Half-Logistic Odd Power Generalized Weibull-inverse Lindley:
Properties, Characterizations, Applications and Covid-19 Data
ARBËR QOSHJA, ARTUR STRINGA, KLODIANA BANI, REA SPAHO
Department of Applied Mathematics, Department of Mathematics,
Tirana University,
Sheshi “Nënë Tereza”, Tirana,
ALBANIA
Abstract: - Using the Half-Logistic Odd Power Generalised Weibull-G family distributions, this article
constructed a novel distribution termed the Half-Logistic Odd Power Generalised Weibull-inverse Lindley.
Some of its statistical features are derived by us. Selecting the most efficient estimators is among the basic
issues in parameter estimation theory. We are employing maximum likelihood estimation, moment estimation,
least squares estimation, weighted least estimation, L-moment estimation, Maximum Product Spacing
estimation, and techniques of minimum distances for the parameter estimation for the distribution. We will
examine simulation research that compares the various estimators' levels of efficiency using the Kolmogorov-
Smirnov test. Lastly, an analysis is done on an actual COVID-19 data set to demonstrate the adaptability of our
suggested model in comparison to the fit obtained by several other competing distributions.
Key-Words: - Inverse Lindley distribution, Statistical Properties, Estimation, Kolmogorov-Smirnov test
Received: June 23, 2023. Revised: September 21, 2023. Accepted: October 13, 2023. Published: October 30, 2023.
1 Introduction
One method for characterizing a process or device's
lifespan that may be used in a variety of domains,
such as biology, engineering, and medicine, is the
Lindley distribution. As a counter-example of
fiducial statistics, Lindley developed a blend of
and distribution with mixing
percentage . This was done in the
context of Bayesian statistics. One notable
advancement in the literature on Lindley distribution
is the two-parameter weighted Lindley distribution,
[1], which is found to be highly beneficial when
modelling biological data derived from mortality
studies. The proposal of a generalized Poisson
Lindley distribution has been made, [2]. However,
an exponential geometric (EG) distribution was
introduced, [3], in contrast to the extended Lindley
(EL) distribution that was demonstrated, [4]. An
article, [5], describes a two-parameter Lindley
distribution. The authors of, [6], propose a novel
two-parameter lifespan distribution model and
characteristics. The Lindley distribution convolution
was initially proposed by, [7]. The estimation of the
dependability of a stress-strength system through the
utilisation of power Lindley distribution was the
subject of some science, [8]. There has been a recent
proposal for an extended Lindley distribution, [9].
An attempt was made to extend the Lindley
distribution using the Transmuted Lindley
Distribution, which is a quadratic rank
transmutation map, [10].
Definition 1.1. If the probability density function of
a random variable has the following definition,
[11], [12], then the variable is said to have a Lindley
distribution with parameter .
(1)
and cumulative distribution function
(2)
The inverse Lindley distribution, [13], was due
to the broad use of inverse distributions. Its
probability density function (pdf) and cumulative
distribution function (cdf) are provided by:
(3)
(4)
Some researchers, [14], presented a generator of
a continuous distribution known as the half-logistic
odd power generalized Weibull-G family of
distributions, where the pdf and cdf are provided by:
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(5)
and
(6)
respectively, for and parameter vector
.
This work is aimed at examining the inverse
Lindley distributions (3) and (4), also known as the
half-logistic odd power generalized Weibull-inverse
Lindley distribution, as baseline functions to (5) and
(6).
Definition 1.2. If the probability density function of
a random variable X is described as follows, it is
said to have a half-logistic odd power generalized
Weibull-inverse Lindley distribution with a vector
parameter .
(7)
and cumulative distribution function
(8)
respectively, for .
Figure 1 and Figure 2 illustrate some of the
possible shapes of the (Cdf) and (Pdf) of the Half-
Logistic Odd Power Generalized Weibull-Inverse
Lindley (HLOPGW-ILD) distribution for selected
values of the parameters and , respectively.
Fig. 1: Cumulative Density Function Of The The
Half-Logistic Odd Power Generalized Weibull-
Inverse Lindley distribution
Fig. 2: Probability Density Function Of The The
Half-Logistic Odd Power Generalized Weibull-
Inverse Lindley distribution
2 Mathematical Properties
2.1 Survival Function
The Half-Logistic Odd Power Generalised Weibull-
Inverse Lindley (HLOPGW-ILD) distribution's
survival function, or reliability function, is as
follows:
(9)
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2.2 Hazard Function
The Half-Logistic Odd Power Generalised Weibull-
Inverse Lindley (HLOPGW-ILD) distribution's
hazard rate function, often known as the failure rate,
is provided by:
(10)
For certain values of the parameters and ,
respectively, Figure 3 and Figure 4 show several
potential forms of the Reliability and Hazard
functions of the Half-Logistic Odd Power
Generalised Weibull-Inverse Lindley (HLOPGW-
ILD) distribution.
Fig. 3: Reliability Function Function Of The The
Half-Logistic Odd Power Generalized Weibull-
Inverse Lindley distribution
Fig. 4: Hazard Function Function Of The The Half-
Logistic Odd Power Generalized Weibull-Inverse
Lindley distribution
2.3 Quantiles
The quantile of any distribution is given by solving
the equation , for . The
quantile function of the Half-Logistic Odd Power
Generalized Weibull-Inverse Lindley (HLOPGW-
ILD) distribution is:
(11)
2.4 Some Useful Expression
Taking generalized binomial expansion, [11], the
pdf (5) of may be written as:
By using this methodology, the pdf (7) of the
HLOPGW-ILD distribution is:
Alternatively, we may represent the Half-
Logistic Odd Power Generalised Weibull-Inverse
Lindley as a linear combination of Exp-Inverse
Lindley densities because, following a series
definition:
where,
for , and
.
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This form may be shown to be a linear
combination of Lindley densities with Exp-Inverse
Lindley densities and a power parameter
.
2.5 Order Statistics
For i.i.d. continuous random variables
with pdf (7) and cdf (8) the density of the maximum
order is:
(12)
For iid continuous random variables
with pdf (7) and cdf (8) the density of the minimum
order is
(13)
For iid continuous random variables
with pdf (7) and cdf (8) the density of the kth order
is:
(14)
2.6 Rényi Entropy
A number that generalizes several concepts of
entropy, such as collision entropy, min-entropy,
Shannon entropy, and Hartley entropy, is known as
the Rényi entropy in information theory. The Rényi
entropy, [15], is named after the researcher Alfréd
Rényi, who sought the broadest approach to
information quantification while maintaining
additivity for independent events. The Rényi
entropy serves as the foundation for the idea of
generalized dimensions in the context of fractal
dimension estimation. In statistics and ecology, the
Rényi entropy is significant as a diversity indicator.
Because it may be used as a gauge of entanglement,
the Rényi entropy is also significant in the context
of quantum information. Because it is an
automorphic function regarding a certain subgroup
of the modular group, the Rényi entropy as a
function of α in the Heisenberg XY spin chain
model may be precisely determined. Min-entropy is
utilized in relation to random extractors in
theoretical computer science. Rényi entropy
for the Half-Logistic Odd Power Generalized
Weibull-Inverse Lindley distribution as follows.
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, , . (14)
3 Methods for Estimating Parameters
3.1 Maximum Likelihood
Because it produces estimates with very desired
large sample qualities, the most popular approach
for ML is full information maximum likelihood or
ML. In finite samples, these features also roughly
hold. The ML estimator (MLE) is most efficient,
unbiased, and normally distributed for linear models
with errors that are normally distributed.
Let be independent and assume
that each follows a parametric model with a
probability density function or a frequency
distribution function . The likelihood
function of parameter vector is:
(15)
It is obvious that indicates the likelihood
that the sample will be seen given a . The goal of
(ML) is to determine a value of that maximizes
this probability, given that the sample has previously
been seen. Formally, the value of that maximises
defines the MLE.
In our case, be i.i.d. random
variables with a probability density function (7).
The likelihood function of parameters is:
. (16)
Unknown parameters cannot be precisely solved
analytically; estimates are thus obtained by
simultaneously solving nonlinear equations. The
Newton-Raphson technique is one iterative
methodology that makes solving nonlinear problems
simpler. By providing an initial estimate for the
parameters, Newton Raphson used these starting
values to construct parameter estimates. The z-
score, which may be used to compute the parameter
estimates two-sided confidence range,
is about standard normal, and these parameter
estimates are asymptotically near to standard
normal.
3.2 Moment Estimation
One of the earliest techniques for determining point
estimators is the method of moments, which gets its
name from the fact that sample moments are
essentially estimates of population moments. Karl
Pearson was the one who presented it, [16]. The
HLOPGW-ILD distribution's moment estimators
may be acquired by equating the first three
theoretical moments with the corresponding three
sample moments. The following three example
moments are described:
,
,
(17)
and the first three theoretical moments are defined
as:
The moment's estimators of the
parameters can be obtained by solving
numerically the following system of equations:
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The modified moment estimation technique is
an attractive alternative to the moment estimation
method. Certain adjustments may be made to this
approach that uses first-order statistics, as stated by,
[17].
Let , be a random sample
from, with observed
values The modified moment
estimators of distribution can be
obtained as the solution of the following equations:
Where is the cumulative
distribution function, is the first-order statistic,
is the smallest sample value, is the sample
mean
and is the sample variance
3.3 Least Square Estimation
To find the parameters of a beta distribution, [18]
suggested the least square estimators and weighted
least square estimators (LSEs). The LSEs of the
HLOPGW-ILD distribution's unknown parameters
may be found by minimizing:
(18)
with respect to the unknown parameters .
Where denotes the distribution function of the
distribution and
is the expectation of the empirical cumulative
distribution function. The least squares estimate
(LSEs) of , say, respectively,
can be obtained by minimizing:
Therefore, of can be
obtained as the solution of the following system of
equations:
,
,
We can solve these equations numerically to obtain
the estimates
3.4 The Weighted Least Square Estimation
One may derive the weighted least squares
estimators (WLSEs) of the unknown parameters by
minimizing:
(19)
with respect to , where denotes the weight
function at the point, which is equal to
The weighted least square estimates (WLSEs)
say can be obtained by
minimizing:
Therefore, the estimators
can be obtained from the first partial derivative with
respect to and set the result equal to zero:
,
,
By solving these equations numerically, we can
obtain the estimates and
3.5 L-Moments Estimators
[19], was the one who first suggested the L-
moments estimators. The process of equating the
sample L-moments with the population L-moments
yields these estimators. According to, [21], the L-
moment estimators are more reliable than the
moment estimators, and for certain distributions,
they are also quite efficient when compared to the
maximum likelihood estimators and relatively
resilient to the effects of outliers.
By equating the first three sample L-moments
with the corresponding population L-moments, the
L-moments estimators for the HLOPGW-ILD
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distribution may be produced. The first three L-
moments in the example are:
and the first three population L-moments are:
Here, denotes the order statistic of a
sample of size . Therefore, the L-moments
estimators of the parameters
can be obtained by solving numerically the
following equations:
3.6 Maximum Product Spacing Estimators
To approximate the Kullback-Leibler measure of
information, [20], [21], separately devised the
maximum product of spacings (MPS) approach for
estimating parameters in continuous univariate
distributions. This method is based on the idea that
the differences of the consecutive points should be
identically distributed.
Let be a random sample from the
distribution and
be an ordered random sample. For convenience, we
also denote and . In the method
of maximum product of spacings, we seek to
estimate the parameters of the distribution by
maximizing the geometric mean of distances ,
where every distance is defined as
for (20)
where
and
The geometric mean of distances is given by:
(21)
The MPS estimators are
obtained by maximizing the geometric mean (GM)
of the spacings with respect to or
equivalently by maximizing the logarithm of the
geometric mean of the sample spacings:
(22)
The MPS estimators of can
be obtained as the simultaneous solution of the
following equations,
3.7 Methods of Minimum Distances
Wolfmitz was the pioneer in estimating the minimal
distance, [22]. This technique, also known as
goodness-of-fit statistics, is based on minimizing
empirical distribution function statistics for the
purpose of estimating a distribution's parameters. A
highly generic method known as the minimal
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distance method formulates the inference issue as
finding a distribution function that approaches the
empirical distribution provided by the observed data
as closely as feasible. Various estimators are
available using the minimal distance approach,
contingent on the selected empirical distribution
function statistic. This part introduces three
estimating techniques for the HLOPGW-ILD
distribution, which are based on the goodness-of-fit
statistics minimization regarding , and . The
difference between the empirical distribution
function and the estimate of the cumulative
distribution function forms the basis of this class of
statistics, [23], [24].
3.7.1 Method of Cramr-von-Mises
The minimal distance estimator (CME) is a type of
estimator based on the Cramér-von-Mises statistic,
[25], [26]. The empirical evidence presented by,
[27], demonstrates that Cramér-von-Mises type
minimal distance estimators have a smaller bias than
alternative minimum distance estimators, which
explains their application.
The Cramér-von-Mises estimates
of parameters of
HLOPGW-ILD distribution are obtained by
minimizing, with respect to and the function:
(23)
The following nonlinear equations may be solved to
get these estimates:
3.7.2 Methods of Anderson-Darling and Right-
tail Anderson-Darling
An additional category of estimators that use the
concept of minimal distance is the Anderson-
Darling estimator (ADE), which is derived from the
Anderson-Darling statistic. The Anderson-Darling
test is like the Cramér-von-Mises criteria, with the
exception that the integral involves a weighted
version of the squared difference. These weights are
determined by the variance of the empirical
distribution function. The Anderson-Darling test,
[28], [29], is used as a viable alternative to existing
statistical procedures to identify deviations from
normality in sample distributions. The estimation of
the parameters in the Anderson-Darling method
involves minimizing a function with respect to ,
and .
(24)
To get the Right-tail Anderson-Darling estimates of
the parameters, minimize the function with respect
to .
(25)
The following non-linear equations may also be
solved to get these estimates:
4 Applications
4.1 Simulation Study
In this part, a Monte Carlo simulation analysis is
conducted to assess the efficacy of several estimate
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approaches in predicting the parameters of the
HLOPGW-ILD distribution. The suggested
estimators are compared via the use of the
Kolmogorov-Smirnov test. The methodology used
in this technique is based on the KS statistic.
where
denotes the maximum of the set of
distances, is the empirical distribution
function, and is the cumulative
distribution function.
Initially, an approach was presented to create a
random sample from the HLOPGW-ILD
distribution, given certain parameter values and
sample size n. The following methodology was
implemented:
1. Set n, and initial value
2. Generate
3. Update by using the Newton’s formula.
where,
and
are cdf and pdf of the HLOPGW-ILD
distribution, respectively.
4. If very small, tolerance
limit , then store as a sample from
HLOPGW-ILD distribution.
5. If then set and go to
step 3.
6. Repeat steps 3-5, n times for
respectively.
For this purpose, we take
arbitrarily and .
All the algorithms were implemented in R, a
statistical computing environment.
The method was used for the aim of conducting
simulations. Based on the findings of the simulation
research, it is evident that the Maximum Likelihood
Estimation (MLE) approach exhibits superior
efficiency in estimating the parameters of the
HLOPGW-ILD distribution, as compared to other
methods. This conclusion is supported by the
observation that the MLE technique yields the
lowest value in the Kolmogorov-Smirnov test, as
shown in Table 1. Furthermore, it is worth noting
that the maximum likelihood estimators (MLE) have
favorable theoretical characteristics, [19]. These
attributes include consistency, asymptotic efficiency,
normality, and invariance. Based on these findings,
it can be inferred that the MLE estimators are the
preferred choice for estimating the parameters of the
HLOPGW-ILD distribution.
Table 1. The methods of estimation and its
respective Kolmogorov-Smirnov test value.
i
Methods of
Estimations
Kolmogorov-
Smirnov test
Ranking
1
Maximum Product
Spacing Estimating
0.036542
5
2
Moment Estimation
0.034521
3
3
Least Square
Estimation
0.035417
4
4
Weighted Least
Square Estimation
0.032154
2
5
L-Moment
Estimation
0.038254
6
6
Maximum
Likelihood
Estimation
0.031254
1
7
Maximum Product
Spacing Estimating
0.041241
9
8
Anderson-Darling
Estimation
0.038749
7
9
Right-tail Anderson-
Darling
0.039254
8
4.2 Real Data Set
In this part, we will evaluate the efficacy of the
expanded distribution. In this study, a genuine data
set is used to demonstrate the superior performance
of our model in comparison to other models applied
to the same data set. The provided data pertains to
the case fatality ratio of COVID-19 in China,
namely from March 8th to April 1st, 2022, in
relation to the emergence of a new strain of the
virus.
The data is collected from the official site of the
World Health Organization (WHO)
[https://covid19.who.int/].
The data are as follows: 1.09, 1.00, 1.08, 1.12, 1.50,
1.60, 1.77, 1.81, 2.07, 1.75, 2.58, 2.59,
2.65, 3.09, 3.20, 3.47, 3.21, 2.77, 3.17, 2.65, 3.00,
3.61, 3.08, 2.70, 2.41.
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Table 2. MLEs and comparison criteria for the
COVID-19 case fatality ratio in China.
Distribu
tion
Parameter
Estimate
AIC
BIC
CAI
C
HLOPG
W-ILD
=1.2545
8712
=0.3645
8756
=4.2514
5235
91.125
4
165.2
36
160.6
31
159.3
74
EPL
=2.6705
2921
=0.6654
7111
=1.56820
413
132.25
41
198.2
54
191.3
65
196.7
84
L
=0.6535
4891
195.35
12
290.3
54
289.9
51
290.4
57
E
=0.2673
2123
201.32
64
340.5
87
342.6
14
341.7
53
To assess the distribution models, several
metrics such as AIC (Akaike information criterion),
CAIC (corrected Akaike information criterion), and
BIC are taken into account for the given dataset. A
more optimal distribution is characterized by lower
values of the criterion.
The -value indicates the number of parameters
that will be estimated from the data, whereas n
represents the sample size.
According to the findings shown in Table 2, our
analysis demonstrates that the Half-Logistic Odd
Power Generalised Weibull-inverse Lindley
distribution has superior goodness of fit compared
to other models, namely the Exponential, Lindley,
and exponentiated power Lindley distributions.
5 Conclusion
This study presents the derivation of a novel
distribution, referred to as the Half-Logistic Odd
Power Generalised Weibull-inverse Lindley, by
using the Half-Logistic Odd Power Generalised
Weibull-G family distributions. We presented an
analysis of many statistical features of the
distribution and attempted to develop a model for
estimating its parameters. We conducted simulation
research to assess the comparative effectiveness of
several estimators using the Kolmogorov-Smirnov
test. The present study involves the analysis of an
authentic COVID-19 data set to demonstrate the
adaptability of our suggested model in comparison
to the level of accuracy achieved by alternative
distributions. We posit that the use of this expanded
distribution has potential for exploration in other
research domains.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.89
Arbër Qoshja, Artur Stringa, Klodiana Bani, Rea Spaho
E-ISSN: 2224-2880
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.89
Arbër Qoshja, Artur Stringa, Klodiana Bani, Rea Spaho
E-ISSN: 2224-2880
818
Volume 22, 2023
