New Statistical Approaches for Modeling the COVID-19 Data Set: A Case Study in the Medical Sector

Department of Mathematics, College of Sciences and Arts (Muhyil), King Khalid University, Muhyil 61421, Saudi Arabia Department of Mathematics and Computer, College of Sciences, Ibb University, Ibb 70270, Yemen Foundation University Medical College, Foundation University School of Health Sciences, DHA-I, Islamabad 44000, Pakistan Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt Department of Statistics, Jahangirnagar University, Savar, Dhaka 1342, Bangladesh Department of Accounting, College of Business Administration in Hawtat Bani Tamim, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia Mathematics Department, College of Science, Jouf University, P. O. Box 2014, Sakaka, Saudi Arabia Department of Mathematics, Faculty of Science, Minia University, Minia 61519, Egypt


Introduction
In the practice of distribution theory, one of the important tasks is to devise an efficient statistical model for real phenomena of nature. Generally, the statistical distributions are implemented to analyze real-life situations that are uncertain and endangered. For example, the probability distributions are frequently applied to analyze data in (i) engineering and related sectors [1], (ii) healthcare engineering [2], (iii) the economic and financial sector [3], (iv) hydrology [4], (v) education [5], (vi) metrology [6], (vii) biological sector [7], and (viii) sports [8].
Due to the applicability of the probability distributions in applied areas/sectors, numerous approaches (probability models) have been proposed and studied. For example, Afify et al. [9] proposed the MOPG-Weibull distribution for analyzing the engineering data set. For further studies related to the engineering sector (i.e., data modeling in the engineering-related area), we refer to studies by Almarashi et al. [10] and Strzelecki [11].
Klakattawi [12] implemented a new extended Weibull (NE-Weibull) model for statistical analysis of the data sets related to cancer patients. For more studies related to the biomedical/healthcare data sets (i.e., data modeling in the biomedical-related area), we refer to studies by Ahmad et al. [13]; Plana et al. [14]; Xin et al. [15]; and Martinez et al. [16].
Tung et al. [17] proposed the arcsine-Weibull (ASin-Weibull) distribution for analyzing data sets in the business and financial sectors. For more studies related to the financial data sets (i.e., data modeling in the financial-related area), we refer to studies by Zhao et al. [18]; Alfaro et al. [19]; Abubakar and Sabri [20]; and Rana et al. [21].
Bakouch et al. [22] implemented the Gumbel model for analyzing the hydrology data set. Singh et al. [23] provided the assessment of groundwater quality data in Nigeria. Hassan et al. [24] implemented the truncated power Lomax (TP-Lomax) distribution for analyzing the flood data set. For other studies related to the hydrology data sets, we refer to studies by Karahacane et al. [25]; Dodangeh et al. [26]; and Tegegne et al. [27].
Maurya et al. [33] proposed a new method called the logarithm transformed (LT) family for introducing flexible probability distributions. Let X has the LT family, if its DF (distribution function) R(x; ψ) is where x ∈ R and M(x; ψ) is a baseline DF. Liu et al. [34] introduced a useful method, namely, a FRL-X (flexible reduced logarithmic-X) family for obtaining the modified versions of the existing distributions. Let X has the FRL-X distributions, if its DF R(x; β, ψ) is where β ∈ R + is an additional parameter. Ahmad et al. [35] proposed another new class of probability distributions, called the weighted T-X (WT-X) family of distributions. e DF R(x; ψ) of the WT-X distributions is with PDF r(x; ψ) given by where m(x; ψ) � d/dxM(x; ψ). Wang et al. [36] studied a NG-X (new generalized-X) family with DF R(x; ψ, θ), provided by where θ ∈ R + is the additional parameter. Bo et al. [37] proposed another useful method, namely, the APTEx-X (alpha power-transformed extended-X) family of distributions. e DF R(x; ψ, α 1 ) of the APTEx-X family is where α 1 ≠ 1, α 1 ∈ R + is an additional parameter.
In the next section, we obtain different modifications of the inverse Weibull (IW) distribution by implementing the approaches defined in Eqs. (1)- (6). For every new modified form of the IW model, the plots of the PDF are also obtained.

Some New Modifications of the Inverse
Weibull Distribution is section offers some new different extensions of the IW distribution by incorporating the well-known approaches described in Section 1. Consider the DF M(x; ψ), PDF m(x; ψ), SF (survival function) S(x; ψ), HF (hazard function) h(x; ψ), and cumulative HF H(x; ψ) of the IW distribution (with parameters α ∈ R + and ψ ∈ R + ) given by respectively, where ψ � (α, ψ).

e Logarithm Transformed-Inverse Weibull Distribution.
Here, we implement the LT family approach (see (1)) to introduce a new version of the IW model. e new version of the IW model is called the logarithmic transformed-inverse Weibull (LT-IW) distribution. e DF of the LT-IW distribution is obtained by using (7) in (1). Let X has the LT-IW model, if its DF is expressed by Associating to Eq. (9), the PDF r(x; ψ), SF R(x; ψ), and HF h(x; ψ) of the LT-IW model are given by 2 Complexity respectively. e PDF plots of the LT-IW model are provided in Figure 1. e plots of the LT-IW model in Figure 1 are obtained

A Flexible Reduced Logarithmic-Inverse Weibull
Distribution. Here, we use the FRL-X approach (see (2)) to introduce a novel generalized version of the IW distribution.
e new updated form of the IW distribution is called the FRL-IW distribution. e DF of the FRL-IW model is obtained by using Eq. (7) in (2). Let X has the FRL-IW distribution, if its DF is given by Corresponding to Eq. (12), the PDF r(x; β, ψ), SF R(x; β, ψ), and HF h(x; β, ψ) of the FRL-IW model are given by respectively. Different plots of r(x; β, ψ) of the FRL-IW distribution are presented in Figure 2. e plots of r(x; β, ψ) in Figure 2 are

e Weigted TX-Inverse Weibull Distribution.
In this section, we apply the WT-X distribution approach to propose a modified version of the IW distribution, called the weighted TX-inverse Weibull (WT X-IW) distribution. e DF of the WT X-IW distribution is obtained by using Eq. (7) in (3). Let X has the WT X-IW model, if its DF is In link to (15), the PDF r(x; ψ), SF R(x; ψ), and HF h(x; ψ) of the WT X-IW model are given by respectively.

A New Generalized-Inverse Weibull Distribution.
In this section, we incorporate a NG-X method and introduce another extended form of the IW distribution.
Here, we consider four frequently used analytical measures (statistical tests or statistical procedures) to show which probability distribution better fits the biomedical data. ese measures are given by the following: (i) e AIC: (ii) e BIC: (iii) e CAIC: (iv) e HQIC: In a general sense, the above-mentioned analytical measures are used for comparative analysis. A statistical model that has smaller values of the statistical tests is considered the most suitable model among other competing statistical models. Table 1 gives the MLEs (α MLE , ψ MLE , β MLE , θ MLE , α 1MLE ) of the competitive probability models using the COVID-19 data set. e analytical measures for the COVID-19 data using the considered probability models are presented in Table 2.
Based on the reported results in As we have seen that the NG-IW model provides a close fit to the biomedical data. erefore, we provide the profiles of the log-likelihood function (LLF) of the NG-IW distribution. Based on the α MLE , ψ MLE , and θ MLE , the LLF profiles of the NG-IW distribution are obtained in Figure 7. e graphs in Figure 7 confirm the unique values of the α MLE , ψ MLE , and θ MLE .
After the numerical illustration of the NG-IW model using the COVID-19 data set (see Table 2), next we show visually that the NG-IW model provides the best fit to the COVID-19 data set. For the visual illustration of the NG-IW model, the plots of the fitted PDF r(x; θ, ψ), DF R(x; θ, ψ), SF R(x; θ, ψ), HF h(x; θ, ψ), cumulative HF H(x; θ, ψ), probability-probability (PP), and QQ (quantile-quantile) are obtained in Figure 8. respectively.
Complexity 7 e empirical and fitted plots in Figure 8 reveal that the NG-IW distribution provides a close fit to the COVID-19 data set.

Concluding Remarks
In recent times, statistical models have been frequently used to analyze data in applied sectors, such as engineering, hydrology, education, finance, and biomedical sectors. To provide the best description of the phenomena under consideration, a number of statistical models have been introduced and implemented. Among these models, the IW distribution has received considerable attention. erefore, numerous modifications of the IW distribution have been proposed and applied. In this paper, we introduced five different modifications of the IW distribution for modeling real-life data sets. Finally, the new modified forms of the IW distribution were applied to real-life data taken from the biomedical sector. e practical application showed that the NG-IW distribution was the best candidate model for analyzing the COVID-19 data set.
In the future, we are motivated to implement the LT-IW, FRL-IW, WTX-IW, NG-IW, and APTE-IW models in other applied sectors. Furthermore, the bivariate extensions of the LT-IW, FRL-IW, WTX-IW, NG-IW, and APTE-IW models can also be introduced to deal with the bivariate data sets. Bayesian estimation of the LT-IW, FRL-IW, WTX-IW, NG-IW, and APTE-IW models using different types of censored samples can be discussed [39].

Data Availability
All data are included in the paper.

Conflicts of Interest
e authors declare no conflicts of interest.