Estimating the Confidence Intervals for the Coefficients of Variation
Delta-Gamma Distributions
WANSIRI KHOORIPHAN, SA-AAT NIWITPONG, SUPARAT NIWITPONG
Department of Applied Statistics,
Faculty of Applied Science,
King Mongkut’s University of Technology North Bangkok, Bangkok,
THAILAND
Abstract: - A common application of the coefficient of variation (CV), which is the ratio of the population
standard deviation to the population mean, is frequently used to assess quality control and economic processes,
among others. The fiducial quantity approach, Bayesian confidence intervals (CIs) using the Jeffreys, uniform,
or normal-gamma-beta (NGB) priors, and highest posterior density (HPD) intervals using the Jeffreys, uniform,
or NGB priors were used to provide estimators for the CI for the ratio of CV of two delta-gamma distributions.
An evaluation of their performance in terms of average length and coverage probability was carried out using
Monte Carlo simulations. The results of this study indicate that the HPD using the Jeffreys prior and fiducial
quantity methods are the best for estimating the CI for the ratio of the CV of two delta-gamma distributions.
Rainfall data from Mae Hong Son province in Thailand was used to illustrate their practicability when
analyzing real-life processes.
Key-Words: - Delta-gamma distribution, Highest posterior density, Jeffreys prior, Normal-Gamma-Beta prior,
Simulation, Uniform prior, Fiducial quantities.
Received: March 22, 2024. Revised: August 14, 2024. Accepted: September 16, 2024. Published: October 24, 2024.
1 Introduction
Thailand’s rainy season lasts from mid-May to mid-
October, with the southwest monsoon bringing an
abundance of annual rainfall. For the majority of the
country, the greatest rainfall occurs from August to
September whereas January and December are
exceptionally dry. Thus, rainfall data in Thailand
and other nations typically includes zero readings at
certain times of the year, which should be
considered when researching rainfall. Aitchison
presented a model for situations in which there are
zero observations by assigning a probability of δ
that the dataset contains zero observations and 1 − δ
as the residual probability for the positive
observations, [1]. The delta-lognormal distribution
first suggested by Aitchison and Brown, includes a
random variable with a lognormal distribution for
the positive observations and a binomial distribution
for the number of zero observations, [2].
The CIs for the parameters of the delta-gamma
distribution and other related distributions have been
determined by numerous researchers using various
methods for statistical inference. For instance,
Muralidharan and Kale created the CIs for the mean
of the mixed distribution and a modified gamma
distribution that includes a singularity at zero, [3].
Lecomte et al. suggested applying the compound
Poisson-gamma and delta-gamma distributions to
handle zero-inflated continuous data inside the
variable sampling volume regime, [4].
The population CV can be defined as the ratio of
the population mean to the population standard
deviation, [5]. Among other fields, biology,
economics, and quality control all frequently use the
CV, [6]. CIs have been provided by many
researchers for the CV of different distributions. For
instance, Methods for estimating the CI for the ratio
of the CVs within two gamma distributions were
proposed in [7]. In [8], it is proposed various methods
to estimate the CI for the ratio of the CVs in two
inverse gamma distributions.
To compare CVs in two populations, one
appropriate method is to use the ratio of the CVs of the
two populations of rainfall data that contain zero
observations. This ratio can be represented by utilizing
the two delta-gamma distributions. So far, no
publications have been forthcoming on estimating the
CIs for the ratio of the CVs within delta-gamma
distributions. In the examination, we used the fiducial
quantity (FQ) and six Bayesian-based approaches to
estimate the CI for this scenario. The Bayesian
methods are Bayesian CIs based on the Jeffreys
(B.Jef), uniform (B.Uni), or NGB (B.NGB) priors and
three HPD based on the Jeffreys (H.Jef), uniform
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(H.Uni), or NGB (H.NGB) priors. Furthermore, we
demonstrate their practicability for real-life situations
by applying them to analyze rainfall data from Mae
Hong Son province in Thailand.
2 Preliminary
The delta-gamma distribution function can be
specified as follows:
0,
;
; , , 0.
1 ; , ;
iij
ij i i i ij
i i ij i i
x
Gx x
Fx
(1)
The gamma cumulative distribution function can
be indicated as
;,
ij i i
Fx
.
Moreover,
ii
and
2
ii
are the respective
means and variances of gamma
,
ii
distribution
with shape parameter
i
and scale parameter
i
. For the zero and positive observations,
identified by
(0)i
n
and
,(1)i
n
respectively, where
,(0) ,(1)i i i
n n n
, the positive observations, follow a
gamma distribution and the zero observations follow
a binomial distribution.
The maximum likelihood estimators (MLEs) of
i
,
i
, and
i
are respectively described as:
,(0)
ˆ/
i i i
nn
,
1
1
ˆ,
2 log log /
i
in
ij ij i
i
X X n
and
ˆˆ
/,
i ij i
X
.
where
ij
X
represents the sample mean of
ij
X
[11].
Following [1], the population mean and variance
of
ij
X
are defined as follows:
( ) 1
ij i i i
EX
(2)
and
22
( ) = (1 ) ( ) (1 ) ( )
ij i i i i i i i
Var X
(3)
Consequently, the CV of
ij
X
can be expressed as
1
1
ii
ij i
ii
CV X v
. (4)
The ratio of their CVs is given by
1
2
v
v
. (5)
The methods to construct the CI for
are
provided in the next subsections.
2.1 The Fiducial Quantity Method
Krishnamoorthy and Wang derived an FQ based on cube
root-transformed samples, [9]. Let
12
, , , ; 1,2
i
ij i i in
X X X X i
and
1,2, , i
jn
be a
sample from a delta-gamma distribution with shape
parameter
i
a
and scale parameter
i
b
. For
1
3; 1, 2
ij ij
Y X i
and
1,2, , i
jn
, then
ij
Y
is
approximately normally distributed with means
i
and variances
2
i
provided by
2
13
2
3
1
3
1
( ) 1 and
99
i
i i i i
ii
b
ba aa
. (6)
The respective FQs of
i
and
2
i
are
2
2
22
11
1( 1)
and ,
ii
ii
ii i i i
ij
nn
i
Zn s n s
Q x Q
n
(7)
where
ij
x
and
i
s
are the observed values of
ij
X
and
i
S
, respectively;
0,1
i
ZN
; and
2
1ni
is an
independent random variable from a Chi-squared
distribution. The FQs for the shape parameter are
1
22
22
22
1
= 1 0.5 1 0.5 1
9
ii
i
ii
QQ
QQQ
. (8)
Then, the FQs for
i
are as follows, [10]
,(1) ,(0) ,(1) ,(0)
11
Beta( , 1) Beta( 1, ).
22
i i i i
i
Q n n n n
:
(9)
Therefore, the FQs for
i
of a delta-gamma
distribution have the following specifications:
1
(1 )
ii
i
ii
v
QQ
QQQ
. (10)
Now, the FQs for
is given by
1
2
v
v
Q
QQ
. (11)
Subsequently, the equal-tailed
100(1 )%
FQ
interval for
can be derived as
( / 2), (1 / 2)
FQ
CI Q Q
, (12)
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where
( / 2)Q
and
(1 / 2)Q
are the
( / 2)100th
and
(1 / 2)100th
percentiles of the distribution of
Q
, respectively.
The CI for
can be obtained by executing
Algorithm 1.
Algorithm 1.
1. Utilizing
( , , )
ij i i i
X
:
, compute
ij
x
and
2
i
s
of
the cube root-transformed samples.
2. Generate
(0,1)
i
ZN:
and
2
1ni
.
3. Generate
,(1) ,(0)
Beta( , 1)
ii
nn
and
,(1) ,(0)
Beta( 1, )
ii
nn
.
4. Calculate
i
Q
and
2
i
Q
using Equation (7).
5. Calculate
i
Q
and
i
Q
using Equations (8) and
(9).
6. Calculate
i
Q
and
Q
using Equations (10) and
(11).
7. Steps 2-6 are repeated 5,000 times to obtain the
Q
.
8. Calculate the 95% CIs for
using Equation (12).
9. To acquire the average lengths (ALs) and
coverage probabilities (CPs), steps 1 through 8 are
repeated 10,000 times.
2.2 The Bayesian Methods
The posterior distribution is used to determine
confidence intervals (CIs) based on the Bayesian
technique for the parameter of interest. [11],
whereas posterior distributions are used to create the
HPD intervals using a Bayesian approach. The
parameter values with the highest posterior density
make up the HPD, whereas the narrowest interval
that can be discovered for a parameter of interest
with probability
100(1 )%
is the HPD interval. [5],
[12]. The HPD was first provided by [13].
Let
( | )py
be a posterior density function. A
region R in the parameter space of
is called a
HPD region of content
(1 )
if
(i)
( | ) = 1Pr R y
,
(ii) For
1R
and
2 1 2
, ( | ) ( | )R p R y p R y
.
2.1.1 Jeffreys Prior
Jeffreys presented a prior created from the square
root of the Fisher information matrix characterized
as
( ) | ( ) |pI
, [14]. The Jeffreys prior for
of
a binomial distribution is characterised as
11
22
( ) ( ) (1 )
i i i
p
, from which the marginal
posterior distribution of
i
is obtainable in the
manner described below:
( ) ,(0) ,(1)
13
|,
22
i jef ij i i
x Beta n n
:
. (13)
Subsequently, the Jeffreys prior for
2
i
in a
lognormal distribution is
22
()
ii
p
. Hence, the
respective marginal posterior distributions of
2
i
are
as follows:
2
,(1)
2=1
()
()
|,
22
ni
ij i
ii
i jef ij
x
n
x IG
:
. (14)
Likewise, the following are the marginal
posterior distributions of
i
:
22
( ) ( ) ,(1)
| , ( , / )
ij
i jef i i jef i
x N x n
:
. (15)
Additionally, we may use
2
()
|,
i jef i x
and
2
()
|
i jef ij
x
to get the gamma distribution's mean and
variance in the following ways:
3
2
( ) ( ) 2
( . ) ( )
=22
i jef i jef
i B Jef i jef
M
, (16)
4
22
( ) ( ) ( )
( . ) 1/4 2 1/4
()
4
=2(9 )( )
i jef i jef i jef
i B Jef
i jef
V
. (17)
Then
( . )
( . )
( . )
=i B Jef
i B Jef
i B Jef
V
M
. (18)
So that
1( . )
.
2( . )
=B Jef
B Jef
B Jef
. (19)
Based on the B.Jef and H.Jef methods, the CI
and HPD intervals for
of a delta-gamma
distribution are defined as
..
.= [ ( / 2), (1 / 2)]
B Jef B Jef
B Jef
CI
. (20)
2.2.2 Uniform Prior
The prior probability is a constant function, [15],
that certainly sets a prior for all possible values,
[16]. The uniform prior for
i
of a binomial
distribution is
( ) 1
i
p
, [12], from which the
marginal posterior distribution of
i
can be obtained
as follows:
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( ) ,(0) ,(1)
| ( 1, 1)
i unif ij i i
x Beta n n
:
. (21)
Kalkur and Rao indicated the uniform prior for
2
i
is
21
i
, [17].
Consequently, the following is the marginal
posterior distribution of
2
i
:
2
,(1)
2=1
()
()
2
|,
22
ni
ij i
ii
i unif ij
x
n
x IG
:
. (22)
Likewise, the respective marginal posterior
distribution of
i
are as follows:
22
( ) ( ) ,(1)
| , ( , / )
ij
i unif i ij i unif i
x N x n
:
. (23)
Additionally, we may use
2
()
|,
i unif i ij
x
and
2
()
|
i unif ij
x
to calculate the gamma distribution's mean
and variance in the following ways:
3
2
( ) ( ) 2
( . ) ( )
=22
i unif i unif
i B Uni i unif
M
, (24)
4
22
( ) ( ) ( )
( . ) 1/4 2 1/4
()
4
=2(9 )( )
i unif i unif i unif
i B Uni
i unif
V
, (25)
Then
( . )
( . )
( . )
=i B Uni
i B Uni
i B Uni
V
M
. (26)
So that
1( . )
.
2( . )
=B Uni
B Uni
B Uni
. (27)
Based on the B.Uni and H.Uni methods, the CI
and HPD intervals for
of a delta-gamma
distribution are defined as
..
.= [ ( / 2), (1 / 2)
B Uni B Uni
B Uni
CI
. (28)
2.2.3 Normal-Gamma-Beta Prior
Maneerat and Niwitpong recommended employing
the H.NGB interval to calculate the common mean
for several delta-lognormal distributions, which
worked well on small-to-large sample sizes, [18].
Let
= log ; = 1, 2
ii
Y W i
be a random variable of a
normal distribution with mean
i
and precision
i
where
( , )
i i i
W LN
:
and
2
=
ii
. The H.NGB for
= ( , , )
i i i i
is indicated as
1/2
1
( ) (1 )
i i i i
p
, where
,
ii
has the
normal-gamma distribution, and
i
has the beta
distribution. Hence, the marginal posterior
distributions of
i
,
2
i
, and
i
are as follows:
( ) ,(0) ,(1)
11
|,
22
i NGB ij i i
x Beta n n
:
, (29)
,(1)
2
,(1)
2=1
()
()
1
|,
22
ni
ij i
ii
i NGB ij
x
n
x IG
:
, (30)
2
=1
, 2( 1)
,(1) ,(1) ,(1)
()
|,
( 1)
ni
ij
ij
i
ij
i NGB ij niii
xx
x t x nn
:
. (31)
Additionally, we may use
()
|
i NGB ij
x
and
2
()
|
i NGB ij
x
to calculate the gamma distribution's mean
and variance in the following ways:
3
2
( ) ( ) 2
( . ) ( )
=,
22
i NGB i NGB
i B NGB i NGB
M
(32)
4
22
( ) ( ) ( )
( . ) 1/4 2 1/4
()
4
=2(9 )( )
i NGB i NGB i NGB
i B NGB
i NGB
V
. (33)
Then
( . )
( . )
( . )
=i B NGB
i B NGB
i B NGB
V
M
. (34)
So that
1( . )
.
2( . )
=B NGB
B NGB
B NGB
. (35)
Based on the B.NGB and H.NGB methods, the
CI and HPD intervals for
of a delta-gamma
distribution are defined
as
..
.= [ ( / 2), (1 / 2)].
B NGB B NGB
B NGB
CI
(36)
Algorithm 2.
1. Utilizing
( , , )
ij i i i
X
:
, compute
ij
x
and
2
i
s
of
the cube root-transformed samples.
2. Generate
|
i ij
x
using Equations (13), (21) and
(29).
3. Generate
2|
i ij
x
using Equations (14), (22), and
(30).
4. Generate
2
|,
i i ij
x
using Equations (15), (23)
and (31).
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5. Calculate the mean and variance using Equations
(16), (17), (24), (25), (32) and (33).
6. Calculate
i
and
using Equations (18), (19),
(26), (27), (34), and (35).
7. Calculate the 95% CIs and HPD for
using
Equations (20), (28) and (36).
8. Steps 1-7 are repeated 10,000 times to obtain the
CPs and ALs.
3 The Monte Carlo Simulation Study
The performances of the CI estimators for the ratio
of CVs of two delta-gamma distributions
constructed with FQ, B.Jef, H.Jef, B.Uni, H.Uni,
B.NGB, and H.NGB were compared in terms of
their CPs and ALs, with the most effective one for a
given situation providing the CP close to or above
0.95 and the shortest AL. There were 10,000
repetitions (M) used in the Monte Carlo simulation
and 5,000 replicates (m) for FQ with a nominal
confidence level of 0.95 employing R statistics
software (version 4.1.0). The data were generated
for
( , , ); 1,2
ij i i i
Xi
:
and
1, 2,..., i
jn
. We
chose (15,15), (25,25), (50,50), or (100,100) for
equal sample sizes
12
( = )nn
and (15,25), (25,50), or
(50,100) for unequal sample sizes
12
( = )nn
. The
probabilities of zeros
12
( , )
were set as (0.2,0.2),
(0.4,0.4), or (0.6,0.6), the shape parameters
12
( , )
as (0.05,0.05), (0.05,0.06), (0.06,0.05) or
(0.06,0.06), and the rate parameter
12
( , )
as (2,2).
The performance of the different techniques for
estimating the nominal
95%
CIs for the ratio of the
CV of two delta-gamma distributions are shown in
Table 1, Table 2 and Figure 1, Figure 2, Figure 3
and Figure 4 in Appendix. The simulation results are
reported in Table 1 and Table 2, and the CPs and
ALs from Table 1 and Table 2 are compiled in
Figure 1, Figure 2, Figure 3 and Figure 4 in
Appendix.
4 Application of the Methods to Real-
World Data Situations
The Upper Northern Region Irrigation Hydrology
Center in Mae Hong Son province, Thailand,
provided monthly rainfall data that were utilized to
compare the CI estimators' performances.
Initially, we employed four models: Cauchy,
Normal, Gamma, and Lognormal to find the best
fitting distribution for the positive rainfall data using
the Akaike information criterion (AIC). AIC is
defined as AIC = −2 lnL+ 2k where L is the
likelihood function and k is the number of
parameters and n be the number of recorded
measurements. From the results in Table 3
(Appendix) ; it can be seen that Data fitting to a
gamma distribution produced the lowest AIC values,
so it was deemed to be the most appropriate.
4.1 The CI for the Ratio of the CVs with
Equal Sample Sizes
The monthly rainfall data from Mueang district,
Mae Hong Son province, in February from 1987 to
2022 and December from 1987 to 2022 were used as
the datasets. The summary statistics in February
were
1
x
= 14.1461,
1
n
= 36,
1,(1)
n
= 13,
1,(0)
n
= 23 and
the MLEs for
1 1 1
,,
, and
1
were
1
ˆ
=
1.0676,
1
ˆ
= 0.64,
1
ˆ
= 13.2501, and
1
ˆ
= 2.0888,
respectively. The summary statistics in the
December dataset were
2
x
= 26.10,
2
n
= 36,
2,(1)
n
=
17,
2,(0)
n
= 19 and the MLEs for
2 2 2
,,
, and
2
were
2
ˆ
= 0.6115,
2
ˆ
= 0.52,
2
ˆ
= 42.6811, and
2
ˆ
=
0.9760, respectively. The 95% CIs estimates for
are shown in Table 4 (Appendix).
From the simulation study results for
12
,nn
= 25
and
12
,
= 0.6, although all of the techniques
achieved CPs close to 0.95, H.Jef obtained the
shorter AL. Therefore, H.Jef is the most effective
technique for creating the CI for the ratio of CVs of
rainfall datasets from the Mueang district in Mae
Hong Son province for February from 1987 to 2022
and December from 1987 to 2022.
4.2 The CI for the Ratio of the CVs with
Unequal Sample Sizes
The monthly rainfall data from Mueang district,
Mae Hong Son province, for January from 2000 to
2022 and November from 1992 to 2022 were used
as the datasets. The summary statistics in the
January dataset from January were
1
x
= 20.48,
1
n
=
23,
1,(1)
n
= 15,
1,(0)
n
= 8 and the MLEs for
1 1 1
,,
,
and
1
were
1
ˆ
= 0.9641,
1
ˆ
= 0.35,
1
ˆ
= 21.2435,
and
1
ˆ
= 1.4573, respectively. The summary
statistics in November dataset were
2
x
= 45.8519,
2
n
= 30,
2,(1)
n
= 27,
2,(0)
n
= 3 and the MLEs for
2 2 2
,,
, and
2
were
2
ˆ
= 1.2111,
2
ˆ
= 0.1,
2
ˆ
=
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37.8569, and
2
ˆ
= 1.0141, respectively. The
95%
CIs estimates for
are showed in Table 5
(Appendix).
From the simulation study results for
1
n
= 15,
2
n
= 25, and
12
,
= 0.2, the FQ, H.Jef, H.Uni,
B.NGB, and H.NGB techniques achieved CPs close
to 0.95 but H.Jef obtained the shorter AL.
Therefore, H.Jef is the most effective technique for
creating the CI for the ratio of CVs of rainfall data
from the Mueang district in Mae Hong Son province
for January from 2000 to 2022 and November from
1992 to 2022.
5 Conclusions
We produced estimators for the CI for the ratio of
the CVs of two delta-gamma distributions by
utilizing the FQ, B.Jef, H.Jef, B.Uni, H.Uni,
B.NGB, and H.NGB techniques. To assess their CPs
and ALs, a Monte Carlo simulation was run.
Following that, monthly rainfall data from
Thailand's Mae Hong Son province were used to test
the proposed approaches. The findings indicate that
the H.Jef and FQ methods are the best for estimating
the CI for the ratio of the CVs of two delta-gamma
distributions.
Acknowledgment:
The first author would like to express gratitude to
the Thai Scientific Achievement Scholarship
(SAST) for financial assistance.
References:
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Positive Random Variable Having a Discrete
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10.1080/01621459.1955.10501976
[2] J. Aitchison and J.A.C. Brown, The
Lognormal Distribution: With Special
Reference to Its Uses in Economics London,
Cambridge University Press: UK, 1963.
[3] K. Muralidharan and B.K. Kale, “Modified
Gamma Distributions with Singularity at
Zero,” Communications in Statistics-
Simulation and Computation, vol. 31, no. 1,
pp. 143-158, 2002.
[4] J.B. Lecomte, H.P. Benot, S. Ancelet, M.P.
Etienne, L. Bel, and E. Parent, “Compound
Poisson-Gamma vs. Delta-Gamma to Handle
Zero-Inflated Continuous Data Under a
Variable Sampling Volume,” Methods in
Ecology and Evolution, vol. 4, no. 12, pp.
1159-1166, 2013. DOI: 10.1111/2041-
210X.12122
[5] G. Casella and R.L. Berger, Statistical
Inference, Cengage Learning: Boston, 2001.
[6] T. Holgersson, P. Karlsson, and R. Mansoor,
“Estimating Mean-Standard Deviation Ratios
of Financial Data,” Journal of Applied
Statistics, vol. 39, no. 3, pp. 657-671, 2013.
[7] P. Sangnawakij, S.A. Niwitpong, and S.
Niwitpong, “Confidence Intervals for the Ratio
of Coefficients of Variation of the Gamma
Distributions,” In Lecture Notes in Computer
Science, pp. 193-203, 2015. DOI: 10.1007/978-
3-319-25135-6_19
[8] T. Kaewprasert, S.A. Niwitpong, and S.
Niwitpong, “Confidence Interval for the Ratio
of the Coefficient of Variation of Inverse
Gamma Distributions,” Applied Science and
Engineering Progres, vol. 16, no. 1, pp. 1-11,
2022. DOI: 10.14416/j.asep.2021.12.002
[9] K. Krishnamoorthy and X. Wang, “Fiducial
Confidence Limits and Prediction Limits for a
Gamma Distribution: Censored and
Uncensored Cases,” Environmetrics, vol. 27,
no. 8, pp. 479-493, 2016. DOI:
10.1002/env.2408
[10] X. Li, X. Zhou, and L. Tian, “Interval
Estimation for the Mean of Lognormal Data
with Excess Zeros,” Statistics Probability
Letters, vol. 83, no. 11, pp. 2447-2453, 2013.
[11] A. Gelman, J.B. Carlin, H.S. Stern, D.B.
Dunson, A. Vehtari, and D.B. Rubin,
Bayesian Data Analysis, Chapman and
Hall/CRC: United States, 2013. DOI:
10.1201/b16018
[12] W.M. Bolstad and J.M. Curran, Introduction
to Bayesian Statistics, Wiley: Hoboken, 2016.
[13] G.E.P. Box and G.C. Tiao, Bayesian Inference
in Statistical Analysis, Wiley Classics: New
York, 1973.
[14] H. Jeffreys, Theory of Probability, Oxford
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[15] J.V. Stone, Bayes’ Rule: A Tutorial
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Models in Cognitive Neuroscience: A
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Cognitive Neuroscience, pp. 179-197, 2015.
[17] T.A. Kalkur and A. Rao, “Bayes Estimator for
Coefficient of Variation and Inverse
WSEAS TRANSACTIONS on SYSTEMS
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Coefficient of Variation for the Normal
Distribution,” International Journal of
Statistics and Systems, vol. 12, no. 4, pp. 721-
732, 2017.
[18] P. Maneerat and S.A. Niwitpong, “Estimating
the Average Daily Rainfall in Thailand Using
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PeerJ, vol. 9, no. e10758, 1-28, 2021. DOI:
10.7717/peerj.10758
APPENDIX
Fig. 1: Charts of lines of the CPs provided by the
techniques for the CI of the ratio of the CVs for
various probabilities of zero values (
12
nn
)
Fig. 2: Charts of lines of the ALs provided by the
techniques for the CI of the ratio of the CVs for
various probabilities of zero values (
12
nn
)
Fig. 3: Charts of lines of the CPs provided by the
techniques for the CI of the ratio of the CVs for
various probabilities of zero values (
12
nn
)
Fig. 4: Charts of lines of the ALs provided by the
techniques for the CI of the ratio of the CVs for
various probabilities of zero values (
12
nn
)
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2024.23.29
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Suparat Niwitpong
E-ISSN: 2224-2678
268
Volume 23, 2024
Table 1. The CPs and the ALs of the different techniques for estimating the nominal 95% two-sided CI for
(
12
nn
)
12
,nn
12
,
12
,
CPs
(ALs)
FQ
B.Jef
H.Jef
B.Uni
H.Uni
B.NGB
H.NGB
15,15
0.2,0.2
0.04,0.04
0.9707
0.9605
0.9667
0.9538
0.9627
0.9951
0.9945
(1.4425)
(1.3376)
(1.2569)
(1.4186)
(1.3244)
(2.2530)
(2.0220)
0.05,0.05
0.9677
0.9554
0.9653
0.9461
0.9584
0.9960
0.9958
(1.3944)
(1.2003)
(1.1374)
(1.3217)
(1.2400)
(2.1218)
(1.9129)
0.06,0.06
0.9649
0.9538
0.9622
0.9456
0.9577
0.9948
0.9929
(1.3384)
(1.0984)
(1.0471)
(1.2464)
(1.1725)
(1.9911)
(1.8044)
0.4,0.4
0.04,0.04
0.9851
0.9954
0.9927
0.9872
0.9850
0.9985
0.9973
(1.3342)
(1.4070)
(1.3211)
(1.6104)
(1.4756)
(1.8695)
(1.7111)
0.05,0.05
0.9833
0.9927
0.9927
0.9851
0.9853
0.9977
0.9965
(1.3373)
(1.3354)
(1.2569)
(1.5878)
(1.4513)
(1.8376)
(1.6825)
0.06,0.06
0.9859
0.9939
0.9945
0.9878
0.9863
0.9986
0.9981
(1.3259)
(1.2748)
(1.2035)
(1.5582)
(1.4238)
(1.7969)
(1.6460)
0.6,0.6
0.04,0.04
0.9940
0.9992
0.9981
0.9946
0.9905
0.9992
0.9972
(0.9942)
(1.6701)
(1.5299)
(2.5930)
(2.1140)
(2.1456)
(1.8853)
0.05,0.05
0.9959
0.9993
0.9991
0.9961
0.9927
0.9991
0.9988
(1.7455)
(1.6467)
(1.5066)
(2.7124)
(2.1831)
(2.1908)
(1.9151)
0.06,0.06
0.9949
0.9997
0.9985
0.9956
0.9899
0.9994
0.9976
(1.7861)
(1.6212)
(1.4821)
(2.7826)
(2.2238)
(2.2108)
(1.9253)
25,25
0.2,0.2
0.04,0.04
0.9688
0.9273
0.9358
0.9240
0.9345
0.9976
0.9982
(1.0913)
(0.8949)
(0.8677)
(0.9215)
(0.8921)
(1.7492)
(1.6234)
0.05,0.05
0.9665
0.9273
0.9345
0.9238
0.9314
0.9972
0.9971
(1.0313)
(0.8046)
(0.7838)
(0.8428)
(0.8189)
(1.5958)
(1.4897)
0.06,0.06
0.9630
0.9255
0.9328
0.9229
0.9309
0.9958
0.9960
(0.9725)
(0.7484)
(0.7312)
(0.7946)
(0.7737)
(1.4561)
(1.3664)
0.4,0.4
0.04,0.04
0.9857
0.9885
0.9901
0.9836
0.9848
0.9994
0.9992
(0.9662)
(0.9784)
(0.9458)
(1.0214)
(0.9833)
(1.4278)
(1.3504)
0.05,0.05
0.9799
0.9836
0.9854
0.9772
0.9809
0.9993
0.9988
(0.9496)
(0.9285)
(0.8994)
(0.9844)
(0.9489)
(1.3682)
(1.2963)
0.06,0.06
0.9810
0.9852
0.9878
0.9800
0.9838
0.9991
0.9989
(0.9274)
(0.8899)
(0.8636)
(0.9547)
(0.9210)
(1.3090)
(1.2430)
0.6,0.6
0.04,0.04
0.9922
0.9985
0.9983
0.9938
0.9952
0.9997
0.9987
(1.0493)
(1.1545)
(1.1031)
(1.2878)
(1.2098)
(1.4506)
(1.3636)
0.05,0.05
0.9913
0.9979
0.9975
0.9953
0.9942
0.9995
0.9993
(1.0616)
(1.1243)
(1.0750)
(1.2872)
(1.2062)
(1.4412)
(1.3540)
0.06,0.06
0.9804
0.9852
0.9866
0.9782
0.9817
0.9992
0.9988
(0.9255)
(0.8878)
(0.8616)
(0.9538)
(0.9200)
(1.3032)
(1.2377)
50,50
0.2,0.2
0.04,0.04
0.9701
0.9007
0.9050
0.9003
0.9042
0.9992
0.9993
(0.7754)
(0.5754)
(0.5671)
(0.5829)
(0.5743)
(1.2727)
(1.2147)
WSEAS TRANSACTIONS on SYSTEMS
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Suparat Niwitpong
E-ISSN: 2224-2678
269
Volume 23, 2024
Table 1. continued.
12
,nn
12
,
12
,
CPs
(ALs)
FQ
B.Jef
H.Jef
B.Uni
H.Uni
B.NGB
H.NGB
0.05,0.05
0.9627
0.8895
0.8948
0.8872
0.8913
0.9968
0.9971
(0.7201)
(0.5239)
(0.5171)
(0.5337)
(0.5266)
(1.1243)
(1.0790)
0.06,0.06
0.9598
0.8948
0.8931
0.8943
0.8949
0.9955
0.9955
(0.6650)
(0.4882)
(0.4825)
(0.4994)
(0.4933)
(0.9931)
(0.9577)
0.4,0.4
0.04,0.04
0.9831
0.9770
0.9768
0.9730
0.9740
0.9995
0.9998
(0.6788)
(0.6408)
(0.6300)
(0.6491)
(0.6379)
(1.0479)
(1.0128)
0.05,0.05
0.9794
0.9737
0.9741
0.9703
0.9703
0.9994
0.9993
(0.6569)
(0.6110)
(0.6014)
(0.6218)
(0.6116)
(0.9731)
(0.9431)
0.06,0.06
0.9787
0.9754
0.9765
0.9725
0.9734
0.9986
0.9989
(0.6307)
(0.5877)
(0.5790)
(0.6007)
(0.5913)
(0.9054)
(0.8796)
0.6,0.6
0.04,0.04
0.9916
0.9960
0.9948
0.9934
0.9930
0.9999
0.9998
(0.6987)
(0.7463)
(0.7304)
(0.7651)
(0.7478)
(1.0328)
(0.9985)
0.05,0.05
0.9882
0.9946
0.9944
0.9906
0.9926
0.9999
0.9997
(0.6917)
(0.7268)
(0.7119)
(0.7504)
(0.7337)
(0.9925)
(0.9609)
0.06,0.06
0.9895
0.9955
0.9949
0.9930
0.9927
0.9998
0.9996
(0.6799)
(0.7120)
(0.6978)
(0.7390)
(0.7227)
(0.9530)
(0.9240)
100,100
0.2,0.2
0.04,0.04
0.9694
0.8822
0.8806
0.8839
0.8827
0.9995
0.9995
(0.5520)
(0.3915)
(0.3880)
(0.3939)
(0.3904)
(0.9206)
(0.8944)
0.05,0.05
0.9656
0.8740
0.8718
0.8726
0.8717
0.9983
0.9985
(0.5064)
(0.3569)
(0.3540)
(0.3597)
(0.3567)
(0.7881)
(0.7693)
0.06,0.06
0.9587
0.8707
0.8694
0.8721
0.8688
0.9963
0.9971
(0.4619)
(0.3321)
(0.3296)
(0.3355)
(0.3329)
(0.6765)
(0.6629)
0.4,0.4
0.04,0.04
0.9851
0.9695
0.9705
0.9686
0.9691
0.9999
0.9999
(0.4854)
(0.4403)
(0.4359)
(0.4423)
(0.4379)
(0.7673)
(0.7509)
0.05,0.05
0.9816
0.9673
0.9679
0.9661
0.9672
0.9994
0.9992
(0.4658)
(0.4199)
(0.4159)
(0.4224)
(0.4183)
(0.6994)
(0.6862)
0.06,0.06
0.9803
0.9693
0.9699
0.9687
0.9684
0.9990
0.9991
(0.4426)
(0.4042)
(0.4005)
(0.4072)
(0.4034)
(0.6359)
(0.6251)
0.6,0.6
0.04,0.04
0.9909
0.9927
0.9922
0.9916
0.9900
0.9999
0.9998
(0.4941)
(0.5074)
(0.5012)
(0.5115)
(0.5052)
(0.7568)
(0.7411)
0.05,0.05
0.9884
0.9913
0.9900
0.9889
0.9889
0.9997
0.9998
(0.4849)
(0.4954)
(0.4895)
(0.5001)
(0.4942)
(0.7115)
(0.6978)
0.06,0.06
0.9894
0.9932
0.9921
0.9910
0.9899
0.9997
0.9998
(0.4726)
(0.4864)
(0.4807)
(0.4916)
(0.4859)
(0.6721)
(0.6602)
WSEAS TRANSACTIONS on SYSTEMS
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Suparat Niwitpong
E-ISSN: 2224-2678
270
Volume 23, 2024
Table 2. The CPs and the ALs of the different techniques for estimating the nominal 95% two-sided CI for
(
12
nn
)
12
,nn
12
,
12
,
CPs
(ALs)
FQ
B.Jef
H.Jef
B.Uni
H.Uni
B.NGB
H.NGB
15,25
0.2,0.2
0.04,0.04
0.9674
0.9405
0.9645
0.9288
0.9660
0.9960
0.9993
(1.3107)
(1.1344)
(1.0835)
(1.2516)
(1.1762)
(1.9902)
(1.8374)
0.05,0.05
0.9628
0.9350
0.9584
0.9243
0.9601
0.9945
0.9987
(1.2418)
(1.0178)
(0.9732)
(1.1579)
(1.0855)
(1.8367)
(1.7028)
0.06,0.06
0.9582
0.9325
0.9570
0.9203
0.9589
0.9925
0.9980
(1.1907)
(0.9500)
(0.9075)
(1.1098)
(1.0369)
(1.7087)
(1.5895)
0.4,0.4
0.04,0.04
0.9817
0.9888
0.9958
0.9779
0.9945
0.9980
0.9994
(1.2286)
(1.2319)
(1.1644)
(1.4611)
(1.3313)
(1.7007)
(1.5738)
0.05,0.05
0.9808
0.9882
0.9954
0.9757
0.9942
0.9979
0.9992
(1.2233)
(1.1702)
(1.1047)
(1.4355)
(1.2991)
(1.6635)
(1.5376)
0.06,0.06
0.9821
0.9878
0.9952
0.9777
0.9943
0.9978
0.9994
(1.2086)
(1.1223)
(1.0579)
(1.4205)
(1.2767)
(1.6130)
(1.4894)
0.6,0.6
0.04,0.04
0.9896
0.9985
0.9981
0.9904
0.9973
0.9990
0.9984
(1.5507)
(1.4614)
(1.3389)
(2.3973)
(1.9161)
(1.9469)
(1.7062)
0.05,0.05
0.9907
0.9982
0.9994
0.9910
0.9978
0.9988
0.9990
(13.9172)
(4.3773)
(3.3945)
(61.8114)
(20.9648)
(11.1420)
(7.1361)
0.06,0.06
0.9919
0.9990
0.9993
0.9915
0.9978
0.9992
0.9993
(1.6515)
(1.4261)
(1.2965)
(2.6117)
(2.0281)
(2.0214)
(1.7451)
25,50
0.2,0.2
0.04,0.04
0.9635
0.9069
0.9222
0.8999
0.9215
0.9944
0.9989
(0.9494)
(0.7537)
(0.7342)
(0.7905)
(0.7671)
(1.4994)
(1.4314)
0.05,0.05
0.9609
0.9039
0.9214
0.8931
0.9198
0.9919
0.9984
(0.8858)
(0.6849)
(0.6667)
(0.7286)
(0.7057)
(1.3358)
(1.2815)
0.06,0.06
0.9574
0.9018
0.9173
0.8958
0.9192
0.9907
0.9985
(0.8250)
(0.6386)
(0.6212)
(0.6865)
(0.6641)
(1.1938)
(1.1513)
0.4,0.4
0.04,0.04
0.9788
0.9764
0.9882
0.9656
0.9862
0.9974
0.9997
(0.8556)
(0.8362)
(0.8107)
(0.8906)
(0.8566)
(1.2629)
(1.2124)
0.05,0.05
0.9751
0.9754
0.9860
0.9649
0.9844
0.9963
0.9999
(0.8318)
(0.7939)
(0.7692)
(0.8567)
(0.8222)
(1.1896)
(1.1438)
0.06,0.06
0.9771
0.9799
0.9887
0.9672
0.9871
0.9967
0.9996
(0.8050)
(0.7630)
(0.7390)
(0.8312)
(0.7966)
(1.1226)
(1.0810)
0.6,0.6
0.04,0.04
0.9886
0.9950
0.9990
0.9888
0.9991
0.9990
0.9999
(0.9379)
(0.9913)
(0.9476)
(1.1353)
(1.0568)
(1.2998)
(1.2281)
0.05,0.05
0.9879
0.9958
0.9988
0.9889
0.9982
0.9990
0.9999
(0.9371)
(0.9617)
(0.9180)
(1.1246)
(1.0414)
(1.2760)
(1.2040)
0.06,0.06
0.9893
0.9970
0.9993
0.9911
0.9985
0.9992
0.9999
(0.9368)
(0.9443)
(0.8997)
(1.1295)
(1.0399)
(1.2523)
(1.1795)
50,100
0.2,0.2
0.04,0.04
0.9666
0.8846
0.8909
0.8817
0.8905
0.9973
0.9991
(0.6665)
(0.4914)
(0.4841)
(0.5013)
(0.4937)
(1.0882)
(1.0572)
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2024.23.29
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Suparat Niwitpong
E-ISSN: 2224-2678
271
Volume 23, 2024
Table 2. continued.
12
,nn
12
,
12
,
CPs
(ALs)
FQ
B.Jef
H.Jef
B.Uni
H.Uni
B.NGB
H.NGB
0.05,0.05
0.9612
0.8844
0.8907
0.8802
0.8883
0.9939
0.9967
(0.6155)
(0.4501)
(0.4435)
(0.4614)
(0.4543)
(0.9396)
(0.9186)
0.06,0.06
0.9553
0.8866
0.8888
0.8825
0.8889
0.9915
0.9959
(0.5640)
(0.4195)
(0.4134)
(0.4316)
(0.4249)
(0.8153)
(0.8012)
0.4,0.4
0.04,0.04
0.9803
0.9707
0.9762
0.9652
0.9727
0.9987
0.9995
(0.5936)
(0.5519)
(0.5431)
(0.5630)
(0.5536)
(0.9203)
(0.8989)
0.05,0.05
0.9751
0.9653
0.9722
0.9592
0.9691
0.9969
0.9988
(0.5703)
(0.5267)
(0.5183)
(0.5394)
(0.5302)
(0.8426)
(0.8253)
0.06,0.06
0.9774
0.9744
0.9767
0.9669
0.9742
0.9974
0.9990
(0.5431)
(0.5059)
(0.4978)
(0.5198)
(0.5108)
(0.7713)
(0.7575)
0.6,0.6
0.04,0.04
0.9872
0.9912
0.9952
0.9869
0.9935
0.9993
0.9999
(0.6144)
(0.6424)
(0.6291)
(0.6644)
(0.6490)
(0.9169)
(0.8930)
0.05,0.05
0.9872
0.9926
0.9952
0.9878
0.9947
0.9986
0.9998
(0.6031)
(0.6248)
(0.6117)
(0.6492)
(0.6338)
(0.8698)
(0.8480)
0.06,0.06
0.9873
0.9936
0.9962
0.9901
0.9952
0.9991
0.9996
(0.5918)
(0.6142)
(0.6010)
(0.6421)
(0.6262)
(0.8296)
(0.8095)
∗ Italicize the shortest AL and bold the CPs greater than the nominal confidence level of 0.95.
Table 3. AIC results to check the distributions of the rainfall datasets.
Rainfall Station
Cauchy
Normal
Lognormal
Gamma
Mueang (February)
104.4263
105.8415
101.4244
98.6124
Mueang (December)
164.8593
166.1968
151.0278
147.6980
Mueang (January)
134.2037
136.2560
124.8226
124.4980
Mueang (November)
269.6906
266.3669
273.2349
263.1739
Table 4. The 95% CIs for the ratio of CV of rainfall data in Mueang district, Mae Hong Son province (
12
nn
)
Methods
CI for
Length of intervals
Lower bound
Upper bound
FQ
0.4551
1.6011
1.1460
B.Jef
0.4779
1.6134
1.1356
H.Jef
0.4297
1.5260
1.0963
B.Uni
0.4162
1.6564
1.2402
H.Uni
0.3693
1.5682
1.1989
B.NGB
0.4439
1.6068
1.1629
H.NGB
0.4057
1.5104
1.1047
Table 5. The 95% CIs for the ratio of CV of rainfall data in Mueang district, Mae Hong Son province (
12
nn
)
Methods
CI for
Length of intervals
Lower bound
Upper bound
FQ
0.0278
0.5704
0.5426
B.Jef
0.0175
0.5525
0.5350
H.Jef
0.0000
0.5058
0.5058
B.Uni
0.0223
0.5921
0.5698
H.Uni
0.0009
0.5370
0.5361
B.NGB
0.0233
0.5669
0.5436
H.NGB
0.0003
0.5168
0.5165
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2024.23.29
Wansiri Khooriphan, Sa-Aat Niwitpong,
Suparat Niwitpong
E-ISSN: 2224-2678
272
Volume 23, 2024
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Wansiri Khooriphan: performed the experiments,
analyzed the data, authored or reviewed drafts of
the paper; Sa-Aat Niwitpong: concived and
designed the experiments, approved the final draft;
Suparat Niwitpong: contributed analysis tools,
prepared tables.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research was funded by Faculty of Applied
Science, King Mongkut’s University of
Technology North Bangkok, Thailand. (Contract
no. 671181)
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.e
n_US
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2024.23.29
Wansiri Khooriphan, Sa-Aat Niwitpong,
Suparat Niwitpong
E-ISSN: 2224-2678
273
Volume 23, 2024
