Confidence Intervals for the Mean and Difference of Means of
Birnbaum-Saunders Distributions with Application to Wind Speed Data
NATCHAYA RATASUKHAROM1, SA-AAT NIWITPONG1, SUPARAT NIWITPONG1,*
1Department of Applied Statistics, Faculty of Applied Science,
King Mongkut’s University of Technology North Bangkok,
Bangkok 10800,
THAILAND
*Corresponding Author
Abstract: - This paper proposes the confidence intervals for the mean and difference of means of Birnbaum-
Saunders (BirSau) distributions based on the Bootstrap confidence interval (BCI), Percentile bootstrap
confidence interval (PBCI), Generalized confidence interval (GCI), Bayesian credible interval (BayCrI) and the
highest posterior density (HPD). The simulation study used R statistical software to evaluate the coverage
probabilities and average lengths. The concerning results of the mean suggest that HPD is the recommended
method for constructing confidence intervals in the BirSau distributions, except for small sample sizes where
the GCI method proves more efficient. For the difference of means, PBCI emerges as the preferred way to
construct confidence intervals, except in some cases where small sample sizes with the HPD method are more
efficient. Moreover, the average lengths of these proposed confidence intervals decreased as both sample size
and shape parameters increased. To illustrate the effectiveness of the suggested confidence intervals, we
applied them to wind speed datasets collected in Ayutthaya and Ratchaburi provinces, Thailand.
Key-Words: - Confidence interval, Birnbaum-Saunders distribution, Mean, Bootstrap confidence interval,
Generalized confidence interval, Bayesian credible interval, Wind speed.
Received: October 17, 2023. Revised: May 16, 2024. Accepted: July 8, 2024. Published: September 2, 2024.
1 Introduction
In 1969, [1], initially concentrated on creating a
model to describe the lifespan of material samples
during fatigue and establishing the corresponding
fatigue-life distribution. In this scenario, the
proposed fatigue-life distribution introduced by [1],
was formulated using a model that outlines the
entire period until accumulated damage, arising
from the formation and expansion of the primary
crack, surpasses a defined threshold, leading to the
material failure. In addition to being utilized in the
fatigue of materials, the Birnbaum-Saunders
(BirSau) distribution has also found application in
other fields, such as [2], [3], examined the incidence
of chronic cardiac diseases and diverse forms of
cancer arising from the cumulative harm inflicted by
various risk factors, ultimately resulting in
degradation and giving rise to a fatigue process. The
findings study of [4], constructing a framework to
handle intangibles within the software execution
process gives rise to cumulative damage that
degrades its performance and ultimately culminates
in failure. The study [5], [6], found that the
disruption in the renewal process causes the death of
small-diameter trees at chest height. According to
the study, [7], [8], [9], employed the BirSau
distribution to evaluate air quality, accounting for
the buildup of pollutants in the air. Research by
[10], also applied the BirSau distribution to explore
wind energy flow patterns and climatic conditions.
Due to the widespread application of the BirSau
distribution, particularly in environmental contexts,
this study is interested in exploring the BirSau
distribution.
Wind energy is a clean and renewable energy
source derived from nature, free from pollutants.
Currently, Thailand is placing more importance and
interest in developing renewable energy. Wind
energy has been utilized to reduce the combustion of
fossil fuels for electricity production, thereby
mitigating the problem of global warming caused by
the consequent carbon dioxide emission, [11]. Due
to this reason, we are interested in utilizing wind
speed data in this study. However, the inherent
natural variability in wind speed introduces
uncertainty. Therefore, we focus on estimating the
mean wind speed and the difference in mean wind
speeds using the BirSau distribution.
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Several researchers have contributed to
developing parameter estimation methods for
BirSau distribution. In 1969, [12], introduced the
maximum likelihood estimators (MLEs) for the
shape (
) and scale (
) parameters. In a
subsequent study [13], demonstrated the asymptotic
joint distribution of these MLEs, establishing their
asymptotic independence. Findings by [14], devised
modified moment estimators (MMEs) and a
straightforward bias correction technique to enhance
the MLEs and MMEs. The work of [15],
investigated the asymptotic confidence ellipses of
parameters for the BirSau distribution. Research by
[16], contributed to the field by presenting
percentile bootstrap and generalized pivotal
processes that create confidence intervals (CIs) for
the
and
parameters of the BirSau distribution.
The study of [17], formulated a bootstrap method
for forecasting intervals related to the BirSau
distribution in a different approach and based on the
research by [18], compared CIs for a population
mean obtained using the dependent bootstrap
procedure to those generated using the independent
bootstrap procedure. As shown, the study by [19],
developed a high-order likelihood asymptotic-based
for the parameters. Research by [20], focused on
determining CIs for fundamental reliability
measures through generalized interval estimation.
Results from [21], extended the discourse by
considering Bayesian inference for the parameters
of the BirSau distribution. They based their
methodology on inverse-gamma priors and
computed Bayesian estimates. In recent research,
the parameters of the BirSau distributions have been
estimated using environmental data. As per the
study by [22], [23], contributed by presenting CIs
for variance, the difference of variances, and the
coefficients of variation of PM 2.5 concentration
data when the data have BirSau distributions.
Lastly, [24], proposed a multivariate generalization
of BirSau distribution based on the multivariate
skew-normal distribution, presenting distributional
properties and an EM algorithm for parameter
estimation.
This paper focuses on the mean of a random
variable or expected value in statistical inference. It
represents the long-term average value of random
variables obtained by integrating the product of the
variable with its probability distribution. Since the
mean is the most widely used statistical measure,
our interest lies in constructing CIs to estimate the
population mean and the difference of means
between two populations. CIs for the mean and the
difference between the two means have applications
in various fields. For instance, in medicine [25],
compared outpatient costs before and after a
Medicaid policy change in Indiana, United States. In
environmental science [26], analyzed monthly
rainfall totals in Bloemfontein and Kimberley in
South Africa. Several studies have delved into CIs
for means, offering valuable insights into statistical
analysis. The study by [27], suggested CIs for both
the mean and coefficient of variation (CV) in a two-
parameter exponential distribution. Furthermore,
[28], introduced the concept of generalized
inference and the method of variance estimates
recovery to constructing the CIs, applicable to the
common mean of several gamma distributions.
Research by [29], proposed the robust CI estimation
for the mean of Poisson distribution. The
investigation into parameter estimation for the
BirSau distribution and interval estimation for the
parameter mean showed that prior studies have not
delved into creating CIs for both the mean and the
difference between the means of BirSau
distributions. Consequently, we propose the
introduction of CIs for both the mean and the
difference between the two means of BirSau
distributions. Therefore, this study aims to compare
the efficiency of methods for estimating the CIs of
the mean and the difference between the means
when the population follows a BirSau distribution.
The methods utilized include the bootstrap
confidence interval (BCI), percentile bootstrap
confidence interval (PBCI), generalized confidence
interval (GCI), Bayesian credible interval (BayCrI),
and the highest posterior density interval (HPD). To
demonstrate the effectiveness of these proposed
methodologies, we have also applied them to wind
speed data from Ayutthaya and Ratchaburi
provinces in Thailand, collected between February
and April 2022.
2 The CI for the Mean of a BirSau
Distribution
A random variable
X
is said to follow the two-
parameter BirSau distribution with parameters
and
, where
0x
,
0
, and
0
. This
distribution is represented as
~ ( , )X BirSau
. The
probability density function is given by:
13
22
2
11
( , , ) 2 .
22 2
x
f x exp
x x x
(1)
The expected value and variance of
X
are
expressed as
2
1
( ) (1 )
2
EX
and
22
5
( ) ( ) 1 4
Var X
,
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respectively. Therefore, in this study, the focus is on
the parameter mean, denoted as
, and is defined
as:
2
1
1.
2
(2)
2.1 BCI
The bootstrap method, introduced by [30], utilizes
resampling techniques to mitigate bias in MLE. In
the context of the BirSau distribution, this method
facilitates the estimation of parameters
and
through a procedure developed by [31]. This
procedure explicitly addresses correcting biases in
the MLE of distribution parameters by leveraging
bootstrap techniques.
Let
12
( , ,..., )T
n
x x xx
represent a random sample
of size
n
from
( , )BirSau
. The MLEs for
and
are
ˆ()r
x
and
ˆ()s
x
, respectively. Next, let
B
represent a bootstrap sample that is created
independently from the initial sample
x
, with
*1 *2 *
( , ,..., )
B
x x x
. The respective bootstrap replications
of
ˆ
and
ˆ
are indicated as
*1 *2 *
ˆ ˆ ˆ
( , ,..., )
B
and
*1 *2
ˆˆ
( , ,...,
*
ˆ)
B
, where
**
ˆ()
by
s
x
and
**
ˆ()
yy
r
x
,
for
1,2,...,yB
. The approximate bootstrap estimator
are calculated by the mean
*(.) *
1
ˆˆ
1/
B
y
y
B
and
*(.) *
1
ˆˆ
1/
B
y
y
B
.
The estimates of bootstrap bias based on
B
replications of
ˆ
and
ˆ
are
*( )
ˆˆˆ
( , ) ( )Bs
x
and
*( )
ˆˆ
ˆ( , ) ( ).Br
x
(3)
The correct estimates for
*
ˆ
and
*
ˆ
using the
idea of constant-bias-correction (CBC) estimates
proposed by [32], can be obtained as follows:
*ˆ
ˆˆ
2 ( , )
yy
B
and
*
ˆˆ
ˆ
2 ( , ).
yy
B
(4)
The percentile bootstrap estimates for
*
ˆ
and
*
ˆ
are:
*ˆ
2
yy
and
*ˆ
2.
yy
(5)
Consequently, it can construct the bootstrap
estimator of the mean as:
2
ˆ(1 ).
2
y
yy
(6)
and the percentile bootstrap estimator of the mean
can be found as:
*2
**(1 ).
2
y
yy
(7)
Hence, the approximated
100 1 %
CI for
based
on BCI and PBCI, it becomes:
ˆˆ
[ ( / 2), (1 / 2)].
BCI
CI
(8)
**
ˆˆ
[ ( / 2), (1 / 2)].
PBCI
CI
(9)
where
ˆ( / 2)
and
*
ˆ( / 2)
denote the
100( / 2)
-th
percentile of bootstrap and percentile bootstrap
distribution of
ˆ
and
*
ˆ
, respectively.
2.2 GCI
In 1993, [33], introduced a method to construct the
GCI by applying the Generalized Pivotal Quantity
(GPQ) concept. Following, [22], the GPQ for
was established by [34], as follows:
12
12
max( , ), 0
: ( ) min( , ), 0.
;J
JJ J
T
X
(10)
where
1
and
2
are the two solutions of the
following equation:
22 0,U V W
(11)
where
22
[( 1) (1/ ) ]U n H n IJ
,
2
[( 1) (1 ) ]V n KH KH J
,
22
( 1) (1/ )W n K n LJ
,
1
1
n
i
i
K n X
,
1
1
1/
n
i
i
H n X
,
2
1
( ) ,
n
i
i
L X K
2
1
(1/ )
n
i
i
I X H
and
~ ( 1)J t n
.
The GPQ for
was subsequently developed by
[20], then the GPQ for
becomes
1/2
2
21
2
: ( ; , ) ,[]
s J nJ s
J J J J t
t
x
(12)
where
1
1
n
i
i
sx
and
2
1
1
n
i
i
sx
and
2
~n
t
. By substituting
J
and
J
into Equation (5), the GPQ of the mean
as:
2
(1 ).
2
J
JJ
(13)
Hence, the approximated
100 1 %
CI for
based on GCI, it becomes
[ ( / 2), (1 / 2)],
GCI
CI J J
(14)
where
( / 2)J
denotes the
100( / 2)
-th percentile of
J
.
2.3 BayCrI
The study conducted by [21], use specific priors
with known values to ensure the accuracy of the
resulting posteriors. They assume an inverse gamma
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(IG) distribution for
, marked as
11
( | , )IG d e
, and
do the same for
2
as
22
2
( | , )IG d e
.
The posterior distribution of
given the data
and the posterior distribution of
given
and the
data are outlined as follows:
1
2
1/2 3/ 2
( 1) 1
1
( 1)/2
2
1
( | ) exp
12.
2
n
nd
iii
nd
ni
ii
e
xx
xe
x
x
(15)
2
22
1
1
( | , ) , 2 .
22
ni
ii
nx
IG d e
x
x
(16)
The sample from Equations (15) and (16) are
obtained using Markov Chain Monte Carlo
techniques. Research by [21], produced posterior
samples using the extended ratio-of-uniforms
technique will be discussed in more detail in the
next section. Alternatively, obtaining the posterior
samples of
2
is straightforward using the
LearnBayes package in the R software suite.
Consequently,
equals the square root of
2
.
2.3.1 The Generalized Ratio-of-uniforms Method
According to the study by [35], an effective
sampling approach for posterior simulation from
Equation (15) using the generalized ratio-of-
uniforms method was created. A summary of the
algorithm is presented as follows: Supposed a pair
of random variables
( , )rs
follows a uniform
distribution over the specified region.
1/( 1)
( ) ( , ) : 0 | , 0,
t
t
s
K t r s r t
r
x
(17)
where
t
is constant and
( | )x
is given by using
Equation (15). Therefore, the density of
/t
sr
is
( | ) / ( | )d
xx
.
To generate random samples uniformly
distributed within the region
()Kt
, random variables
( , )rs
are generated with a uniform distribution
across the one-dimensional rectangle
[0, ( )] [ ( ), ( )]a t b t b t
, where:
1/( 1)
0
( ) sup{[ ( | )] }
t
at
x
(18)
/( 1)
0
( ) inf { [ ( | )] }
tt
bt
x
(19)
and
/( 1)
0
( ) sup{ [ ( | )] }
tt
bt
x
(20)
According to research by [21], both
()at
and
()bt
assume finite values with
()bt
equating to
zero. Consequently, the potential variate
/t
sr
is
deemed acceptable if
1/( 1)
[ ( | )] t
r
x
; should this
not be the case, the process is reiterated. By
executing these steps, the BayCrI for the mean of
the BirSau distribution can be derived
(1) Indicate the values of
1 1 2 2
, , ,e d ed
and
t
then
calculate
()at
and
()bt
.
(2)
i
th iteration:
a. Generate
r
and
s
from
(0, ( ))Unif a t
and
(0, ( ))Unif b t
, respectively, then compute
/t
sr
.
b. Set
()i
if
1/( 1)
[ ( | )] t
r
x
if the value
of
is acceptable; if not, repeat the previous
step.
c. Create
2
i
using
()
22
()
1
1
( , ( 2) )
22
ni
i
ij
j
nx
IG d e
x
then set
()
2
ii
.
(3) Compute the Bayesian estimator of the mean by
2
()
*
( ) ( ) (1 )
2
i
ii
. (21)
(4) Go through steps (2) and (3)
M
times.
(5) Compute the approximated
100(1 )%
BayCrI
for
the
as
**
[ ( / 2), (1 / 2)],
BayCrI
CI
(22)
where
*( / 2)
denotes the
100( / 2)
-th
percentile of
*
.
To create the HPD interval for the mean, we
utilized the hdi function provided by the HDInterval
package in the R software suite. This step was
performed after obtaining the Bayesian mean
estimate in step 4.
3 The CIs for the Difference between
the Means of BirSau Distributions
The concepts of GCI, BCI, PBCI, BayCrI, and HPD
presented in the previous section are expanded upon
in this section, focusing on new CI to determine the
difference between the two means. In a statistical
model, the difference between the means results
from subtracting or comparing two means. Usually,
this is done to compare two quantitative datasets
when it is necessary to fit the BirSau distribution
closely. Let
12
, ,..., n
X X X X
and
12
, ,..., m
Z Z Z Z
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be independent random samples from the BirSau
distribution, with sample sizes of
n
and
m
,
respectively (referred to as
, BirSau
,
,
zz
BirSau
). Consequently, the mean of
Z
becomes
2
1
1.
2
z z z
(23)
Given the independence of
X
and
Z
, the
difference between the means (represented by
)
can be expressed as
22
11
1 1 .
22
z z z
(24)
3.1 BCI
Let
12
, ,..., T
m
z z zz
be a random sample of size
m
from generated by
,
zz
BirSau
. The MLEs of
z
and
z
, denoted as
ˆ()
zs
z
and
ˆ()
zr
z
,
respectively. Given that
*1 *2 *
, ,..., B
z z z
represents
bootstrap samples generated independently from the
original sample
z
. The corresponding bootstrap
replications for
are represented as
*1 *2 *
ˆ ˆ ˆ
, , , B
z z z
,
and for
, they are denoted as
*1 *2 *
ˆ ˆ ˆ
, , , .
B
z z z
where
**
ˆ()
yy
zs
z
and
**
ˆ()
yy
zr
z
,
1,2,...,yB
. The
approximate bootstrap estimators are calculated by
*. *
1
ˆˆ
(1/ )
By
zz
y
B
and
*. *
1
ˆˆ
(1/ )
By
zz
y
B
.
The bootstrap bias estimate based on
B
replications
of
ˆz
and
ˆz
are obtained as:
*.
ˆˆˆ
, ( )
z z z
Br
z
and
*.
ˆˆ
ˆ, ( ).
z z z
Bs
z
(25)
Therefore, the corrected estimate for
*
ˆz
and
*
ˆz
can be written as:
*
, , , ,
ˆ
ˆˆ
2 ( , )
z y z y z y z y
B
and
*
, , , ,
ˆ ˆ ˆ
ˆ
2 ( , ).
z y z y z y z y
B
(26)
The percentile bootstrap estimators for
*
ˆz
and
*
ˆz
are:
*
,,
ˆ
2
z y z y
and
*
,,
.
ˆ
2
z y z y
(27)
Therefore, the bootstrap estimator of the difference
of means can be obtained as:
22
,,
11
ˆ1 1 .
22
y y y z y z y
(28)
and the percentile bootstrap estimator of the
difference of means can be obtained as:
* * *2 * *2
,
,
11
ˆ
1 1 .
22
y y y z y
zy
(29)
Thus, the approximated
100 1 %
CI for
based
on BCI and PBCI, it becomes:
ˆˆ
[ ( / 2), (1 / 2)].
BCI
d
CI
(30)
**
ˆˆ
[ ( / 2), (1 / 2)].
PBCI
d
CI
(31)
where
ˆ( / 2)
and
*
ˆ( / 2)
denote the
100( / 2)
-th
percentile of bootstrap and percentile bootstrap
distribution of
ˆ
and
*
ˆ
, respectively.
3.2 GCI
The GPQ of
z
can be defined by utilizing the
random variable
Z
as follows:
,1 ,2
,1 ,2
, , 0
,
, ,
;
0
:z
zz
z
zz
z
z
z
max if J
J J J min if J
Z
(32)
where
~1
z
J t m
, and
,1z
and
,2z
are the two
solutions for:
22 0,
z z z z z
U V W
(33)
where
22
[( 1) (1/ ) ]
z z z z
U m H m I J
,
2
[( 1) (1 ) ]
z z z z z z
V m K H K H J
,
22
( 1) (1/ )
z z z z
W m K m L J
,
1
1
m
zi
i
K m Z
,
1
1
1/
m
zi
i
H m Z
,
2
1
()
m
z i z
i
L Z K
and
2
1
(1/ )
m
z i z
i
I Z H
.
For
z
, the GPQ is provided by
1/2
2
,2 ,1
2
: ; , ,
zz
yz
z
zz
zz z
s J mJ s
J J t J Jt
Z
(34)
where
,1 1
m
zi
i
sz
,
,2 1
1
m
zii
sz
and
2
~
z
tm
.
Therefore, the GPQ for the difference of means can
be obtained as:
22
11
(1 ) (1 ).
22
zz
J J J J J
(35)
Therefore, the approximated
100 1 %
CI for
based on GCI, it becomes:
/ 2 , 1 / 2 ,
d
GCI
CI J J
(36)
where
( / 2)J
denotes the
100( / 2)
-th percentile of
J
.
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3.3 BayCrI
In the Bayesian method, for
~ ( , )
zz
Z BirSau
the IG
priors of
z
and
2
z
, are denoted as
11
|,
z
IG f g
and
2
22
|,
z
IG f g
, respectively. Thus, the marginal
distribution of
z
becomes:
1
13
22
11
1
|
m
mf
i
y
zz
z
ii
zz
eg
z
xp
z
2
1
2
2
1
2 .
1
2
mg
mi z
z
ii
z
zf
(37)
The posterior conditional distribution of
2
z
given
z
becomes:
222
1
1
| , , 2 .
22
m
iz
zz izi
z
m
z IG f g
z
(38)
The procedure for utilizing BayCrI to estimate
the difference between the means can be condensed
into the following steps: Initially, determine the
values of
1 1 2 2
, , ,f g f g
and
z
t
, with the condition that,
as a constant. Subsequently, calculate
z
at
and
z
bt
, where
z
at
and
z
bt
are defined as
follows:
1/ 1
0
sup | ,
z
z
t
zz
a t z
(39)
/1
0
sup | .
zz
z
tt
z z z
b t z
(40)
Second, generate
z
r
and
z
s
from
~ (0, ( ))
zz
r Unif a t
and
~ (0, ( )),
zz
s Unif b t
then compute
/z
t
z z z
sr
.
If
1/ 1
|z
t
zz
rz
, set
,z i z
; otherwise,
generate
z
r
and
z
s
again. Next, generate
2
, 2 2
1
1
~ , 2
22
mz
j
zi jj
z
z
m
IG f g
z
and we can find the
2
,,z i z i
. Hence, the Bayesian estimator of the
difference between the means is denoted as
*
, is
given by
* 2 2
,,
11
1 1 , 1,2, , ,
22
i i i z i z i iM
(41)
where
M
is the number of iterations, and the last
calculates the
100 1 %
CI for
by applying:
**
/ 2 , 1 / 2 ,
d
BayCrI
CI
(42)
when
*( / 2)
denotes the
100( / 2)
-th percentile of
*
. The confidence of
was determined using the
R package HDInterval for the HPD interval
calculation.
4 Simulation Studies
Five approaches were examined in a Monte Carlo
simulation using the R statistical software: GCI,
BCI, PBCI, BayCrI, and HPD. The purpose of the
simulation was to create new CIs for the mean and
the difference between the means of two BirSau
distributions. The coverage probabilities (CPs) and
average lengths (ALs) of the five suggested
approaches were compared to evaluate them. Two
crucial factors were considered when selecting a
preferred method: the CPs should be at least or close
to the nominal confidence level of 0.95, and the
shortest AL. The simulation settings consist of the
number of replications of 5,000, with 5,000 pivotal
quantities for GCI,
B
= 500 for BCI and PBCI, and
M
= 1,000 for BayCrI and HPD interval. For a
single mean of BirSau, the sample size was set
n
=
10, 20, 30, 50 or 100 with shape parameters
=
0.10, 0.25, 0.50, 0.75 or 1.00. The sample sizes for
the difference between the means of the two BirSau
distributions, however, were set as
( , )nm
= (10,10),
(20,20), (30,30), (50,50), (100,100), (10,20),
(20,30), (30,50) or (50,100) with shape parameter
2
,
= (0.25,0.25), (0.25,0.50), (0.25,0.75),
(0.25,1.00), (0.50,0.50), (0.50,0.75), (0.50,1.00),
(0.75,0.75), (0.75,1.00) or ( 1.00,1.00 ). The values
for the scale parameters
and
2
were fixed at 1
for all cases. In the case of BayCrI and HPD, we
examined the parameter
t
=
z
t
= 2 and the suggested
hyperparameter
1 2 1 2 1 2
, , , , , ,d d e e f f
4
12
, 10gg
as
proposed by [21].
For the single mean of a BirSau distribution,
according to the simulation results shown in Table 1
(Appendix), when dealing with small sample sizes (
n
=10), the GCI method performs the best in CP and
AL. However, for another medium (
n
=30,50) and
large sample sizes (
n
=100), we observed that GCI,
BayCrI, and HPD gave CPs greater than or close to
the nominal confidence level of 0.95. Among these,
HPD provided the shortest AL. In contrast, although
PBCI had the shortest ALs, its CPs were the lowest
and under the nominal confidence level of 0.95, but
it improved as
n
increased. When considering the
ALs of the other methods, they exhibited similar
trends. Additionally, the ALs of all five methods
tended to decrease as the sample size increased, as
shown in Figure 1.
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When considering differences between the
means of BirSau distributions, we generated data
from two independent BirSau distributions. The CPs
and ALs of the 95% CI for the difference between
the means, with equal and unequal sample sizes, are
listed in Table 2 and Table 3 in Appendix,
respectively. For equal sample sizes, we consistently
observed that the CPs of the GCI, BayCrI, and HPD
methods were greater than or close to the nominal
confidence level of 0.95. Notably, the HPD method
produced the shortest AL, except for one instance
with a large sample size where PBCI outperformed
in terms of both CP and AL. In the case of unequal
sample sizes, the results indicated that the GCI,
BayCrI, HPD, and PBCI methods all provided CPs
greater than or very close to the nominal confidence
level of 0.95. Regarding AL, HPD resulted in the
shortest AL for small sample sizes, while PBCI was
associated with the shortest AL for other cases.
Moreover, the performances of all five methods in
terms of AL improved as sample sizes
( , )nm
increased, as shown in Figure 2.
Fig. 1: Comparison of the CPs and ALs for
estimating the 95% CI for the mean of BirSau
distribution at
0.5
Fig. 2: Comparison of the CPs and ALs for
estimating the 95% CI for the difference between
the means of BirSau distributions with equal sample
sizes at
2
, (0.5,0.5)
5 An Empirical Application
Wind energy is an eco-friendly and sustainable
power source, untainted by carbon emissions or
pollution, [11]. According to [36], the BirSau
distribution is the optimal method for estimating
wind speed distribution. We employed datasets
containing daily wind speed records from Ayutthaya
and Ratchaburi provinces, Thailand, [37], to
demonstrate the efficiency of CIs for the mean and
difference of the means of BirSau distributions
obtained through methods such as GCI, BCI, PBCI,
BayCrI, and HPD. These data sets were gathered
from February to April 2022 as detailed in Table 4
(Appendix).
As the data comprises positive values, it is
feasible to fit it into various distributions such as
Cauchy, logistic, exponential, Weibull, normal, or
BirSau distributions. Therefore, we tested the
distributions of positive wind speed datasets using
the Akaike information criterion (AIC) and the
Bayesian information criterion (BIC). The data
presented in Table 5 (Appendix) indicates that the
wind speed datasets from Ayutthaya and Ratchaburi
province conform to a BirSau distribution, as
supported by the smallest values of AIC and BIC.
Table 6 (Appendix) presents the basic statistics
computed for the daily wind speed data. The mean
and difference of the means are accompanied by
two-sided CIs derived from the GCI, BCI, PBCI,
BayCrI, and HPD, as outlined in Table 7
(Appendix), respectively. In terms of the mean, as
per the simulation results, PBCI consistently yielded
the shortest ALs. It is worth mentioning, however,
that, similar to the simulation results, the CP of
PBCI was lower and fell below 0.95, while both
GCI and HPD exhibited CPs greater than or close to
the nominal confidence level of 0.95. Overall, HPD
stands out as the most suitable method for
constructing a CI for the mean of the BirSau
distribution.
Regarding the two-sided CIs for the difference of
means, the results align with the simulation results.
For large sample sizes, GCI, BayCrI, HPD, and
PBCI consistently achieved CPs greater than or
close to the nominal confidence level of 0.95, with
PBCI offering the shortest AL. Therefore, we
recommend using the PBCI method for constructing
CIs for the difference of means in wind speed data
collected between February and April 2022 in
Ayutthaya and Ratchaburi, Thailand.
6 Discussion
Wind energy is a reusable, environmentally friendly
resource. Transforming the speed of wind into
kinetic energy can generate electricity.
Nevertheless, wind speed can vary beyond typical
fluctuations, leading to unpredictability. The
research work conducted by [38], highly
recommended how these factors are essential in the
business strategy and how to manage this
sustainable resource. For this reason, estimating CI
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values for the mean and differences between
average wind speeds in different areas is essential.
Suppose we know the average wind speed estimate
and whether there is a consistent wind speed in each
area. It will help in deciding on the area to have a
wind farm. This is because the wind speed used to
produce electricity is consistent and is required to
produce electricity efficiently.
The study by [22], utilized the Highest Posterior
Density (HPD) method for both the variance and the
difference in variances of the BirSau distribution to
enhance the CIs. In some cases, the mean might be
considered a better method than the variance. In this
study, we were focusing on estimating the CIs of the
mean and mean difference for the BirSau
distributions. The results suggest that for the mean,
HPD was the most consistent with the best CP and
AL. It provides the same findings as the CIs for the
variance, and the difference between two variances,
[22] and CV [23], of BirSau distributions. For the
difference between the means, the results suggest
that the PBCI is the best approach for constructing
CIs, which is consistent with the estimated
simultaneous CIs for pairwise comparisons of the
means of delta-lognormal distributions, [39].
7 Conclusions
In this study, the CIs for the parameters mean and
difference between two means of the BirSau
distributions are proposed. The results suggest that
the HPD method effectively estimates the CI of the
mean of the BirSau distribution. The GCI approach
is more effective than other methods with small
sample sizes. Regarding the difference of means in
BirSau distributions, the data shows that the PBCI is
a preferred method for constructing CIs. Once
again, the exception is noticeable for smaller data
sets. Having CPs above or close to 0.95 with the
shortest ALs proves the HPD method is more
effective. The methods illustrated using real wind
speed datasets. It was found that the results
corresponded with the simulation results. This
research investigates how to estimate CIs, for the
parameters in the BirSau distribution, which works
well for positive data. However, if the dataset
includes both zeros and positive values it might be
better to look into the delta BirSau distribution. In
future work, we would like to extend our work to
estimating the CIs for parameters in either the
BirSau distribution covering zeros well or the delta-
BirSau distribution.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of funding for research presented in a
scientific article or scientific article itself
This research was funded by King Mongkut’s
University of Technology North Bangkok, Contract
no. KMUTNB-67-BASIC-22.
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
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APPENDIX
Table 1. The CPs and ALs of the 95% CIs for the mean of a BirSau distribution
n
CP (AL)
GCI
BCI
PBCI
BayCrI
HPD
10
0.1
0.9498
0.9010
0.8988
0.9362
0.9318
(0.1409)
(0.1139)
(0.1136)
(0.1314)
(0.1297)
0.25
0.9478
0.8994
0.8960
0.9360
0.9350
(0.3841)
(0.3000)
(0.2960)
(0.3582)
(0.3476)
0.5
0.9460
0.8998
0.8748
0.9326
0.9386
(0.9601)
(0.6782)
(0.6534)
(0.8885)
(0.8203)
0.75
0.9482
0.8964
0.8644
0.9364
0.9366
(1.9315)
(1.2010)
(1.1361)
(1.7485)
(1.5230)
1
0.9486
0.8988
0.8536
0.9366
0.9444
(3.5720)
(1.8871)
(1.7530)
(3.0451)
(2.5058)
20
0.1
0.9440
0.9196
0.9226
0.9348
0.9340
(0.0933)
(0.0842)
(0.0840)
(0.0903)
(0.0894)
0.25
0.9484
0.9222
0.9210
0.9404
0.9372
(0.2452)
(0.2185)
(0.2170)
(0.2373)
(0.2335)
0.5
0.9518
0.9302
0.9158
0.9448
0.9466
(0.5741)
(0.4917)
(0.4823)
(0.5553)
(0.5348)
0.75
0.9470
0.9260
0.9058
0.9434
0.9414
(1.0477)
(0.8503)
(0.8296)
(1.0031)
(0.9402)
1
0.9442
0.9272
0.8998
0.9406
0.9456
(1.7447)
(1.3235)
(1.2947)
(1.6330)
(1.4923)
30
0.1
0.9510
0.9312
0.9336
0.9452
0.9426
(0.0748)
(0.0698)
(0.0698)
(0.0733)
(0.0726)
0.25
0.9482
0.9276
0.9262
0.9416
0.9362
(0.1957)
(0.1809)
(0.1801)
(0.1915)
(0.1889)
0.5
0.9482
0.9324
0.9224
0.9448
0.9448
(0.4454)
(0.4016)
(0.3972)
(0.4351)
(0.4237)
0.75
0.9438
0.9258
0.9150
0.9394
0.9398
(0.7929)
(0.6901)
(0.6796)
(0.7687)
(0.7360)
1
0.9476
0.9354
0.9166
0.9456
0.9480
(1.2953)
(1.0804)
(1.0649)
(1.2363)
(1.1632)
50
0.1
0.9498
0.9370
0.9358
0.9436
0.9404
(0.0569)
(0.0545)
(0.0545)
(0.0562)
(0.0557)
0.25
0.9544
0.9432
0.9414
0.9500
0.9500
(0.1477)
(0.1407)
(0.1403)
(0.1455)
(0.1439)
0.5
0.9456
0.9342
0.9314
0.9420
0.9404
(0.3331)
(0.3125)
(0.3104)
(0.3281)
(0.3220)
0.75
0.9506
0.9344
0.9284
0.9474
0.9470
(0.5884)
(0.5395)
(0.5347)
(0.5759)
(0.5595)
1
0.9426
0.9328
0.9216
0.9418
0.9424
(0.9373)
(0.8327)
(0.8258)
(0.9039)
(0.8688)
100
0.1
0.9474
0.9414
0.9398
0.9444
0.9414
(0.0398)
(0.0388)
(0.0388)
(0.0394)
(0.0391)
0.25
0.9506
0.9464
0.9442
0.9500
0.9462
(0.1031)
(0.1002)
(0.1001)
(0.1022)
(0.1013)
0.5
0.9510
0.9434
0.9378
0.9470
0.9450
(0.2299)
(0.2216)
(0.2208)
(0.2273)
(0.2244)
0.75
0.9504
0.9446
0.9394
0.9464
0.9474
(0.3998)
(0.3801)
(0.3782)
(0.3934)
(0.3862)
1
0.9454
0.9328
0.9300
0.9430
0.9404
(0.6262)
(0.5841)
(0.5812)
(0.6085)
(0.5943)
Notes: Bold represents values that satisfy criteria and the best-performing method
Table 2. The CPs and ALs of the 95% CIs for the difference between the means of BirSau distributions with
equal sample sizes
2.nn
2
,nn
2
,
CP (AL)
GCI
BCI
PBCI
BayCrI
HPD
(10,10)
(0.25,0.25)
0.9590
0.9170
0.9244
0.9464
0.9522
(0.5548)
(0.4304)
(0.4240)
(0.5158)
(0.5086)
(0.25,0.50)
0.9556
0.9148
0.9068
0.9460
0.9528
(1.0491)
(0.7468)
(0.7216)
(0.9716)
(0.9217)
(0.25,0.75)
0.9500
0.9006
0.8796
0.9356
0.9448
(2.0096)
(1.2534)
(1.1876)
(1.8220)
(1.6156)
(0.25,1.00)
0.9490
0.8966
0.8640
0.9396
0.9482
(3.6056)
(1.9102)
(1.7739)
(3.0600)
(2.5581)
(0.50,0.50)
0.9626
0.9320
0.9460
0.9530
0.9674
(1.4558)
(0.9956)
(0.9510)
(1.3422)
(1.3062)
(0.50,0.75)
0.9620
0.9258
0.9326
0.9500
0.9690
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.54
Natchaya Ratasukharom, Sa-Aat Niwitpong, Suparat Niwitpong
E-ISSN: 2224-2880
525
Volume 23, 2024
2
,nn
2
,
CP (AL)
GCI
BCI
PBCI
BayCrI
HPD
(2.3069)
(1.4255)
(1.3424)
(2.0865)
(1.9603)
(0.50,1.00)
0.9598
0.9186
0.9236
0.9482
0.9650
(2.3086)
(1.4261)
(1.3423)
(2.0850)
(1.9601)
(0.75,0.75)
0.9604
0.9360
0.9578
0.9512
0.9728
(3.0699)
(1.7875)
(1.6613)
(2.7530)
(2.6378)
(0.75,1.00)
0.9548
0.9264
0.9416
0.9450
0.9704
(4.4769)
(2.3392)
(2.1378)
(3.8568)
(3.6018)
(1.00,1.00)
0.9564
0.9390
0.9648
0.9438
0.9764
(5.8327)
(2.8448)
(2.5647)
(4.8971)
(4.6398)
(20,20)
(0.25,0.25)
0.9556
0.9358
0.9400
0.9510
0.9490
(0.3506)
(0.3125)
(0.3098)
(0.3395)
(0.3362)
(0.25,0.50)
0.9516
0.9302
0.9290
0.9422
0.9474
(0.6296)
(0.5410)
(0.5317)
(0.6091)
(0.5938)
(0.25,0.75)
0.9500
0.9300
0.9160
0.9442
0.9478
(1.0847)
(0.8827)
(0.8617)
(1.0408)
(0.9854)
(0.25,1.00)
0.9510
0.9276
0.9008
0.9442
0.9480
(1.7581)
(1.3371)
(1.3080)
(1.6437)
(1.5124)
(0.50,0.50)
0.9510
0.9342
0.9442
0.9448
0.9526
(0.8364)
(0.7079)
(0.6918)
(0.8082)
(0.7971)
(0.50,0.75)
0.9534
0.9344
0.9366
0.9468
0.9574
(1.2329)
(0.9978)
(0.9721)
(1.1811)
(1.1469)
(0.50,1.00)
0.9506
0.9330
0.9224
0.9442
0.9548
(1.8769)
(1.4270)
(1.3858)
(1.7577)
(1.6560)
(0.75,0.75)
0.9518
0.9380
0.9532
0.9444
0.9606
(1.5709)
(1.2401)
(1.2000)
(1.5014)
(1.4734)
(0.75,1.00)
0.9484
0.9342
0.9438
0.9404
0.9572
(2.1472)
(1.6205)
(1.5646)
(2.0200)
(1.9559)
(1.00,1.00)
0.9508
0.9428
0.9620
0.9460
0.9654
(2.6561)
(1.9363)
(1.8649)
(2.4707)
(2.4134)
(30,30)
(0.25,0.25)
0.9522
0.9378
0.9406
0.9476
0.9446
(0.2767)
(0.2563)
(0.2551)
(0.2707)
(0.2682)
(0.25,0.50)
0.9476
0.9324
0.9306
0.9412
0.9454
(0.4892)
(0.4429)
(0.4375)
(0.4781)
(0.4692)
(0.25,0.75)
0.9398
0.9294
0.9210
0.9370
0.9384
(0.8262)
(0.7203)
(0.7096)
(0.8024)
(0.7726)
(0.25,1.00)
0.9492
0.9378
0.9280
0.9452
0.9520
(1.3142)
(1.0980)
(1.0831)
(1.2579)
(1.1882)
(0.50,0.50)
0.9554
0.9468
0.9540
0.9538
0.9564
(0.5397)
(0.4966)
(0.4920)
(0.5299)
(0.5248)
(0.50,0.75)
0.9532
0.9432
0.9438
0.9492
0.9524
(0.7805)
(0.7028)
(0.6930)
(0.7624)
(0.7491)
(0.50,1.00)
0.9490
0.9326
0.9272
0.9434
0.9468
(1.1407)
(0.9938)
(0.9805)
(1.1004)
(1.0648)
(0.75,0.75)
0.9516
0.9444
0.9530
0.9472
0.9580
(0.9700)
(0.8626)
(0.8479)
(0.9442)
(0.9333)
(0.75,1.00)
0.9532
0.9436
0.9468
0.9478
0.9574
(1.2964)
(1.1226)
(1.1046)
(1.2485)
(1.2257)
(1.00,1.00)
0.9524
0.9466
0.9594
0.9516
0.9618
(1.5691)
(1.3373)
(1.3118)
(1.5059)
(1.4860)
(50,50)
(0.25,0.25)
0.9484
0.9406
0.9370
0.9460
0.9454
(0.2096)
(0.1995)
(0.1992)
(0.2066)
(0.2048)
(0.25,0.50)
0.9538
0.9462
0.9454
0.9528
0.9520
(0.3662)
(0.3442)
(0.3417)
(0.3606)
(0.3556)
(0.25,0.75)
0.9526
0.9438
0.9352
0.9488
0.9510
(0.6082)
(0.5592)
(0.5531)
(0.5952)
(0.5805)
(0.25,1.00)
0.9492
0.9396
0.9316
0.9456
0.9466
(0.9483)
(0.8445)
(0.8365)
(0.9151)
(0.8821)
(0.50,0.50)
0.9542
0.9448
0.9500
0.9508
0.9526
(0.4754)
(0.4452)
(0.4405)
(0.4681)
(0.4637)
(0.50,0.75)
0.9510
0.9412
0.9394
0.9476
0.9494
(0.6820)
(0.6257)
(0.6191)
(0.6678)
(0.6577)
(0.50,1.00)
0.9548
0.9430
0.9366
0.9484
0.9488
(0.9984)
(0.8917)
(0.8830)
(0.9670)
(0.9405)
(0.75,0.75)
0.9554
0.9490
0.9542
0.9532
0.9574
(0.8456)
(0.7684)
(0.7582)
(0.8266)
(0.8181)
(0.75,1.00)
0.9486
0.9414
0.9458
0.9466
0.9552
(1.1273)
(1.0015)
(0.9882)
(1.0899)
(1.0725)
(1.00,1.00)
0.9512
0.9510
0.9526
0.9472
0.9602
(1.3599)
(1.1904)
(1.1732)
(1.3075)
(1.2923)
(100,100)
(0.25,0.25)
0.9516
0.9444
0.9468
0.9506
0.9488
(0.1458)
(0.1419)
(0.1416)
(0.1445)
(0.1433)
(0.25,0.50)
0.9452
0.9384
0.9396
0.9434
0.9410
(0.2522)
(0.2436)
(0.2427)
(0.2496)
(0.2469)
(0.25,0.75)
0.9512
0.9452
0.9382
0.9480
0.9486
(0.4134)
(0.3936)
(0.3915)
(0.4068)
(0.4003)
(0.25,1.00)
0.9542
0.9452
0.9386
0.9506
0.9508
(0.6339)
(0.5914)
(0.5885)
(0.6158)
(0.6024)
(0.50,0.50)
0.9510
0.9434
0.9464
0.9480
0.9494
(0.3271)
(0.3151)
(0.3134)
(0.3234)
(0.3207)
(0.50,0.75)
0.9530
0.9472
0.9476
0.9504
0.9504
(0.4633)
(0.4407)
(0.4389)
(0.4563)
(0.4512)
(0.50,1.00)
0.9578
0.9482
0.9486
0.9528
0.9542
(0.6696)
(0.6266)
(0.6231)
(0.6530)
(0.6416)
(0.75,0.75)
0.9458
0.9428
0.9448
0.9438
0.9480
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.54
Natchaya Ratasukharom, Sa-Aat Niwitpong, Suparat Niwitpong
E-ISSN: 2224-2880
526
Volume 23, 2024
2
,nn
2
,
CP (AL)
GCI
BCI
PBCI
BayCrI
HPD
(0.5727)
(0.5421)
(0.5385)
(0.5636)
(0.5586)
(0.75,1.00)
0.9502
0.9462
0.9498
0.9480
0.9510
(0.7526)
(0.7021)
(0.6978)
(0.7337)
(0.7251)
(1.00,1.00)
0.9522
0.9498
0.9536
0.9516
0.9554
(0.9008)
(0.8328)
(0.8260)
(0.8747)
(0.8667)
Notes: Bold represents values that satisfy criteria and the best-performing method
Table 3. The CPs and ALs of the 95% CIs for the difference between the means of BirSau distributions with
unequal sample sizes
2.nn
2
,nn
2
,
CP (AL)
GCI
BCI
PBCI
BayCrI
HPD
(10,20)
(0.25,0.25)
0.9622
0.9262
0.9304
0.9512
0.9540
(0.4582)
(0.3732)
(0.3689)
(0.4318)
(0.4246)
(0.25,0.50)
0.9592
0.9314
0.9320
0.9490
0.9540
(0.7037)
(0.5811)
(0.5706)
(0.6720)
(0.6604)
(0.25,0.75)
0.9504
0.9224
0.9162
0.9446
0.9510
(1.1405)
(0.9138)
(0.8911)
(1.0870)
(1.0413)
(0.25,1.00)
0.9476
0.9320
0.9126
0.9426
0.9484
(1.8044)
(1.3666)
(1.3327)
(1.6900)
(1.5707)
(0.50,0.50)
0.9572
0.9242
0.9298
0.9452
0.9588
(1.1619)
(0.8565)
(0.8259)
(1.0868)
(1.0473)
(0.50,0.75)
0.9660
0.9420
0.9560
0.9564
0.9702
(1.5135)
(1.1224)
(1.0808)
(1.4236)
(1.3907)
(0.50,1.00)
0.9572
0.9334
0.9350
0.9522
0.9644
(2.1095)
(1.5225)
(1.4714)
(1.9592)
(1.8899)
(0.75,0.75)
0.9526
0.9266
0.9412
0.9406
0.9606
(2.3624)
(1.5271)
(1.4422)
(2.1522)
(2.0269)
(0.75,1.00)
0.9584
0.9424
0.9608
0.9492
0.9698
(2.8777)
(1.8691)
(1.7684)
(2.6158)
(2.5222)
(1.00,1.00)
0.9564
0.9330
0.9482
0.9458
0.9706
(4.3828)
(2.4446)
(2.2580)
(3.8109)
(3.5095)
(20,30)
(0.25,0.25)
0.9566
0.9392
0.9418
0.9498
0.9498
(0.3157)
(0.2857)
(0.2839)
(0.3069)
(0.3038)
(0.25,0.50)
0.9572
0.9390
0.9382
0.9532
0.9522
(0.5130)
(0.4610)
(0.4548)
(0.5005)
(0.4926)
(0.25,0.75)
0.9484
0.9328
0.9250
0.9446
0.9486
(0.8379)
(0.7282)
(0.7177)
(0.8122)
(0.7850)
(0.25,1.00)
0.9464
0.9298
0.9150
0.9412
0.9418
(1.3119)
(1.0921)
(1.0776)
(1.2510)
(1.1844)
(0.50,0.50)
0.9532
0.9390
0.9490
0.9486
0.9558
(0.7392)
(0.6411)
(0.6286)
(0.7174)
(0.7069)
(0.50,0.75)
0.9566
0.9436
0.9504
0.9506
0.9602
(1.0113)
(0.8640)
(0.8457)
(0.9785)
(0.9628)
(0.50,1.00)
0.9512
0.9342
0.9344
0.9422
0.9532
(1.4436)
(1.1953)
(1.1729)
(1.3796)
(1.3351)
(0.75,0.75)
0.9534
0.9420
0.9548
0.9482
0.9630
(1.3756)
(1.1252)
(1.0918)
(1.3219)
(1.2947)
(0.75,1.00)
0.9460
0.9366
0.9508
0.9426
0.9576
(1.7480)
(1.4022)
(1.3660)
(1.6633)
(1.6329)
(1.00,1.00)
0.9562
0.9438
0.9542
0.9470
0.9632
(2.2937)
(1.7479)
(1.6926)
(2.1532)
(2.0993)
(30,50)
(0.25,0.25)
0.9536
0.9386
0.9404
0.9490
0.9472
(0.2456)
(0.2301)
(0.2292)
(0.2410)
(0.2387)
(0.25,0.50)
0.9462
0.9336
0.9374
0.9428
0.9426
(0.3894)
(0.3637)
(0.3610)
(0.3826)
(0.3780)
(0.25,0.75)
0.9542
0.9456
0.9382
0.9514
0.9508
(0.6229)
(0.5706)
(0.5660)
(0.6091)
(0.5954)
(0.25,1.00)
0.9426
0.9324
0.9236
0.9440
0.9428
(0.9534)
(0.8497)
(0.8418)
(0.9202)
(0.8886)
(0.50,0.50)
0.9516
0.9392
0.9462
0.9480
0.9526
(0.5635)
(0.5139)
(0.5068)
(0.5518)
(0.5452)
(0.50,0.75)
0.9510
0.9460
0.9494
0.9468
0.9542
(0.7507)
(0.6787)
(0.6697)
(0.7333)
(0.7246)
(0.50,1.00)
0.9486
0.9382
0.9398
0.9440
0.9504
(1.0473)
(0.9284)
(0.9166)
(1.0126)
(0.9907)
(0.75,0.75)
0.9506
0.9408
0.9474
0.9460
0.9552
(1.0207)
(0.8944)
(0.8785)
(0.9929)
(0.9771)
(0.75,1.00)
0.9518
0.9412
0.9508
0.9480
0.9572
(1.2632)
(1.0950)
(1.0773)
(1.2201)
(1.2046)
(1.00,1.00)
0.9476
0.9400
0.9488
0.9408
0.9550
(1.6514)
(1.3822)
(1.3539)
(1.5796)
(1.5489)
(50, 100)
(0.25,0.25)
0.9498
0.9398
0.9410
0.9484
0.9466
(0.1809)
(0.1736)
(0.1733)
(0.1787)
(0.1770)
(0.25,0.50)
0.9528
0.9490
0.9454
0.9488
0.9490
(0.2745)
(0.2636)
(0.2625)
(0.2712)
(0.2686)
(0.25,0.75)
0.9564
0.9484
0.9448
0.9526
0.9522
(0.4278)
(0.4067)
(0.4043)
(0.4211)
(0.4146)
(50,100)
(0.25,1.00)
0.9500
0.9452
0.9448
0.9496
0.9496
(0.6457)
(0.6021)
(0.5996)
(0.6280)
(0.6150)
(0.50,0.50)
0.9452
0.9406
0.9436
0.9434
0.9462
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.54
Natchaya Ratasukharom, Sa-Aat Niwitpong, Suparat Niwitpong
E-ISSN: 2224-2880
527
Volume 23, 2024
2
,nn
2
,
CP (AL)
GCI
BCI
PBCI
BayCrI
HPD
(0.4081)
(0.3855)
(0.3833)
(0.4022)
(0.3980)
(0.50,0.75)
0.9450
0.9400
0.9428
0.9444
0.9466
(0.5264)
(0.4963)
(0.4924)
(0.5178)
(0.5130)
(0.50,1.00)
0.9492
0.9428
0.9436
0.9472
0.9492
(0.7145)
(0.6640)
(0.6596)
(0.6958)
(0.6866)
(0.75,0.75)
0.9480
0.9386
0.9428
0.9438
0.9458
(0.7182)
(0.6622)
(0.6555)
(0.7036)
(0.6943)
(0.75,1.00)
0.9488
0.9450
0.9492
0.9474
0.9538
(0.8728)
(0.8004)
(0.7917)
(0.8505)
(0.8421)
(1.00,1.00)
0.9464
0.9432
0.9462
0.9424
0.9502
(1.1483)
(1.0289)
(1.0154)
(1.1102)
(1.0927)
Notes: Bold represents values that satisfy criteria and the best-performing method.
Table 4. The daily wind speed data is from the Ayutthaya and Ratchaburi provinces in Thailand.
Provinces
Wind Speed (knots)
Ayutthaya
6.6
4.7
4.2
9
9
4.4
2.8
2.8
4.4
3.9
3.5
3.2
5.3
4.2
6.4
3.6
2.9
5.4
8.7
10.2
14.4
9.6
16.3
16.5
14
9.7
7.1
3.3
3.9
4.7
3.3
5.9
7.5
7.2
9.1
6.7
3.5
4.8
4.4
3.9
5.7
7.8
6.8
7.3
5.9
8.1
4.7
6.3
6.9
6.1
4.7
6
7.2
6.6
6.5
6.2
6.3
4.1
5.1
14.6
22
18.8
13.2
8.6
9.7
9.7
7.5
3.9
5.1
7.9
7.9
7.4
6.4
7.5
7.4
6.2
7.4
4.5
3.8
6.4
7.2
6.8
7.6
7.4
9
5
9.1
5.2
5.2
Ratchaburi
5.2
5.2
2.6
5.2
6.6
3.9
5.0
4.9
5.5
4.6
5.1
3.9
5.1
4.8
4.7
3.1
3.9
4.7
5.6
5.6
5.8
6.4
7.7
10.4
9.9
7.7
2.6
2.4
3.3
3.9
4.2
3.0
3.3
7.0
6.2
3.8
2.1
3.7
4.0
4.1
4.2
6.8
6.5
6.5
7.1
6.3
5.3
6.6
7.8
7.3
4.9
4.5
6.0
5.5
5.1
6.7
5.1
4.5
6.3
8.0
16.8
12.9
7.8
5.1
3.7
6.5
4.5
2.3
3.4
2.5
4.4
5.3
5.6
4.7
6.0
5.1
7.5
4.4
5.1
6.1
5.5
5.9
6.0
5.7
4.8
5.1
10.7
5.8
4.2
Table 5. AIC and BIC values of Ayutthaya and Ratchaburi provinces are used to fit seven asymmetric
distributions
Provinces
Crite-
ria
Distributions
BirSau
Cauchy
Logistic
Exponential
Weibull
Normal
Ayutthaya
AIC
439.34
466.13
463.15
527.88
462.85
480.17
BIC
446.81
471.11
468.12
530.37
467.83
485.15
Ratchaburi
AIC
369.28
379.92
375.47
484.03
391.33
396.34
BIC
376.76
384.89
380.44
486.52
396.30
401.32
Table 6. Descriptive statistics for the wind speed data.
Province
n
Min
Median
Mean
Max
Variance
Ayutthaya
89
2.8
6.4
7.0579
22
12.4740
Ratchaburi
89
2.1
5.1
5.5179
16.8
4.8635
Table 7. The 95% CIs for the mean and difference of two means of wind speed for the Ayutthaya and
Ratchaburi datasets
Methods
Ayutthaya
Ratchaburi
Ayutthaya-Ratchaburi
Interval
Length
Interval
Length
Interval
Length
GCI
6.4217-7.716
1.3499
5.1138-5.9871
0.8733
0.7408-2.3596
1.6188
BCI
6.3935-7.8827
1.4892
5.0861-6.0388
0.9526
0.7311-2.3936
1.6625
PBCI
6.4494-7.7233
1.2739
5.1258-5.9420
0.8162
0.7622-2.3227
1.5605
BayCrI
6.5072-7.8466
1.3394
5.1583-5.9922
0.8338
0.7847-2.4473
1.6626
HPD
6.4378-7.7552
1.3174
5.1585-5.9926
0.8342
0.7232-2.3612
1.6308
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.54
Natchaya Ratasukharom, Sa-Aat Niwitpong, Suparat Niwitpong
E-ISSN: 2224-2880
528
Volume 23, 2024
