Numerical Estimation Method for the Generalized Weibull
Distribution Parameters
M. MASWADAH
Department of Mathematics, Faculty of Science,
Aswan University, Aswan,
EGYPT
Abstract: - In this study, a new estimation method using the Runge-Kutta iteration technique is presented to
improve point estimation methods. The improved method has been applied to the generalized Weibull
distribution, which is a member of a family of distributions (T-X family). The estimates of the generalized
Weibull model parameters were derived using the Runge-Kutta and Bayesian estimation methods based on the
generalized progressive hybrid censoring scheme, via a Monte Carlo simulation. The simulation results
indicated that the Runge-Kutta estimation method is highly efficient and outperforms the Bayesian estimation
method based on the informative and kernel priors. Finally, two real data sets were studied to ensure the Runge-
Kutta estimation method can be used more effectively than the most popular estimation methods in fitting and
analyzing real lifetime data.
Key-Words: - Bayesian inference, Generalized progressive hybrid censoring scheme, Informative prior, Kernel
prior, Rung-Kutta method, Weibull model
Received: August 11, 2022. Revised: February 12, 2023. Accepted: March 9, 2023. Published: April 5, 2023.
1 Introduction
The maximum likelihood estimation (MLE) method
is the most popular point estimation method in
statistical inference, and it is widely used in the
social sciences and psychology, though it is biased
in situations where sample sizes are small or data is
heavily censored. These biases can mislead
subsequent inferences, and it is not as effective as
the Bayesian estimation method. Furthermore, in
some distributions, it contains nonlinear equations
that require numerical techniques. Therefore, the
main objective of this study is to introduce an
improvement in point estimation methods such as
the MLE method by using the Runge–Kutta
iteration technique. The simulation results indicated
that the improved estimation method is highly
efficient and outperforms the Bayesian estimation
method based on the informative and kernel priors
for different loss functions. Thus, the statistical
significance of this method is its efficiency
compared to the most popular point estimation
methods in statistical inference and it is reliable and
easy to apply, especially for social sciences and
psychology researchers.
In the last decade, extensive efforts have been
made to present new models in distribution theory
and related statistical applications. Some of the new
distributions were developed as generalizations or
modifications of the Weibull distribution and have
been extensively used for data modelling in many
fields, such as engineering and medical science. For
a review of some generalized Weibull distributions,
one may refer to [8]. [5], created a new class of
distributions known as the (T-X family), which is
defined as, for given a random variable with a
cumulative distribution function and a
generator random variable defined on with
and as the probability density function
"PDF" and the cumulative distribution function
"CDF," respectively. As a result, the generalised (T-
X family) CDF is given by:
.
Some lifetime distributions are extended by this
family, including the generalised Weibull, Weibull,
Weibull extension, Lomax, Logistic, and Log-
Logistic distributions. Several new distributions
within this family, including the Weibull-Pareto
distribution, have been introduced and studied in
[6], the Gamma-Pareto distribution in [4], and the
Gamma-Normal distribution in [7]. For a review of
this family and other distributions, [3].
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
1
Volume 5, 2023
For deriving the generalized Weibull
distribution, let be the PDF of the Weibull
distribution, which is defined as follows:
, .
Then, we have the Weibull (T-X family) with CDF
defined as follows:
.
Letting , be the
CDF of the exponential extension model. Thus, the
CDF of the generalized Weibull distribution (GWD)
can be derived as follows:
, .
(1)
The corresponding PDF is given as
αβγγαγβγα
(2)
where and are scale parameters, and is a
shape parameter.
This distribution has higher skewness compared
with the Weibull, inverse Weibull, and Log-Logistic
distributions and therefore it is more suitable to
model heavily skewed data that often arise in
reliability and survival analysis, [3], [8]. The GWD
has some special cases when the random variable
, we get has the two-parameter
Weibull distribution with and are the scale and
shape parameters respectively. When α, we get
the exponential extension model, which is a special
case of the Weibull extension model, [32]. For
comparison between the two estimation methods,
the generalized Weibull distribution parameters
have been estimated using the Runge-Kutta (R-K)
and Bayesian estimation methods based on the
informative and informative kernel priors with
different loss functions based on the generalized
progressive hybrid censoring scheme.
In reliability analysis, the progressive Type-II
censoring scheme is the most applicable in life test
experiments, it is useful for both industrial life test
applications and clinical trials and allows removing
some of the surviving experimental units at various
stages before testing is terminated. [9], [10],
presented comprehensive studies on the topic of
progressive censoring and its applications.
However, the trial time can be quite long due to
some highly reliable units. Thus, [20], recently
proposed a censoring scheme called the Type-II
progressive hybrid censoring scheme.
However, the progressive hybrid censoring scheme
has the disadvantage of having very few failures
before time point To provide a guarantee of the
number of failures observed as well as the time to
complete the test, [15], [16], proposed the
generalized progressive hybrid censoring scheme
(GPHCS) that modifies the progressive hybrid
censoring scheme. If the number of failures is less
than a specified number of , it allows the
experiment to continue beyond time to observe at
least the number of failures . The GPHCS has been
described in [24], [25].
Thus, given a generalized progressive hybrid
censored sample, the likelihood function can be
written in a unified form as follows:
θ
,
(3)
δ
δ
δ
where , is the sample size
and is the number of surviving units that are
removed at the stopping time
.
The GPHCS has been applied to some
distributions such as the Weibull distribution, [16],
the inverse Weibull distribution, [24], [27], the
exponential distribution, [15], [17], the Rayleigh
distribution, [14], the shape-scale family, [25], and
the generalized shape-scale family, [26].
2 Estimation Methods
2.1 Runge-Kutta Method
The MLE θ
θ
of θ is the solution of the
stationary equation,
, which is a
function of and θ
, where θ
is the log-
likelihood function that depends on the unknown
parameter θ
αβγ and the data
. Applying the implicit function
theorem to the stationary equation by considering all
partial derivatives, as well as the total derivatives,
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
2
Volume 5, 2023
are assumed to be evaluated at some known value of
θ
θ, say. Taking the total derivative for the
stationary equation with respect to , [28], we
obtain.
. (4)
Solving (4) we obtain the first derivative with
respect to x for at θθ
as follows:
. (5)
Thus, we can write (5) as the first-order ordinary
differential equation in the maximum likelihood
estimator as
. (6)
where
and
are defined and
continuous functions at all points θ
, which
ensures the existence of a unique solution for (6).
Using any numerical technique, such as the fourth-
order Runge-Kutta, we can find the approximate
solution given a trial set of parameter values and
initial conditions. If the initial conditions are
unavailable, they must be appended to the parameter
θ
as quantities concerning which the fit is optimized.
Thus, the R-K recurrence solution for (6) can be
obtained as
, (7)
for ,
where
, ,
,
.
Here is a small known value (say, 1E-02) and
, is the initial value for .
For the generalized Weibull lifetime model (1)
and (2), the log-likelihood function based on the
GPHCS (3) can be derived as follows:
,
(8)
K is the normalizing constant. The derivatives of
(8) have been derived in Appendix A.
Thus, using (6) with the corresponding
derivatives, we can find the point estimates for as
with the fourth-order Runge–Kutta
method using the recurrence relations (7).
2.2 Bayesian Method
In this section, the Bayes estimations will be derived
using gamma and kernel prior distributions based on
two loss functions:
Firstly, the squared error loss function (SLF),
. For this loss function, the
Bayes estimator that minimizes the risk function is
given by .
Secondly, the compound LINEX loss function is
defined as follows:
It is named LINEX-based loss function, [31], where
, .
is the LINEX loss function (LLF) that has been
introduced in [30], [32]. The Bayes estimator of the
parameter θ that minimizes the risk function can be
derived as follows:
22.1 Informative Prior
We consider the unknown parameters , and to
have independent gamma prior distributions with
the joint probability density function, which is given
by:
, (9)
where the hyper-parameter , and are
assumed to be known, positive, and chosen to reflect
the prior belief about the unknown parameters.
2.2.2 Kernel Prior
For deriving the kernel prior, we introduce the
trivariate kernel density estimator for the unknown
probability density function with support
on ( ∞), which is defined as
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
3
Volume 5, 2023
, (10)
are called the bandwidths or smoothing
parameters, which are chosen such that and
∞ as ∞, where is the sample size.
The influence of the smoothing parameter is
critical because it determines the amount of
smoothing. However, the optimal choice for ,
which minimizes the mean squared errors given by
, and is the sample standard
deviation. The optimal choice for the kernel
function can be used as the trivariate standard
normal distribution for the parameters ,, and .
The basic elements associated with the kernel
density function have been studied extensively by
[18, 19]. Based on the properties of the MLEs of the
parameters, which are converging in probability to
the original parameters, the kernel prior estimate has
been derived, [26]. It is worthwhile to mention that
this kernel prior has been used for some
distributions, [2], [21], [22], [23], [24], [25].
Thus, using the joint priors (9) and (10) with the
likelihood function of the GPHCS (3) the posterior
density for the parameters , , and can be
written in a unified form as follows:
, where
is the general prior distribution function with
for the informative prior (9), and
, , and
for the kernel prior (10).
Thus, the posterior density can be written as
(11)
Thus, based on (11) we can use the Tierney and
Kadane approximation method to approximate all
the Bayes estimators for the unknown parameters.
[29], introduced an easily computable
approximation for the posterior mean and variance
of a non-negative parameter or more generally, of a
smooth function of the parameter that is non-zero on
the interior of the parameter space. For details, let
αβγ be a smooth, positive function in the
parameter space. The posterior expectation of
αβγ can be obtained as
(12)
where , and
.
For αβγ the Bayes estimator using Tierney and
Kadane approximation for αβγ can be obtained
as
where αβ
γ and maximize the
αβ
γ and , respectively
, and
denote the minus of inverse of Hessians of
and at and
respectively.
Thus, we can define αβγ, with the log
likelihood of the posterior (11) as follows:
.
The derivatives of and αβγ have
been derived in Appendix B.
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
4
Volume 5, 2023
3 Simulation Study
The purpose of the simulation study is to compare
the performance of the estimates using the Runge-
Kutta and Bayes methods based on the informative
and the informative kernel priors with two different
loss functions, through two criteria: the average bias
(AVB) and the root mean squared error (RMSEs) as
given by:
and
,
is the estimate of and is the number of
replications.
In the simulation study, we choose different
combinations for the hyperparameters of αand β:
, , ,
, and . Thus, we can
generate from the gamma distribution two values for
each parameter α, β
and . Using the above parameter
values for generating different samples from the
generalized Weibull distribution with sizes N = 20,
40, and 60 to represent small, moderate, and large
sizes. To assess the performance of these estimates,
the average Bias (AVB) and the RMSEs for each
were calculated using 1000 replicates.
An algorithm for generating the generalized
progressive hybrid censoring scheme has been
written in [24], [25]. Some of the points are quite
clear based on these estimates from the simulation
results in Table 2, Table 3, Table 4, Table 5, Table
6, and Table 7, and the others have been
summarized in the following main points:
i. It is clear that generally, for both parameters β
and the average Bias values based on the
Runge-Kutta method outperform the
corresponding values based on Bayes’ method
for the different loss functions. However, for
the parameter , the Runge-Kutta and Bayes
methods have almost the same average bias
especially based on the LINEX-based loss
function.
ii. In terms of the RMSE values, we can easily see
that the R-K method has the smallest RMSE
values compared with their counterparts that
are based on the Bayes’ method, but for the
parameter , the Runge-Kutta and Bayes
methods have the same RMSEs approximately.
iii. It is evident that the estimated AVB and RMSE
values decrease with increasing the
hyperparameters, the termination time of the
experiment T, and the sample sizes as expected
for all methods.
iv. For the parameter α, the estimated RMSE
values increase with increasing the value of α,
while decreasing as the value of β and
γincrease.
v. For the parameters βand γthe estimated
RMSE values increase with increasing values
of β and γ while decreasing as the value of α
increases
vi. In general, the estimated RMSE values for the
Bayes method based on the LINEX-based loss
function are less than those based on the
squared error loss function.
In conclusion, the R-K estimates compete and
outperform the Bayesian estimation method based
on the informative and kernel priors.
4 Real Data Analysis
In this section, we studied two real data sets to
demonstrate the performance of the proposed
methods on the generalized Weibull model, which is
suitable for fitting several types of data and can be
adapted to fit the data set with the monotone hazard
rate function. It also demonstrates that the GW
distribution can be used in many applications in
reliability engineering and new fields such as
biomedical sciences and survival analysis to
describe the age of specific mortality rates and
failure rates. As a result, for a significance level of
0.05, we fitted these datasets using the goodness of
fit tests such as the Kolmogorov-Smirnov (K-S) and
Anderson-Darling (A-D) tests. [12], [13], presented
a comprehensive study of these tests.
4.1 Vinyl Chloride Data Application
Since vinyl chloride is a known human carcinogen,
exposure to this compound should be avoided to the
maximum practicable extent, and levels should be
kept as low as technically possible. Whereas, it is
known that the concentration of vinyl chloride in
drinking water of 0.5 mg/liter was calculated to be
associated with an increased risk of liver and brain
tumors for exposure starting from adulthood and
would double the risk of developing cancer from
continuous exposure from birth. Therefore, we
consider the dataset used by [11], which represents
34 data points in mg/L of vinyl chloride obtained
from clean upgrade monitoring wells, as follows:
5.1, 1.2, 1.3, 0.6, 0.5, 2.4, 0.5, 1.1, 8.0, 0.8, 0.4, 0.6,
0.9, 0.4, 2.0, 0.5, 5.3, 3.2, 2.7, 2.9, 2.5, 2.3, 1.0, 0.2,
0.1, 0.1, 1.8, 0.9, 2.0, 4.0, 6.8, 1.2, 0.4, 0.2.
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
5
Volume 5, 2023
We found that the generalized Weibull model is
a good fit for this dataset as shown in Figure (1 a).
Also, the goodness of fit tests such as the K-S test
has a value of 0.7349, which is less than the critical
value of 0.8579 with a P-value of 0.2389. The A-D
test has a value of 0.7170, which is less than the
critical value of 0.7464 with a P-value of 0.1583.
To study the concentration of vinyl chloride in
the water of these wells based on this dataset, we
find the estimates of the parameters that represent
the scale and shape of the concentration using our
model to determine the average concentration in the
water. We observed that the R-K and Bayes
estimates
are very close to one, indicating that
this dataset is right-skewed, which means that the
concentration decreases with increasing time, see
Figure (1 b). Also, the R-K and Bayes estimates for
βand γare close to 2, ensuring that the dataset is
right-skewed, which means the vinyl chloride
concentration will decrease with increasing time, so
monitoring these wells is very important.
4.2 Leukemia Data Application
In healthcare, Leukemia affects blood status and can
be detected with a Blood Cell Counter (CBC).
Mostly, leukemia patients undergo chemotherapy.
Therefore, we study the effect of this treatment on
leukemia patients based on a dataset collected by the
Ministry of Health Hospital in Saudi Arabia and
used in [1], which indicates the lifetimes in days for
forty-three blood patients with leukemia after
chemotherapy:
115, 181, 255, 418, 441, 461, 516, 739, 743, 789,
807, 865, 924, 983, 1025, 1062, 1063, 1165, 1191,
1222, 1222, 1251, 1277, 1290, 1357, 1369, 1408,
1455, 1478, 1549, 1578, 1578, 1599, 1603, 1605,
1696, 1735, 1799, 1815, 1852, 1899, 1925, 1965.
We found that the generalized Weibull model is
more fitting for this dataset than the Vinyl chloride
data as shown in Figure (2 a). Also, the goodness of
fit tests such as the K-S test has a value of 0.5402,
which is less than the critical value of 0.8666 with a
P-value of 0.6512. The A-D test has a value of
0.4421, which is less than the critical value of
0.7462 with a P-value of 0.3044.
To study the effect of chemotherapy on patients
based on this dataset, we estimated the distribution
parameters, which represent the scale and shape of
the lifetime. We observed that the estimates of the
R-K and Bayes methods for β and γare greater than
one and close to 2, while for αis less than one in
most cases, which indicates a decreasing hazard rate
and the graph is approximately symmetric, see
Figure (2 b). Thus, the parameter estimates indicate
the decreasing hazard rate for cancer and that means
the longer the patient survives, the more likely they
are to reach the upper limit of their natural lifespan.
So overall, this dataset indicates that the patient's
lifespan is more stable and lives longer due to the
chemotherapy dose, and it is highly effective in
giving patients more antibodies against cancer.
Figure (1 a) and Figure (2 a) display the
empirical CDF and the CDF of the GWD
distribution for these data sets, which confirm the
goodness of fit tests. The results in Table 1 indicated
that the R-K estimates of the parameters have AVB
and MSEs have values lower than the Bayesian
estimates based on the gamma prior and almost as
close as to those based on the informative kernel
prior. For both sets of data the MSE values for the
parameters
,
and
decrease as the T values
increase.
5 Conclusion
In this study, we applied the Runge-Kutta estimation
and Bayesian estimation methods for estimating the
generalized Weibull distribution parameters, as a
new lifetime distribution. The simulation results
indicated that the average Bias and RMSEs of the
parameters based on the Runge-Kutta method are
more efficient than the Bayesian estimation method
based on the informative and kernel priors using two
different loss functions, based on the generalized
progressive hybrid censored scheme. However,
Bayes estimates based on the informative kernel
prior are more efficient than those based on the
informative prior and are close to those based on the
Runge-Kutta estimates. Thus, the statistical
significance of the Runge-Kutta method is its
efficiency compared to most popular methods of
estimation, it is a viable point estimation method for
effectively any lifetime model and is reliable and
easy to apply especially for medical, biological,
social, psychological, and engineering researchers.
Acknowledgment:
The author is very grateful to the anonymous
referees and the Editor for their constructive efforts
and Comments that improved this work.
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
6
Volume 5, 2023
References:
[1] Abd-Elrahman A.M. and Sultan K.S.,
Reliability estimation based on general
progressive censored data from the Weibull
model: Comparison between Bayesian and
Classical approaches. METRON-International
Journal of Statistics, LX2, 2007, pp.239-257.
[2] Ahsanullah M., Maswadah M. and Seham
A.M., Kernel Inference on the Generalized
Gamma Distribution based on Generalized
Order Statistics. Journal of Statistical Theory
and Applications, Vol.12, No. 2, 2003,
pp.152-172.
[3] Almheidat M., Lee C. and Famoye F. A.,
generalization of the Weibull Distribution
with Applications. Journal of Modern Applied
Statistical Methods, Vol. 15, No. 2, 2016,
pp.788-820.
[4] Alzaatreh A., Famoye F. and Lee C., Gamma-
Pareto distribution and its applications,
Journal of Modern Applied Statistical
Methods, Vol. 11, No.1, 2012, pp.78-94.
[5] Alzaatreh A., Lee C. and Famoye F., A new
method for generating families of continuous
distributions, METRON-International Journal
of Statistics, Vol. 71, No. 1, 2013a, pp. 1:63-
79.
http://dx.doi.org/10.1007/s40300-013-0007-y
[6] Alzaatreh A., Famoye F. and Lee C., Weibull-
Pareto distribution and its applications,
Communications in Statistics-Theory and
Methods, Vol. 42, No. 9, 2013b, pp. 1673-
1691.
http://dx.doi.org/10.1080/03610926.2011.599
002
[7] Alzaatreh A., Famoye F. and Lee C., The
gamma-normal distribution: Properties and
applications, Journal of Computational
Statistics and Data Analysis. Vol.69, No. 1,
2014, pp. 67-80.
[8] Alzaatreh A., Lee C., Famoye F., On
generating T-X family of distributions using
quantile functions, Journal of Statistical
Distributions and Applications. Vol. I, No. 2,
2014, pp. 1-17.
[9] Balakrishnan N, and Aggarwala R.,
Progressive Censoring: Theory, Methods and
Applications. Birkhãuser Publishers, Boston,
2000.
[10] Balakrishnan N. and Cramer E., The art of
Progressive Censoring: Applications to
Reliability and Quality, Statistics for Industry
and Technology. Springer, New York, 2014.
[11] Bhaumik D.K. and Kapur K., Gibbons RD.
Testing Parameters of a Gamma Distribution
for Small Samples, Technometrics, Vol. 51,
No.3, 2009, pp. 326-334.
[12] Bush J. G., Woodruff B.W., Moore A., H.,
and Dunne, E. J., Modified Cramervon Mises
and Anderson-Darling tests for Weibull
distributions with unknown location and scale
parameters. Communications in Statistics,
Part A-Theory and Methods, Vol. 12: 1983,
pp. 2465-2476.
[13] Chandra M., Singpurwalla N., D. and
Stephens M., A., Kolmogorov statistics for
tests of fit for the extreme-value and Weibull
distribution, Journal of the American
Statistics Association, Vol. 76, No. 375. 1981,
pp. 729-731.
[14] Cho Y., Sun H., and Lee K., An estimation of
the entropy for a Rayleigh distribution based
on doubly generalized Type-II hybrid
censored samples, Entropy, Vol. 16, 2014, pp.
3655–3669.
[15] Cho Y, Sun H, Lee K. Exact likelihood
inference for an exponential parameter under
generalized progressive hybrid censoring
scheme, Stat. Method. 2015a; 23:18–34.
[16] Cho Y., Sun H. and Lee K. Estimating the
Entropy of a Weibull Distribution under
Generalized Progressive Hybrid Censoring,
Entropy., Vol. 17, 2015b, pp.102-122.
doi:10.3390/e17010102.
[17] Gurney J., E. and Cramer E., Exact likelihood
inference for exponential distribution under
generalized progressive hybrid censoring
schemes, Statistical Methodology, Vol. 29,
2016, pp. 70- 94.
[18] Guillamon A. J., Navarro J. and Ruiz J., M.,
Kernel density estimation using weighted
data. Commun. Statist. -Theory Meth., Vol.
27, No. 9, 1988, pp. 2123-2135.
[19] Guillamon A. J., Navarro J. and Ruiz J., M., A
note on kernel estimators for positive valued
random variables, Sankhya: The Indian
Journal of Statistics. Vol. 6, No. A, 1999,
pp.:276-281.
[20] Kundu D. and Joarder A., Analysis of Type-II
progressive hybrid censored data, Comput.
Stat Data Anal., Vol. 50, 2006, pp. 2509–
2528.
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
7
Volume 5, 2023
[21] Maswadah M., Kernel inference on the
inverse Weibull distribution. The Korean
Communications in Statistics. Vol. 13, No. 3,
2006, pp. 503-512.
[22] Maswadah M., Kernel inference on the
Weibull distribution, Proc. Of the Third
National Statistical Conference. Lahore,
Pakistan. Vol. 14, 2007, pp. 77-86.
[23] Maswadah M., Kernel inference on the type-II
Extreme value distribution, Proceedings of the
Tenth Islamic Countries Conference on
Statistical Sciences (ICCS-X), Lahore,
Pakistan. Vol. II, 2010, pp. 870-880.
[24] Maswadah M., An optimal point estimation
method for the inverse Weibull model
parameters using the Runge-Kutta method,
Aligarh Journal of Statistics (AJS). Vol. 48,
No. 5, pp.1-22.
[25] Maswadah M., Improved maximum
likelihood estimation of the shape-scale
family based on the generalized progressive
hybrid censoring scheme. Journal of Applied
Statistics. (JAS). Vol. 49, No. 11, 2022, pp.
2825-2844.
DOI/10.1080/02664763.2021.1924638.
[26] Maswadah, M., Conditional Inference on the
Generalized Shape-Scale Family,
International Journal of Applied Mathematics,
Computational Science and Systems
Engineering, 2022, DOI:
10.37394/232026.2022.4.14.
[27] Mohie El-Din M. M. and Nagy M.,
Estimation of the Inverse Weibull distribution
under Generalized Progressive Hybrid
Censoring Scheme, J. Stat. Appl. Pro. Lett.,
Vol. 4, No. 3, 2017, pp. 97-107.
[28] Ramsay J. O, Hooker G., Campbell D. and
Cao J., Parameter estimation for differential
equations: a generalized smoothing approach,
J. R. Statist. Soc. B., Vol. 69, No. 5, 2007, pp.
741–796.
[29] Tierney, L. and Kadane, J. B., Accurate
Approximations for Posterior Moments and
Marginal Densities. Journal of the American
Statistical Association, Vol 81, No. 393, 1986,
pp. 82-86.
[30] Wei C.D., Wei S. and Su H., Bayes estimation
and application of Poisson distribution
parameter under compound LINEX
symmetric loss [J], Statistics and Decision,
Vol. 7, 2010, pp. 156-157. (In Chinese).
[31] Xie M., Tang Y. and Goh T., N., A modified
Weibull extension with bathtub shaped failure
rate function, Reliability Engineering and
System Safety, Vol. 76, 2002, pp.279-285.
[32] Xiuchun L., Yimin S., Jieqiong W. and Jian
C., Empirical Bayes estimators of reliability
performances using LINEX loss under
progressively Type-II censored samples,
Mathematics and Computers in Simulation.
Vol. 73, No. 5, 2007, pp. 320–326.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in 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
No funding was received for conducting this study.
Conflict of Interest
The author declares no potential conflicts of interest
with respect to the research, authorship, and/or
publication of this article.
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.en
_US
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
8
Volume 5, 2023
Appendix A:
The log-likelihood function of (8) can be derived as
,
β
γαγ
δγαγ,
ααβ
γαγ
+,
α γγ
γ
γβα
γαγγ
,
ββ
γαδγα
ββ,
βαγ
γαγ.
βα
γαγ
δγαγ
γ
γαγγγ
γ
βαα
γαγ
δγαγ
Appendix B:
The log of the posterior density function (9) can be
derived as
β
′β
ββ
γαδ
γα
ββ
β
β
β
β
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
9
Volume 5, 2023
δ
γαγ
γ
′γ
γγ
α γ
γ
αβ
γγα
δ
γγα
,
.
γ
αγγγ
γ
αβ
γγα
αγγα
δ
γγα
αγγα,
where the derivative of the kernel density estimation
can be defined as
(*)
where r=0,1,2,3,-----.
Using the Gaussian kernel and (*), we have
,
′α
παα
αα
″α
παα
αα
Similarly for the kernel priors and
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
10
Volume 5, 2023
Table 1. The estimate, AVB, and the MSEs in parentheses for the parameters , , and based on the R-K and
Bayes methods using the gamma and kernel priors under the squared error loss function based on the
GHPCS for . The hyperparameters are: .
Samples
T
Par.
R-K Estimate
Gamma Prior
Kernel Prior
Estimate
AVB(MSE)
Estimate
AVB (MSE)
Estimate
AVB (MSE)
The vinyl
Chloride data
N=34
0.75
0.6189
0.065(0.0043)
0.5965
0.088(0.0078)
0.5956
0.089(0.0079)
0.8503
0.101(0.0101)
0.8512
0.099(0.0099)
0.8473
0.103(0.0108)
0.2964
0.036(0.0013)
0.2958
0.036(0.0013)
0.2951
0.037(0.0014)
3.5
0.6179
0.067(0.0045)
0.6071
0.078(0.0061)
0.6066
0.078(0.0061)
0.8527
0.099(0.0096)
0.8494
0.101(0.0103)
0.8479
0.103(0.0106)
0.2982
0.034(0.0012)
0.2979
0.034(0.0012)
0.2979
0.034(0.0012)
The Leukemia
Data
N=43
500
0.6157
0.069(0.0048)
0.6159
0.069(0.0048)
0.6159
0.069(0.0048)
0.8440
0.106(0.0113)
0.8110
0.139(0.0195)
0.7729
0.178(0.0316)
0.2985
0.034(0.0012)
0.2985
0.034(0.0011)
0.2985
0.034(0.0011)
850
0.6161
0.069(0.0047)
0.6161
0.069(0.0047)
0.6161
0.069(0.0047)
0.8445
0.106(0.0113)
0.7764
0.174(0.0303)
0.7522
0.206(0.0423)
0.2988
0.033(0.0011)
0.2987
0.033(0.0011)
0.2988
0.033(0.0011)
Table 2. The Average Bias (AVB) and Root Mean Squared Errors (RMSEs) in parentheses for GWD
parameter using the R-K and Bayes methods with m = (n/2 and 3n/4) and k=(m/2 and 3m/4) at T=0.75 and
for LINEX loss function.
N
m
k
R-K
Method
Gamma Prior
Kernel Prior
SQEL
LNXL
SQEL
LNXL
20
10
5
0.59
0.79
0.97
0.027(0.0414)
0.070(0.0895)
0.034(0.0401)
0.065(0.0806)
0.034(0.0393)
0.92
1.32
0.035(0.0544)
0.076(0.1285)
0.034(0.0432)
0.068(0.1050)
0.033(0.0411)
1.11
0.79
0.97
0.032(0.0358)
0.136(0.1432)
0.084(0.0861)
0.126(0.1306)
0.087(0.0891)
0.92
1.32
0.028(0.0342)
0.127(0.1364)
0.076(0.0792)
0.115(0.1220)
0.080(0.0824)
8
0.59
0.79
0.97
0.025(0.0389)
0.062(0.0861)
0.035(0.0404)
0.059(0.0765)
0.035(0.0398)
0.92
1.32
0.030(0.0460)
0.061(0.1003)
0.031(0.0384)
0.057(0.0841)
0.031(0.0372)
1.11
0.79
0.97
0.035(0.0394)
0.127(0.1310)
0.088(0.0892)
0.122(0.1248)
0.090(0.0918)
0.92
1.32
0.028(0.0344)
0.115(0.1214)
0.078(0.0808)
0.111(0.1151)
0.082(0.0841)
15
8
0.59
0.79
0.97
0.026(0.0420)
0.066(0.1003)
0.036(0.0426)
0.062(0.0855)
0.036(0.0417)
0.92
1.32
0.028(0.0428)
0.058(0.0932)
0.031(0.0373)
0.055(0.0811)
0.030(0.0364)
1.11
0.79
0.97
0.035(0.0373)
0.126(0.1294)
0.088(0.0894)
0.122(0.1242)
0.091(0.0922)
0.92
1.32
0.030(0.0397)
0.120(0.1364)
0.080(0.0832)
0.114(0.1215)
0.084(0.0858)
11
0.59
0.79
0.97
0.020(0.0276)
0.055(0.0611)
0.036(0.0397)
0.054(0.0592)
0.036(0.0400)
0.92
1.32
0.023(0.0344)
0.052(0.0611)
0.032(0.0358)
0.050(0.0581)
0.032(0.0357)
1.11
0.79
0.97
0.036(0.0388)
0.120(0.1228)
0.091(0.0920)
0.119(0.1205)
0.093(0.0944)
0.92
1.32
0.030(0.0331)
0.111(0.1146)
0.083(0.0846)
0.110(0.1126)
0.086(0.0877)
40
20
10
0.59
0.79
0.97
0.022(0.0329)
0.048(0.0576)
0.028(0.0323)
0.046(0.0549)
0.028(0.0323)
0.92
1.32
0.027(0.0376)
0.042(0.0538)
0.023(0.0279)
0.040(0.0507)
0.023(0.0276)
1.11
0.79
0.97
0.029(0.0321)
0.111(0.1139)
0.084(0.0851)
0.108(0.1108)
0.086(0.0874)
0.92
1.32
0.022(0.0251)
0.101(0.1050)
0.076(0.0776)
0.099(0.1019)
0.079(0.0804)
15
0.59
0.79
0.97
0.022(0.0309)
0.043(0.0501)
0.028(0.0316)
0.042(0.0484)
0.028(0.0317)
0.92
1.32
0.026(0.0372)
0.040(0.0539)
0.024(0.0290)
0.039(0.0510)
0.024(0.0287)
1.11
0.79
0.97
0.030(0.0325)
0.105(0.1079)
0.084(0.0859)
0.105(0.1070)
0.087(0.0883)
0.92
1.32
0.024(0.0277)
0.097(0.1007)
0.076(0.0783)
0.097(0.0996)
0.080(0.0811)
30
15
0.59
0.79
0.97
0.021(0.0300)
0.043(0.0526)
0.028(0.0321)
0.042(0.0507)
0.028(0.0323)
0.92
1.32
0.026(0.0373)
0.040(0.0543)
0.024(0.0292)
0.039(0.0513)
0.024(0.0289)
1.11
0.79
0.97
0.032(0.0337)
0.108(0.1105)
0.087(0.0879)
0.108(0.1094)
0.089(0.0901)
0.92
1.32
0.024(0.0271)
0.097(0.0999)
0.077(0.0786)
0.096(0.0992)
0.080(0.0814)
23
0.59
0.79
0.97
0.014(0.0186)
0.044(0.0467)
0.034(0.0367)
0.044(0.0466)
0.035(0.0371)
0.92
1.32
0.017(0.0241)
0.037(0.0413)
0.026(0.0298)
0.037(0.0409)
0.027(0.0302)
1.11
0.79
0.97
0.037(0.0388)
0.108(0.1087)
0.092(0.0928)
0.108(0.1086)
0.094(0.0944)
0.92
1.32
0.029(0.0315)
0.099(0.1003)
0.084(0.0847)
0.099(0.1008)
0.086(0.0869)
0.59
0.79
0.97
0.019(0.0269)
0.037(0.0442)
0.024(0.0279)
0.036(0.0431)
0.025(0.0282)
0.92
1.32
0.027(0.0347)
0.033(0.0408)
0.019(0.0237)
0.032(0.0393)
0.020(0.0236)
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
11
Volume 5, 2023
60
30
15
1.11
0.79
0.97
0.028(0.0297)
0.101(0.1036)
0.084(0.0846)
0.101(0.1027)
0.086(0.0866)
0.92
1.32
0.020(0.0231)
0.091(0.0942)
0.075(0.0761)
0.091(0.0934)
0.077(0.0785)
23
0.59
0.79
0.97
0.018(0.0244)
0.036(0.0408)
0.026(0.0291)
0.036(0.0404)
0.026(0.0295)
0.92
1.32
0.023(0.0311)
0.031(0.0387)
0.020(0.0238)
0.031(0.0377)
0.020(0.0240)
1.11
0.79
0.97
0.029(0.0310)
0.099(0.1009)
0.085(0.0861)
0.100(0.1011)
0.087(0.0880)
0.92
1.32
0.022(0.0241)
0.090(0.0924)
0.077(0.0781)
0.091(0.0930)
0.079(0.0805)
45
23
0.59
0.79
0.97
0.018(0.0242)
0.035(0.0402)
0.025(0.0285)
0.035(0.0398)
0.026(0.0289)
0.92
1.32
0.024(0.0325)
0.032(0.0392)
0.020(0.0241)
0.031(0.0381)
0.020(0.0242)
1.11
0.79
0.97
0.029(0.0315)
0.099(0.1011)
0.085(0.0861)
0.100(0.1012)
0.087(0.0881)
0.92
1.32
0.021(0.0235)
0.088(0.0911)
0.076(0.0772)
0.090(0.0917)
0.079(0.0796)
34
0.59
0.79
0.97
0.012(0.0162)
0.039(0.0415)
0.031(0.0340)
0.039(0.0416)
0.032(0.0344)
0.92
1.32
0.016(0.0211)
0.032(0.0360)
0.025(0.0275)
0.032(0.0360)
0.025(0.0280)
1.11
0.79
0.97
0.036(0.0375)
0.103(0.1035)
0.091(0.0919)
0.103(0.1038)
0.093(0.0933)
0.92
1.32
0.028(0.0299)
0.094(0.0951)
0.083(0.0839)
0.095(0.0959)
0.085(0.0857)
Table 3. The Average Bias (AVB) and Root Mean Squared Errors (RMSEs) in parentheses for GWD parameter
using the R-K and Bayes methods with m = (n/2 and 3n/4) and k=(m/2 and 3m/4) at T=1.5 and for
LINEX loss function.
N
m
k
R-K
Method
Gamma Prior
Kernel Prior
SQEL
LNXL
SQEL
LNXL
20
10
5
0.59
0.79
0.97
0.024(0.0366)
0.060(0.0776)
0.034(0.0398)
0.058(0.0712)
0.035(0.0395)
0.92
1.32
0.032(0.0485)
0.062(0.0874)
0.031(0.0393)
0.058(0.0783)
0.031(0.0378)
1.11
0.79
0.97
0.034(0.0387)
0.128(0.1332)
0.086(0.0883)
0.122(0.1259)
0.089(0.0910)
0.92
1.32
0.029(0.0340)
0.119(0.1265)
0.079(0.0811)
0.113(0.1184)
0.082(0.0844)
8
0.59
0.79
0.97
0.024(0.0357)
0.058(0.0738)
0.035(0.0398)
0.056(0.0682)
0.035(0.0396)
0.92
1.32
0.029(0.0466)
0.063(0.1841)
0.032(0.0400)
0.058(0.1259)
0.031(0.0387)
1.11
0.79
0.97
0.036(0.0389)
0.123(0.1265)
0.088(0.0900)
0.120(0.1223)
0.091(0.0926)
0.92
1.32
0.029(0.0329)
0.113(0.1178)
0.080(0.0822)
0.111(0.1140)
0.084(0.0856)
15
8
0.59
0.79
0.97
0.025(0.0381)
0.061(0.0773)
0.036(0.0406)
0.059(0.0715)
0.036(0.0401)
0.92
1.32
0.028(0.0445)
0.060(0.0903)
0.031(0.0387)
0.056(0.0780)
0.031(0.0376)
1.11
0.79
0.97
0.035(0.0387)
0.124(0.1297)
0.087(0.0892)
0.120(0.1237)
0.091(0.0919)
0.92
1.32
0.029(0.0346)
0.116(0.1219)
0.079(0.0814)
0.112(0.1159)
0.083(0.0847)
11
0.59
0.79
0.97
0.022(0.0315)
0.056(0.0693)
0.036(0.0402)
0.055(0.0647)
0.036(0.0401)
0.92
1.32
0.025(0.0384)
0.054(0.0724)
0.032(0.0380)
0.052(0.0665)
0.032(0.0375)
1.11
0.79
0.97
0.037(0.0396)
0.120(0.1224)
0.090(0.0919)
0.118(0.1198)
0.093(0.0942)
0.92
1.32
0.030(0.0335)
0.111(0.1144)
0.082(0.0842)
0.110(0.1124)
0.086(0.0874)
40
20
10
0.59
0.79
0.97
0.019(0.0281)
0.044(0.0534)
0.030(0.0335)
0.043(0.0515)
0.030(0.0337)
0.92
1.32
0.025(0.0358)
0.040(0.0522)
0.025(0.0295)
0.039(0.0495)
0.025(0.0293)
1.11
0.79
0.97
0.032(0.0340)
0.107(0.1092)
0.087(0.0879)
0.107(0.1084)
0.089(0.0901)
0.92
1.32
0.025(0.0290)
0.097(0.1000)
0.077(0.0791)
0.097(0.0993)
0.080(0.0818)
15
0.59
0.79
0.97
0.018(0.0259)
0.043(0.0491)
0.029(0.0328)
0.042(0.0481)
0.030(0.0331)
0.92
1.32
0.023(0.0329)
0.039(0.0468)
0.024(0.0283)
0.038(0.0451)
0.024(0.0284)
1.11
0.79
0.97
0.032(0.0340)
0.106(0.1083)
0.087(0.0881)
0.106(0.1079)
0.089(0.0903)
0.92
1.32
0.025(0.0277)
0.097(0.0996)
0.078(0.0801)
0.097(0.0994)
0.081(0.0828)
30
15
0.59
0.79
0.97
0.018(0.0254)
0.042(0.0479)
0.029(0.0327)
0.042(0.0470)
0.030(0.0331)
0.92
1.32
0.025(0.0348)
0.038(0.0479)
0.023(0.0279)
0.037(0.0459)
0.024(0.0279)
1.11
0.79
0.97
0.031(0.0330)
0.105(0.1075)
0.085(0.0866)
0.105(0.1070)
0.088(0.0889)
0.92
1.32
0.024(0.0276)
0.096(0.0995)
0.077(0.0790)
0.097(0.0990)
0.080(0.0818)
23
0.59
0.79
0.97
0.015(0.0199)
0.045(0.0477)
0.034(0.0368)
0.045(0.0475)
0.035(0.0372)
0.92
1.32
0.018(0.0252)
0.037(0.0427)
0.027(0.0304)
0.037(0.0422)
0.027(0.0307)
1.11
0.79
0.97
0.037(0.0387)
0.108(0.1089)
0.092(0.0929)
0.108(0.1088)
0.094(0.0944)
0.92
1.32
0.030(0.0320)
0.099(0.1006)
0.084(0.0850)
0.099(0.1010)
0.086(0.0871)
30
15
0.59
0.79
0.97
0.015(0.0210)
0.036(0.0402)
0.027(0.0306)
0.036(0.0401)
0.028(0.0310)
0.92
1.32
0.023(0.0313)
0.032(0.0394)
0.021(0.0250)
0.031(0.0384)
0.021(0.0251)
1.11
0.79
0.97
0.030(0.0322)
0.100(0.1013)
0.086(0.0871)
0.100(0.1016)
0.088(0.0889)
0.92
1.32
0.022(0.0247)
0.089(0.0918)
0.077(0.0785)
0.091(0.0926)
0.080(0.0808)
23
0.59
0.79
0.97
0.016(0.0227)
0.037(0.0416)
0.028(0.0307)
0.037(0.0413)
0.028(0.0311)
0.92
1.32
0.023(0.0301)
0.031(0.0372)
0.021(0.0247)
0.031(0.0365)
0.021(0.0249)
1.11
0.79
0.97
0.031(0.0325)
0.100(0.1016)
0.087(0.0877)
0.101(0.1019)
0.089(0.0895)
0.92
1.32
0.023(0.0255)
0.090(0.0919)
0.077(0.0789)
0.091(0.0927)
0.080(0.0812)
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
12
Volume 5, 2023
60
45
23
0.59
0.79
0.97
0.014(0.0194)
0.038(0.0418)
0.029(0.0319)
0.038(0.0417)
0.030(0.0324)
0.92
1.32
0.022(0.0299)
0.031(0.0382)
0.021(0.0252)
0.031(0.0374)
0.022(0.0254)
1.11
0.79
0.97
0.032(0.0336)
0.101(0.1024)
0.088(0.0887)
0.102(0.1027)
0.090(0.0904)
0.92
1.32
0.023(0.0261)
0.091(0.0928)
0.079(0.0797)
0.092(0.0936)
0.081(0.0819)
34
0.59
0.79
0.97
0.012(0.0151)
0.039(0.0421)
0.032(0.0345)
0.040(0.0422)
0.033(0.0350)
0.92
1.32
0.015(0.0204)
0.032(0.0357)
0.024(0.0275)
0.032(0.0358)
0.025(0.0280)
1.11
0.79
0.97
0.036(0.0377)
0.103(0.1035)
0.091(0.0920)
0.103(0.1039)
0.093(0.0933)
0.92
1.32
0.028(0.0300)
0.095(0.0959)
0.084(0.0844)
0.096(0.0967)
0.086(0.0862)
Table 4. The Average Bias (AVB) and Root Mean Squared Errors (RMSEs) in parentheses for the GWD
parameter using the R-K and Bayes methods with m = (n/2 and 3n/4) and k=(m/2 and 3m/4) at T=0.75 and
for LINEX loss function.
N
m
K
R-K
Method
Gamma Prior
Kernel Prior
SQEL
LNXL
SQEL
LNXL
20
10
5
0.59
0.79
0.97
0.058(0.0582)
0.141(0.1432)
0.095(0.0955)
0.131(0.1331)
0.093(0.0932)
0.92
1.32
0.083(0.0837)
0.181(0.1832)
0.125(0.1256)
0.165(0.1664)
0.119(0.1194)
1.11
0.79
0.97
0.053(0.0528)
0.145(0.1468)
0.090(0.0901)
0.130(0.1316)
0.088(0.0885)
0.92
1.32
0.069(0.0695)
0.181(0.1828)
0.112(0.1118)
0.159(0.1597)
0.108(0.1081)
8
0.59
0.79
0.97
0.050(0.0505)
0.106(0.1071)
0.089(0.0885)
0.103(0.1038)
0.087(0.0874)
0.92
1.32
0.067(0.0669)
0.133(0.1338)
0.110(0.1103)
0.127(0.1279)
0.107(0.1075)
1.11
0.79
0.97
0.049(0.0492)
0.106(0.1073)
0.087(0.0868)
0.102(0.1025)
0.086(0.0858)
0.92
1.32
0.063(0.0628)
0.133(0.1338)
0.106(0.1057)
0.125(0.1258)
0.104(0.1036)
15
8
0.59
0.79
0.97
0.050(0.0504)
0.106(0.1075)
0.088(0.0885)
0.103(0.1040)
0.087(0.0874)
0.92
1.32
0.067(0.0666)
0.133(0.1336)
0.110(0.1100)
0.127(0.1277)
0.107(0.1073)
1.11
0.79
0.97
0.049(0.0491)
0.108(0.1085)
0.087(0.0867)
0.103(0.1034)
0.086(0.0858)
0.92
1.32
0.063(0.0628)
0.133(0.1337)
0.106(0.1057)
0.125(0.1256)
0.104(0.1036)
11
0.59
0.79
0.97
0.047(0.0474)
0.092(0.0923)
0.086(0.0857)
0.091(0.0911)
0.085(0.0851)
0.92
1.32
0.061(0.0608)
0.112(0.1128)
0.105(0.1047)
0.110(0.1106)
0.103(0.1031)
1.11
0.79
0.97
0.047(0.0473)
0.093(0.0931)
0.085(0.0851)
0.091(0.0913)
0.085(0.0845)
0.92
1.32
0.059(0.0595)
0.114(0.1141)
0.103(0.1027)
0.111(0.1108)
0.101(0.1013)
40
20
10
0.59
0.79
0.97
0.057(0.0573)
0.114(0.1147)
0.095(0.0948)
0.111(0.1110)
0.093(0.0933)
0.92
1.32
0.079(0.0796)
0.146(0.1459)
0.122(0.1219)
0.139(0.1394)
0.118(0.1182)
1.11
0.79
0.97
0.052(0.0520)
0.112(0.1120)
0.089(0.0893)
0.107(0.1070)
0.088(0.0883)
0.92
1.32
0.068(0.0679)
0.141(0.1409)
0.110(0.1104)
0.132(0.1324)
0.108(0.1080)
15
0.59
0.79
0.97
0.052(0.0522)
0.098(0.0979)
0.090(0.0902)
0.096(0.0965)
0.089(0.0893)
0.92
1.32
0.069(0.0695)
0.123(0.1228)
0.113(0.1128)
0.120(0.1199)
0.111(0.1106)
1.11
0.79
0.97
0.049(0.0495)
0.097(0.0970)
0.087(0.0871)
0.095(0.0949)
0.086(0.0864)
0.92
1.32
0.064(0.0635)
0.120(0.1198)
0.106(0.1064)
0.116(0.1161)
0.105(0.1048)
30
15
0.59
0.79
0.97
0.052(0.0521)
0.097(0.0976)
0.090(0.0900)
0.096(0.0962)
0.089(0.0891)
0.92
1.32
0.069(0.0696)
0.122(0.1226)
0.113(0.1128)
0.120(0.1197)
0.111(0.1106)
1.11
0.79
0.97
0.050(0.0496)
0.096(0.0967)
0.087(0.0871)
0.094(0.0946)
0.086(0.0865)
0.92
1.32
0.063(0.0635)
0.120(0.1201)
0.106(0.1063)
0.116(0.1163)
0.105(0.1048)
23
0.59
0.79
0.97
0.048(0.0478)
0.086(0.0864)
0.086(0.0861)
0.086(0.0860)
0.086(0.0857)
0.92
1.32
0.061(0.0614)
0.105(0.1049)
0.105(0.1054)
0.104(0.1041)
0.104(0.1042)
1.11
0.79
0.97
0.047(0.0471)
0.086(0.0859)
0.085(0.0849)
0.085(0.0854)
0.084(0.0845)
0.92
1.32
0.059(0.0591)
0.103(0.1034)
0.102(0.1024)
0.102(0.1025)
0.101(0.1015)
60
30
15
0.59
0.79
0.97
0.056(0.0559)
0.104(0.1042)
0.094(0.0935)
0.102(0.1023)
0.092(0.0925)
0.92
1.32
0.078(0.0782)
0.135(0.1346)
0.121(0.1207)
0.131(0.1307)
0.118(0.1179)
1.11
0.79
0.97
0.052(0.0517)
0.103(0.1034)
0.089(0.0891)
0.100(0.1004)
0.088(0.0883)
0.92
1.32
0.067(0.0675)
0.128(0.1283)
0.110(0.1100)
0.123(0.1234)
0.108(0.1082)
23
0.59
0.79
0.97
0.052(0.0517)
0.093(0.0934)
0.090(0.0897)
0.093(0.0926)
0.089(0.0890)
0.92
1.32
0.069(0.0688)
0.117(0.1167)
0.112(0.1122)
0.115(0.1149)
0.110(0.1105)
1.11
0.79
0.97
0.049(0.0493)
0.092(0.0923)
0.087(0.0869)
0.091(0.0911)
0.086(0.0864)
0.92
1.32
0.063(0.0632)
0.113(0.1134)
0.106(0.1061)
0.111(0.1113)
0.105(0.1049)
45
23
0.59
0.79
0.97
0.052(0.0517)
0.093(0.0933)
0.090(0.0897)
0.092(0.0924)
0.089(0.0890)
0.92
1.32
0.069(0.0689)
0.117(0.1169)
0.112(0.1123)
0.115(0.1152)
0.111(0.1106)
1.11
0.79
0.97
0.049(0.0493)
0.092(0.0923)
0.087(0.0869)
0.091(0.0912)
0.086(0.0864)
0.92
1.32
0.063(0.0632)
0.113(0.1133)
0.106(0.1061)
0.111(0.1112)
0.105(0.1049)
0.59
0.79
0.97
0.048(0.0479)
0.085(0.0855)
0.086(0.0863)
0.085(0.0852)
0.086(0.0859)
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
13
Volume 5, 2023
34
0.92
1.32
0.062(0.0616)
0.104(0.1039)
0.105(0.1055)
0.103(0.1033)
0.105(0.1046)
1.11
0.79
0.97
0.047(0.0471)
0.085(0.0848)
0.085(0.0850)
0.085(0.0845)
0.085(0.0846)
0.92
1.32
0.059(0.0592)
0.102(0.1021)
0.102(0.1025)
0.102(0.1015)
0.102(0.1017)
Table 5. The Average Bias (AVB) and Root Mean Squared Errors (RMSEs) in parentheses for GWD parameter
using the R-K and Bayes methods with m = (n/2 and 3n/4) and k=(m/2 and 3m/4) at T=1.5 and for LINEX
loss function.
N
m
k
R-K
Method
Gamma Prior
Kernel Prior
SQEL
LNXL
SQEL
LNXL
20
10
5
0.59
0.79
0.97
0.050(0.0503)
0.105(0.1067)
0.088(0.0884)
0.102(0.1034)
0.087(0.0873)
0.92
1.32
0.070(0.0705)
0.146(0.1469)
0.114(0.1136)
0.137(0.1384)
0.110(0.1102)
1.11
0.79
0.97
0.050(0.0500)
0.116(0.1174)
0.088(0.0876)
0.109(0.1102)
0.086(0.0865)
0.92
1.32
0.064(0.0645)
0.144(0.1454)
0.107(0.1072)
0.133(0.1342)
0.105(0.1047)
8
0.59
0.79
0.97
0.049(0.0490)
0.099(0.1000)
0.087(0.0872)
0.097(0.0978)
0.086(0.0863)
0.92
1.32
0.064(0.0642)
0.124(0.1247)
0.108(0.1078)
0.120(0.1205)
0.106(0.1056)
1.11
0.79
0.97
0.048(0.0484)
0.101(0.1020)
0.086(0.0861)
0.098(0.0985)
0.085(0.0853)
0.92
1.32
0.061(0.0615)
0.125(0.1255)
0.104(0.1045)
0.119(0.1196)
0.103(0.1027)
15
8
0.59
0.79
0.97
0.049(0.0495)
0.101(0.1019)
0.088(0.0877)
0.099(0.0994)
0.087(0.0867)
0.92
1.32
0.067(0.0667)
0.133(0.1337)
0.110(0.1100)
0.127(0.1278)
0.107(0.1074)
1.11
0.79
0.97
0.049(0.0491)
0.108(0.1088)
0.087(0.0867)
0.103(0.1037)
0.086(0.0858)
0.92
1.32
0.063(0.0628)
0.133(0.1338)
0.106(0.1057)
0.125(0.1258)
0.104(0.1036)
11
0.59
0.79
0.97
0.047(0.0474)
0.092(0.0923)
0.086(0.0858)
0.091(0.0912)
0.085(0.0851)
0.92
1.32
0.061(0.0608)
0.112(0.1127)
0.105(0.1048)
0.110(0.1105)
0.103(0.1031)
1.11
0.79
0.97
0.047(0.0473)
0.094(0.0939)
0.085(0.0851)
0.092(0.0920)
0.084(0.0845)
0.92
1.32
0.060(0.0595)
0.113(0.1137)
0.103(0.1027)
0.110(0.1105)
0.101(0.1013)
40
20
10
0.59
0.79
0.97
0.051(0.0506)
0.093(0.0933)
0.089(0.0887)
0.092(0.0923)
0.088(0.0879)
0.92
1.32
0.068(0.0678)
0.119(0.1189)
0.111(0.1112)
0.116(0.1165)
0.109(0.1092)
1.11
0.79
0.97
0.049(0.0491)
0.095(0.0948)
0.087(0.0867)
0.093(0.0930)
0.086(0.0861)
0.92
1.32
0.063(0.0627)
0.117(0.1167)
0.106(0.1057)
0.114(0.1136)
0.104(0.1042)
15
0.59
0.79
0.97
0.050(0.0505)
0.093(0.0928)
0.089(0.0886)
0.092(0.0918)
0.088(0.0879)
0.92
1.32
0.066(0.0665)
0.115(0.1156)
0.110(0.1100)
0.114(0.1136)
0.108(0.1082)
1.11
0.79
0.97
0.049(0.0487)
0.092(0.0925)
0.086(0.0864)
0.091(0.0911)
0.086(0.0858)
0.92
1.32
0.062(0.0620)
0.114(0.1138)
0.105(0.1050)
0.111(0.1112)
0.104(0.1037)
30
15
0.59
0.79
0.97
0.050(0.0503)
0.092(0.0924)
0.088(0.0884)
0.091(0.0915)
0.088(0.0877)
0.92
1.32
0.068(0.0680)
0.119(0.1192)
0.111(0.1114)
0.117(0.1168)
0.109(0.1094)
1.11
0.79
0.97
0.049(0.0491)
0.095(0.0948)
0.087(0.0867)
0.093(0.0931)
0.086(0.0861)
0.92
1.32
0.063(0.0627)
0.116(0.1164)
0.106(0.1057)
0.113(0.1134)
0.104(0.1042)
23
0.59
0.79
0.97
0.048(0.0478)
0.086(0.0864)
0.086(0.0861)
0.086(0.0861)
0.086(0.0856)
0.92
1.32
0.061(0.0615)
0.105(0.1049)
0.105(0.1054)
0.104(0.1041)
0.104(0.1043)
1.11
0.79
0.97
0.047(0.0471)
0.086(0.0857)
0.085(0.0849)
0.085(0.0853)
0.084(0.0845)
0.92
1.32
0.059(0.0591)
0.103(0.1035)
0.102(0.1023)
0.103(0.1025)
0.101(0.1014)
60
30
15
0.59
0.79
0.97
0.050(0.0499)
0.089(0.0889)
0.088(0.0880)
0.088(0.0885)
0.087(0.0875)
0.92
1.32
0.067(0.0670)
0.113(0.1132)
0.110(0.1105)
0.112(0.1118)
0.109(0.1090)
1.11
0.79
0.97
0.049(0.0488)
0.090(0.0902)
0.086(0.0865)
0.089(0.0893)
0.086(0.0860)
0.92
1.32
0.062(0.0622)
0.110(0.1102)
0.105(0.1052)
0.109(0.1086)
0.104(0.1041)
23
0.59
0.79
0.97
0.050(0.0501)
0.089(0.0894)
0.088(0.0882)
0.089(0.0889)
0.088(0.0877)
0.92
1.32
0.066(0.0661)
0.112(0.1116)
0.110(0.1097)
0.110(0.1104)
0.108(0.1083)
1.11
0.79
0.97
0.049(0.0486)
0.089(0.0892)
0.086(0.0863)
0.088(0.0884)
0.086(0.0858)
0.92
1.32
0.062(0.0618)
0.109(0.1089)
0.105(0.1048)
0.107(0.1074)
0.104(0.1038)
45
23
0.59
0.79
0.97
0.049(0.0495)
0.088(0.0884)
0.088(0.0877)
0.088(0.0879)
0.087(0.0872)
0.92
1.32
0.065(0.0654)
0.110(0.1102)
0.109(0.1090)
0.109(0.1090)
0.108(0.1077)
1.11
0.79
0.97
0.048(0.0484)
0.088(0.0883)
0.086(0.0861)
0.088(0.0877)
0.086(0.0856)
0.92
1.32
0.061(0.0615)
0.108(0.1078)
0.105(0.1045)
0.106(0.1065)
0.104(0.1035)
34
0.59
0.79
0.97
0.048(0.0479)
0.085(0.0855)
0.086(0.0862)
0.085(0.0853)
0.086(0.0858)
0.92
1.32
0.062(0.0615)
0.104(0.1039)
0.105(0.1055)
0.103(0.1033)
0.105(0.1046)
1.11
0.79
0.97
0.047(0.0471)
0.085(0.0849)
0.085(0.0849)
0.085(0.0846)
0.085(0.0846)
0.92
1.32
0.059(0.0592)
0.102(0.1022)
0.102(0.1025)
0.102(0.1016)
0.102(0.1018)
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
14
Volume 5, 2023
Table 6. The Average Bias (AVB) and Root Mean Squared Errors (RMSEs) in parentheses for the GWD
parameter using the R-K and Bayes methods with m = (n/2 and 3n/4) and k=(m/2 and 3m/4) at T=0.75 and
for LINEX loss function.
N
m
k
R-K
Method
Gamma Prior
Kernel Prior
SQEL
LNXL
SQEL
LNXL
20
10
5
0.59
0.79
0.97
0.077(0.0772)
0.101(0.1015)
0.123(0.1233)
0.103(0.1030)
0.113(0.1127)
0.92
1.32
0.139(0.1392)
0.138(0.1435)
0.200(0.2000)
0.146(0.1464)
0.161(0.1608)
1.11
0.79
0.97
0.060(0.0602)
0.103(0.1032)
0.107(0.1074)
0.103(0.1028)
0.106(0.1057)
0.92
1.32
0.091(0.0906)
0.144(0.1446)
0.154(0.1540)
0.143(0.1431)
0.148(0.1480)
8
0.59
0.79
0.97
0.062(0.0616)
0.101(0.1011)
0.109(0.1086)
0.101(0.1011)
0.105(0.1051)
0.92
1.32
0.098(0.0979)
0.141(0.1412)
0.160(0.1605)
0.141(0.1407)
0.147(0.1473)
1.11
0.79
0.97
0.056(0.0561)
0.102(0.1016)
0.104(0.1035)
0.101(0.1012)
0.103(0.1028)
0.92
1.32
0.082(0.0818)
0.141(0.1411)
0.146(0.1456)
0.140(0.1400)
0.143(0.1428)
15
8
0.59
0.79
0.97
0.062(0.0616)
0.101(0.1011)
0.109(0.1086)
0.101(0.1010)
0.105(0.1051)
0.92
1.32
0.097(0.0975)
0.141(0.1415)
0.160(0.1602)
0.141(0.1407)
0.147(0.1471)
1.11
0.79
0.97
0.056(0.0561)
0.101(0.1014)
0.104(0.1035)
0.101(0.1011)
0.103(0.1028)
0.92
1.32
0.082(0.0818)
0.141(0.1412)
0.146(0.1457)
0.140(0.1401)
0.143(0.1429)
11
0.59
0.79
0.97
0.056(0.0556)
0.100(0.0999)
0.103(0.1031)
0.100(0.0997)
0.102(0.1018)
0.92
1.32
0.083(0.0833)
0.139(0.1388)
0.147(0.1471)
0.138(0.1379)
0.141(0.1415)
1.11
0.79
0.97
0.054(0.0541)
0.100(0.1004)
0.102(0.1017)
0.100(0.1002)
0.101(0.1013)
0.92
1.32
0.078(0.0776)
0.139(0.1387)
0.142(0.1417)
0.138(0.1380)
0.140(0.1401)
40
20
10
0.59
0.79
0.97
0.075(0.0755)
0.107(0.1070)
0.122(0.1218)
0.107(0.1066)
0.114(0.1144)
0.92
1.32
0.130(0.1298)
0.152(0.1540)
0.191(0.1910)
0.152(0.1521)
0.164(0.1643)
1.11
0.79
0.97
0.059(0.0592)
0.104(0.1040)
0.107(0.1065)
0.104(0.1036)
0.106(0.1055)
0.92
1.32
0.089(0.0886)
0.146(0.1461)
0.152(0.1521)
0.144(0.1445)
0.148(0.1484)
15
0.59
0.79
0.97
0.064(0.0642)
0.104(0.1042)
0.111(0.1113)
0.104(0.1037)
0.108(0.1080)
0.92
1.32
0.103(0.1033)
0.148(0.1479)
0.166(0.1659)
0.146(0.1458)
0.153(0.1532)
1.11
0.79
0.97
0.056(0.0565)
0.102(0.1022)
0.104(0.1039)
0.102(0.1019)
0.103(0.1033)
0.92
1.32
0.083(0.0827)
0.143(0.1427)
0.147(0.1465)
0.142(0.1417)
0.144(0.1443)
30
15
0.59
0.79
0.97
0.064(0.0641)
0.104(0.1043)
0.111(0.1111)
0.104(0.1037)
0.108(0.1078)
0.92
1.32
0.103(0.1033)
0.148(0.1480)
0.166(0.1659)
0.146(0.1459)
0.153(0.1532)
1.11
0.79
0.97
0.056(0.0565)
0.102(0.1023)
0.104(0.1039)
0.102(0.1020)
0.103(0.1034)
0.92
1.32
0.083(0.0827)
0.143(0.1426)
0.146(0.1465)
0.142(0.1416)
0.144(0.1443)
23
0.59
0.79
0.97
0.055(0.0551)
0.101(0.1006)
0.103(0.1029)
0.100(0.1004)
0.102(0.1020)
0.92
1.32
0.082(0.0820)
0.140(0.1404)
0.146(0.1463)
0.139(0.1394)
0.143(0.1427)
1.11
0.79
0.97
0.054(0.0537)
0.100(0.1003)
0.101(0.1014)
0.100(0.1002)
0.101(0.1011)
0.92
1.32
0.077(0.0767)
0.139(0.1388)
0.141(0.1410)
0.138(0.1383)
0.140(0.1401)
60
30
15
0.59
0.79
0.97
0.073(0.0728)
0.108(0.1086)
0.119(0.1192)
0.108(0.1076)
0.114(0.1139)
0.92
1.32
0.127(0.1268)
0.156(0.1573)
0.188(0.1882)
0.154(0.1544)
0.166(0.1664)
1.11
0.79
0.97
0.059(0.0590)
0.104(0.1040)
0.106(0.1063)
0.104(0.1037)
0.106(0.1055)
0.92
1.32
0.088(0.0880)
0.147(0.1466)
0.152(0.1515)
0.145(0.1452)
0.149(0.1486)
23
0.59
0.79
0.97
0.063(0.0635)
0.105(0.1048)
0.111(0.1105)
0.104(0.1042)
0.108(0.1080)
0.92
1.32
0.102(0.1018)
0.149(0.1497)
0.164(0.1645)
0.147(0.1472)
0.154(0.1543)
1.11
0.79
0.97
0.056(0.0563)
0.102(0.1022)
0.104(0.1038)
0.102(0.1020)
0.103(0.1033)
0.92
1.32
0.082(0.0824)
0.143(0.1428)
0.146(0.1462)
0.142(0.1419)
0.145(0.1445)
45
23
0.59
0.79
0.97
0.064(0.0636)
0.105(0.1049)
0.111(0.1106)
0.104(0.1042)
0.108(0.1080)
0.92
1.32
0.102(0.1020)
0.149(0.1490)
0.165(0.1647)
0.147(0.1469)
0.154(0.1544)
1.11
0.79
0.97
0.056(0.0564)
0.102(0.1022)
0.104(0.1038)
0.102(0.1020)
0.103(0.1033)
0.92
1.32
0.082(0.0824)
0.143(0.1428)
0.146(0.1462)
0.142(0.1419)
0.145(0.1445)
34
0.59
0.79
0.97
0.055(0.0554)
0.101(0.1010)
0.103(0.1032)
0.101(0.1007)
0.102(0.1024)
0.92
1.32
0.082(0.0825)
0.141(0.1413)
0.147(0.1468)
0.140(0.1403)
0.144(0.1437)
1.11
0.79
0.97
0.054(0.0538)
0.100(0.1004)
0.101(0.1015)
0.100(0.1003)
0.101(0.1013)
0.92
1.32
0.077(0.0770)
0.139(0.1390)
0.141(0.1412)
0.139(0.1386)
0.140(0.1404)
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
15
Volume 5, 2023
Table 7. The Average Bias (AVB) and Root Mean Squared Errors (RMSEs) in parentheses for the GWD
parameter using The R-K and Bayes methods with m = (n/2 and 3n/4) and k=(m/2 and 3m/4) at T=1.5 and
for LINEX loss function.
N
m
k
R-K
Method
Gamma Prior
Kernel Prior
SQEL
LNXL
SQEL
LNXL
20
10
5
0.59
0.79
0.97
0.061(0.0614)
0.101(0.1011)
0.108(0.1084)
0.101(0.1010)
0.105(0.1049)
0.92
1.32
0.106(0.1065)
0.140(0.1412)
0.169(0.1686)
0.142(0.1418)
0.150(0.1504)
1.11
0.79
0.97
0.057(0.0571)
0.102(0.1020)
0.104(0.1045)
0.102(0.1016)
0.104(0.1035)
0.92
1.32
0.084(0.0839)
0.142(0.1421)
0.148(0.1476)
0.141(0.1409)
0.144(0.1442)
8
0.59
0.79
0.97
0.059(0.0587)
0.100(0.1005)
0.106(0.1060)
0.100(0.1004)
0.104(0.1035)
0.92
1.32
0.091(0.0915)
0.140(0.1406)
0.155(0.1546)
0.140(0.1396)
0.145(0.1448)
1.11
0.79
0.97
0.055(0.0553)
0.101(0.1010)
0.103(0.1028)
0.101(0.1008)
0.102(0.1022)
0.92
1.32
0.080(0.0801)
0.140(0.1402)
0.144(0.1441)
0.139(0.1393)
0.142(0.1418)
15
8
0.59
0.79
0.97
0.060(0.0597)
0.101(0.1007)
0.107(0.1069)
0.101(0.1006)
0.104(0.1041)
0.92
1.32
0.097(0.0975)
0.141(0.1415)
0.160(0.1601)
0.141(0.1407)
0.147(0.1471)
1.11
0.79
0.97
0.056(0.0561)
0.101(0.1014)
0.104(0.1035)
0.101(0.1011)
0.103(0.1027)
0.92
1.32
0.082(0.0818)
0.141(0.1411)
0.146(0.1456)
0.140(0.1400)
0.143(0.1428)
11
0.59
0.79
0.97
0.056(0.0556)
0.100(0.0998)
0.103(0.1032)
0.100(0.0997)
0.102(0.1018)
0.92
1.32
0.083(0.0834)
0.139(0.1387)
0.147(0.1471)
0.138(0.1378)
0.141(0.1415)
1.11
0.79
0.97
0.054(0.0541)
0.100(0.1003)
0.102(0.1017)
0.100(0.1001)
0.101(0.1013)
0.92
1.32
0.078(0.0776)
0.139(0.1388)
0.142(0.1417)
0.138(0.1381)
0.140(0.1401)
40
20
10
0.59
0.79
0.97
0.061(0.0609)
0.103(0.1029)
0.108(0.1081)
0.102(0.1025)
0.106(0.1058)
0.92
1.32
0.099(0.0989)
0.147(0.1469)
0.162(0.1618)
0.145(0.1448)
0.151(0.1511)
1.11
0.79
0.97
0.056(0.0560)
0.102(0.1019)
0.103(0.1035)
0.102(0.1016)
0.103(0.1030)
0.92
1.32
0.082(0.0817)
0.142(0.1420)
0.146(0.1455)
0.141(0.1410)
0.144(0.1436)
15
0.59
0.79
0.97
0.061(0.0607)
0.103(0.1031)
0.108(0.1080)
0.103(0.1026)
0.106(0.1057)
0.92
1.32
0.095(0.0955)
0.145(0.1456)
0.159(0.1587)
0.144(0.1438)
0.150(0.1495)
1.11
0.79
0.97
0.056(0.0556)
0.102(0.1016)
0.103(0.1031)
0.101(0.1014)
0.103(0.1026)
0.92
1.32
0.081(0.0807)
0.141(0.1414)
0.145(0.1447)
0.141(0.1405)
0.143(0.1429)
30
15
0.59
0.79
0.97
0.060(0.0603)
0.103(0.1027)
0.108(0.1076)
0.102(0.1023)
0.105(0.1055)
0.92
1.32
0.099(0.0991)
0.146(0.1465)
0.162(0.1620)
0.145(0.1447)
0.151(0.1513)
1.11
0.79
0.97
0.056(0.0560)
0.102(0.1019)
0.103(0.1035)
0.102(0.1016)
0.103(0.1030)
0.92
1.32
0.082(0.0816)
0.142(0.1420)
0.146(0.1455)
0.141(0.1411)
0.144(0.1436)
23
0.59
0.79
0.97
0.055(0.0551)
0.101(0.1006)
0.103(0.1029)
0.100(0.1004)
0.102(0.1020)
0.92
1.32
0.082(0.0821)
0.140(0.1403)
0.146(0.1465)
0.139(0.1394)
0.143(0.1428)
1.11
0.79
0.97
0.054(0.0537)
0.100(0.1003)
0.101(0.1014)
0.100(0.1002)
0.101(0.1011)
0.92
1.32
0.077(0.0767)
0.139(0.1387)
0.141(0.1410)
0.138(0.1382)
0.140(0.1400)
60
30
15
0.59
0.79
0.97
0.059(0.0595)
0.103(0.1032)
0.107(0.1069)
0.103(0.1027)
0.105(0.1053)
0.92
1.32
0.097(0.0968)
0.147(0.1473)
0.160(0.1599)
0.145(0.1454)
0.152(0.1517)
1.11
0.79
0.97
0.056(0.0557)
0.102(0.1018)
0.103(0.1032)
0.102(0.1016)
0.103(0.1029)
0.92
1.32
0.081(0.0811)
0.142(0.1419)
0.145(0.1450)
0.141(0.1411)
0.144(0.1435)
23
0.59
0.79
0.97
0.060(0.0600)
0.103(0.1034)
0.107(0.1073)
0.103(0.1029)
0.106(0.1057)
0.92
1.32
0.094(0.0946)
0.147(0.1467)
0.158(0.1578)
0.145(0.1448)
0.150(0.1505)
1.11
0.79
0.97
0.055(0.0555)
0.102(0.1016)
0.103(0.1030)
0.101(0.1014)
0.103(0.1026)
0.92
1.32
0.080(0.0805)
0.142(0.1415)
0.144(0.1444)
0.141(0.1408)
0.143(0.1431)
45
23
0.59
0.79
0.97
0.059(0.0588)
0.103(0.1027)
0.106(0.1062)
0.102(0.1023)
0.105(0.1048)
0.92
1.32
0.093(0.0926)
0.146(0.1459)
0.156(0.1560)
0.144(0.1441)
0.149(0.1494)
1.11
0.79
0.97
0.055(0.0552)
0.101(0.1015)
0.103(0.1028)
0.101(0.1013)
0.102(0.1024)
0.92
1.32
0.080(0.0800)
0.141(0.1412)
0.144(0.1440)
0.141(0.1405)
0.143(0.1427)
34
0.59
0.79
0.97
0.055(0.0553)
0.101(0.1010)
0.103(0.1031)
0.101(0.1008)
0.102(0.1024)
0.92
1.32
0.082(0.0824)
0.141(0.1413)
0.147(0.1468)
0.140(0.1402)
0.144(0.1436)
1.11
0.79
0.97
0.054(0.0538)
0.100(0.1004)
0.101(0.1015)
0.100(0.1003)
0.101(0.1013)
0.92
1.32
0.077(0.0770)
0.139(0.1390)
0.141(0.1412)
0.139(0.1386)
0.140(0.1404)
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
16
Volume 5, 2023
Fig. 1: a) The Empirical CDF and the fitted CDF for the Vinyl Chloride Data.
b) The Histogram and the fitted PDF for the Vinyl Chloride Data.
Fig. 2: a) The Empirical CDF and the fitted CDF for the Leukemia Data.
b) The Histogram and the fitted PDF for the Leukemia Data.
0
0,3
0,6
0,9
1,2
0 2 4 6 8 10
F(x)
X-Values
a) Vinyl data
0
3
6
9
12
0 2 4 6 8 10
f(x)
X- Valus
b)- Vinyl data
0
0,3
0,6
0,9
1,2
0 1000 2000 3000
F(x)
X-Values
(a)-Leukemia data
0
1,5
3
4,5
0 1000 2000 3000
f(x)
X- Values
b)- Leukemia data
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2023.5.1
M. Maswadah
E-ISSN: 2766-9823
17
Volume 5, 2023
