Received 30 June 2015

Accepted 9 August 2015

Evaluating the Three Methods of Goodness of Fit Test for Frequency Analysis 

Xiankui Zeng, Dong Wang
*
, Jichun Wu 

Key Laboratory of Surficial Geochemistry, Ministry of Education, School of Earth Sciences and Engineering, 
Nanjing University, Nanjing, China. 

*Email: wangdong@nju.edu.cn 
 

 

Abstract 

In hydrological statistics, the traditional assessment of goodness of fit test is interested in the testing precision to the sample 
generated from supposed PDF. However, the ability to reject the hypotheses when the supposed PDF is different from real 
PDF should be also emphasized. In addition, the sensitivity of test method to series length is important in hydrological 
analysis. Three methods of goodness of fit test that include Chi-Square (C-S), Kolmogorov-Smirnov (K-S), and 
Anderson-Darling (A-D) tests are applied in this study. The results of power test indicate that the most powerful tests for 
normal, uniform, P3, and Weibull distribution are K-S, C-S, and A-D tests, respectively. The test method with the best 
comprehensive power is C-S test, followed by K-S and A-D test. 

Keywords: Test method, Frequency analysis, Chi-Square, Kolmogorov-Smirnov, Anderson-Darling 

1 Introduction 

Frequency analysis is an important method in 
hydrological statistics, which has been applied in 
uncertainty assessment of hydrological process 
extensively, such as flood control, drought resistance, 
and safety design of water conservancy project (Raju 
and Kumar, 2011; Saf, 2009; Xu et al., 2010; Lee et 
al., 2012). This method serves many purposes in 
statistical data analysis by fitting a probability 
distribution function (PDF) to a series of observations. 
Then, the fitted PDF provides summary information 
about the data, and allows predictions beyond the 
range of data used for parameter estimation. This 
prediction is based on the information of past events 
in order to define the probabilities of future events. 
Therefore, the prediction is attached a risk of failure. 
The frequency analysis does not predict the future 
with certainty, but it provides good models for 
explaining and making efficient use of the events that 
had occurred in the past (Khaliq et al., 2006; 
Smakhtin, 2001; Yang et al., 2010). 

There are three steps in frequency analysis (Khaliq 
et al., 2006): (i) selecting a suitable probability 
distribution function (PDF); (ii) estimating the 
parameters of PDF based on samples; (iii) assessing 
the uncertainty of objective of interest in prescribed 
confidence level. Hereinto, how to select a suitable 

PDF is the most crucial step for frequency analysis, 
because the following parameter estimation is 
influenced by the supposed PDF. Therefore, the 
probability distribution characteristics of objective 
and its uncertainty assessment are determined by the 
selected PDF. Thus, plenty of studies focus on 
determining the PDF that fits best the observed 
hydrological series. Beard (1974) estimated the flood 
records at 300 stations in the USA by 8 different 
models, and the results showed that log-P3 and 
log-normal fitted the observations best. By applying 
the available methods to flood records which lengths 
varied between 44 and 97 year, the US Water 
Resources Council (1976) recommended the use of 
log-P3 distribution, with a regional skew coefficient 
in the case of short records. The Flood Studies Report 
of the NERC (1975) showed that the generalized 
extreme value (GEV) distribution was recommended 
for the use in regional flood frequency analysis. 
Vogel et al., (1993) analyzed the flood data at 61 
stations in Australia with records of 20 years or more. 
They found the best PDF at one region was even 
different for winter and summer-dominated rainfall 
regimes. Recently, Shabri and Ariff (2009) found that 
the generalized logistic distribution was the most 
suitable PDF to fit the data of maximum daily 
rainfalls for stations in Selangor and Kuala Lumpur. 

Journal of Risk Analysis and Crisis Response, Vol. 5, No. 3 (October 2015), 178-187

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mailto:wangdong@nju.edu.cn


Song and Singh (2010) analyzed the joint probability 
distribution of different hydrological periods in the 
Wei River basin, China. The results showed that the 
drought duration and inter-arrival time followed the 
Weibull distribution, and the drought severity 
followed the gamma distribution. Similar to 
mentioned literatures, many others researchers made 
the goodness of fit tests for different types of 
hydrological series, and the best PDFs were not the 
same at different regions or periods (e.g., Gunasekara 
and Cunnane,1991; Onoz and Bayazit, 1995; Onoz 
and Bayazit, 2001; Bi et al., 2010; Lansigan, 2009; 
Sandoval and Raynal-Villasenor, 2008; Shao et al., 
2004). Therefore, there is not universal applicability 
to the selection of best PDF, and it should be selected 
based on the practical observations and effective 
comparison and testing. 

The choice of the suitable probability distribution 
function (PDF) is almost arbitrary as no physical 
basis is available to rationalize the use of any 
particular function. A common way is selecting one 
from several often-used probability distribution 
functions (PDFs) by goodness of fit test. As a result, 
the result of frequency analysis is affected by the 
power of the goodness of fit test. In the context of 
probability statistics, there is a wealth of the 
literatures on the goodness of fit test, such as Pombo 
et al. (2015), Durot and Reboul (2010), Goegebeur 
and Guillou (2010), Lee (2010), Saldana-Zepeda et al. 
(2010), Stephens (1974a), Stephens (1974b), 
Woodruff et al. (1984). Nevertheless, for hydrological 
frequency analysis, except for the precision and 
efficiency of the goodness of fit test, some issues 
should also be given special consideration. Generally, 
there are two key points should be included. (i) The 
influence of series length on the goodness of fit test. 
It is hard to ensure the length of observations in 
hydrological process, the insufficiency of the sample 
size makes it difficult to choose the best PDF; (ii) the 
PDF for fitting observations is unknown in the 
ordinary course of events, it should be selected from 
several alternative PDFs. Therefore, not only is the 
precision important for goodness of fit test, but also 
the accuracy that selecting the suitable PDF and 
rejecting others from several alternative PDF is 
crucial. 

The assessment to the goodness of fit test is 
common in hydrological literatures. According to the 
flood studies report of the NERC (1975), the 
conventional Chi-Square (C-S) and 
Kolmogorov-Smirnov (K-S) goodness of fit tests 
were insensitive to departures from the assumed 
distribution for the samples size of 30 to 60. Ahmad 

et al. (1988) assessed the K-S, Anderson-Darling 
(A-D), and Cramer-von Mises (C-vM) goodness of fit 
tests by testing gamma distribution from several 
alternative distributions, which were gamma, weibull, 
normal, log-normal, uniform, and beta distribution. 
The sample size for each distribution was varied from 
5 to 30, each test was repeated for 5000 times. The 
power of goodness of fit test was defined as the 
proportion of rejection to the hypothesis that the 
samples used for test were not from gamma 
distribution, but from another distribution. The 
testing result showed that C-vM test was most 
powerful, followed by A-D test, but the differences 
between these tests were very slight. Nevertheless, 
the sample sizes of that study (5-30) was not widely 
representative in hydrological series, and the 
definition of power of test was not reasonable by only 
considering the proportion of rejections.  

Onoz and Bayazit (1995) studied the flood data of 
19 stations with record lengths in the range 60 to 165, 
several test methods were applied to search for the 
best probability distribution of flood series. They 
found that C-S test was highly sensitive to the 
location of the data near the class limits, small errors 
in the parameters of the distribution may have 
significant effects on the test results, and the data 
inside the class intervals was insignificant. Thus, C-S 
was not suitable for application in flood flows. In 
addition, A-D test was only limit to test GEV 
distribution, as the critical values were not available 
for other distributions. Therefore, this study was not a 
comprehensive assessment to the power of goodness 
of fit tests. Shahabuddin et al. (2009) investigated the 
power of several goodness of fit test such as K-S, 
A-D, C-vM, etc., and the critical values of each test 
were obtained by Monte Carlo simulation. By 
comparing with the power, the modified K-S test was 
found to be the best. In that study, however, the 
power of test was defined as the proportion of 
hypotheses that were rejected. In addition, the 
parameters of PDF were assumed known in 
hypothesis, and the samples used for testing were 
generated from same distribution, only with small 
difference in location or scale parameter. For 
hydrological series, however, the parameters of 
supposed distribution are unknown, and should be 
estimated by observations, and the observation data 
may be from the supposed PDF, or from other 
distributions. Therefore, the conclusions from 
Shahabuddin et al. (2009) may be not applicable for 
frequency analysis in hydrological series.  

As the special requirement in hydrologic frequency 
analysis, the supposed probability distribution 

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function (PDF) is arbitrary chosen from several 
common used functions. Moreover, the parameters of 
PDF should be estimated, and the length of series 
cannot be guaranteed. Therefore, the assessment of 
goodness of fit test should consider not only the 
testing precision to the sample from known 
distribution function, but also emphasis on the ability 
to reject the hypotheses when the supposed PDF is 
different from the real one. In this paper, three often 
used goodness of fit test methods, C-S, K-S, and A-D 
tests were assessed. The samples used for test were 
generated from 4 common functions: normal, P3, 
Weibull, and uniform. 

The paper is organized as follows: An introduction 
is first provided for the frequency analysis and 
goodness of fit test in hydrologic frequency analysis. 
It follows the description to the methods, including 
C-S, K-S, and A-D tests, the probability distribution 
functions and parameters estimation, and samples 
generation. Next, the statistic results of test, power 
comparison of the three test methods, and a 
discussion on the application of the goodness of fit 
test in hydrologic frequency analysis are presented. 
Finally, the main conclusions drawn from the analysis 
are presented. 

2 Methods 

The methods used in this paper include goodness 
of fit tests, C-S, K-S, A-D tests, parameter estimation 
for probability distribution function (PDF), and 
samples generation. 

2.1 Goodness of fit test 

Assuming (x1, x2, …, xn) is the samples from 
population X, making a hypothesis: H0: F(x) = F0(x), 
where F0(x) is the alternative PDF (supposed PDF) 
with the parameters estimated by samples. 
Chi-Square test 

Chi-Square (C-S) test is a simple and convenient 
method for hypothesis test, it is related to the overall 
fit, the process can be written as follows (Zhang, Luo, 
2000): 
(1) Choosing k-1 numbers: -∞ < t1 < t2 < … < tk-1 < 
+∞, k ≈ 1.87(n-1)0.4, and the number axis is 
partitioned into k intervals, (-∞, t1], (t1, t2], …, (tk-2, 
tk-1], (tk-1, +∞]. 
(2) Collecting the number of samples dropped into 
the i-th interval ni, i=1, 2, … , k, and then calculating 
the probability of the population which obeys 
alternative PDF fallen into the i-th interval: 

1 1 0 1

2 1 2 0 2 0 1

1 2 1 0 1 0 2

1 0 1

( ) ( )

( ) ( ) ( )

......

( ) ( ) ( )

( ) 1 ( )

k k k k k

k k k

p P X t F t

p P t X t F t F t

p P t X t F t F t

p P t X F t

    

 

  


    



     


   

    (1) 

(3) Constructing a statistics:  

2
2

1

( )k
i i

i i

n np

np





                (2) 

which obeys Chi-square distribution with the degree 
of freedom m, m=k-1, or m=k-1-r when there are r 
independent parameters of F0(x) need to be estimated 
by samples. And specifying a significance level α, if 
p (χ

2 ≥ χ2 1-α) ≥ α, then accept the hypothesis, otherwise 
reject it. 
Kolmogorov-Smirnov test 

Kolmogorov-Smirnov (K-S) test measures the 
greatest discrepancy between the observed and 
hypothesized distribution. The process can be 
summarized as follows (Melo et al., 2009; Wang, 
Wang, 2010): 
(1) Sorting the samples X (x1, x2, …, xn) by ascending 
order, and storing it to a new vector X′ (x′1, x

′
2, …, x

′
n), 

calculating the empirical distribution function: 

1

1

0,

( ) ,

1,

n k k

n

x x

k
F x x x x

n

x x



 



     


 

         (3) 

(2) Calculating the K-S statistics D(n): 

1

( )

0

0 0
1

max ( ) ( )

1
max ( ) , ( )

n

n

n
x x x

i i
i n

D F x F x

i i
F x F x

n n

   

 

  


   

       (4) 

Specifying a significance level α, if p (D(n) ≥ D(n)(1-α)) 
≥ α, then accept the hypothesis, otherwise reject it. 
Anderson-Darling test 

Anderson-Darling (A-D) test emphasizes 
discrepancies in both tails of the distribution, and that 
is often of prime importance in hydrologic frequency 
analysis. The process can be written as follows 
(Coronel-Brizio, Hernandez-Montoya, 2010; Zhang 
et al., 2009): 
(1) Sorting the samples X (x1, x2, …, xn) by ascending 
order, and storing it to a new vector X′ (x′1, x

′
2, …, x

′
n), 

calculating the empirical distribution function (Eq. 3). 
(2) Calculating the A-D statistics A2 n : 

2

1

1
[(2 1) log ( )

+ (2 1 2 ) log(1 ( )) 1, 2,...,

n

n i

i

i

A n i F x
n

n i F x i n



   

   


   (5) 

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And specifying a significance level α, if A2 n  < A2 n (1-α), 
then accept the hypothesis, otherwise reject it. 

However, the critical values of the distributions in 
this study are unavailable directly at the whole range 
of series length, and so, they are obtained by 
simulations, for which the following simulation steps 
are taken: 
(1) Generating a sample of length n from the selected 
distribution, and estimating the parameters based on 
sample. 
(2) Calculating the A-D statistics A2 n . 
(3) Based on Monte Carlo simulation, step (1) – step 
(2) are repeated 10000 times, and the A-D statistics 
are sorted by ascending order, the critical values at 
the significance level α = 0.01, 0.05, 0.1 are the 
values at 9900th, 9500th, 9000th of the statistics series. 

2.2 Probability distribution function and parameters 

estimation 

Four probability distribution functions are chosen 
for goodness of fit test in this study. Two of this have 
two parameters (normal, uniform), and other two 
have three parameters (P3, Weibull). These 
distributions have been selected because P3 and 
Weibull distribution are currently applied in the 
hydrologic frequency analysis in various countries 
frequently. Moreover, normal and uniform 

distributions have a significant different 
characteristics of probability distribution compared 
with other two distributions (Fig. 1), in addition, they 
are often used in other field, especially in 
hydrogeology field (Hassan et al., 2009; Kuczera, 
Parent, 1998). Therefore, the power of the goodness 
of fit test can be assessed comprehensively. 

Parameters of the P3 distribution are estimated by 
the principle of maximum entropy (Chen et al., 2002; 
Singh, 1987; Singh, Singh, 1985), and maximum 
likelihood method is used to estimate the parameters 
of Weibull distribution (Qiao and Tsokos, 1994; Qiao 
and Tsokos, 1995). 

2.3 Samples generation 

  The sample is fundamental to the power test of 
goodness of fit test in this paper. The assessment of 
the test methods is influenced by the characteristics 
of samples. Therefore, the characteristics of 
generated samples should be keeping identical with 
their probability density function as much as possible. 
The samples of 4 probability density functions 
(normal, P3, Weibull, and uniform distribution) are 
generated based on the procedure of random number 
generation of IMSL (IMSL, 1997). 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 1. The probability density curves of normal, uniform, P3, and Weibull distributions. 

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3 Results and discussion 

The procedure of this power test is described as 
Fig. 2. The four samples with length n were 
generated from normal, P3, Weibull, and uniform 
distribution, and the supposed probability density 
function (PDF) in null hypothesis was set as one of 
four above functions respectively. After that, 
according to the hypothesis, estimating the 
parameters of PDF by sample, and testing all the 
hypotheses by C-S, K-S, and A-D tests in turn. The 
length of series n was varied from 20 to 300 with 
interval 10, each sample with length n was repeated 
for 2000 times based on Monte Carlo simulation, and 
the sample was generated randomly for each running. 

The significance levels of Chi-Square (C-S), 
Kolmogorov-Smirnov (K-S), and Anderson-Darling 
(A-D) test are all set as 0.05, the acceptance 
proportion for each hypothesis is denoted as follows: 

_ _ _ ( ) _ _ ( ) / 2000P M S H i M S H i     (6) 
Which M is the method used for goodness of fit test, 
S is the function used for generating samples, H is the 
supposed function in hypothesis, i = 1, 2, 3, …29, 
indicates the length of series varies from 20 to 300, 
M_S_H(i) is the accepting times for the specific 
hypothesis. 

As showed by Fig. 3 – Fig. 5, for the three test 
methods, in most instances, the accepting proportion 
is increase with the length of series when the sample 

 

Fig. 2. The flow chart of this study. 

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is generated from supposed probability density 
function (PDF), and decrease when the sample is not 
generated from supposed function. In fact, it is 
attribute to the characteristics of sample which is 
closer to its PDF when the sample size is increased. 
However, when samples are generated from normal 
distribution, the hypotheses with Weibull as the 
assumed function have very high accepting 
proportion for all three goodness of fit tests. The 
accepting proportions of the hypotheses in which the 
actual and supposed PDF both are P3 distribution are 
low for C-S and K-S tests. K-S and A-D tests are easy 
to accept the hypotheses when the supposed function 
is Weibull distribution and the actual function is P3 
distribution. Furthermore, the K-S test is ineffective 
for testing the normal function from the samples 
generated by Weibull distribution, and A-D test is 
very easy to accept the hypotheses when the 
supposed function is P3 and the samples are 
generated from Weibull or uniform distribution. 

The ideal goodness of fit test should not only keep 
high testing precision for the samples generated from 
the supposed PDF, but also control the false accepting 
proportion when the samples are generated from 
other functions. Therefore, a formula is proposed for 
describing the power of goodness of fit test for a 
specific PDF: 

_ _ ( ) _ _ _ ( )

(1 _ _ _ ( ))

*
3

S H

S H

Power M S i P M S H i

P M S H i







      (7) 

The symbols in Eq. (7) are same as in Eq. (6). 
    Fig. 6 exhibits the powers of the three goodness of 
fit test methods for 4 specific PDFs. The power is
increase with the length of series for each test method. 
Moreover, A-D test is faster to get a stable power 
value and the starting power is higher for each PDF 
(except uniform distribution) than C-S and K-S tests. 
For the 4 PDFs, the most powerful goodness of fit 
test for normal, uniform, P3, and Weibull distribution 
are K-S, C-S, A-D, and A-D test respectively. For the 
test methods, C-S test is good at testing uniform, 
normal, and Weibull distribution, but bad at P3 
distribution. K-S test is powerful for testing normal 
and Weibull distribution, and weak at P3 distribution. 
Different from other two test methods, A-D test is 
good at P3, normal, and Weibull distribution, and 
ineffective to uniform distribution. 

Considering the powers for different probability 
density functions, the comprehensive power for each 
goodness of fit test is described as follows: 
 

 

 

 

Fig. 3. The accepting proportion of hypotheses by Chi-Square test. 

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Fig. 4. The accepting proportion of hypotheses by Kolmogorov-Smirnov test. 

 

 

 

Fig. 5. The accepting proportion of hypotheses by Anderson-Darling test. 

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Fig. 6. The powers of Chi-square, Kolmogorov-Smirnov, and Anderson-darling tests for specific probability 
density function. 

 

Fig. 7. The comprehensive powers of Chi-square, Kolmogorov-Smirnov, and Anderson-darling tests.

_ _

_
4

S

Power M S

Power M 


         (8) 

The symbols in Eq. (7) are same as in Eq. (6). 
Fig. 7 shows that the comprehensive powers for 

the tests are all increase with the series length. The 
goodness of fit test with the best performance is C-S 
test, the second is K-S test, and the last is A-D test, 
nevertheless, the difference among the powers of 3 
test methods is not so significant. 

From the testing results above, it is concluded that 
the three test methods used in this study measure the 
different aspects of the goodness of fit. The C-S test 
is related to the overall fitting. Therefore, it is 
appropriate to select C-S as the goodness of fit test 
when nothing is known about the samples. In 
addition, C-S test is powerful for distinguishing 
uniform distribution from other distributions and 
distinguishing other distributions when the sample is 
generated from uniform distribution. The power of 
C-S test is not so sensitive to the length of series (Fig. 
3). The K-S test measures the greatest discrepancy 
between the sample and supposed probability density 

function (PDF). It has the highest precision for 
testing the samples which generated from supposed 
function, and there is no other advantage to others 
except this. Moreover, the K-S test is most insensitive 
to the length of series (Fig. 4). The A-D test is 
emphasis on the differences at the both tails. Thus, 
the power for specific PDF is most sensitive to the 
length of series, and it reach to a stable value quickly 
(Fig. 5). In addition, the A-D test is extremely 
inefficient for testing PDFs when the samples are 
generated from uniform distribution. 

According to Fig. 3 – Fig. 5, it is obvious to found 
that the power for testing normal distribution from 
Weibull samples, and testing Weibull distribution 
from normal sample are fairly low for all the 3 test 
methods. Referring to the probability density 
functions of normal and Weibull distribution: 

 
2

2

2

1 ( )
( ; , ) exp[ ]

22

x
f x


 




         (9) 

1
( ; , , ) ( ) exp[ ( ) ], 0

x
f x x x

 



 
   

 

 
     

 (10) 

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In fact, the above 2 distributions can be both 
transformed from exponential gamma distribution: 

1
1

( ; , , , ) ( ) exp( ( ) )
( )

aa
b

f x a b c x c b x c
a

 




   


 (11) 

when the following conditions are satisfied 
respectively: 

1 1 1
, , 0,

2 2 2
a b c               (12) 

1
1, , ,a b c


  




            (13) 

Therefore, the normal and Weibull distribution can be 
mutual transformed in some special condition, and it 
is hard to test them from each other with high 
precision. 

To sum up the above arguments, each goodness of 
fit test has its own advantage and disadvantage. As a 
result of the power test in this paper adopted the 
different evaluation criterion from other studies, 
which considering the ability to distinguish the 
sample from both real and false distributions. 
Therefore, the A-D test is not always the best test 
method as reported in previous literatures. In order to 
acquire best testing effect in hydrological frequency 
analysis, it is beneficial to collect the prior 
information about the research variable as much as 
possible, and then select corresponding method. For 
example, from previous literatures, the observations 
always represent normal, Weibull, and P3 distribution, 
and then A-D test should be selected for testing, 
because the A-D test has greatest comprehensive 
power when uniform distribution is excluded. In 
addition, the A-D test should be adopted when the 
length of series is not so large. When nothing is 
known about the probability distribution 
characteristics of samples, the C-S test is preferred. In 
addition, parameters estimation of probability 
distribution functions is always included in the 
procedure of goodness of fit test. Thus, the methods 
used for estimating parameters may have influence 
on the results of hypothesis tests. Moreover, how to 
effectively distinguish normal distribution (or 
Weibull distribution) from Weibull distribution (or 
normal distribution) is the focus of future work. 

4 Conclusions 

By choosing Chi-Square (C-S), Kolmogorov- 
Smirnov (K-S), and Anderson-Darling (A-D) test as 
the methods of goodness of fit test, the supposed PDF 
in hypothesis and the function used for generating 
sample are selected as normal, uniform, P3, and 
Weibull distributions. Based on the comparison of 

goodness of fit test methods, following conclusions 
are demonstrated: 

(1) Each goodness of fit test has its own advantage 
and disadvantage. The most powerful goodness of fit 
test method for normal, uniform, P3, and Weibull 
distribution are K-S, C-S, A-D, and A-D test 
respectively. C-S test is good at testing uniform, 
normal, and Weibull distribution, but bad at P3 
distribution. K-S test is powerful for testing normal 
and Weibull distribution, and weak at P3 distribution. 
A-D test is good at P3, normal, and Weibull 
distribution, and ineffective to uniform distribution.  

(2) The goodness of fit test with the best 
comprehensive power is C-S test, the second is K-S 
test, and the last is A-D test. It is beneficial to search 
for the prior information about the research variables 
as much as possible, and then select corresponding 
method. 

Acknowledgements 

This study was supported by the National Basic 
Research Program (2013CB956503), National 
Natural Science Fund of China (No. 41302181, 
41172207, 51190091 and 51409161), and Program 
for New Century Excellent Talents in University 
(NCET-12-0262), Science and Technology program 
of Jiangsu Province (No. BK20140080). 

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