CHEN Yuanfang ， XU Shengbin, WANG Ling
Department of Hydrology and Water resources, Hohai, University ， Nanjing 210098,China
E-mail:yfchen@mail.edu.cn
VAN GELDER, Pieter
Faculty of Civil Engineering and Geosciences, Delft 2600 GA, The Netherlands
A defect of a weighted function moment method for Pearson –III distribution developed by Chinese hydrologist MA Xiufeng is that the parameter of Cv is negatively biased, especially when the population parameter Cv0 is large. In order to overcome it, a new weighted function moment method is proposed, where the parameter Cv is estimated by the L-moment method. Monte-Carlo calculations show that the new weighted function moment method based on L-moment is better than the conventional one with respect to the bias of the quantiles and the parameter Cv, but the effectiveness of the quantiles and the parameter Cv of above two estimation methods are almost the same.
Weighted function moment method; L-moment; bias, efficiency; Pearson-III distribution.
1 Introduction
The weighted function moment method is first developed by a Chinese hydrologist--Ma Xiufeng in
1984[1]. It may overcome a defect of conventional moment method in the estimation of the parameter Cs (Skewness coefficient)in a Pearson-III distribution. The defect is that Cs is negatively biased due to the existing of error for calculation of moments, especially for higher moment. So
Chinese hydrologists are paying a good deal of attention to its development, and some researches have been carried out for modifying it. The use of Pearson-III distribution (or its log-version) in hydrological modelling is well known, as is reviewed for instance by Bobee [9]. Rasmussen et al
[10], Durrans [11], and Wu et al [12] have studied the L-Moments method as an estimation method for the unknown parameters of the Pearson-III distribution. On the basis of Ma Xiufeng ‘s research,
Liu Zhizhong and Liu Guangwen [2][3] developed two new modified weighted function moment methods, called single weighted function moments through numerical integral, and two weighted function moment method .By using the evaluation standard of ideal sample fitting, they have obtained a conclusion that two new weighted function methods are better than original one. In terms of Monte-Carlo experiment by Chen Y.F. [4][6], the conclusion of the comparison of above weighted function moment methods are contrary to that of [2][3]. In other words, the above two modified weighted function moment methods are not better than the original one. In this case, Chen
(1992,1994) developed an effective weighted function moment method based on the original one which may consider historical flood series. Therefore, the application scope have been widely extended. Now the problem of weighted function moment method still exists, i.e. the parameter
Cv(coefficient of variation) is estimated by conventional moment method, which causes a negatively bias of the estimation of Cv.
1

In order to overcome it, a weighted function moment method based on L-moment is proposed in the paper. According to the Monte-Carlo experiment, the new one of weighted function moment is better than the original one.
2 Weighted function moments method (WF1)
The principle and derivation of weighted function moments method with a simple sample refers to
[1], the parameter estimation formulae with the consideration of historical flood information developed by Chen [4][6] are listed as follows:
Let X represent a series with historical floods in which the maximum return period is N ; the length of recorded series (systematic) is n , the number of historical floods is a , the number of historical floods in the recorded series is l , { x ,m=1,2,…,(n+l-a)} is the series in descending order (from highest to m lowest).
EX

1
N

 a  m

1 x m

N n

 a
 m
 n a 
 a



1 x m

 (1)
C
V

N
1

1 a  m

1
( x m
EX

1 )
2 
N
 a n
  a
 n
   m
 a

1
( x m
EX

1 )
2
1
2
(2)
C s

4

E G (3) where E

1
N

 m a 

1

EX
 x m x m

N n

 l a m
 n a 
 a
 l 

1
EX
 x m x m

 (4)
G

1
N

 a  m

1

EX
 x m x m

N n

 l a a m

 n  a
 l 
EX

1
 x m x m

 (5)

(x) is the probability density function of the normal distribution with mean E X and standard deviation

.
 
EXC v
.
When the value of Cs calculated by equation (3) is less than 4, the final value of Cs is estimated by equation (6)
E

2 2
1
 
( r )
( 2 r / e ) r
*

 r

( r
2
)

2

( r

2
1
)

 (6)
Where r = 4/ C
2
,e=2.71828
， E is calculated by equation (4).
S
The above method is still called conventional weighted function moment method.
3 Weighted function moments method based on L-moment (WF2)
In order to overcome the defect of WF1 in the estimation of parameter C
V
, the parameter C
V is not yet estimated by the moment method in equation (2),but it is estimated by L-moment[7], because the estimation of Cv by L-moment has a very good unbiased feature.
Estimation formula of C
V by L-moment is as follows:
2

Let X represent a series with historical flood in which the maximum return period is N , the length of recorded series (systematic) is n, the number of historical floods is a , the number of historical floods in recorded series is  . The samples
 x ' m
, m

1, 2,..., n l a

is the series in increasing order (from lowest to highest) b o

1
N


N n

  a

 n  m

1 x
 m
 m


 n  n
 a
  
1 x
 m

 (7) b
1

1
N 


N n

  a n
   m

1 n m
 

1

1
N
N
 a

1

1 x
 m
 n
 m

  a  n
  
1
N
 n
   a

N

1 m

1 x
 m 


(8) b
2
 m
1
N 


N n


 a n
 m
 

1
( n
( m



1 )( m

1 )( n



2 )

2 ) n

 n
 
 a
  
1
( N
 n
   a

( m
N

1 )( N

1 )( N
( N
 a
( N

1 )( N

1 )( N

 a
2 )

2 ) x
 m

(9)

 n
2 )
   a
 m

2 ) x
 m 


When b
0
, b
1
, b
2
are estimated by using historical flood sample,

1
,

2
,

3
,

3 are calculated by equation( 10)

1
= b
0
,

2
= 2b
1
- b
0
,

3
= 6b
2
- 6b
1
+ b
0
,

3
=

3
/

2
(10)
With

1
,

2
,

3
,C
V
is estimated by following equations:
if
 
3

1/3, then
Z

3

3
2
 
Z
1

0 .
2906 Z

0 .
1882 Z
2 
0 .
0442 Z
3
(11)
if 1/3
 
3

1,then
Z

1
 
3
 
0
1

.
36067
2 .
Z
78861 Z


0 .
59567 Z
2
2 .
56096 Z
2

0 .
25361 Z
3

0 .
77045 Z
3
(12)
C
V
 
2
 1
2
 1
2

(

) /

(
 
1
2
)
(13)
EX, C
S
are still estimated by WF1 method .
4.
Monte-Carlo experiment
The statistical performance of new weighted function moment method is compared with the conventional one by Monte-Carlo method in the consideration of simple sample and the sample with historical flood information.
4.1 The standard of evaluation
The performance of a parameter estimation method is evaluated by calculating the bias and efficiency of the parameter and quantiles. The bias and efficiency of the parameter are indicated by the mean and root mean square error(r.m.s.e) of Ns Monte-Carlo generating samples respectively. For clearness, the bias and efficiency of the quantiles are expressed as follows by Sxp and Bxp ([4],[8]).
S
X p
 i
N
S 

1

ˆ i p
 x
0 p
 
N
S
 x
0 2 p


100%
3

B
X p

Ns  i

1
 x i p
 x o p

N x o s p

100%
In which p is the design probability ( where P ( X
 x
0 p
) = p , p p
1
=0.1%, p
2
=1%), x
0 p
is the population quantile with p , x i
is the quantile estimated by the i th generating sample ( i = 1, 2, …, Ns ), Ns is the number of generating samples, in this paper, Ns=1000.
If Bx p
is positive, it means that the quantile estimated is positively biased, else negatively biased. The larger

Bx p

,the more serious of bias of quantile. For Sx p
, the smaller of Sx p
, the better of quantile estimation. In practical evaluation, one good estimation method is required to have : (1) Bx p
is larger than 0,but

Bx p

is not exceeded 5%,(2) Sx p
is as smaller as possible .
4.2 Monte-Carlo experiment scheme design
4.2.1 A simple sample n = 30, 50, EXo =1.0,Cvo = 0.3, 0.5, 1.0, Cso/Cvo = 3, 4. There are in total 12 schemes.
4.2.2 A sample with historical flood
N = 100, n = 30, 50, a =1, 3, EXo =1.0 Cvo =0.3, 0.5, 1.0, Cso/Cvo is set to 3, 4, there are totally 24 schemes.
Table 1 Results of the bias and the efficiency of parameter and quantiles (partial schemes)
Population parameters
Estimation method
EC
V
EC
S
SC
V
SC
S
BX p1
BX p2
SX p1
SX p2
C
Vo
C
So
N n a
0.3 0.9 30 30 0
0.3 0.9 30 30 0
0.5 2.0 50 50 0
0.5 2.0 50 50 0
0.5 2.0 30 30 0
0.5 2.0 30 30 0
1.0 4.0 50 50 0
1.0 4.0 50 50 0
1.0 3.0 100 50 1
1.0 3.0 100 50 1
1.0 3.0 100 50 3
1.0 3.0 100 50 3
0.5 2.0 100 50 1
0.5 2.0 100 50 1
0.3 0.9 100 30 3
0.3 0.9 100 30 3
WF1
WF2
WF1
WF2
WF1
WF2
WF1
WF2
WF1
WF2
WF1
WF2
WF1
WF2
WF1
WF2
0.30 0.88 0.04 0.60 -0.31 0.04 13.23 18.46
0.30 0.88 0.04 0.60 0.67 1.26 13.58 19.07
0.49 2.04 0.08 0.56 -0.75 -0.15 17.23 21.73
0.50 2.04 0.07 0.56 0.75 1.64 17.62 22.44
0.48 2.05 0.08 0.74 -1.38 -0.39 22.41 29.30
0.50 2.05 0.08 0.74 1.10 2.55 23.14 30.37
0.93 3.94 0.24 1.18 -4.45 -3.66 32.71 40.06
1.00 3.94 0.23 1.18 0.93 2.31 33.78 41.80
0.96 2.90 0.14 0.77 -4.80 -3.65 20.34 24.16
0.99 2.90 0.13 0.77 -1.72 -1.84 21.42 25.67
0.98 2.94 0.14 0.77 -2.52 -2.62 19.31 23.29
1.00 2.94 0.12 0.77 -0.35 -0.20 19.62 23.68
0.49 2.06 0.06 0.61 -0.62 0.06 14.71 19.29
0.50 2.06 0.06 0.61 0.83 1.81 15.54 20.52
0.30 0.90 0.03 0.53 -0.85 -0.69 9.19 13.69
0.30 0.90 0.04 0.53 0.14 0.51 10.08 14.48
Note: EC
V
, SCv is the mean and standard deviation of parameter Cv in Ns generating samples,
,
ECs
,
SC
S is the mean and standard deviation of parameter Cs .
4

A lot of calculation of Monte-Carlo experiment( refers to table [1]) shows : The new weighted function moment method based on L-moment may not only overcome the defect of parameter Cv estimated by WF1 ,but also may have almost non-negatively biased quantile estimation, and

Bx p
 is usually less than 2%. So WF2 is much better than WF1 in the bias of Parameter Cv and quantiles, but the effectiveness of the quantiles for above two methods are almost the same .
5 CONCLUSION
According to the calculation and analysis, our conclusion is as follows:
The new weighted function moment method (WF2) developed by us may really overcome the defect of negative bias for parameter Cv and quantiles. Therefore it is recommended that the WF2 may be used in practical flood frequency analysis for Pearson-III population distributions.
REFERENCES
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[7] Chen Y.F. Sha Z.G. et al , Study on L-Moment Estimation Method for P-III Distribution With
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[10] Rasmussen, P.F., Bobée, B., and Bernier, J., 1994. Parameter estimation for the Pearson type 3 distribution using order statistics (Durrans, 1992)---Comment, J. Hydrol., 153, 417-424.
[11] Durrans, S.R., 1992. Parameter estimation for the Pearson type 3 distribution using order statistics , J. Hydrol ., 133, 215-232.
[12] Wu B., Hou Y., and Ding J., 1991. Method of lower-bound to estimate the parameters of a
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