Abstract
Since December 2019, the COVID-19 outbreak has touched every area of everyday life and caused immense destruction to the planet. More than 150 nations have been affected by the coronavirus outbreak. Many academics have attempted to create a statistical model that may be used to interpret the COVID-19 data. This article extends to probability theory by developing a unique two-parameter statistical distribution called the half-logistic inverse moment exponential (HLIMExp). Advanced mathematical characterizations of the suggested distribution have explicit formulations. The maximum likelihood estimation approach is used to provide estimates for unknown model parameters. A complete simulation study is carried out to evaluate the performance of these estimations. Three separate sets of COVID-19 data from Al Bahah, Al Madinah Al Munawarah, and Riyadh are utilized to test the HLIMExp model’s applicability. The HLIMExp model is compared to several other well-known distributions. Using several analytical criteria, the results show that the HLIMExp distribution produces promising outcomes in terms of flexibility.
1. Introduction
In recent years, many various of statisticians have been attracted by create new families of distributions for example; exponentiated generalized-G in [1], logarithmic-X family of distributions [2], sine-G in [3], odd Perks-G in [4], odd Lindley-G in [5], truncated Cauchy power-G in [6], truncated Cauchy power Weibull-G-G in [7], Topp-Leone-G in [8], odd Nadarajah–Haghighi-G in [9], the Marshall–Olkin alpha power-G in [10], T-X generator studied in [11], type I half-logistic Burr X-G in [12], KM transformation family in [13], (DUS) transformation family in [14], arcsine exponentiated-X family in [15], Marshall-Olkin odd Burr III-G family in [16], among others.
Reference [17] investigates the half-logistic-G (HL-G) family, a novel family of continuous distributions with an additional shape parameter . The HL-G cumulative distribution function (cdf) is supplied via
The HL-G family’s density function (pdf) is described as respectively. A random variable (R.v)has pdf (2) which would be specified as
Reference [18] presented the moment exponential (MExp) model by allocating weight to the exponential (Exp) model. They established that the MExp distribution is more adaptable than the Exp model. The cdf and pdf files are available. respectively, where is a scale parameter.
The inverse MExp (IMExp) distribution was presented in reference [19], and it is produced by utilizing the R.v , where is as follows (4). The cdf and pdf files in the IMExp distribution are specified as
In this research, we propose an extension of the IMExp model, which is built using the HL-G family and the IMExp model, known as the half-logistic inverse moment exponential (HLIMExp) distribution.
The aim goal of this article can be considered in the following items: (i)To introduce a new two-parameter lifetime model which is called the HLIMExp(ii)The new model is very flexible, and the pdf can take different shapes such as unimodal, right skewness, and heavy tail. Also, the hrf can be increasing, upside-down, and J-shaped(iii)Many numerical values of the moments are calculated in Table 1. And we can note from it that () whenandis increasing, then the numerical values of,,,, variance, skewness (SK), and kurtosis (KU) are decreasing but the numerical values of harmonic mean () are increasing(iv)The simulation study is carried out to assess the behavior of parameters, and the numerical results are mentioned in Tables 2–5. From these tables, we can note that when the value of is increased, the value of and is decreased(v)Three separate sets of COVID-19 data from Al Bahah, Al Madinah Al Munawarah, and Riyadh are utilized to test the HLIMExp model’s applicability. The HLIMExp model is compared to several other well-known distributions. Using several analytical criteria, the results show that the HLIMExp distribution produces promising outcomes in terms of flexibility
The following is an outline of the remainder of this article: Section 2 discusses the construction of the HLIMExp model. Section 3 calculates the basic properties of the distribution, including the linear representation of HLIMExp pdf, order statistics, moments, moment generating function, and conditional moment. In contrast, Section 4 discusses parameter estimation using the maximum likelihood (ML) estimation method. Section 5 employs Monte Carlo simulation techniques. In Section 6, we investigated the potentiality of the HLIMExp using three different metrics of goodness of fit such as the Akaike Information Criterion (IC) (), Consistent AIC (), Bayesian IC (), Hannan-Quinn IC (), Kolmogorov–Smirnov () test, and value (). Finally, Section 7 mentions the conclusion.
2. The New Two-Parameter Statistical Model
A nonnegative R.v with the HLIMExp model is constructed by putting (5) and (6) in (1) and (2), respectively; we should get cdf and pdf.
The survival function (sf) is provided by
The hrf or failure rate and reversed hrf for the HLIMExp are calculated as follows:
Different shapes of the pdf and hrf of HLIMExp with different parameter values are mentioned in Figures 1 and 2.
3. Statistical Properties
We discussed certain HLIMExp distribution features in this part, including linear representation of HLIMExp pdf, moments (Mo), the harmonic mean (), moment generating function (MoGF), and conditional moment (CoMo).
3.1. Linear Representation
A linear form of the pdf and cdf is offered in this part to introduce statistical properties of the HLIMExp distribution. Using the following binomial expansion, where and is a positive real noninteger. By applying (10) in the next term, we get
Inserting the previous equation in (7), we have
Again, applying the general binomial theorem, we get
Inserting the previous equation in (7), we have Again, using the binomial expansion, we get where
3.2. Moments
The Mos of the HLIMExp distribution are discussed in this subsection. Moments are essential in any statistical study, but especially in applications, it can be used to investigate the main properties and qualities of a distribution (e.g., tendency, dispersion, skewness, and kurtosis). The Mo of Z denoted by may be calculated using (8). then,
The inverse Mo of denoted by may be calculated using (8). then,
The harmonic mean of is given by then,
MoGFs are useful for several reasons, one of which is their application to analysis of sums of random variables. The MoGF of is deduced from (7) as
Numerical values for specific values of parameters of the first four ordinary Mos, , , , , variance , skewness (SK), and kurtosis (KU) of the HLIMExp model are reported in Table 1.
3.3. The Conditional Moment
For empirical intents, the shapes of various distributions, such as income quantiles and Lorenz and Bonferroni curves, can be usefully described by the first incomplete moment, which plays a major role in evaluating inequality. These curves have a variety of applications in economics, reliability, demographics, insurance, and medical. Let denote a R.v with the pdf given in (7). The upper incomplete Mo say could be expressed with
Similarly, the lower incomplete Mo function is provided through
4. Method of Maximum Likelihood
Let be a random sample of size from the HLIMExp model with two parameters and ; the log-likelihood function is
For calculation MLE estimation, we need partial derivatives of by parameters where and As result, estimations of the parameters can be found and the solution of the two equations and by using software Mathematica (9).
5. Simulation Results
A simulation result is included in this section to analyze the behavior of estimators in the presence of complete samples by using the Newton-Raphson iteration method and by using Mathematica (8) software. Mean square errors (), lower and upper bound (and ) of confidence interval (CIn), and average length () of 90% and 95% are computed using Mathematica 9. The accompanying algorithm is constructed in the next part: (i)5000 RS of size , 50, 100, 300, 400, and 500 are generated from the HLIMExp model(ii)The parameters’ exact values are chosen(iii)The ML estimates (MLEs), s, , , and for selected values of parameters are computed(iv)Tables 2–5 provide the numerical outputs based on the entire data set
6. Applications
This section concerned with three important real data sets. The data called Saudi Arabia Coronavirus cases (COVID-19) situation in Al Bahah, Al Madinah Al Munawarah and Riyadh regions from January 2022 to May 2022.
The three data sets were obtained from the following electronic address: https://datasource.kapsarc.org/explore/dataset/saudi-arabia-coronavirus-disease-COVID-19-situation/. The data sets are reported in Table 6. The descriptive analysis of the three data sets is reported in Table 7.
Here, in this section, the three data sets mentioned below are examined to demonstrate how the HLIMExp distribution outperforms alternative models, comparing the new model to some models, namely, type II Topp-Leone inverse Rayleigh (TIITOLIR) distribution by [20], half-logistic inverse Rayleigh (HLOIR) distribution by [21], beta transmuted Lindley (BTLi) distribution by [22], the transmuted modified Weibull (TMW) distribution by [23], and the weighted Lindley (W-Li) distribution by [24]. We calculate the model parameters’ MLEs and standard errors (SEs). To evaluate distribution models, we use criteria such as the , , , , , and tests. In contrast, the wider distribution relates to smaller , , , , and and the highest value of . The MLEs of the eight fitted models and their SEs and the numerical values of , , , , , and for the three data sets are presented in Tables 8–10. We find that the HLIMExp distribution with two parameters provides a better fit than seven models. It has the smallest values of , , , , and and the greatest value of among those considered here. Moreover, the plots of empirical cdf, empirical pdf, and PP plots of our competitive model for the three data sets are displayed in Figures 3–5, respectively. The HLIMExp model clearly gives the best overall fit and so may be picked as the most appropriate model for explaining data.
7. Conclusion
We propose a novel two-parameter distribution called the half-logistic inverted moment exponential distribution in this research. HLIMExp’s pdf may be written as a linear combination of IMExp densities. We compute explicit formulas for several of its statistical features, such as HLIMExp pdf linear representation, OS, Moms, MoGF, and CoMo. The greatest likelihood estimate is investigated. The accuracy and performance of estimations are evaluated using simulation results. Three separate sets of COVID-19 data from Al Bahah, Al Madinah Al Munawarah, and Riyadh are utilized to test the HLIMExp model’s applicability. The HLIMExp model is compared to several other well-known distributions. Using several analytical criteria, the results show that the HLIMExp distribution produces promising outcomes in terms of flexibility. In the future works, we can use the new suggested model in many works such as (a) using it to study the statistical inference of the suggested model under different censored schemes, (b) using it to study the statistical inference of the suggested model under different ranked set sampling, (c) accelerated lifetime test can be studied for the new model, and (d) the statistical inference of stress strength model for the new suggested model can be studied.
Data Availability
All data are mentioned in this article.
Conflicts of Interest
The authors declare no conflict of interest.
Acknowledgments
This research was supported by the Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University (IMSIU), Saudi Arabia, Grant No. 21-13-18-034.