# A New Generalized-X Family for Analyzing the COVID-19 Data Set: a Case Study

Authors: Yue Xin; Yinghui Zhou; Getachew Tekle Mekiso
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Nowadays, researchers in applied sectors are highly motivated to propose and study new generalizations of the existing distributions to provide the best fit to data. To provide a close fit to data in numerous sectors, a series of new distributions have been proposed. In this study, we propose a new family called the new generalized-X (for short, “NG-X”) family of distributions. Based on the NG-X method, a novel modification of the Weibull model called the new generalized-Weibull (for short, “NG-Weibull”) distribution is studied. The heavy-tailed characteristics of the NG-X distributions are derived. The maximum likelihood estimators of the NG-X distributions are also obtained. Based on the special case (i.e., NG-Weibull) of the NG-X family, a simulation study is provided. The practical performance of the new NG-Weibull model is assessed by analyzing the COVID-19 data set. The fitting results of the NG-Weibull model are compared with three other competing models. Based on certain statistical measures, it is observed that the NG-Weibull model is the best competitive model.


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## Body

## 1. Introduction

In the practice of applied statistics, statisticians are highly motivated and attracted to introduce new probability distributions. This motivation has gained popularity in the recent decade. In this credit, the development and introduction of new families of distribution have received much attention. The new families are developed by introducing new parameters [1–4].Theoretically and practically, some distributions do not have enough flexibility to counter the complex forms of the data sets. For example, the classical models do not provide a reasonable fit to the extreme value data [5]. This is the main attracting reason to go for the development of new statistical distributions. The development of the modified forms of the existing distributions helps to improve the fitting power of the existing distributions [6].Recently, Chipepa et al. [7] introduced a new generalized family of distributions called the OGHLW-G (odd generalized half-logistic Weibull-G) family of distributions. They have obtained some structural properties of the OGHLW-G family. They have also discussed some special models of the OGHLW-G family to show the importance of the new models. Chipepa et al. [8] also proposed the Burr III-Topp-Leone-G family of distributions for analyzing (i) carbon fibres data, (ii) fracture toughness of silicon nitride data, and (iii) turbocharger failure times data.Ahmad and Hussain [9] introduced a NEF-Weibull (new extended flexible-Weibull) distribution to model the lifetime data with bathtub-shaped failure rates. Some other modified and extended distributions include the beta-Weibull (B-Weibull) distribution of Famoye et al. [10]; the flexible Weibull (FWEx) distribution of Bebbington et al. [11]; the generalized modified Weibull (GM-Weibull) distribution proposed by Carrasco et al. [12]; the Kumaraswamy Weibull (K-Weibull) distribution proposed by Cordeiro et al. [13]; the beta modified Weibull (BM-Weibull) distribution of Silva et al. [14]; the exponentiated modified Weibull extension (EMWE) distribution introduced by Sarhan and Apaloo [15]; the Gumbel-Lomax (GL) distribution suggested by Tahir et al. [16]; and the Marshall–Olkin inverse Lomax distribution (MO-ILD) studied by Maxwell et al. [17].Another prominent approach for extending and updating the existing distributions is called the new exponential-X family of Ahmad et al. [5]. They introduced the new exponential-X family by combining the exponential distribution with the T-X approach of Alzaatreh et al. [18].Eliwa et al. [19] further contributed to the literature by introducing the exponentiated form of the Chen-G family of distributions. For other recent contributions to the development of new approaches for updating the existing distributions, we refer to (i) the truncated burr-XG family [20], (ii) the Fréchet Topp Leone-G family [21], (iii) the transmuted alpha power-G family [22], (iv) the Marshall–Olkin Zubair-G family [23], (v) a new family of continuous (NFC) distributions [24], and (vi) the Topp-Leone Odd Burr III-G (TLOBIII-G) family [25].In this article, we propose a NG-X family of distributions to obtain the modified and updated versions of the existing distributions. The NG-X method can be implemented to obtain new flexible distributions for modeling data in various sectors such as reliability, survival analysis, healthcare, biomedical engineering, and lifetime studies. The next section is devoted to introduce the proposed NG-X family of distributions.


## 2. A New Generalized-X Family of Distributions: the Proposed Method

In this section, the probability density function (PDF), cumulative distribution function (CDF), survival function (SF), hazard function (HF), and cumulative HF (CHF) for the NG-X family are computed.Definition: A random variableX has a NG-X family, if it is CDF Gx;α,δ,Ξ is given by(1)Gx;α,δ,Ξ=1−1−1−δ2Fx;Ξ1−δFx;Ξ2α,α>0,δ∈0,1,x∈ℝ.The function defined in (1) is a valid DF, if and only if, δ∈0,1. Corresponding to Gx;α,δ,Ξ, the PDF gx;α,δ,Ξ is given by(2)gx;α,δ,Ξ=α1−δ2fx;Ξ1+δFx;Ξ1−δFx;Ξ2α+1F¯x;Ξ1−δ2Fx;Ξα−1.Furthermore, in link toGx;α,δ,Ξ and gx;α,δ,Ξ, the G¯x;α,δ,Ξ, HF hx;α,δ;Ξ, and CHF Hx;α,δ;Ξ are given by(3)G¯x;α,δ,Ξ=1−1−δ2Fx;Ξ1−δFx;Ξ2α,hx;α,δ,Ξ=α1−δ2fx;Ξ1+δFx;Ξ1−δFx;ΞF¯x;Ξ1−δ2Fx;Ξ−1,Hx;α,δ,Ξ=−αlog1−1−δ2Fx;Ξ1−δFx;Ξ2,respectively.In this paper, we implement the NG-X family approach to introduce an updated version of the Weibull distribution called the NG-Weibull (new generalized Weibull) distribution. In the next section, we provide the expressions for the PDF, CDF, SF, and HF of the NG-Weibull distribution. Furthermore, the plots for the PDF, CDF, SF, and HF of the NG-Weibull distribution are also provided.


## 3. A NG-Weibull Distribution: a Special Model

With parametersθ>0 and γ>0, consider the CDF Fx;Ξ of the Weibull distribution is given by(4)Gx;Ξ=1−e−γxθ,x≥0.The respective PDFgx;Ξ, SF Sx;Ξ, and HF hx;Ξ are, respectively, given by(5)gx;Ξ=θγxθ−1e−γxθ,x>0,Sx;Ξ=e−γxθ,x>0,hx;Ξ=θγxθ−1,x>0,where Ξ=α,ϑ.Using (4) in (1), we get the CDF of the NG-Weibull distribution, given by(6)Gx=1−1−1−δ21−e−γxθ1−δ1−e−γxθ2α,x≥0.The PDF of the NG-Weibull distribution is given by(7)gx=α1−δ2θγxθ−1e−αγxθ1+δ1−e−γxθ1−δ1−e−γxθ2α+11−δ21−e−γxθα−1.Some possible behaviors forgx of the NG-Weibull are presented in Figure 1. These plots are obtained for (i) α=2.9,δ=0.8,γ=2.5,θ=4.9 (red curve), (ii) α=2.7,δ=0.7,γ=0.6,θ=4.1 (blue curve), (iii) α=3.1,δ=0.7,γ=4.6,θ=0.9 (gold curve), (iv) α=3.8,δ=0.1,γ=3.2,θ=3.5 (green curve), and (iv) α=0.8,δ=0.4,γ=3.8,θ=0.5 (black curve).Figure 1
The PDF plots of the NG-Weibull distribution.Furthermore, the SF and HF of the NG-Weibull distribution are given by(8)G¯x=1−1−δ21−e−γxθ1−δ1−e−γxθ2α,hx=α1−δ2θγxθ−1e−αγxθ1+δ1−e−γxθ1−δ1−e−γxθ1−δ21−e−γxθ−1,respectively.Forα=1.2,δ=0.99,γ=7.7, and θ=0.6, the plots of CDF Gx and SF G¯x=1−Gx of the NG-Weibull distribution are provided in Figure 2.Figure 2
The plots of the CDF and SF of the NG-Weibull distribution.Furthermore, some possible behaviors forhx of the NG-Weibull are presented in Figure 3. These plots are obtained for (i) α=0.8,δ=0.5,γ=2.8,θ=0.5 (red curve), (ii) α=1.8,δ=0.2,γ=2.1,θ=1.2 (green curve), and (iii) α=5.2,δ=0.8,γ=1.1,θ=0.2 (blue curve).Figure 3
The plots of the HF of the NG-Weibull distribution.The NG-Weibull distribution is an updated version of the Weibull distribution. It has certain advantages over some other updated versions of the Weibull distribution.(i)
The NG-Weibull distribution has a closed-form CDF, which makes it easier to generate random numbers for the simulation study.(ii)
The NG-Weibull is capable of capturing the bathtub shape of the HF. Statistical models with a bathtub shape of the HF are very useful for modeling the healthcare engineering data sets.(iii)
The NG-Weibull distribution provides a close fit to the COVID-19 data set. Therefore, the implementation of the NG-Weibull distribution could be a useful choice for modeling data in health and other related sectors.Besides the abovementioned advantages, the NG-Weibull distribution also has certain limitations. For example,(i)
The NG-Weibull distribution is a continuous model, and it is employed to analyze the mortality rates of COVID- 19 infected people. Therefore, it could not be implemented to analyze the data sets that are not continuous in nature, such as (i) the number of COVID-19 daily registered cases, (ii) the number of COVID-19 confirmed deaths, and (iii) the number of COVID-19 recovered cases.(ii)
Since the PDF of the NG-X distributions has a complicated form, more computational work would be required to derive the distributional properties.


## 4. The HT Characteristics

The statistical distributions that possess HT (heavy-tailed) behavior are very useful for dealing with extreme value phenomena. In this section, we prove mathematically the HT characteristics of the NG-X distributions.


### 4.1. The Regularly Varying Tail Behavior

This subsection is devoted to proving the RVTB (regularly varying tail behavior) of the NG-X distributions. According to Karamata’s theorem [26], in terms of SF G¯x;α,δ;Ξ, we haveTheorem IfF¯x;Ξ is the SF of the regular varying (RVa) distribution, then G¯x;α,δ;Ξ is also an RVa distribution.Proof.
Let assumelimx⟶∞F¯λx;Ξ/F¯x;Ξ=fλ is finite but nonzero ∀λ>0. Using (3), we have(9)limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−1−δ2Fx;Ξ1−δFx;Ξ2α/1−1−δ2Fx;Ξ1−δFx;Ξ2α.
Usingα=1 in (9), we get(10)limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−1−δ2Fλx;Ξ1−δFλx;Ξ2/1−1−δ2Fx;Ξ1−δFx;Ξ2.
On simplification, we get(11)limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−δ2Fλx;Ξ1−δFλx;Ξ2/1−Fx;Ξ1−δ2Fx;Ξ1−δFx;Ξ2,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−δ2Fλx;Ξ1−Fx;Ξ1−δ2Fx;Ξ1−δFx;Ξ21−δFλx;Ξ2,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δFλx;Ξ21−δ2Fx;Ξ1−δFx;Ξ21−δ2Fλx;Ξ.
Since,Fx;Ξ is a CDF. Therefore, we have(12)limx⟶∞Fx;Ξ=1.
Hence, from (11), we get(13)limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ21−δ21−δ21−δ2,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=fλ.
which is finite but nonzero∀u>0; thus, G¯x;α,δ;Ξ is an RVa distribution.
Now, we prove the HT behavior of the special case of the NG-X family. Using (4) in (11), we get(14)limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ1−e−γλxθ21−δ21−e−γxθ1−δ1−e−γxθ21−δ21−e−γλxθ,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ1−e−γλ.∞θ21−δ21−e−γ.∞θ1−δ1−e−γ.∞θ21−δ21−e−γλ.∞θ,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ1−e−∞21−δ21−e−∞1−δ1−e−∞21−δ21−e−∞,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ1−1/e∞21−δ21−1/e∞1−δ1−1/e∞21−δ21−1/e∞,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ1−1/∞21−δ21−1/∞1−δ1−1/∞21−δ21−1/∞,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ1−021−δ21−01−δ1−021−δ21−0,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ21−δ21−δ21−δ2,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=fλ,∂∂ΞLx;α,δ,Ξ=0.


### 4.2. A Supportive Application of the RVTB

Consider the distribution ofX has the power-law behavior (PLB), then we have(15)F¯x;Ξ=ℙX>x∼x−λ.By incorporating Karamata’s theorem, we can writeF¯x;Ξ as(16)F¯x;Ξ=x−λLx;Ξ,where Lx;Ξ stands for the slowly varying function. From (3), we have(17)G¯x;α,δ;Ξ=1−1−δ2Fx;Ξ1−δFx;Ξ2α.Usingα=1 in (17), we get(18)G¯x;α,δ;Ξ=1−1−δ2Fx;Ξ1−δFx;Ξ2,G¯x;α,δ;Ξ=1−Fx;Ξ1−δ2Fx;Ξ1−δFx;Ξ2.SinceF¯x;Ξ=1−Fx;Ξ=x−λ. Therefore, we can write (18), as follows:(19)G¯x;α,δ;Ξ=x−λ1−δ2Fx;Ξ1−δFx;Ξ2,G¯x;α,δ;Ξ=x−λLx;Ξ,where(20)Lx;Ξ=1−δ2Fx;Ξ1−δFx;Ξ2.IfLx;Ξ is a slowly varying function, then the result obtained in (19) is true. According to Resnick [27], for all λ>0, we have to prove(21)limx⟶∞Ltx;ΞLx;Ξ=1.By incorporating (19), we get(22)Ltx;ΞLx;Ξ=1−δ2Ftx;Ξ1−δFtx;Ξ2×1−δFx;Ξ21−δ2Fx;Ξ,Ltx;ΞLx;Ξ=1−δ21−tx−λ1−δ1−tx−λ2×1−δ1−x−λ21−δ21−x−λ,Ltx;ΞLx;Ξ=1−δ21−t−λx−λ1−δ1−t−λx−λ2×1−δ1−x−λ21−δ21−x−λ,Ltx;ΞLx;Ξ=1−δ21−t−λ/xλ1−δ1−t−λ/xλ2×1−δ1−x−λ21−δ21−x−λ,Ltx;ΞLx;Ξ=1−δ21−t−λ/xλ1−δ1−t−λ/xλ2×1−δ1−1/xλ21−δ21−1/xλ.Applyinglimx⟶∞ to both sides of (22), we get(23)limx⟶∞Ltx;ΞLx;Ξ=limx⟶∞1−δ21−t−λ/xλ1−δ1−t−λ/xλ2×1−δ1−1/xλ21−δ21−1/xλ,limx⟶∞Ltx;ΞLx;Ξ=1−δ21−t−λ/∞1−δ1−t−λ/∞2×1−δ1−1/∞21−δ21−1/∞,limx⟶∞Ltx;ΞLx;Ξ=1−δ21−01−δ1−02×1−δ1−021−δ21−0,limx⟶∞Ltx;ΞLx;Ξ=1−δ21−δ2×1−δ21−δ2,limx⟶∞Ltx;ΞLx;Ξ=1.


## 4.1. The Regularly Varying Tail Behavior

This subsection is devoted to proving the RVTB (regularly varying tail behavior) of the NG-X distributions. According to Karamata’s theorem [26], in terms of SF G¯x;α,δ;Ξ, we haveTheorem IfF¯x;Ξ is the SF of the regular varying (RVa) distribution, then G¯x;α,δ;Ξ is also an RVa distribution.Proof.
Let assumelimx⟶∞F¯λx;Ξ/F¯x;Ξ=fλ is finite but nonzero ∀λ>0. Using (3), we have(9)limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−1−δ2Fx;Ξ1−δFx;Ξ2α/1−1−δ2Fx;Ξ1−δFx;Ξ2α.
Usingα=1 in (9), we get(10)limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−1−δ2Fλx;Ξ1−δFλx;Ξ2/1−1−δ2Fx;Ξ1−δFx;Ξ2.
On simplification, we get(11)limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−δ2Fλx;Ξ1−δFλx;Ξ2/1−Fx;Ξ1−δ2Fx;Ξ1−δFx;Ξ2,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−δ2Fλx;Ξ1−Fx;Ξ1−δ2Fx;Ξ1−δFx;Ξ21−δFλx;Ξ2,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δFλx;Ξ21−δ2Fx;Ξ1−δFx;Ξ21−δ2Fλx;Ξ.
Since,Fx;Ξ is a CDF. Therefore, we have(12)limx⟶∞Fx;Ξ=1.
Hence, from (11), we get(13)limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ21−δ21−δ21−δ2,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=fλ.
which is finite but nonzero∀u>0; thus, G¯x;α,δ;Ξ is an RVa distribution.
Now, we prove the HT behavior of the special case of the NG-X family. Using (4) in (11), we get(14)limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ1−e−γλxθ21−δ21−e−γxθ1−δ1−e−γxθ21−δ21−e−γλxθ,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ1−e−γλ.∞θ21−δ21−e−γ.∞θ1−δ1−e−γ.∞θ21−δ21−e−γλ.∞θ,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ1−e−∞21−δ21−e−∞1−δ1−e−∞21−δ21−e−∞,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ1−1/e∞21−δ21−1/e∞1−δ1−1/e∞21−δ21−1/e∞,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ1−1/∞21−δ21−1/∞1−δ1−1/∞21−δ21−1/∞,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ1−021−δ21−01−δ1−021−δ21−0,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ×1−δ21−δ21−δ21−δ2,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=limx⟶∞1−Fλx;Ξ1−Fx;Ξ,limx⟶∞G¯λx;α,δ;ΞG¯x;α,δ;Ξ=fλ,∂∂ΞLx;α,δ,Ξ=0.


## 4.2. A Supportive Application of the RVTB

Consider the distribution ofX has the power-law behavior (PLB), then we have(15)F¯x;Ξ=ℙX>x∼x−λ.By incorporating Karamata’s theorem, we can writeF¯x;Ξ as(16)F¯x;Ξ=x−λLx;Ξ,where Lx;Ξ stands for the slowly varying function. From (3), we have(17)G¯x;α,δ;Ξ=1−1−δ2Fx;Ξ1−δFx;Ξ2α.Usingα=1 in (17), we get(18)G¯x;α,δ;Ξ=1−1−δ2Fx;Ξ1−δFx;Ξ2,G¯x;α,δ;Ξ=1−Fx;Ξ1−δ2Fx;Ξ1−δFx;Ξ2.SinceF¯x;Ξ=1−Fx;Ξ=x−λ. Therefore, we can write (18), as follows:(19)G¯x;α,δ;Ξ=x−λ1−δ2Fx;Ξ1−δFx;Ξ2,G¯x;α,δ;Ξ=x−λLx;Ξ,where(20)Lx;Ξ=1−δ2Fx;Ξ1−δFx;Ξ2.IfLx;Ξ is a slowly varying function, then the result obtained in (19) is true. According to Resnick [27], for all λ>0, we have to prove(21)limx⟶∞Ltx;ΞLx;Ξ=1.By incorporating (19), we get(22)Ltx;ΞLx;Ξ=1−δ2Ftx;Ξ1−δFtx;Ξ2×1−δFx;Ξ21−δ2Fx;Ξ,Ltx;ΞLx;Ξ=1−δ21−tx−λ1−δ1−tx−λ2×1−δ1−x−λ21−δ21−x−λ,Ltx;ΞLx;Ξ=1−δ21−t−λx−λ1−δ1−t−λx−λ2×1−δ1−x−λ21−δ21−x−λ,Ltx;ΞLx;Ξ=1−δ21−t−λ/xλ1−δ1−t−λ/xλ2×1−δ1−x−λ21−δ21−x−λ,Ltx;ΞLx;Ξ=1−δ21−t−λ/xλ1−δ1−t−λ/xλ2×1−δ1−1/xλ21−δ21−1/xλ.Applyinglimx⟶∞ to both sides of (22), we get(23)limx⟶∞Ltx;ΞLx;Ξ=limx⟶∞1−δ21−t−λ/xλ1−δ1−t−λ/xλ2×1−δ1−1/xλ21−δ21−1/xλ,limx⟶∞Ltx;ΞLx;Ξ=1−δ21−t−λ/∞1−δ1−t−λ/∞2×1−δ1−1/∞21−δ21−1/∞,limx⟶∞Ltx;ΞLx;Ξ=1−δ21−01−δ1−02×1−δ1−021−δ21−0,limx⟶∞Ltx;ΞLx;Ξ=1−δ21−δ2×1−δ21−δ2,limx⟶∞Ltx;ΞLx;Ξ=1.


## 5. Estimation and Simulation Study

In this section, the procedure of the maximum likelihood estimation for the model parameters of the NG-X distributions is implemented. Furthermore, a simulation study for different values of the model parameters is discussed.


### 5.1. Maximum Likelihood Estimation

In this subsection, the computation of the maximum likelihood estimators (MLEs) for the parametersα,δ,Ξ of NG-X is provided. Consider x1,x2,…,xn be a set of observed values of a sample randomly selected from NG-X distributions with parameters α,δ, and Ξ. In link to the PDF of the NG-X distributions given by (2), the LF (likelihood function) is(24)Lx;α,δ,Ξ=∏i=1nα1−δ2fxi;Ξ1+δFxi;Ξ1−δFxi;Ξ2α+1F¯xi;Ξ1−δ2Fxi;Ξα−1.Corresponding toLx;α,δ,Ξ, the log LF (LLF) is given by(25)Lx;α,δ,Ξ=nlogα+2nlog1−δ+∑i=1nlogfxi;Ξ+∑i=1nlog1+δFxi;Ξ+α−1∑i=1nlogF¯xi;Ξ1−δ2Fxi;Ξ−2α+1∑i=1nlog1−δFxi;Ξ.On behalf of the parametersα,δ,Ξ of the NG-X distributions, the partial derivative of the LLF are, respectively, given by(26)∂∂αLx;α,δ,Ξ=nα+∑i=1nlogF¯xi;Ξ1−δ2Fxi;Ξ−2∑i=1nlog1−δFxi;Ξ,∂∂δLx;α,δ,Ξ=−2n1−δ+∑i=1nFxi;Ξ1+δFxi;Ξ+2α+1∑i=1nFxi;Ξ1−δFxi;Ξ−2α−1∑i=1nδF¯xi;ΞFxi;ΞF¯xi;Ξ1−δ2Fxi;Ξ,∂∂ΞLx;α,δ,Ξ=∑i=1n∂/∂Ξfxi;Ξfxi;Ξ+δ∑i=1n∂/∂ΞFxi;Ξ1+δFxi;Ξ+δ2α+1∑i=1n∂/∂ΞFxi;Ξ1−δFxi;Ξ+α−1∑i=1n1−δ2Fxi;Ξ∂/∂ΞF¯xi;Ξ−δF¯xi;Ξ∂/∂ΞFxi;ΞF¯xi;Ξ1−δ2Fxi;Ξ.The MLEsα^MLE,δ^MLE,Ξ^MLE of the parameters α,δ,Ξ can be obtained by solving(27)∂∂αLx;α,δ,Ξ=0,∂∂δLx;α,δ,Ξ=0,∂∂ΞLx;α,δ,Ξ=0.


### 5.2. Simulation Study

In this subsection, the performance of MLEsα^MLE,δ^MLE,Ξ^MLE of α,δ,Ξ is evaluated by conducting a simulation study. The process is carried out as follows:(i)
A sequence of RS (random sample), sayX1,X2,…,Xn of sizes, n=25,50,…,975, and 1000 are obtained from the NG-Weibull distribution.(ii)
Two statistical quantities (MSE and bias) are considered to evaluateα^MLE,δ^MLE, and Ξ^MLE. These quantities are, respectively, given by(28)Biasν^=11000∑i=11000ν^−ν,MSEν^=11000∑i=11000ν^−ν2.(iii)
Two different cases are considered for the four parameters of the NG-Weibull distribution.(iv)
One thousand repetitions are done to compute the bias and the MSE forα^MLE,δ^MLE, and Ξ^MLE.The numerical results are provided in Tables1–3. From the numerical illustration presented in Tables 1–3, it can be observed that as sample size n increases (i) the values of the MLEs α^MLE,δ^MLE,Ξ^MLE get closer and closer to the parameters α,δ,Ξ, and (ii) the values of the MSE and bias of α^MLE,δ^MLE, and Ξ^MLE tend to zero.Table 1
Forα=0.4,δ=0.5,θ=0.6,γ=0.5, the simulation results of the NG-Weibull model.
nParametersMLEsMSEsBiases25α1.37764775.37898760.9706542δ0.61287570.17387650.1129876θ0.58678680.09129870.1398765γ1.22289702.63498070.7226754100α0.86476652.00334520.4642937δ0.56609880.15132870.0963454θ0.58334520.07329380.1019876γ0.95709871.22320930.4578769375α0.54065571.24098760.2409865δ0.49920980.10798710.0665438θ0.60234520.04090900.0876548γ0.70620930.93987820.2062983525α0.47809870.74887610.0788967δ0.49209090.46987620.0198764θ0.60128370.03876580.0209873γ0.61535460.16098720.1152342750α0.42323410.48767540.0238972δ0.473–0980.26098720.0098766θ0.607–0920.01987650.0088735γ0.56527890.09998760.0659897950α0.41276580.05509870.0120981δ0.48309870.01889760.0077653θ0.60509560.00150980.0059887γ0.56823760.08765430.04987061000α0.40428330.03008950.0040892δ0.50102930.00296740.0038765θ0.60854120.00108760.0024498γ0.51823980.01987630.0286710Table 2
Forα=0.5,δ=0.5,θ=0.6,γ=1, the simulation results of the NG-Weibull model.
nParametersMLEsMSEsBiases25α1.53988754.76609231.0399876δ0.53995850.17512340.1295234θ0.59965390.31329380.1089097γ1.51909872.22709840.5199870100α0.96987682.07675890.4698795δ0.521–0980.10642560.1087653θ0.59487640.17865670.0987609γ1.45130981.56812340.4517563375α0.62887650.60175410.1280987δ0.470–0980.09543780.0865309θ0.60932450.13987640.0554233γ1.16387690.50441220.2798734525α0.58209870.35998720.0974076δ0.47265430.06543690.0680969θ0.60909870.10754890.0187653γ1.10998750.32541650.1094234750α0.53420980.13509870.0347865δ0.47323900.01745780.0260987θ0.60727600.04987630.0076538γ1.04912340.17219820.0493452950α0.50620910.04778920.0068759δ0.47123420.01045680.0284327θ0.60820950.00267890.0064231γ1.03476540.10542170.03468991000α0.50504710.02789010.0047887δ0.49765400.00856740.0206541θ0.60798760.00095630.0030982γ1.01987690.06809800.0156983Table 3
Forα=0.2,δ=0.5,θ=0.6,γ=0.3, the simulation results of the NG-Weibull model.
nParametersMLEsMSEsBiases25α0.64898761.91809870.4488765δ0.63987650.16609890.1399780θ0.57065430.12654390.0892543γ1.13878902.88098760.8380987100α0.39676540.67287650.1966543δ0.61509870.12964530.1158987θ0.56354320.10109870.0667890γ0.84767091.54134580.5478654375α0.24975860.10576540.1094410δ0.52909870.09589090.0997981θ0.58957990.09665490.0409783γ0.45787900.21165430.1578091525α0.23098760.09909870.0800876δ0.51369870.07709900.0536543θ0.59309890.03275430.0166541γ0.43065430.16078900.1309876750α0.21687610.01876540.0460987δ0.51675680.02209870.0267659θ0.59598760.01745690.0090987γ0.39509870.08809870.0954569950α0.21398010.01064320.0139878δ0.50598750.02020980.0090645θ0.59654320.00157890.0060989γ0.32954900.05167480.05954371000α0.21005610.01006580.0101432δ0.49109820.00912760.0028976θ0.59965340.00100190.0019909γ0.31097550.03409870.0287654


## 5.1. Maximum Likelihood Estimation

In this subsection, the computation of the maximum likelihood estimators (MLEs) for the parametersα,δ,Ξ of NG-X is provided. Consider x1,x2,…,xn be a set of observed values of a sample randomly selected from NG-X distributions with parameters α,δ, and Ξ. In link to the PDF of the NG-X distributions given by (2), the LF (likelihood function) is(24)Lx;α,δ,Ξ=∏i=1nα1−δ2fxi;Ξ1+δFxi;Ξ1−δFxi;Ξ2α+1F¯xi;Ξ1−δ2Fxi;Ξα−1.Corresponding toLx;α,δ,Ξ, the log LF (LLF) is given by(25)Lx;α,δ,Ξ=nlogα+2nlog1−δ+∑i=1nlogfxi;Ξ+∑i=1nlog1+δFxi;Ξ+α−1∑i=1nlogF¯xi;Ξ1−δ2Fxi;Ξ−2α+1∑i=1nlog1−δFxi;Ξ.On behalf of the parametersα,δ,Ξ of the NG-X distributions, the partial derivative of the LLF are, respectively, given by(26)∂∂αLx;α,δ,Ξ=nα+∑i=1nlogF¯xi;Ξ1−δ2Fxi;Ξ−2∑i=1nlog1−δFxi;Ξ,∂∂δLx;α,δ,Ξ=−2n1−δ+∑i=1nFxi;Ξ1+δFxi;Ξ+2α+1∑i=1nFxi;Ξ1−δFxi;Ξ−2α−1∑i=1nδF¯xi;ΞFxi;ΞF¯xi;Ξ1−δ2Fxi;Ξ,∂∂ΞLx;α,δ,Ξ=∑i=1n∂/∂Ξfxi;Ξfxi;Ξ+δ∑i=1n∂/∂ΞFxi;Ξ1+δFxi;Ξ+δ2α+1∑i=1n∂/∂ΞFxi;Ξ1−δFxi;Ξ+α−1∑i=1n1−δ2Fxi;Ξ∂/∂ΞF¯xi;Ξ−δF¯xi;Ξ∂/∂ΞFxi;ΞF¯xi;Ξ1−δ2Fxi;Ξ.The MLEsα^MLE,δ^MLE,Ξ^MLE of the parameters α,δ,Ξ can be obtained by solving(27)∂∂αLx;α,δ,Ξ=0,∂∂δLx;α,δ,Ξ=0,∂∂ΞLx;α,δ,Ξ=0.


## 5.2. Simulation Study

In this subsection, the performance of MLEsα^MLE,δ^MLE,Ξ^MLE of α,δ,Ξ is evaluated by conducting a simulation study. The process is carried out as follows:(i)
A sequence of RS (random sample), sayX1,X2,…,Xn of sizes, n=25,50,…,975, and 1000 are obtained from the NG-Weibull distribution.(ii)
Two statistical quantities (MSE and bias) are considered to evaluateα^MLE,δ^MLE, and Ξ^MLE. These quantities are, respectively, given by(28)Biasν^=11000∑i=11000ν^−ν,MSEν^=11000∑i=11000ν^−ν2.(iii)
Two different cases are considered for the four parameters of the NG-Weibull distribution.(iv)
One thousand repetitions are done to compute the bias and the MSE forα^MLE,δ^MLE, and Ξ^MLE.The numerical results are provided in Tables1–3. From the numerical illustration presented in Tables 1–3, it can be observed that as sample size n increases (i) the values of the MLEs α^MLE,δ^MLE,Ξ^MLE get closer and closer to the parameters α,δ,Ξ, and (ii) the values of the MSE and bias of α^MLE,δ^MLE, and Ξ^MLE tend to zero.Table 1
Forα=0.4,δ=0.5,θ=0.6,γ=0.5, the simulation results of the NG-Weibull model.
nParametersMLEsMSEsBiases25α1.37764775.37898760.9706542δ0.61287570.17387650.1129876θ0.58678680.09129870.1398765γ1.22289702.63498070.7226754100α0.86476652.00334520.4642937δ0.56609880.15132870.0963454θ0.58334520.07329380.1019876γ0.95709871.22320930.4578769375α0.54065571.24098760.2409865δ0.49920980.10798710.0665438θ0.60234520.04090900.0876548γ0.70620930.93987820.2062983525α0.47809870.74887610.0788967δ0.49209090.46987620.0198764θ0.60128370.03876580.0209873γ0.61535460.16098720.1152342750α0.42323410.48767540.0238972δ0.473–0980.26098720.0098766θ0.607–0920.01987650.0088735γ0.56527890.09998760.0659897950α0.41276580.05509870.0120981δ0.48309870.01889760.0077653θ0.60509560.00150980.0059887γ0.56823760.08765430.04987061000α0.40428330.03008950.0040892δ0.50102930.00296740.0038765θ0.60854120.00108760.0024498γ0.51823980.01987630.0286710Table 2
Forα=0.5,δ=0.5,θ=0.6,γ=1, the simulation results of the NG-Weibull model.
nParametersMLEsMSEsBiases25α1.53988754.76609231.0399876δ0.53995850.17512340.1295234θ0.59965390.31329380.1089097γ1.51909872.22709840.5199870100α0.96987682.07675890.4698795δ0.521–0980.10642560.1087653θ0.59487640.17865670.0987609γ1.45130981.56812340.4517563375α0.62887650.60175410.1280987δ0.470–0980.09543780.0865309θ0.60932450.13987640.0554233γ1.16387690.50441220.2798734525α0.58209870.35998720.0974076δ0.47265430.06543690.0680969θ0.60909870.10754890.0187653γ1.10998750.32541650.1094234750α0.53420980.13509870.0347865δ0.47323900.01745780.0260987θ0.60727600.04987630.0076538γ1.04912340.17219820.0493452950α0.50620910.04778920.0068759δ0.47123420.01045680.0284327θ0.60820950.00267890.0064231γ1.03476540.10542170.03468991000α0.50504710.02789010.0047887δ0.49765400.00856740.0206541θ0.60798760.00095630.0030982γ1.01987690.06809800.0156983Table 3
Forα=0.2,δ=0.5,θ=0.6,γ=0.3, the simulation results of the NG-Weibull model.
nParametersMLEsMSEsBiases25α0.64898761.91809870.4488765δ0.63987650.16609890.1399780θ0.57065430.12654390.0892543γ1.13878902.88098760.8380987100α0.39676540.67287650.1966543δ0.61509870.12964530.1158987θ0.56354320.10109870.0667890γ0.84767091.54134580.5478654375α0.24975860.10576540.1094410δ0.52909870.09589090.0997981θ0.58957990.09665490.0409783γ0.45787900.21165430.1578091525α0.23098760.09909870.0800876δ0.51369870.07709900.0536543θ0.59309890.03275430.0166541γ0.43065430.16078900.1309876750α0.21687610.01876540.0460987δ0.51675680.02209870.0267659θ0.59598760.01745690.0090987γ0.39509870.08809870.0954569950α0.21398010.01064320.0139878δ0.50598750.02020980.0090645θ0.59654320.00157890.0060989γ0.32954900.05167480.05954371000α0.21005610.01006580.0101432δ0.49109820.00912760.0028976θ0.59965340.00100190.0019909γ0.31097550.03409870.0287654


## 6. Analyzing the COVID-19 Data Set

This section is devoted to demonstrating the usefulness of the NG-X method by applying its special model (NG-Weibull distribution) to analyze the COVID-19 data set. The data set represents the mortality rates of COVID-19 infected person in Canada. Corresponding to the COVID-19 data set, the fitting results of the NG-Weibull distribution are compared with three other competing models. These models are given by(i)
Weibull distribution(29)Gx;Ξ=1−e−γxθ,x≥0,whereθ>0,γ>0.(ii)
K-Weibull distribution(30)Gx;a,b,Ξ=1−1−1−e−γxθab,x≥0,wherea>0,b>0,θ>0,γ>0.(iii)
Alpha power transformed Weibull (APT-Weibull) distribution(31)Gx;α1Ξ=α11−e−γxθ−1α1−1,4x≥0,whereα1>0,α1≠1,θ>0,γ>0.Some commonly used analytical goodness of fit measures are considered to show which distribution better fits the data. These measures are given by(32)AIC=2p−2ℓΞ,BIC=plogn−2ℓΞ,CBIC=2npn−p−1−2ℓΞ,HQIC=2ploglogn−2ℓΞ,AD=−n−1n∑u=1n2u−1logGxu+log1−Gxn−u+1,CM=112n+∑u=1n2u−12n−Gxu2.Corresponding to the COVID-19 data set, Table4 gives the values of the MLEs with standard errors enclosed in parentheses. However, Table 5 gives the analytical measures of the competing models for the COVID-19 data set.Table 4
The values of the MLEs along with standard errors in parentheses using the COVID-19 data.
Modelsαδγθabα1NG-Weibull0.7358 (0.5199)0.9864 (0.0230)1.5708 (1.1629)1.0923 (0.4809)———Weibull——0.0139 (0.0075)3.3089 (0.3591)———K-Weibull——0.7011 (1.8876)1.0555 (1.9107)13.1558 (40.5978)2.1563 (6.6188)—APT-Weibull——0.0675 (0.0509)2.5153 (0.4365)——7.2916 (7.7550)Table 5
The analytical measures of the NG-Weibull and other competitive models for the COVID-19 data.
ModelsAICBICCAICHQICCMADNG-Weibull102.7614109.0955104.0518104.97220.06470.3741Weibull106.9487110.1157107.3123108.05410.17260.9904K-Weibull104.0038110.3379105.2941106.21450.09240.5373APT-Weibull107.8743112.6249108.6243109.53240.15310.8723Based on the results in Table4, it is clear that the implementation of the NG-Weibull model is a suitable choice to use for modeling the COVID-19 data set. Furthermore, for supporting the numerical results obtained in Table 4, the estimated PDF, CDF, SF, and PP (probability-probability) plots of the NG-Weibull model are presented in Figure 4. The plots in Figure 4 visually confirm the close-fitting of the NG-Weibull distribution.Figure 4
The fitted PDF, CDF, SF, and PP plots of the NG-Weibull distribution.


## 7. Concluding Remarks

In this paper, a new family of distributions called the NG-X family was introduced. The NG-X method was introduced using the exponential distribution in combination with the T-X approach. The purpose of introducing the NG-X family was to update the distributional flexibility of the classical/modified distributions for modeling data in applied sectors. Based on the NG-X distributions approach, an updated/modified form of the Weibull model called the NG-Weibull was studied in depth. The HT characteristics of the NG-Weibull distribution were proved empirically. Based on the NG-Weibull distribution, a simulation study was also conducted. The simulation study showed that the values of MLEs are quite steady and are close to the true parameters’ values as the sample gets increased. Finally, to establish the applicability of the NG-Weibull distribution, a data set related to the mortality rates of COVID-19 infected people was analyzed. The performance of the NG-Weibull distribution was compared with three other competitors. Based on the different six criteria, it is shown that the NG-Weibull distribution outperforms the competitive models [5].



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*Source: 1901526-2022-04-28.xml*