Application of Odd Chen-Log-Logistic Distribution to Covid-19 Data
KLODIANA BANI, ARBËR QOSHJA, AURORA SIMONI, MARSELINA UZUNI
Department of Applied Mathematics,
Tirana University,
Sheshi “Nënë Tereza”, Tirana,
ALBANIA
Abstract: - This article created the Odd Chen-Log-Logistic distribution from Odd Chen-G family distributions.
We derive various statistical features. The parameter estimation theory focuses on selecting the best estimators.
We estimate distribution parameters using maximum likelihood, moment, least squares, weighted least, L-
moment, maximum product spacing, and minimal distance methods. We will examine Kolmogorov-Smirnov
simulation studies that compare estimator efficiency. Finally, we analyze a genuine COVID-19 data set to
demonstrate the flexibility of our model and its accuracy compared to other distributions.
Key-Words: - Log-Logistic distribution, Hazard Function, Maximum Likelihood, Moment Estimation,
Simulations, COVID-19.
1 Introduction
The log-logistic distribution, also known as the Fisk
distribution, is a massive continuous probability
distribution with a huge tail. Indeed, it has a singular
form parameter and a single scale or rate. This
distribution uses a non-negative random variable
whose logarithm has the common logistic
distribution, [1]. The log-logistic distribution
enables the closed representation of the cumulative
distribution and helps one to estimate incomplete (or
censored) data. Indeed, in the domains of business,
medicine, economics, income, wealth, and the social
sciences, the log-logistic distribution models do find
their use. They help to depict data with a substantial
degree of fluctuation.
In several areas, the log-logistic distribution
differs from many parametric distributions used in
survival and reliability research.
Many disciplines, including demography [2],
economics [3], engineering [4], and hydrology [5],
use the log-logistic distribution as a basic yet
effective parametric model.
Definition 1.1. A random variable has a Log-
Logistic distribution with shape parameter
and scale parameter , based on the probability
density function:
(1)
and cumulative distribution function
(2)
According to [6], the Odd Chen-G Family of
distributions generator provides the pdf and cdf of a
continuous distribution:
(3)
and
(4)
respectively, for and parameter
vector .
This study examines Log-Logistic (3) and (4),
often known as the Odd Chen distribution, as
baseline functions for (5) and (6).
Definition 1.2. The probability density function of a
random variable X with a vector parameter is called
an Odd Chen Log-Logistic distribution .
Received: May 9, 2024. Revised: September 23, 2024. Accepted: October 15, 2024. Published: November 8, 2024.
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(5)
and cumulative distribution function
(6)
respectively, for .
Some alternative shapes of the Odd Chen Log-
Logistic distribution (OC-LL) distribution for given
values of λ,β,θ, and μ are shown in Figure 1 and
Figure 2.
Fig. 1: CDF
Fig. 2: PDF
2 Some Properties
2.1 Survival Function
Survival function or reliability function of the Odd
Chen Log-Logistic distribution:
(7)
2.2 Hazard Function
Hazard rate function, or failure rate, of the Odd
Chen Log-Logistic distribution is:
(8)
Figure 3 and Figure 4 display the potential
Reliability and Hazard functions of the Odd Chen
Log-Logistic (OC-LL) distribution for specific
values and .
Fig. 3: Reliability Function
Fig. 4: Hazard Function
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2.3 Quantiles
Solving equation , we get the quantile of
some distribution, for . Odd Chen Log-
Logistic distribution quantile function is:
(9)
2.4 Some Useful Expression
Extension of Taylor series, [7], the pdf (7) of
becomes:
Generalized binomial expansion:
2.5 Order Statistics
The maximum order density for i.i.d. continuous
random variables with pdf (7) and cdf
(8) is:
(10)
The minimum order density for i.i.d. continuous
random variables with pdf (7) and cdf
(8) is:
(11)
The minimum order density for i.i.d. continuous
random variables with pdf (7) and cdf
(8) is:
(12)
2.6 Rényi Entropy
Information theory's Rényi entropy generalises
collision, min-entropy, Shannon, and Hartley
entropies. The Rényi entropy studies the largest
method for information quantification that maintains
additivity for separate events, [8]. In fractal
dimension estimation, generalized dimensions are
based on Rényi entropy. Rényi entropy is a measure
of diversity in statistics and ecology. The Rényi
entropy is an important entanglement gauge in
quantum information. It is possible to exactly
determine the Rényi entropy as a function of α in
the Heisenberg XY spin chain model, as it is an
automorphic function for a specific subgroup of the
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modular. Rényi entropy for the Odd Chen
Log-Logistic distribution is as follows:
,
, . (14)
3 Approaches to Parameter Estimation
3.1 Maximum Likelihood
The most prevalent ML method is full information
maximum likelihood (ML) because it gives
estimates with desirable large sample quality. In
finite samples, these traits hold roughly. For
independent consider a parametric
model with probability density or frequency
distribution functions . We know that:
(13)
represents the probability of the sample being
observed for . So, the value of that maximises
defines the MLE.
So, be i.i.d. random variables with pdf
(7). The likelihood function of parameters and
is:
(14)
Solving nonlinear equations simultaneously
estimates unknown parameters, which cannot be
solved analytically. Iterative methods like the
Newton-Raphson approach simplify nonlinear
situations. Newton Raphson estimated parameters
using these beginning values. The parameter
estimates for the two-sided confidence
range are asymptotically close to standard normal,
as indicated by the z-score.
3.2 Moment Estimation
Since sample moments are estimates of population
moments, the method of moments is one of the
oldest point estimator methods, [9]. Equalising the
first three theoretical moments with the three sample
moments yields the Odd Chen Log-Logistic
distribution's moment estimators. These four
moments are examples:
(15)
and the first four theoretical moments were
characterised as:
The moment's estimators of
the parameters can be obtained by solving
numerically the following system of equations:
Modified moment estimate is a good alternative
to moment estimation. This first-order statistics
approach can be adjusted, as mentioned by [10].
Let be a sample from Odd Chen
Log-Logistic distribution, with observed values
. Solving the following equations yields
Odd Chen Log-Logistic distribution modified
moment estimators:
where
and is the sample variance
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3.3 Least Square Estimation
Beta distribution parameters can be found using
least square estimators and weighted LSEs, [11].
The Odd Chen Log-Logistic distribution's
unknown parameters' LSEs can be found by
minimizing:
(16)
regarding unknown parameters .
To calculate the least squares estimate (LSE) of
, can be derived by minimizing the
respective values:
Hence, of can be
found by solving the following system of equations:
We can calculate estimates by solving these
equations numerically
3.4 The Weighted Least Square Estimation
The unknown parameters' weighted least squares
estimators (WLSEs) can be calculated by
minimizing:
(17)
consider , and denote the weight function:
The WLSEs say by
minimizing:
We may calculate the estimators , ,
, by taking the first partial derivative
of and setting the result to zero:
We can calculate estimates by solving these
equations numerically , , ,
.
3.5 L-Moments Estimators
Equating sample and population L-moments yields
L-moment estimators, [12]. Equating sample and
population L-moments yields these estimators. The
L-moment estimators are more reliable than the
moment estimators, more immune to outliers, and
more efficient than the maximum likelihood
estimators for specific distributions, [13].
Equating the first three sample L-moments with
the population L-moments yields the Odd Chen
Log-Logistic distribution L-moments estimators.
Example's first three L-moments:
a first three population L-moments of:
is the jth order statistic of an n-sample. To
calculate the L-moments estimators for the
parameters solve the following equations
numerically:
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3.6 Maximum Product Spacing Estimators
For continuous univariate distributions, [13], [14]
developed the maximum product of spacings (MPS)
approach to estimate parameters and approximate
the Kullback-Leibler measure of information. This
approach assumes an equal distribution of
consecutive point discrepancies.
Consider as a random sample from
the Odd Chen Log-Logistic distribution and
as an ordered sample. We refer to
and . I The method of
maximum product of spacings estimates distribution
parameters by maximising the geometric
mean of distances , denoted as:
for (18)
where , and
The geometric mean of distances is expressed as:
(19)
The MPS estimators are
calculated by maximising the geometric mean (GM)
of spacings concerning or the logarithm of
the geometric mean of sample spacings:
(20)
The MPS estimators
can be derived by solving the following equations
simultaneously:
3.7 Methods of Minimum Distances
The goodness-of-fit statistics method minimizes
empirical distribution function statistics to estimate
distribution parameters, [15]. The generic minimal
distance method assumes establishing a distribution
function that closely matches the empirical
distribution of the observed data. The minimal
distance approach has estimators based on the
empirical distribution function statistic. This section
presents three Odd Chen Log-Logistic distribution
estimate methods based on goodness-of-fit statistics
minimization for . This statistical class is
based on the difference between the empirical
distribution function and the cumulative distribution
function estimate, [16], [17].
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
[18], [19]. The real-world data in [20] shows that
Cramér-von-Mises-type minimal distance estimators
are less biased than other minimum distance
estimators, which explains why they are used. The
Cramér-von-Mises estimates
of parameters
of Odd Chen Log-Logistic distribution are obtained
by minimizing, concerning the function:
(21)
To obtain these estimates, you can solve the
following nonlinear equations:
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3.7.2 Anderson-Darling and Right-tail Methods
The Anderson-Darling estimator (ADE) is derived
from the statistic, another minimum distance
estimator. Besides using a weighted squared
difference, the Anderson-Darling test is similar to
the Cramér-von-Mises criterion. These weights are
determined by the deviation from the empirical
distribution function. The Anderson-Darling test is
an alternative to traditional statistical procedures
used to identify deviations from normality in sample
distributions, [21], [22]. We minimize a function to
estimate the Anderson-Darling parameter :
(22)
To obtain right-tail Anderson-Darling
estimations of parameters, minimise function about
.
(23)
The following non-linear equations may be
solved for these estimates:
4 Computer Applications
4.1 Simulation Study
Using Monte Carlo simulation, this section tests
multiple estimation methods for forecasting Odd
Chen Log-Logistic distribution parameters. The
Kolmogorov-Smirnov test compares the
recommended estimators. This approach uses KS
statistics.
where
denotes the maximum of the set of
distances, is the empirical distribution
function, and is the cumulative
distribution function.
We proposed a method to randomly sample the
Odd Chen Log-Logistic distribution given
parameter values and sample size n.
We take
arbitrarily and .
We implemented all techniques in the statistical
computing environment R.
Simulations were done using the approach.
Table 1. Estimation techniques and Kolmogorov-
Smirnov values
i
Methods of
Estimations
Kolmogorov-
Smirnov test
Ranking
1
Maximum Product
Spacing Estimating
0.068542
5
2
Moment Estimation
0.066021
3
3
Least Square
Estimation
0.066254
4
4
Weighted Least
Square Estimation
0.062741
2
5
L-Moment
Estimation
0.070517
6
6
Maximum
Likelihood
Estimation
0.061254
1
7
Maximum Product
Spacing Estimating
0.072749
9
8
Anderson-Darling
Estimation
0.070654
7
9
Right-tail Anderson-
Darling
0.071154
8
The simulation study shows that the Maximum
Likelihood Estimation (MLE) technique estimates
Odd Chen Log-Logistic distribution parameters
more efficiently than other approaches. Table 1
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shows that MLE produces the lowest Kolmogorov-
Smirnov test result. Additionally, maximal
likelihood estimators (MLE) have excellent
theoretical properties.. These are consistency,
asymptotic efficiency, normalcy, and invariance.
These data suggest that MLE estimators are best for
calculating Odd Chen Log-Logistic distribution
parameters.
4.2 Actual Data
Now, we'll assess the enlarged distribution's
effectiveness. In this investigation, our model
outperforms other models on a real data set (Table
2). The data shows the case fatality ratio of COVID-
19 in China from March 8th to April 1st, 2022,
relative to a new strain. Data is obtained from the
WHO website (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.
Table 2. Covid-19 case fatality ratio in China:
MLEs and comparability criteria
Distribu
tion
Parameter
Estimate
AIC
BIC
CAI
C
Chen
Log-
Logistic
=0.5
=0.9
=1.2
=1.7
89.241
9
157.3
26
151.3
22
148.2
31
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
For the dataset, AIC, CAIC, and BIC are used to
evaluate distribution models. Lower criteria values
indicate a better dispersion.
The sample size is n, and the p-value reflects the
number of parameters calculated from the data.
According to our preview work [23], we can say
that our new modified distribution, Odd Chen Log-
Logistic, fits the data better than other models.
5 Conclusion
This paper derives the Odd Chen Log-Logistic
distribution from the Odd Chen Log-G family
distributions. We analyzed various statistical aspects
of the distribution and tried to design a parameter
estimation model. We used Kolmogorov-Smirnov
simulations to compare multiple estimators. To
compare our model's adaptability to other
distributions' correctness, this research analyses an
actual COVID-19 data set. This broader spread may
be useful in different study areas.
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Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
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