Statistical Analysis of COVID-19 Data for Three Different Regions in the Kingdom of Saudi Arabia: Using a New Two-Parameter Statistical Model

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.

Reference [17] investigates the half-logistic-G (HL-G) family, a novel family of continuous distributions with an additional shape parameter θ > 0. 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)Zhas pdf (2) which would be specified asZ~HL − Gðz ; ωÞ: 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 β > 0 is a scale parameter. The inverse MExp (IMExp) distribution was presented in reference [19], and it is produced by utilizing the R.v Z = 1/T, where T 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: The following is an outline of the remainder of this article: Section 2 discusses the construction of the HLIMExp

The New Two-Parameter Statistical Model
A nonnegative R.v Z 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:

Statistical Properties
We discussed certain HLIMExp distribution features in this part, including linear representation of HLIMExp pdf, moments (Mo), the harmonic mean (H), moment generating function (MoGF), and conditional moment (CoMo).

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|z | <1 and b is a positive real noninteger. By applying (10) in the next term, we get    (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 r th 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 r th Mo of Z denoted by μ r may be calculated using (8). then, The r th inverse Mo of Z denoted by μ r may be calculated using (8). then, The harmonic mean of Z 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 ZM z ðtÞ is deduced from (7) as Numerical values for specific values of parameters of the first four ordinary Mos, EðZÞ, EðZ 2 Þ, EðZ 3 Þ, EðZ 4 Þ, variance ðσ 2 Þ, 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 Z denote a R.v with the pdf given in (7). The s th Similarly, the s th lower incomplete Mo function is provided through

Method of Maximum Likelihood
Let z 1 , z 2 , ⋯, z n be a random sample of size n from the HLI-MExp model with two parameters β and θ; the loglikelihood function is For calculation MLE estimation, we need partial derivatives of LðZ | β, θÞ by parameters where As result, estimations of the parameters can be found b β MLE and b θ MLE the solution of the two equations ∂L/∂β = 0 and ∂L/∂θ = 0 by using software Mathematica (9).

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 (Ω1), lower and upper bound (Ω2 and Ω3) of confidence interval (CIn), and average length (Ω4) of 90% and 95% are computed using Mathematica 9. The accompanying algorithm is constructed in the next part:  The three data sets were obtained from the following electronic address: https://datasource.kapsarc.org/explore/ dataset/saudi-arabia-coronavirus-disease-COVID-19situation/. 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   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 V1, V2, V3, V4, and V5 and the greatest value of V6 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.

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.

Data Availability
All data are mentioned in this article.

Conflicts of Interest
The authors declare no conflict of interest.