in the cases of the four parameters for the sample sizes 10 and 300 only (though intermediate values 100 and 200 were also considered).

― 286 ―A Comparison of Diﬀerent Procedures Relating to Parameter Estimation in Extreme Value Type I Distribution Through Simulation

4.1 Sample size (n = 10)
The error distribution of âMOM

The error distribution of b̂MOM

Variance = 0.0231 Skewness = 1.3314
Kurtosis = 2.5973
Test for location
Test
Student’s t
Sign
Signed rankt
M
S

Statistic–2.5680–31–3448

Variance = 0.3749 Skewness = 1.0892
Kurtosis = 3.3117
Test for location

p Value
Pr > |t|0.0110Pr ≥|M| <0.0001
Pr ≥|S|<0.0001

Test
Student’s t
Sign
Signed rankt
M
S

Statistic
3.1078122057

p Value
Pr > |t|0.0022Pr ≥|M|
0.1036
Pr ≥|S|
0.0117Graph 1: Q-Q Plot for error distribution of method of
moments estimates of a

Graph 2: Q-Q Plot for error distribution of method of
moments estimates of b

The error distribution of âML

The error distribution of b̂ML

Variance = 0.0230 Skewness = 1.3257
Kurtosis = 2.5765
Test for location
Test
Student’s t
Sign
Signed rankt
M
S

Statistic–2.5684–27–3429

Variance = 0.0333 Skewness = 0.5650
Kurtosis = 0.6277
Test for location

p Value
Pr > |t|0.0109
Pr≥|M|
0.0002Pr ≥|S|<0.0001

Test
Student’s t
Sign
Signed rankGraph 3: Q-Q Plot for error distribution of maximum
likelihood estimates of at
MS

Statistic
1.59768873

p Value
Pr > |t|0.1117
Pr≥|M|
0.2888Pr ≥|S|
0.2879Graph 4: Q-Q Plot for error distribution of maximum
likelihood estimates of b

― 287 ―

Satyabrata PAL, Sanpei KAGEYAMA and Subhabaha PAL

4.2 Sample size (n = 300)
The error distribution of b̂MOM

The error distribution of âMOM

Variance = 0.0337 Skewness = 9.6596
Kurtosis = 93.9748
Test for location

Variance = 0.0027 Skewness = –3.8523
Kurtosis = 22.8482
Test for location
Test
Student’s t
Sign
Signed rankt
M
S

Statistic
6.5010
63
7471

Test
Student’s t
Sign
Signed rank

p Value
Pr > |t|<.0001
Pr≥|M|<.0001Pr ≥|S|<.0001t
M
S

Statistic
0.6873–30–3695

p Value
Pr > |t|0.4927
Pr≥|M|<0.0001Pr ≥|S|<0.0001

Graph 5: Q-Q Plot for error distribution of method of
moments estimates of a

Graph 6: Q-Q Plot for error distribution of method of
moments estimates of b

The error distribution of âML

The error distribution of b̂ML

Variance = 0.0028 Skewness = –3.8476
Kurtosis = 22.2221
Test for location
Test
Student’s t
Sign
Signed rankt
M
S

Statistic
6.161863
7431

Variance = 0.0034 Skewness = 8.1770
Kurtosis = 75.2613
Test for location

p Value
Pr > |t|<.0001
Pr≥|M|<.0001Pr ≥|S|<.0001

Test
Student’s t
Sign
Signed rankGraph 7: Q-Q Plot for error distribution of maximum
likelihood estimates of at
MS

Statistic
1.417612846

p Value
Pr > |t|0.1579
Pr≥|M|
0.1036
Pr ≥|S|
0.3031Graph 8: Q-Q Plot for error distribution of maximum
likelihood estimates of b

It can be seen from the above Q-Q plots that when n = 10 (the first four graphs), the points are lying moreor less parallel (sometimes coincident) to the straight line, which indicate that the error distributions are― 288 ―A Comparison of Diﬀerent Procedures Relating to Parameter Estimation in Extreme Value Type I Distribution Through Simulationnear normal. But when n = 300, last four graphs, the points do no longer lie parallel to the straight line revealing that the error distributions deviate away from normality, which is also an interesting finding inthis paper. The existence of the above pattern is also observed for values of n = 100 and 200. Thus, as the
sample size increases, the error distributions deviate much away from normality.

5. Findings from the investigation
1) For a < 0.5, when the sample size increases the MSE in case of method of moments estimator isless than the MSE in case of the maximum likelihood estimator.
2) The value of the parameter b does not play any significant role.
3) When a < 0.5, the number of times the method of moments estimator of the parameter a is near tothe original parameter is much more than that in case of the maximum likelihood estimator as the
sample size increases.
4) The investigation has been carried out assigning very small values to ‘a and b’, at first. Some parameter values were taken in the range less than 1 (also greater than 1) for a, and after examining
the outcomes, the latter parameter set values were taken and it was found that a sharp feature(contrast) is visible in the range around 0.5. In the range, a ≤ 0.5, the method of moments estimators are found to be better than the maximum likelihood estimators as the sample size increases,
as was mentioned in Section 3 also. This is evident while examining the tables for values of a ≤ 0.5
as well as a > 0.5.
5) When the sample size is small, the error distributions (obtained by using both methods and forboth a and b) follow near normality.
6) As the sample size increases, the error distributions deviate much away from normality.
7) The observations (graph plots) on the error distribution which have been included for a = 0.5 and b= 10, can be obtained for other parameter set values also. Similar patterns will follow in case ofincrease in sample sizes for other parameter set values.

6. Conclusion in case of EV I(2) distribution
1) Method of moments yields more accurate estimators of the parameters than the maximum likelihood estimators when the values of the parameter a is less than or equal to 0.5.
2) When the sample size is small, the error distribution in parameter estimation follows normality,
but it gradually deviates away from normality with the increase in sample sizes.

References
Gumbel, E. J. (1958). Statistics of Extremes, Columbia Univ. Press, New York, USA.
Kite, G. W. (1977). Frequency and Risk Analysis in Hydrology, Water Resource Publications, Fort Collins,
Colorado, USA.

― 289 ―