Fuzzy System Reliability Analysis for Kumaraswamy Distribution:
Bayesian and Non-Bayesian Estimation with Simulation and an
Application on Cancer Data Set
YASSER S. ALHARBI1, AMR R. KAMEL2*
1Statistical Science, General Studies Department, Technical and Vocational Training Corporation,
Mecca, SAUDI ARABIA
2Department of Applied Statistics and Econometrics, Faculty of Graduate Studies for Statistical
Research (FGSSR), Cairo University, Giza 12613, EGYPT
2Data Processing and Tabulation at Central Agency for Public Mobilization and Statistics (CAPMAS),
Nasser City 2086, EGYPT
Abstract: - This paper proposes the fuzzy Bayesian (FB) estimation to get the best estimate of the unknown
parameters of a two-parameter Kumaraswamy distribution from a frequentist point of view. These estimations
of parameters are employed to estimate the fuzzy reliability function of the Kumaraswamy distribution and to
select the best estimate of the parameters and fuzzy reliability function. To achieve this goal we investigate the
efficiency of seven classical estimators and compare them with FB proposed estimation. Monte Carlo
simulations and cancer data set applications are performed to compare the performances of the estimators for
both small and large samples. Tierney and Kadane approximation is used to obtain FB estimates of traditional
and fuzzy reliability for the Kumaraswamy distribution. The results showed that the fuzziness is better than the
reality for all sample sizes and the fuzzy reliability at the estimates of the FB proposed estimated is better than
other estimators, it gives the lowest Bias and root mean squared error.
Key-Words: - Fuzzy Bayesian estimation, Fuzzy information system, Kumaraswamy Distribution, Maximum
likelihood estimator, Maximum product spacing estimator, Monte Carlo simulation, Reliability analysis,
R software.
Received: May 25, 2021. Revised: March 16, 2022. Accepted: April 16, 2022. Published: June 7, 2022.
1. Introduction
In probability and statistics, the Kumaraswamy
double bounded distribution is a family of
continuous probability distributions defined on the
interval. It is similar to the Beta distribution,
but much simpler to use especially in simulation
studies since its probability density function,
cumulative distribution function and quantile
functions can be expressed in closed form. The
behavior of both distributions is governed by two
shape and two boundary parameters. The
relationships between the distributions possible
shapes and the values of their shape parameters are
qualitatively identical, and both distributions are
special cases of McDonald’s [1] generalized Beta of
the first kind. Most importantly, these two
distributions are very flexible and can take
approximately the same shapes. This distribution
was originally proposed by Poondi Kumaraswamy
in 1980 for variables that are lower and upper
bounded with a zero-inflation. This was extended to
inflations at both extremes.
Over the last few years, there has been a great
interest in studying the Kumaraswamy distribution,
and mixing it with other well-known probability
models to achieve greater flexibility in modeling
several types of real data exhibiting various patterns.
Some of these recent developments in the
Kumaraswamy distribution are: Barreto-Souza and
Lemonte [2] introduced a bivariate Kumaraswamy
distribution for which the marginal distributions are
univariate Kumaraswamy laws. Ghosh and
Nadarajah [3] studied in details, Bayesian inference
for Kumaraswamy distribution based on censored
samples. Al-Fattah et al. [4] introduced the inverted
Kumaraswamy distribution with two positive shape
parameters. Aly et al. [5] studied the estimation of
the bivariate Kumaraswamy lifetime distribution
under progressive Type-I censoring. Sagrillo et al.
[6] discussed new modified Kumaraswamy
distributions for double bounded hydro-
environmental data. Finally, Mohammed et al. [7]
introduced bivariate Kumaraswamy distribution
based on conditional hazard functions.
Moreover, the maximum likelihood (ML) and
method of moments (MM) estimation methods are
known to be traditional estimation methods. Even
though ML is efficient and has good theoretical
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features, there is evidence that it does not perform
effectively, particularly for small samples. The MM
approach is simple to use and frequently produces
explicit forms for unknown parameter estimators. In
other circumstances, however, the MM does not
provide explicit estimators. As a result, different
estimating approaches have been offered in the
literature as alternatives to traditional methods. The
L-moments (LM), ordinary least squares (OLS) and
maximum Product Spacing (MPS) estimators are
frequently recommended among them. These
approaches, in general, do not have good theoretical
properties, but they can yield better estimates of
unknown parameters in specific instances than the
ML and the MM when the data set does not contain
extreme observations. These Traditional techniques
are ineffective when the data set comprises extreme
observations. The parameters must be estimated
using a robust estimator. Many papers in several
models explore many robust estimators; see e.g.
[8-12].
On the other hand, the occurrence of fuzzy
random variable makes the combination of
randomness and fuzziness more persuasive, since
the probability theory can be used to model
uncertainty and the fuzzy sets theory can be used to
model imprecision. In the fuzzy reliability analysis,
the survivors are sometimes unable to be reported
accurately due to unforeseen circumstances. For
example, the item may not have failed fully during
the test, or some failed items may have been
recorded incorrectly due to human error. It claims
that the survival probability cannot be calculated
precisely. As a result, in this case, the survival
probability should be treated as a fuzzy real number.
However, one of the most essential and
successful strategies for evaluating the work of
systems or units is reliability. It is the function that
determines the likelihood of a unit or vehicle
operating without failure for a given period of time.
In its traditional form, many approaches and models
in reliability theory assume that all of the parameters
of the life-time probability function are crisp. In
real-world applications, it is required to
generalize traditional real-number statistical
estimation methods to fuzzy numbers. This is
because, due to flaws in experience, personal
judgment, estimation, or unanticipated conditions,
the parameters of a probability distribution can
occasionally be inaccurately reported. The
parameters in the life distributions are then
ambiguous. As a result, dealing with the function of
traditional dependability may prove problematic for
the system of reliability. As a result, we can deal
with a broader word than the standard concept of
dependability. Zadeh [13] introduced fuzzy logic in
1968, when he used the phrase “fuzzy variables” to
describe approximate or erroneous linguistic
expressions and language. This is the first book to
lay the groundwork for the theory of fuzzy sets. A
fuzzy set is a collection of objects or elements with
varying degrees of membership. The function of
membership to each object in the set distinguishes
them. The degree of membership is usually
somewhere between zero and one.
In recent years, numerous papers on
generalization of classical statistical methods to
analysis of fuzzy data have appeared in the
literature. Wu [14] calculated fuzzy dependability in
the fuzzy environment using the Bayes approach,
assuming a fuzzy treatment of fuzzy variables with
preceding fuzzy distributions. By incorporating the
notion of resolution identity and determining the
degree of membership to any Bayes estimate of
reliability, the classic Bayes estimation method was
applied to develop the fuzzy estimator of reliability.
In 2006, Huang and Zuo [15] looked on the basis
reliability of fuzzy life data. By assuming a new
approach to identify the function of membership
estimation and the reliability function of multi-
parameter life distributions, the Bayes method was
used to estimate fuzzy reliability based on the size
of a small sample.
Pak [16] used the Lindley distribution with one
parameter when the data was available in a fuzzy
data format, using ML estimation and Bayes
estimation by EM-algorithm to determine the MLE
of the parameter and establish confidence limits
using the maximum potential estimator's asymptotic
normality. The researcher determined from the
Monte Carlo investigation that Bayes estimates
based on prior non-informative as well as maximum
likelihood estimates provided identical estimation
outcomes. The Bayes estimate has the lowest
average error squares in the case of previous
information.
The main objective of this paper is to propose the
fuzzy Bayesian (FB) estimation method to get the
best estimate of the unknown two-parameter
Kumaraswamy distribution and reliability function.
To achieve this goal we investigate the efficiency of
seven classical estimators and compare them with
FB proposed estimation. Simulations are used to
compare the performance as it is not possible to
compare all estimators theoretically and also to
determine which method is more efficient according
to the Bias and root mean square error (RMSE). The
uniqueness of this study comes from the fact that
thus far no attempt has been made to compare all
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these estimators for the two-parameter
Kumaraswamy distribution.
The rest of this paper is organized as follows:
After this introduction, Section 2 presents the
Kumaraswamy distribution, definition and
properties. While Section 3, provides fuzzy systems,
basic definitions and fuzzy reliability analysis. In
Section 4, we discuss the seven classical point
estimation methods for the unknown parameters.
The fuzzy Bayesian estimation analyses are
provided in Section 5. A Monte Carlo simulation
study and caner real data application are presented
in Section 6, which provides a comparison of all
estimation procedures developed in this paper.
Finally, conclusions appear in Section 7.
2. The Kumaraswamy Distribution:
Definition and Properties
Kumaraswamy [17] introduced a two parameter
absolutely continuous distribution which compares
extremely favorably, in terms of simplicity, with the
beta distribution. In its general form, the probability
density function of the continuous part of the
distribution Kumaraswamy introduced in his 1980
article can be written as;
(1)
with shape parameters and , and
boundary parameters and . The general form of
the distribution will be denoted by .
Making the transformation
and using the
change of variable theorem, we obtain the standard
form of the Kumaraswamy density function. The
cumulative distribution function (CDF) and the
corresponding probability density function (PDF)
can be expressed as;
(2)
(3)
Which will be denoted by
. In what follows the standard form of
the distribution will be employed unless otherwise
indicated.
For simplicity, we denote Kumaraswamy
distribution with two positive parameters and as
. Based on varying values of and , there
are similar shape properties between Kumaraswamy
distribution and Beta distribution. However, the
former is superior to the latter in some respects:
there are not any special functions involved in
and its quantile function; the generation of
random variables is simple, as L-moments and
moments of order statistics for have simple
formulars. For the PDF of , as shown on the
Figure 1 , when and , it is unimodal;
when and , it is increasing; when
and , it is decreasing; when and ,
it is uniantimodal; when , it is constant.
The CDF of as shown on the Figure 2, has
an explicit expression, while the CDF of the Beta
distribution appears in an integral form. Therefore,
Kumaraswamy distribution is considered as a
substitutive model for Beta distribution in practical
terms.
Fig. 1 The PDF plots of the Kumaraswamy
distribution for different parameter values.
From Eq. (2), it immediately follows that the
quantile function is also available in closed-
form:
(4)
In particular, the median of the Kumaraswamy
distribution can be written as;
(5)
Therefore, the reliability and hazard functions at
an arbitrary time for the Kumaraswamy
distribution are given by;
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(6)
and:
(7)
respectively.
If the random variable is distributed
its moments around zero can be expressed as;
(8)
where
is the Beta function and is
the Gamma function. Thus, the expectation and
variance of are;
Fig. 2 The CDF plots of the Kumaraswamy
distribution for different parameter values.
Many natural phenomena with lower and upper
boundaries, such as individual heights, test scores,
atmospheric temperatures, hydrological data,
economic data (such as unemployment data), etc.,
are suitable to this distribution. Despite its
adaptability, this distribution has received little
statistical attention. However, recently, the genesis
and the basic properties of the Kumaraswamy
distribution were studied by Jones [18]. He noted
that while this distribution has many of the same
properties as the Beta distribution, it has some
advantages in terms of tractability: its quantile
function is simple and does not require any special
functions, random variate generation is simple, L-
moments and moments of order statistics have
simple formulae and statistical meanings for the
parameters, and so on.
Moreover, this distribution has a close relation
with Beta and generalized Beta (first kind) listed
below:
If then .
If then .
If then .
where stands for the generalized Beta
distribution of the first kind.
Nadarajah [19] stated that the Kumaraswamy
distribution can capture the shape of several well-
known distributions such as the uniform
distribution, triangular distribution, or practically
any single modal distribution depending on the
choice of the two shape parameters. The
Kumaraswamy distribution is a specific instance of
a three-parameter Beta distribution.
3. Fuzzy Systems
In the following, at first, we consider the
fundamental notation and some basic definitions of
fuzzy set theory which will be frequently used in
this paper. Consider an experiment characterized by
a probability space , where is
a Borel measurable space and belongs to a
specified family of probability measures
on Assume that the observer
cannot distinguish or transmit with exactness the
outcome in the performance of , but that rather the
available observation may be described in terms of
fuzzy information which is defined as follows, For
details on this topic, see [20].
3.1 Basic Definitions
Definition 3.1: A fuzzy event on , characterized
by a Borel measurable membership function
from to , where represents the “grade
of membership” of to , is called fuzzy
information associated with the experiment . The
set consisting of all observable events from the
experiment determines a fuzzy information
system associated with it, which is defined as
follows.
Definition 3.2: A fuzzy information system
(henceforth, in short FIS) associated with the
experiment is a fuzzy partition with fuzzy events
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on , that is a finite set of fuzzy events on
satisfying the orthogonality condition
for all . Alternatively, according to Zadeh
, given the experiment
and a FIS associated with it, each probability
measure on , induces a probability
measure on defined as follows.
Definition 3.3: The probability distribution on
induced by is the mapping from to
such that;
for . In particular, the conditional density of a
continuous random variable with PDF given
the fuzzy event can be defined as;
For more details about the membership functions
and probability measures of fuzzy sets, one can refer
to Pak et al. [21].
Definition 3.4: A fuzzy number is a subset, denoted
by , of the set of real numbers (denoted by ) and
is characterized by the so called membership
function , satisfying the following constraints:
(i) is Borel measurable.
(ii) For every .
(iii) The usual -cuts , defined as
, are all
closed interval, i.e.,
Some widely known examples of membership
functions to characterize fuzzy numbers are
triangular and trapezoidal fuzzy numbers. For
example, triangular fuzzy number is defined as
with the corresponding membership
function;
Similarly, a trapezoidal fuzzy number can be
defined as with the corresponding
membership function.
Let us again revisit the example as mentioned
earlier in the context of life length of an electric
bulb.
3.2 Fuzzy Reliability Analysis
Reliability was defined as the probability that the
unit or device will remain valid after a period of
time on use. If is a continuous random
variable,, the reliability function is:
Its properties are:
.
.
Now, we can say that the fuzzy reliability
represents the probability of the unit performing the
work required. It is with varying degrees of success
for a specified period of time under normal
conditions and symbolized by, which is a function
in the fuzzy set :
, While then;
We will assume that the values of the random
variable are fuzzy number,
So; the fuzziness is a real
triangular fuzzy number, and:
where; .
If the random variable has a traditional
fractional distribution with , the
correspondingof random variable will have
variable. For each , the
cumulative fuzzy distribution function is;
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(9)
Then, the fuzzy reliability function is:
(10)
4. Classical Point Estimation Methods
In this section, we describe some methods for
estimating the parameters and of the
Kumaraswamy distribution. We assume throughout
that is a random sample of size
from the Kumaraswamy distribution with both
parameters unknown. We let
denote the associated order statistics. The
parameters estimations of the Kumaraswamy
distribution is investigated using seven estimations
methods, namely, the maximum likelihood (ML),
maximum product spacing (MPS), method of
moments (MM), probability weighted moment
(PWM), ordinary least squares (OLS), weighted
least squares (WLS) and Cramér–von Mises (CVM)
will be discussed in details.
4. 1 Maximum Likelihood Estimation
The method of maximum likelihood (ML)
estimation is the most frequently used method of
parameter estimation. Its success stems from its
many desirable properties including consistency,
asymptotic efficiency, invariance property as well as
its intuitive appeal. For the random sample
, the log-likelihood function can be
written as:
. (11)
The ML estimations, and, can be
obtained by maximizing Eq. (11) with respect to
and . The partial derivatives,
and
, are;
Setting these to zero, we obtain the ML
estimation is the solution of:
(12)
The ML estimation , that is:
(13)
In determining the estimation in equations (12)
and (13) of Kumaraswamy distribution by the ML
estimation method which cannot be solved
analytically, this can be solved by numerical
iteration method that is Newton Raphson’s method.
Newton Raphson Algorithm
The steps in Newton Raphson’s (NR) algorithm are:
1. Determining the starting value.
2. Determining the first derivative and the second
derivative of i.e. :
3. Defining as the first gradient vector and
derivative vector of its parameters:
4. Next defining the Hessian matrix where the
Hessian Matrix or second derivative matrix to
its parameters, denoted by are:
5. Iteration will stop when
,
where is the specified error limit.
4.2 Maximum Product of Spacing Estimation
In statistics, maximum product spacing (MPS)
estimator is a method for estimating the parameters
of univariate statistical models. The method requires
maximization of the geometric mean of spacings in
the data, which are the differences between the
values of the cumulative distribution function at
neighbouring data points. One of the most common
methods for estimating the parameters of a
distribution from data, the method of ML estimates,
can break down in various cases, such as involving
certain mixtures of continuous distributions. In these
cases the method of MPS method may be
successful. The MPS method chooses the parameter
values that make the observed data as uniform as
possible, according to a specific quantitative
measure of uniformity, see [22].
The uniform spacing of the random sample
can be defined as:
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;
Clearly,
.
Following Cheng and Amin [22], the MPS
estimation, and , are the values of and
maximizing the geometric mean of the spacing:
or, equivalently, maximizing the function:
The estimators, and, can also be
obtained by solving:
where;
(14)
and;
(15)
Cheng and Amin [22] showed that maximizing
as a method of parameter estimation is as efficient
as ML estimation and that the MPS estimators are
consistent under more general conditions than ML
estimators.
4.3 Method of Moments Estimation
The method of moments (MM) estimation can be
obtained by equating the mean and variance of Eq.
(3) to their sample counterparts, that is;
,
where and are the sample mean and sample
variance, respectively.
4.4 Probability Weighted Moment
Estimation
To estimate the parameters of Kumaraswamy
distribution we have to find the inverse
function of the cumulative distribution function,
while the inverse function as is obtained as follows:
Next to search for probability weighted moment
(PWM) estimation of Kumaraswamy distribution by
searching for the-r moment is as follows:
So the obtained PWM form to estimate
parameter from Kumaraswamy distribution
is:
After obtaining the-r moment then the next
step is to determine the estimators for parameters
and
respectively;
Moreover, after obtaining the respective
estimators from the Kumaraswamy distribution, the
next step is to examine the unbiased characteristic
for estimator a and b with the definition of unbiased
are , and , respectively.
Thus, and are unbiased estimator for and.
4.5 Ordinary Least Squares Estimation
Swain et al. [23] introduced the ordinary least
squares estimator (OLS) method is used to estimate
the parameters of Beta distribution.
Let be the order statistics of
a sample from the Kumaraswamy distribution and
then the OLS of and can be obtained by
minimizing the following function with respect
toand :
with respect to and , where is given by Eq.
(2). Equivalently, they can be obtained by solving:
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where and are given by
equations (14) and (15), respectively.
4.6 Weighted Least Squares Estimation
Swain et al. [23] introduced the weighted least
square (WLS) estimators. We use the WLS
procedure for estimating the parameters of
the Kumaraswamy distribution. The WLS,
and, can be obtained by minimizing the
following function;
These estimators can also be obtained by
solving:
where and are given by
equations (14) and (15), respectively.
4.7 Cramér–von-Mises Estimation
Cramér–von Mises type minimum distance
estimators are based on minimizing the distance
between the theoretical and empirical cumulative
distribution functions. Choi and Bulgren [24]
provided empirical evidence that the bias of these
estimators is smaller than the bias of other minimum
distance estimators. The Cramér–von Mises (CVM)
estimator, and, are the values of and
minimizing the following function;
The estimators can also be obtained by solving:
where and are given by
equations (14) and (15), respectively.
5. Fuzzy Bayesian Estimation
Fuzzy Bayesian (FB) approach has been adopted to
enhance the probability updating process with fuzzy
evidences by utilizing the conditional probability
densities and the membership functions of the
evidence's values. This approach has been widely
applied in structural reliability to access the safety
of the constructed projects, see [14].
This technique can be used to calculate the
likelihood probability and posterior probability for a
given fuzzy value using the likelihood density
function. However, determining the likelihood
density function is difficult. The density functions
are approximated as a certain sort of distribution,
such as Guassian or Weibull, in many of its
applications, and the parameters of the
approximated distributions are calculated using
laboratory tests and statistical methodology. This
estimation is complicated and time consuming.
Furthermore, once we know the likelihood density
function for the continuous valued data, determining
the likelihood probability for the fuzzy valued
evidence is a waste of time. The likelihood density
function for continuous valued evidence should be
calculated using the likelihood probability for fuzzy
valued data.
The Bayes theorem can be expressed as the
conditional distribution of given is given as;
where is the likelihood function of the
distribution, is the prior probability
distribution for the parameter and is the
posterior probability distribution. The initial stage
needs to be done is to determine the prior
distribution. Prior Gamma distribution is used as a
conjugate prior distribution for , see [25]. Suppose
random variable with following prior density
function;
or it can be written
Note that is Gamma distribution with
parameter and . By using the method of
moments will be obtained parameter values as
follows;
and;
.
In the Bayesian estimation unknown parameter is
assumed to behave as random variable with
distribution commonly known as prior probability
distribution. Here, we consider the following
independent gamma priors for all the parameters
and given as follows:
Prior for.
Prior for.
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We make no claims that these hyperparameter
selections are optimum or uniformly best in all
cases. However, we found this to be a fair option in
all of the simulations/examples we explored. Of
course, there could be more. By combining Eq. (11)
with the above set of independent priors, the joint
density functions of the data and the parameters
and becomes;
(16)
Therefore, the marginal posterior density
functions of and respectively given the data
can be obtained as;
Note that the FB estimate of any function of,
say , under squared error loss function is the
posterior mean which is given by;
(17)
and similarly for the other parameter as well.
However, the Equations (16) and (17) are not
available in analytically tractable and closed nice
form due to the complex form of the likelihood
function. Therefore, we use Tierney and Kadane
approximation as well as Markov Chain Monte
Carlo (MCMC) technique for computing the FB
estimate of and, see [26].
First, we rewrite the expression in Eq. (16) as
(for both the parameters and respectively);
(18)
and;
(19)
where;
,
and;
Tierney and Kadane [26] applied Laplaces
method to produce an approximation of Eq. (22) as
follows:
(20)
where and maximize and ,
respectively, and and are minus of the inverse
of the second derivatives of and at
and respectively.
Similar operation will be assumed for the other
parameter as well. Next, we apply this
approximation to obtain the FB estimate of the
parameter. Setting, we have;
(21)
and;
(22)
On substitution of equations (21) and (22) in Eq.
(20), one can obtain the FB estimate of under
squared error loss function (SELF). Similar
approach can also be made to obtain the FB estimate
of under SELF.
6 Numerical Studies
6.1 Simulation Study
In this section, simulation study are conducted to
compare the performances of the different
estimators of unknown parameters and
reliability function for Kumaraswamy distribution
and to illustrate the effect of the estimation method
of the reliability function. Our main objective is to
compare the performances of the seven classical
estimation methods and the proposed method FB
estimates of the unknown parameters and reliability
function for various sample sizes and parameter
values.
All the computations are performed in an R-
programming environment “version 4.1.2”. App For
simulation purposes, we have considered
and
. For each combination of
and , we simulated samples each of
size for each simulation experiment to obtain
homogenous in estimating the reliability function of
the Kumaraswamy distribution. The inversion
method was used. For each, we have generated
random sample from the Kumaraswamy distribution
with different parameter values. Then, using the
method as proposed by Pak et al. [27], each
realization of the generated samples was fuzzified
by employing FIS. The estimates of the parameters
and for the fuzzy sample were computed using
the maximum likelihood method and under the
Bayesian paradigm (with independent priors set up).
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For initial choices of the parameters
required for the ML method, we have taken values
that are wide apart from the actual values of the
parameters. For computing the FB estimates, we
have assumed that both and have independent
gamma priors with specific choices of the
hyperparameters (described earlier). The generation
of a variable follows a uniform distribution
using the rand term. Generate fuzzy
data following the Kumaraswamy distribution by
inverse transformation method using the following
formula:
(23)
The sample is represented by vector of
Kumaraswamy distribution. The -sample vector is
converted to fuzzy using the fuzzy hypothetical
information system as in Figure 3, corresponding to
the following membership functions:
Fig. 3 The FIS hypothetical used in simulating
simulation data.
The estimates of and reliability function for
after the creation of the randomized fuzzy values
of the CDF function according to the size of the
given samples and the default values of initial
parameters according to the formula, the
values of and the initial parameters were
computed according to the functions of the
for each fuzzy unit . Then, extract for each
and find expectation of as follows:
(24)
The parameters and fuzzy reliability
function were estimated by each of the seven
methods and FB estimates for each of the simulated
samples. We compare the performances of the ML,
the MPS, the MM, the PWM, the OLS, the WLS,
the CVM and the FB estimates in terms of Bias and
root mean squared error (RMSE) criteria defined by:
,
,
,
and;
,
,
.
Where , and is the estimated of , and
respectively, at experiment of
Monte Carlo experiments.
The results of the simulation study have been
provided in Appendix A, Tables (A.1-A.5). The
following concluding remakes are noticed based on
these results as follows;
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1. All the estimates reveal the property of
consistency, i.e., the Biases and the RMSE of
, and
always decreased as increased
2. The MPS method has more relative efficiency
than ML, PWM and OLS for most parameters
of Kumaraswamy distribution in all tables.
3. The FB estimation method exceeds the
estimate of the seven classical methods are
used. The fuzzy reliability was estimated using
FB method with the lowest Bias and the least
RMSE.
4. In the FB method, when sample size is larger,
the Bias and the RMSE are reduced to as little
as the sample size is 200.
5. The Biases and the RMSE of and always
appeared smallest for the MPS, WLS and
CVM methods.
6. The parameter estimated by FB method and
MPS and CVM methods with the default
values as well as the fuzzy reliability
converges from the default reliability as the
size of the sample increases.
7. FB estimation method has achieved the lowest
value of the seven classical estimation
methods. This indicates that the duration data
of linear accelerator is more consistent with
the fuzzy Kumaraswamy distribution when
estimating the parameters of this distribution
in Bayes.
6.2 An Application to Cancer Data Set
Cancer is a deadly disease that spreads quickly.
Millions of people are affected with this disease
each year. This lethal disease is a major focus of
scientific research and new breakthroughs in the
field of treatment. As a result of these efforts, new
technology gadgets have been developed that
provide new sources of cancer detection and
therapy. As a result, the significance of gadgets that
show disease has to be addressed. One of the most
important is the linear accelerator device, which is a
modern and advanced device in the detection of
cancer and radiation treatment. These data are taken
from a cancer study described by [28].
6.2.1 Linear Accelerator
The linear accelerator device is one of the most
advanced and cutting-edge devices for detecting and
killing cancer cells using radiation. In the following
situations, this gadget is used:
1. To treat cancer by destroying cancer cells.
2. Control cancer by preventing cancer cells
from growing and spreading.
3. Relieving cancer symptoms such as pain.
The device is one of the most modern medical
devices used to treat tumors in the Babylonian
center for tumor therapy. The center only operates
one piece of equipment for a linear accelerator that
provides continuous service to citizens. However, if
the first device fails or stops working for technical
reasons, the second device is used to offer ongoing
service. However, it should be emphasized that
there is no reliable tracking of the device's operating
and halting times. In the event that the gadget
malfunctions, the operators must have a precise
understanding of the operation periods and holidays.
For example, the device operator must inform the
center's management and the hospital’s management
orally. In turn, management must contact the
device's manufacturer to arrange for the devices
imprecision in recording operating times and
holidays to be repaired. As a result, data on the
linear accelerator’s operation time are fuzzy integers
that belong to fuzzy times with varying degrees of
membership, as shown in [29].
Approximate information was obtained on the
length of operation of the equipment until the work
of the specialists in charge of the device, the
supervising engineers and the administration of the
center. These times were arranged in Table 1,
measured in months for the period from the
beginning of installation of the equipment at the
center.
Table 1
Cancer data set extend the speed of the linear
accelerator system until it stops working in months
1.3, 1.3, 1.4, 1.6, 1.7, 1.7, 1.8, 1.8, 1.9, 2, 2, 2, 2,
2.1, 2.2, 2.2, 2.4, 2.4, 2.4, 2.5, 2.6, 2.8, 3, 3, 3.1, 3.3,
3.3, 3.3, 3.4, 3.5, 3.5, 3.6, 3.7, 3.8, 3.8, 3.9, 4.1, 4.2,
4.5, 4.5, 4.6, 5.1, 5.2, 5.3, 5.5, 5.8, 5.8, 5.9, 6.1, 6.3,
6.5, 6.6, 6.8, 7.7, 8.5, 10.4, 10.6, 10.6, 10.9, 11.4,
12.6, 14, 17.7
6.2.2 Data Fuzzification
The real sample vector was converted to mist
using the FIS as in Figure 4, corresponding to the
following functions;
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Fig. 4 FIS used to process cancer data set.
We have used the relationship between the
arithmetic mean and the variance for the distribution
of the parameter matrix. This is to obtain the initial
value of the algorithms used in the estimation of the
parameters by solving the non-linear equations and
using the NR algorithm. In Table 2, we computed
the Kolmogorov-Smirnov (K-S) distance between
the empirical and the fitted Kumaraswamy
functions.
Table 2
Goodness of-fit for cancer data set
D
0.0648
P-value
0.7513
The distance (D) between the fitted and the
empirical distribution functions for the data is
0.0648 and the corresponding p-value is 0.7513.
Therefore, it indicates that Kumaraswamy
distribution can be fitted to the data set, by use
empirical cumulative distribution function (ecdf) to
obtain a data graph to confirm the accuracy of the
K-S test as Figure 5.
Fig. 5 Kolmogorov-Smirnov test and the plot of
max distance between two ecdf Curves.
The selection of models for specific data is one
of the basic tasks of the scientific study in choosing
a predictive model from a group of candidate
models. Several statistical methods are available to
determine the best method of estimation, where the
most widely used are the Akaike information
criterion (AIC), the correct Akaike information
criterion (CAIC), Bayesian information criterion
(BIC) and Hannan-Quinn information criterion
(HQIC). However, the better estimation method
corresponds to the smaller values of AIC, CAIC,
BIC and HQIC. These methods are determined
according to the following formulas respectively.
The AIC is evaluated as follows:
AIC, (25)
The CAIC is: CAIC
, (26)
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The BIC is given by:
BIC, (27)
The HQIC is:
HQIC. (28)
where is the ML estimation log-likelihood
function value, is parameters count in the
distribution in the proposed distribution , and is
considered as the size of the sample used in
calculations.
Table 3
Parameter estimates and goodness-of-fit measures for cancer data set
Methods
Estimate
Goodness-of-Fit Measures
AIC
BIC
CAIC
HQIC
ML
31.3795
1.9606
0.7165
243.6362
284.2593
257.3357
228.175
MPS
24.1267
1.8244
0.4678
158.7529
178.1246
191.8282
156.247
MM
34.0458
2.0133
0.8305
173.0514
194.3498
184.0239
168.398
PWM
34.3898
1.9509
0.5224
172.6098
139.2743
182.3851
177.436
OLS
27.8196
1.9587
0.9187
241.3962
209.0632
253.1743
219.513
WLS
29.2665
1.9628
0.6153
229.7521
238.6071
233.0747
252.8726
CVM
32.4227
2.0424
0.4321
151.6935
131.3964
172.7658
167.409
FB
22.4979
1.8102
0.2452
132.0517
117.67383
160.9147
145.5824
Table 3, summarizes the estimates of the
methods of Kumaraswamy distribution parameters,
reliability function and the rate of uncertainty
function and the values of goodness-of-fit measures
for cancer data set. We note that from the results in
Table 3, the four varieties of goodness-of-fit
measures for FB estimation method have achieved
the lowest value of the seven classical estimation
methods. This indicates that the duration data of the
linear accelerator is more consistent with the fuzzy
Kumaraswamy distribution when estimating the
parameters of this distribution in FB proposed
estimation method.
7. Conclusions
In this paper, we have discussed several estimation
procedures for the Kumaraswamy distribution. In
particular, we have discussed seven classical
estimation methods and propose a fuzzy Bayesian
procedure to estimate the unknown parameters and
fuzzy reliability function. The seven classical
methods are; maximum likelihood estimation,
maximum product spacing estimation, method of
moments, probability-weighted moment, ordinary
least squares, weighted least squares and Cramér–
von Mises estimation. It is not feasible to compare
these methods theoretically. We have performed an
extensive simulation study to compare these
methods. R software is used to perform this study,
see Appendix B. We have also compared estimators
by cancer data set application; the results show that
the four varieties of goodness-of-fit measures for the
FB estimation method have achieved the lowest
value of the seven classical estimation methods. In
terms of overall comparison (with respect to Bias
and RMSE) the performance of the proposed FB
estimates is generally the best.
Acknowledgement
The authors wish to thank the editor. We also
thank anonymous for their encouragement and
support. The authors are grateful to anyone who
reviewed the paper carefully and for their helpful
comments that improve this paper.
Conflicts of Interest
The authors declare that they have no conflicts of
interest regarding the publication of this paper.
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Authors Biographies
Yasser S. Alharbi
bachelor’s degree in
Mathematics, Umm Al
Qura University,
Makkah and Master
degree in Data science,
LaTrbe University,
Melbourn. His main
research interests are
data science,
mathematical
statistics, neural
networks,
multidimensional systems and algorithms and
applications mappings.
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/dee
d.en_US
Amr R. Kamel is a
Ph.D. student at the
Faculty of Graduate
Studies for Statistical
Research (FGSSR),
Cairo University,
Egypt. He got a
bachelor’s degree in
Statistics in 2015, from
the Faculty of
Commerce at Al-Azhar
University (Egypt). He earned the M.Sc. degree in
Statistics in April 2021 from the Faculty of
Graduate Studies for Statistical Research-Cairo
University. He works as an assistant teacher in some
academies at the Department of Mathematical
Statistics. Also, He works Statistician, Data
Processing and Tabulation at Central Agency for
Public Mobilization and Statistics (CAPMAS) in
Egypt. He worked on Multidimensional poverty
analysis and measurement from the HIECS survey
(2019/2020). He has delivered lectures as a series of
Webinars on selected SDG Indicators for the Arab
Region: SDG 3.d.1. (April 2020). Also, Series of
SDG Webinars for the Arab Region in the World
Bank (July 2021). He has been serves as a reviewer
for many highly-respected international journals in
the frame of mathematical statistics, applied
statistics, applied mathematics, and econometrics.
His main research interests are systems of
regression equations, fuzzy systems, multivariate
regression models, robust regression, probability
and distribution theory, statistical inference and
prediction, Bayesian statistics and R statistical
package.
E-mail: amr_ragab@pg.cu.edu.eg
Orcid Id: https://orcid.org/0000-0002-3451-1517
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Appendices
Appendix A: Simulation Results Table A.1
Bias and RMSE values of the parameters and reliability for and
Estimate
ML
MPS
MM
PWM
OLS
WLS
CVM
FB
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
50
0.2089
0.2382
0.1589
0.1688
0.1559
0.198
0.2038
0.1520
0.1524
0.1042
0.1936
0.1159
0.1252
0.1071
0.0963
0.0816
0.1438
0.1699
0.0702
0.0609
0.1293
0.1243
0.0766
0.0801
0.0907
0.0959
0.1038
0.1092
0.0858
0.0906
0.0742
0.0659
0.1398
0.1650
0.1292
0.1365
0.1522
0.1784
0.1783
0.2029
0.1227
0.1304
0.1728
0.2085
0.1123
0.1210
0.1070
0.0903
75
0.1652
0.1884
0.0762
0.0646
0.1257
0.1335
0.1233
0.1566
0.1531
0.0917
0.1612
0.1202
0.0990
0.0847
0.1205
0.0824
0.1138
0.1344
0.0678
0.0716
0.0555
0.0474
0.0821
0.0864
0.0870
0.0983
0.0718
0.0758
0.0587
0.0521
0.0606
0.0633
0.1411
0.1605
0.1367
0.1586
0.1204
0.1411
0.1106
0.1305
0.0888
0.0957
0.1022
0.1080
0.0970
0.1032
0.0846
0.0714
100
0.0994
0.1056
0.0953
0.0652
0.1307
0.1491
0.1275
0.0951
0.0975
0.1239
0.0783
0.0670
0.1211
0.0725
0.0602
0.0511
0.0900
0.1063
0.0439
0.0375
0.0688
0.0778
0.0649
0.0683
0.0537
0.0567
0.0568
0.0600
0.0479
0.0501
0.0464
0.0412
0.1116
0.1269
0.0703
0.0757
0.1081
0.1254
0.0875
0.1033
0.0952
0.1116
0.0768
0.0816
0.0808
0.0854
0.0669
0.0565
150
0.1034
0.1179
0.0787
0.0835
0.0771
0.0980
0.0620
0.0530
0.0958
0.0573
0.1009
0.0752
0.0754
0.0516
0.0477
0.0404
0.0712
0.0841
0.0424
0.0448
0.0544
0.0615
0.0347
0.0297
0.0449
0.0474
0.0513
0.0540
0.0379
0.0396
0.0367
0.0326
0.0817
0.0855
0.0556
0.0599
0.0753
0.0883
0.0692
0.0883
0.1004
0.0992
0.0607
0.0645
0.0639
0.0675
0.0530
0.0447
200
0.0818
0.0933
0.0597
0.0408
0.0622
0.0661
0.0610
0.0775
0.0798
0.0595
0.0758
0.0454
0.0490
0.0419
0.0377
0.0320
0.0563
0.0665
0.0300
0.0313
0.0430
0.0487
0.0406
0.0427
0.0355
0.0375
0.0336
0.0355
0.0275
0.0235
0.0290
0.0258
0.0596
0.0699
0.0440
0.0474
0.0698
0.0794
0.0547
0.0646
0.0676
0.0785
0.0506
0.0534
0.0480
0.0511
0.0419
0.0353
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Table A.2
Bias and RMSE values of the parameters and reliability for and
Estimate
ML
MPS
MM
PWM
OLS
WLS
CVM
FB
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
50
0.4958
0.5654
0.2971
0.2543
0.3772
0.4006
0.4837
0.3607
0.4594
0.2750
0.3700
0.470
0.3617
0.2473
0.2285
0.1899
0.3414
0.4032
0.1649
0.1423
0.2610
0.2951
0.2463
0.2592
0.2035
0.2150
0.2153
0.2275
0.1760
0.1446
0.1817
0.1900
0.4101
0.476
0.3612
0.4235
0.4233
0.4816
0.3319
0.3917
0.2912
0.3096
0.3066
0.3239
0.2665
0.2873
0.2540
0.1668
75
0.3681
0.4197
0.2685
0.1836
0.2800
0.2973
0.3590
0.2678
0.3410
0.2042
0.2746
0.3489
0.2206
0.1888
0.1697
0.1410
0.2534
0.2993
0.1236
0.1057
0.1938
0.2190
0.1828
0.1924
0.1511
0.1596
0.1598
0.1689
0.1349
0.1411
0.1306
0.1073
0.3142
0.3575
0.2276
0.2405
0.2681
0.3144
0.2161
0.2298
0.2464
0.2908
0.3045
0.3532
0.1978
0.2133
0.1885
0.1238
100
0.2217
0.2528
0.1617
0.1106
0.1687
0.1791
0.1654
0.2101
0.1329
0.1137
0.2054
0.1230
0.2163
0.1613
0.1022
0.0849
0.1101
0.1159
0.0963
0.1017
0.1167
0.1319
0.1527
0.1803
0.0745
0.0637
0.0910
0.0961
0.0813
0.0850
0.0787
0.0646
0.1893
0.2153
0.1371
0.1448
0.1615
0.1894
0.1484
0.1751
0.1834
0.2127
0.1302
0.1384
0.1192
0.1285
0.1136
0.0746
150
0.1506
0.1913
0.1472
0.1007
0.2018
0.2301
0.1968
0.1468
0.1870
0.1119
0.1535
0.1630
0.1209
0.1035
0.0930
0.0773
0.1389
0.1641
0.0828
0.0875
0.1062
0.1201
0.0678
0.0579
0.1002
0.1055
0.0876
0.0926
0.0716
0.0588
0.0740
0.0773
0.1723
0.1960
0.1669
0.1936
0.1470
0.1724
0.1351
0.1594
0.1248
0.1318
0.1185
0.1260
0.1085
0.1169
0.1034
0.0679
200
0.1634
0.1864
0.0979
0.0838
0.1243
0.1320
0.1219
0.1549
0.1514
0.0907
0.1594
0.1189
0.1192
0.0815
0.0753
0.0626
0.0860
0.0973
0.0599
0.0626
0.1125
0.1329
0.0812
0.0854
0.0549
0.0469
0.0671
0.0709
0.0710
0.0750
0.0580
0.0477
0.1191
0.1396
0.0878
0.0947
0.1395
0.1587
0.1352
0.1568
0.1094
0.1291
0.1011
0.1068
0.0960
0.1020
0.0837
0.0550
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2022.19.14
Yasser S. Alharbi, Amr R. Kamel
E-ISSN: 2224-2902
134
Volume 19, 2022
Table A.3
Bias and RMSE values of the parameters and reliability for and
Estimate
ML
MPS
MM
PWM
OLS
WLS
CVM
FB
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
50
0.9418
1.0740
0.7027
0.8927
0.7165
0.7608
0.9187
0.6852
0.8726
0.5224
0.5644
0.4830
0.6870
0.4698
0.4341
0.3607
0.6485
0.7659
0.3164
0.2704
0.4958
0.5605
0.4678
0.4923
0.4090
0.4321
0.3866
0.4083
0.3452
0.3610
0.3343
0.2746
0.8040
0.9147
0.7790
0.9037
0.6861
0.8045
0.6305
0.7440
0.5824
0.6153
0.5531
0.5880
0.5062
0.5457
0.4824
0.3169
75
0.5675
0.6026
0.5565
0.7070
0.7459
0.8506
0.7276
0.5427
0.4470
0.3825
0.6911
0.4137
0.3041
0.2721
0.3438
0.2857
0.5136
0.6066
0.2506
0.2142
0.3926
0.4439
0.3705
0.3899
0.3239
0.3423
0.3062
0.3234
0.2734
0.2859
0.2648
0.2175
0.6368
0.7244
0.6170
0.7157
0.5434
0.6371
0.4380
0.4657
0.4613
0.4873
0.4993
0.5893
0.4009
0.4322
0.3821
0.2510
100
0.5213
0.5945
0.3089
0.2481
0.3966
0.4211
0.5085
0.3793
0.4830
0.2892
0.3124
0.2674
0.3803
0.2600
0.2403
0.1997
0.3590
0.4239
0.1751
0.1497
0.2744
0.3103
0.2589
0.2725
0.2264
0.2392
0.2140
0.2260
0.1850
0.1520
0.1911
0.1998
0.4450
0.5063
0.4312
0.5002
0.3798
0.4453
0.3490
0.4118
0.3224
0.3406
0.3061
0.3255
0.2802
0.3021
0.2670
0.1754
150
0.3456
0.3941
0.2579
0.3276
0.2629
0.2792
0.3202
0.1917
0.3372
0.2515
0.2071
0.1773
0.2521
0.1724
0.1593
0.1324
0.2380
0.2811
0.1161
0.0992
0.1819
0.2057
0.1717
0.1807
0.1419
0.1499
0.1501
0.1586
0.1267
0.1325
0.1227
0.1008
0.2951
0.3357
0.2859
0.3316
0.2518
0.2952
0.2314
0.2730
0.2137
0.2258
0.2030
0.2158
0.1858
0.2003
0.1770
0.1163
200
0.2076
0.2367
0.1549
0.1968
0.2025
0.1510
0.1579
0.1677
0.1244
0.1065
0.1923
0.1151
0.0957
0.0795
0.1514
0.1035
0.1429
0.1688
0.0697
0.0596
0.1093
0.1235
0.1031
0.1085
0.0901
0.0953
0.0852
0.0900
0.0761
0.0796
0.0737
0.0605
0.1772
0.2016
0.1717
0.1992
0.1390
0.1640
0.1512
0.1773
0.1219
0.1296
0.1284
0.1356
0.1116
0.1203
0.1063
0.0698
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2022.19.14
Yasser S. Alharbi, Amr R. Kamel
E-ISSN: 2224-2902
135
Volume 19, 2022
Table A.4
Bias and RMSE values of the parameters and reliability for and
Estimate
ML
MPS
MM
PWM
OLS
WLS
CVM
FB
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
50
0.6267
0.7146
0.4571
0.3126
0.4768
0.5063
0.4676
0.5940
0.5806
0.3476
0.6113
0.4559
0.3755
0.3214
0.2888
0.2449
0.4315
0.5096
0.2572
0.2717
0.3629
0.4298
0.3112
0.3276
0.3299
0.3730
0.2721
0.2875
0.2224
0.1977
0.2105
0.1799
0.5350
0.6086
0.4195
0.4951
0.4565
0.5353
0.5184
0.6013
0.3368
0.3631
0.3875
0.4094
0.3680
0.3913
0.3210
0.2709
75
0.4320
0.4926
0.1991
0.1688
0.3286
0.3490
0.4214
0.3143
0.3223
0.4094
0.2589
0.2215
0.4002
0.2396
0.3151
0.2155
0.2274
0.2571
0.1773
0.1873
0.2974
0.3513
0.2501
0.2963
0.1876
0.1982
0.2145
0.2258
0.1533
0.1363
0.1451
0.1240
0.3688
0.4195
0.2322
0.2503
0.3573
0.4145
0.2892
0.3413
0.3147
0.3690
0.2537
0.2697
0.2671
0.2822
0.2213
0.1867
100
0.2978
0.3396
0.1784
0.1527
0.2692
0.2822
0.2265
0.2406
0.2759
0.1652
0.2905
0.2166
0.2172
0.1485
0.1372
0.1164
0.2050
0.2421
0.1222
0.1291
0.1724
0.2042
0.1567
0.1772
0.1479
0.1557
0.1293
0.1366
0.1057
0.0939
0.1062
0.0855
0.2542
0.2892
0.1600
0.1725
0.2169
0.2543
0.1993
0.2352
0.2463
0.2857
0.1749
0.1859
0.1841
0.1945
0.1525
0.1287
150
0.2053
0.2341
0.1497
0.1024
0.1562
0.1658
0.1531
0.1945
0.1902
0.1139
0.2002
0.1493
0.1230
0.1053
0.0946
0.0802
0.1413
0.1669
0.0690
0.0589
0.1189
0.1408
0.1019
0.1073
0.0891
0.0942
0.1080
0.1222
0.0843
0.0890
0.0729
0.0648
0.1495
0.1753
0.1205
0.1282
0.1752
0.1994
0.1698
0.1969
0.1269
0.1341
0.1374
0.1622
0.1103
0.1189
0.1051
0.0887
200
0.1415
0.1613
0.1032
0.0706
0.1076
0.1143
0.1380
0.1029
0.1056
0.1341
0.0848
0.0726
0.1311
0.0785
0.0652
0.0553
0.0819
0.0970
0.0581
0.0613
0.0974
0.1151
0.0745
0.0842
0.0703
0.0740
0.0614
0.0649
0.0502
0.0446
0.0475
0.0406
0.1208
0.1374
0.0760
0.0820
0.1031
0.1209
0.1170
0.1358
0.0875
0.0924
0.0947
0.1118
0.0831
0.0883
0.0725
0.0612
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2022.19.14
Yasser S. Alharbi, Amr R. Kamel
E-ISSN: 2224-2902
136
Volume 19, 2022
Table A.5
Bias and RMSE values of the parameters and reliability for and
Estimate
ML
MPS
MM
PWM
OLS
WLS
CVM
FB
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
50
0.8140
0.9283
0.4878
0.4175
0.6193
0.6576
0.6073
0.7715
0.7940
0.5922
0.7542
0.4515
0.5938
0.4060
0.3752
0.3181
0.4285
0.4844
0.3341
0.3529
0.5605
0.6619
0.4714
0.5583
0.4043
0.4255
0.3535
0.3735
0.2889
0.2568
0.2734
0.2337
0.6949
0.7906
0.4375
0.4717
0.6733
0.7810
0.5449
0.6431
0.5930
0.6953
0.5034
0.5318
0.4780
0.5082
0.4169
0.3518
75
0.6020
0.6865
0.3608
0.3088
0.4492
0.5706
0.4580
0.4864
0.5578
0.3339
0.5873
0.4380
0.2775
0.2353
0.4392
0.3003
0.3169
0.3583
0.2137
0.1899
0.4145
0.4896
0.3486
0.4129
0.2614
0.2762
0.2990
0.3147
0.2471
0.2610
0.2022
0.1729
0.4030
0.4756
0.3535
0.3759
0.4980
0.5777
0.5139
0.5847
0.4386
0.5142
0.3723
0.3933
0.3236
0.3488
0.3084
0.2602
100
0.3851
0.4391
0.2308
0.1975
0.2929
0.3111
0.2873
0.3650
0.3756
0.2801
0.3568
0.2136
0.2809
0.1921
0.1775
0.1505
0.2651
0.3131
0.1912
0.2013
0.2027
0.2292
0.1672
0.1767
0.2230
0.2641
0.1581
0.1670
0.1294
0.1106
0.1367
0.1215
0.2805
0.3289
0.2070
0.2231
0.3287
0.3740
0.3185
0.3695
0.2381
0.2516
0.2578
0.3042
0.2261
0.2404
0.1972
0.1664
150
0.1941
0.2213
0.1163
0.0995
0.1448
0.1839
0.1476
0.1568
0.1893
0.1412
0.1798
0.1076
0.1416
0.0968
0.0894
0.0758
0.0964
0.1014
0.0843
0.0890
0.1022
0.1155
0.1336
0.1578
0.1124
0.1331
0.0797
0.0841
0.0689
0.0612
0.0652
0.0557
0.1657
0.1885
0.1043
0.1124
0.1605
0.1862
0.1299
0.1533
0.1414
0.1658
0.1206
0.1268
0.1140
0.1212
0.0994
0.0839
200
0.1062
0.1211
0.0775
0.0530
0.0808
0.0858
0.1036
0.0773
0.0793
0.1007
0.0637
0.0545
0.0984
0.0589
0.0490
0.0415
0.0731
0.0864
0.0436
0.0461
0.0615
0.0729
0.0528
0.0555
0.0461
0.0487
0.0559
0.0632
0.0357
0.0305
0.0377
0.0335
0.0774
0.0907
0.0571
0.0615
0.0907
0.1032
0.0711
0.0839
0.0879
0.1019
0.0624
0.0663
0.0657
0.0694
0.0544
0.0459
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2022.19.14
Yasser S. Alharbi, Amr R. Kamel
E-ISSN: 2224-2902
137
Volume 19, 2022
Appendix B: R-Codes
###################################
Parameters Kumaraswamy Distribution
###################################
# theta > 0
# beta > 0
# 0 < x < 1
###################################
## clean up everything
remove(list=objects())
options(warn = -1)
### Packages ###
library("stats4")
library("MASS")
library("bbmle")
library("maxLik")
### PDF ###
dkuma <- Vectorize(function(x,
theta , beta,log = FALSE){
logden <- log(theta ) + log(beta)
+(theta -1)*log(x) + (beta-
1)*log(1-x^ theta )
val<- ifelse(log, logden,
exp(logden))
return(val)
})
### CDF ###
pkuma <-
Vectorize(function(q,theta ,
beta,log.p = FALSE){
cdf <- 1 - (1-q^theta)^beta
val <- ifelse(log.p, log(cdf),
cdf)
return(val)
})
### Quantile function ###
qkuma <- Vectorize(function(u,theta
,Beta){
val <- ( 1 - (1-u)^(1/beta))^(1/
theta )
return(val)
})
### Moments of order n of the
Kumaraswamy(theta,beta)
Distribution ###
mn <- function(theta,beta,n){
log.num <- log(beta) + lgamma(1 +
n/ theta) + lgamma(beta)
log.den <- lgamma(1 + beta + n/
theta)
return(exp(log.num-log.den))
}
### NR-Algorithm ###
rm(list=ls(all=TRUE))
library("rootSolve")
n=100; theta<-1.5; beta<-3;y<-c()
a<-c();b<-c();w<-c();u<-c();H<-c()
y<- rkuma(n,theta,beta)
betahat<-c();thetahat<-c()
for(it in 1:10000){
for(i in 1:n){
a[i]=runif(1,y[i]-1,y[i])
b[i]=runif(1,y[i],y[i]+1)
w[i]<-runif(1)
u[i]<-runif(1,0,1-w[i])
H[i]<-(1+w[i]-u[i])/2}
f<-function(t,theta,beta){
f<-pkuma(t,shape=theta,beta)}
mu<-function(t,i){
if(t>=a[i] & t<=y[i])
return(H[i]*(t-a[i])/(y[i]-a[i]))
if(t>y[i] & t<=b[i])
return(H[i]*(b[i]-t)/(b[i]-y[i]))
else
return(0)}
h1<-function(t,i,theta,beta){
h1<-mu(t,i)*f(t,theta,beta)}
I<-function(z){
theta<-z[1]
beta<-z[2]
i<-z[3]
I<-log(integrate(h1,max(0,a[i]),
b[i],i,theta,beta)$value)}
logl<-function(z){
theta <-z[1]
beta <-z[2]
ss<-0
for(j in 1:n){
c<-c(theta,beta,j)
ss<-ss+I(c)}
logl<--ss}
c<-c(theta,beta)
out<-
suppressWarnings(nlminb(c,logl))
thetahat[it]<-out$par[1]
betahat[it]<-out$par[2]}
mean(thetahat);mean(betahat)
mean(thetahat)-theta;
mean(betahat)- beta
mean((thetahat-theta)ˆ2)
mean((betahat-beta)ˆ2)
x<-5
R<-exp(-((x/theta)ˆbeta))
print(R)
RR<-exp(-((x/thetahat)ˆbetahat))
print(mean(RR))
print(mean(RR-R))
print(mean((RR-R)ˆ2))
### FB-Estimation ####
rm(list=ls(all=TRUE))
library("numDeriv")
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2022.19.14
Yasser S. Alharbi, Amr R. Kamel
E-ISSN: 2224-2902
138
Volume 19, 2022
betahat<-c();thetahat<-c()
n=50;theta<-5;beta<-10;y<-c()
a<-c();b<-c();w<-c();u<-c();H<-c()
thet.<-c();bet.<-c()
for(it in 1:10000){
d1<- rkuma(50,theta,beta)
l.prim<-function(z){
theta<-z[1]
beta<-z[2]
l.prim<--sum(log(pkuma(d1,theta,
beta)))}
c<-c(theta,beta)
out.<-
suppressWarnings(nlm(l.prim,c))
thet.[it]<-out.$estimate[1];
bet.[it]<-out.$estimate[2]}
V1<-mean((thet.-theta)ˆ2)
V2<-mean((bet.-beta)ˆ2)
b1<-mean(thet.)/(V1)
a1<-mean(thet.)*b1/2
a2<-(mean(bet.))ˆ2/(12*V2)
b2<-(mean(bet.))*(a2-1)/12
for(it in 1:10000){
y<- rkumar(n,theta,beta)
for(i in 1:n){
a[i]=runif(1,y[i]-1,y[i])
b[i]=runif(1,y[i],y[i]+1)
w[i]<-runif(1)
u[i]<-runif(1,0,1-w[i])
H[i]<-(1+w[i]-u[i])/2}
f<-function(t,theta,beta){
f<-pkuma(t,shape=theta,beta)}
mu<-function(t,i){
if(t>=a[i] & t<=y[i])
return(H[i]*(t-a[i])/(y[i]-a[i]))
if(t>y[i] & t<=b[i])
return(H[i]*(b[i]-t)/(b[i]-y[i]))
function(t,i){
if(t>=a[i] & t<=y[i])
return(H[i]*(t-a[i])/(y[i]-a[i]))
if(t>y[i] & t<=b[i])
return(H[i]*(b[i]-t)/(b[i]-y[i]))
else
return(0)}
h1<-function(t,i,theta,beta){
mu(t,i)*f(t,theta,beta)}
I1<-function(theta,beta,i){
I1<-integrate(h1,max(0,a[i]),
b[i],i,theta,beta)$value}
I.1<-function(theta,beta){
ss<-0
for(j in 1:n){
ss<-ss+log(I1(theta,beta,j))}
I.1<-ss}
HH<-function(z){
theta<-z[1]
beta<-z[2]
HH<--((1/n)*((n+a1-1)*log(theta)-
b1*theta-(n+a2+1)*log(beta)-
b1/beta+I.1(theta,beta)))}
c<-c(theta,beta)
out<-suppressWarnings(nlminb(c,HH))
c1<-c(out$par[1],out$par[2])
hess1<-suppressWarnings(hessian(
func=HH, x=c1))
sigma1<--solve(hess1)
Hstar1<-function(z){
theta<-z[1]
beta<-z[2]
cstar<-c(theta,beta)
Hstar1<-(-
(1/n)*log(theta)+(HH(cstar)))}
c<-c(theta,beta)
out.star1<-
suppressWarnings(nlminb(c,Hstar1))
c2<-
c(out.star1$par[1],out.star1$par[2)
hess2<-suppressWarnings(hessian(
func=Hstar1, x=c2))
sigma2<--solve(hess2)
Hstar2<-function(z){
theta <-z[1]
beta <-z[2]
cstar<-c(theta,beta)
Hstar2<-(-
(1/n)*log(beta)+(HH(cstar)))}
c<-c(theta,beta)
out.star2<-suppressWarnings(nlminb(
c,Hstar2))
c3<-
c(out.star2$par[1],out.star2$par[2)
hess3<-suppressWarnings(hessian(
func=Hstar2, x=c3))
sigma3<--solve(hess3)
thetahat[it]<-
suppressWarnings(((det(sigma2)
)/(det(sigma1)))ˆ(1/2)*exp(-n*(
Hstar1(c2)-HH(c1))))
mean(thetahat);mean(betahat)
mean(thetahat)-theta;
mean(betahat)-beta
mean((thetahat-theta)ˆ2)
mean((betahat-beta)ˆ2)
x<-5
R<-exp(-((x/theta)ˆbeta))
print(R)
RR<-exp(-((x/betahat)ˆthetahat))
print(mean(RR))
print(mean(RR-R))
print(mean((RR-R)ˆ2)
### END Codes ###
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2022.19.14
Yasser S. Alharbi, Amr R. Kamel
E-ISSN: 2224-2902
139
Volume 19, 2022
