# A New Transmuted Generalized Lomax Distribution: Properties and Applications to COVID-19 Data

Authors: Wael S. Abu El Azm; Ehab M. Almetwally; Sundus Naji AL-Aziz; Abd Al-Aziz H. El-Bagoury; Randa Alharbi; O. E. Abo-Kasem
Journal: Computational Intelligence and Neuroscience
 (2021)

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## Abstract

A new five-parameter transmuted generalization of the Lomax distribution (TGL) is introduced in this study which is more flexible than current distributions and has become the latest distribution theory trend. Transmuted generalization of Lomax distribution is the name given to the new model. This model includes some previously unknown distributions. The proposed distribution's structural features, closed forms for anrth moment and incomplete moments, quantile, and Rényi entropy, among other things, are deduced. Maximum likelihood estimate based on complete and Type-II censored data is used to derive the new distribution's parameter estimators. The percentile bootstrap and bootstrap-t confidence intervals for unknown parameters are introduced. Monte Carlo simulation research is discussed in order to estimate the characteristics of the proposed distribution using point and interval estimation. Other competitive models are compared to a novel TGL. The utility of the new model is demonstrated using two COVID-19 real-world data sets from France and the United Kingdom.


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## Body

## 1. Introduction

Many generators have been studied in recent years by expanding some effective classical distributions. Many applied fields, including dependability, demographics, engineering, economics, actuarial sciences, biological research, hydrology, insurance, medicine, and finance, have employed such created families of distributions for modeling and evaluating lifetime data. However, there are still a lot of real-world data occurrences that do not fit into any of the statistical distributions. Shaw and Buckley [1] introduced a new class of distributions known as transmuted distributions with cumulative distribution function (CDF) as(1)FTx=ℏx1+β−βℏx;β≤1,By differentiating equation (1), we get the probability density function (pdf) as follows:(2)fTx=ℜx1+β−2βℏx,where ℜx and ℏx are the base distribution's pdf and CDF.There are various transmuted distributions suggested. Aryal and Tsokos [2] proposed the transmuted Weibull distribution as a new generalization of the Weibull distribution. Merovci [3] devised and explored the varied structural properties of the transmuted Rayleigh distribution. Khan and King [4] obtained the transmuted modified Weibull distribution. The transmuted Lomax distribution was presented by Ashour and Eltehiwy [5]. Transmuted Pareto distribution is introduced by Merovci and Puka [6]. The transmuted generalized linear exponential distribution was introduced by Elbatal et al. [7]; among others. Poboková and Michalková [8] proposed a transmuted Weibull distribution. Ali and Athar [9] have created a new generalized transmuted family of distributions (TD). They utilized Weibull distribution to generalized transmuted families of distributions (TWDn).The Lomax distribution is a heavy-tail pdf popular in business, economics, and actuarial modeling. In some cases, it is also known as the Pareto Type-II distribution. In the event of a business failure, Lomax used it to fit data. It is essentially a Pareto distribution with a 0-support level. The pdf is as follows:(3)ℜx;γ;δ=γδ1+γx−δ+1,x≥0,δandγ>0.The CDF for (3) is(4)ℏx;γ;δ=1−1+γx−δ,x≥0,δandγ>0,where δ and γ are the shape and scale parameters, respectively.The CDF and pdf of Gompertz-generalized G-family of distribution are given by Alizadeh et al. [10] as(5)Fx;ϑ;π=1−eϑ/π1−1−ℏx−π;x≥0,πandϑ>0,(6)fx;ϑ;π=ϑℜx1−ℏx−π−1eϑ/π1−1−ℏx−π;x≥0,πandϑ>0,where ϑ and π are extra form parameters that change the tail weights. ℜx and ℏx are the parent (or baseline) distribution's pdf and CDF, respectively.Now, if the density from (3) and (4) is replaced into (5) and (6), Oguntunde et al. [11] introduced a novel generalization of the Lomax distribution known as the Gompertz Lomax distribution (GoLom) with vector parameters ƛ where ƛ=δ,γ,ϑ,π; the CDF and pdf are(7)Fx;ƛ=ℏx;ƛ=1−eϑ/π1−1+γxδπ;x≥0,π,δ,γandϑ>0.The corresponding pdf is created by inserting the densities from (3) and (4) into (6) in the following order:(8)fx;ƛ=ℜx;ƛ=ϑδγ1+γxδπ−1eϑ/π1−1+γxδπ;x≥0,π,δ,γandϑ>0.The transmuted generalized Lomax (TGL) distribution is a new five-parameter transmuted generalization of the Lomax distribution presented in this article. This generalization stems from a fundamental motivation as follows:(i)
Providing a very flexible life distribution that includes several new existing distributions as submodels(ii)
Making a significant difference in data modelingOne of the advantages of this distribution is that it works on modeling COVID-19 data. In COVID-19, a new coronavirus disease has expanded worldwide since December 2019, producing over 218 million cases and over 4.5 million deaths, reported by the World Health Organization (WHO). There have been about 6.5 million cases of COVID-19 in France as of September 2021, with over 112850 deaths. There have been about 6.8 million cases of COVID-19 in the United Kingdom as of September 2021, with over 132740 deaths. Therefore, we decided to find the best mathematical-statistical model for modeling the data of the countries of France and the United Kingdom. There were also many researchers who worked on finding a model for these data, such as Almetwally [12], Almetwally [13], Almetwally [14], and others.The following is a representation of how this article is structured. In Section2, we define the new distribution. The new distribution's structural features are discussed in Section 3. The maximum likelihood estimators (MLEs) of parameters under complete and Type-II censored samples are investigated in Section 4. Section 5 describes the various bootstrap confidence intervals. Section 6 describes a Monte-Carlo simulation analysis using entire sample sizes and Type-II censored samples to estimate point and interval estimation of TGL distribution parameters. In Section 7, two real-world data sets are introduced, and at the end of the article, there is a conclusion.


## 2. Transmuted Generalized Lomax Model

The TGL distribution and its submodels are shown here. The CDF of the TGL distribution with vector parametersℤ where ℤ=δ,γ,ϑ,π,β can be derived by substituting (7) and (8) in (1) and (2) as(9)Fx;ℤ=1−eϑ/π1−1+γxδπ1+βeϑ/π1−1+γxδπ,and its pdf is as follows:(10)fx;ℤ=δϑγ1+γxδπ−1eϑ/π1−1+γxδπ1−β+2βeϑ/π1−1+γxδπ;x>0,π,β,δ,γandϑ>0.As a result, the pdf (10) is defined as X ∼ TGL δ,γ,ϑ,π,β. Table 1 lists the TGL distribution's special submodels.Table 1
The TGL distribution's special submodels.
No.DistributionsγϑδβπAuthor1LOMγ1δ01Lomax [15]2GOLOMγϑδ0πOguntunde et al. [11]The survival (reliability) functionF¯x;ℤ and the hazard rate function hx;ℤ have the following definitions:(11)F¯x;ℤ=eϑ/π1−1+γxδπ1+βeϑ/π1−1+γxδπ,hx;ℤ=δϑγ1+γxδπ−11−β+2βeϑ/π1−1+γxδπ1−β+βeϑ/π1−1+γxδπ.For specific parameters selections, the pdf of TGL model is shown in Figure1.Figure 1
Plots of the TGL distribution's pdf.We can deduct from Figure1 plots of the TGL distribution's pdf can be unimodal, normal, or right-skewed.We may derive from Figure2 that the TGL distribution's hazard function can take the form of a decreasing, increasing, or upside-down shape.Figure 2
Plots showing the TGL distribution's hazard function.Lemma 1.
The TGL density function's limit is provided as(12)limx⟶0fx;ℤ=0,limx⟶∞fx;ℤ=0.Proof.
The density function's conclusion is easy to illustrate (10).
Furthermore, the TGL hazard function's limit asx⟶0 is 0 and x⟶ ∞ is ∞ as shown as follows:(13)limx⟶0hx;ℤ=0,limx⟶∞hx;ℤ=∞.
This statement is simple to demonstrate.


## 3. Statistical Properties

The statistical aspects of the TGL distribution are examined in the following subsections, including moments, mode, quantile function, Rényi entropy, and order statistics.


### 3.1. Moments

Therth instant near 0 is calculated. We can write as follows from (10).(14)μr′=δϑγ∫0∞xr1+γxδπ−1eϑ/π1−1+γxδπ1−β+2βeϑ/π1−1+γxδπdx=I1+I2.First, to obtainI1, as a result, we use binomial expansion.(15)I1=γ−r1−β∫0∞πϑy+11/δπ−1re−ydy.So,I1 is given by(16)I1=γ−r1−β∑i=0r∑j=0∞ri−1iπϑ1/δπr−i−jΓ1δπr−i−j+1.In a similar way, I2 is as follows:(17)I2=βγ−r∑i=0r∑j=0∞ri1δπr−ij−1iπ2ϑ1/δπr−i−jΓ1δπr−i−j+1.Then,μr′ can be expressed as(18)μr′=Ai,jγ−r∑i=0r∑j=0∞ri1δπr−ij−1iΓ1δπr−i−j+1whereAi,j=1−βπϑ1/δπr−i−j+βπ2ϑ1/δπr−i−j.


### 3.2. Mode

The mode of the TGL distribution was obtained in this subsection. Thelnfx is as follows:lnfx=lnδγϑ+δπ−1ln1+γx+ϑ/π1−1+γxδπ+ln1−β+2βeϑ/π1−1+γxδπ. The mode is the solution of the following equation:(19)2βαγ−1eϑ/π1−1+γxδπ+δπ−1−δϑ1+γxδπ1−β=0.


### 3.3. Quantile and Median

By inverting cdf (9) as follows, the TGL distribution can be easily simulated: on (0, 1), if U follows a uniform distribution, then(20)q=1−eϑ/π1−1+γxqδπ1+βeϑ/π1−1+γxqδπ.Special Caseβ=1:(21)x0.5=1γ1+π2ϑln21/δπ−1.


### 3.4. Rényi Entropy

The variance of the uncertainty is measured by the Rényi entropy of a pdffx with random variable X of TGL distribution. The Rényi entropy is defined as for any real parameter ψ>0andψ≠1:(22)IRψ=1ψ−1log∫Rfψxdx,ψ>0andψ≠1.We can extract the integrated component using the density function (10) as follows:(23)∫Rfψx=δϑγψ∫0∞1+γxψδπ−1eϑ/πψ1−1+γxδπ1−β+2βeϑ/π1−1+γxδπψdx.The binomial expansion is then used as follows:(24)∫Rfψxdx=∑i,j=0∞∑l=0jψiψa−1+bijjl−1i+jγl+2ψa1+βψ−iβa+bi×∫0∞xψ+le−γxψ+jdx,As a result, the TGL distribution's Rényi entropy is(25)IRψ=δϑγψψ−1∑i=0∞∑j=0∞ψi1−ψ1δπ−1j1−βψ−i2βiπ/ψϑj1+ij+1Γj+1.


### 3.5. Order Statistics

In this part, we will look at single-order statistics for the TGL distribution. Let us sayx1,…,xn; there are n TGL random variables that are both independent and identically distributed. Let x1,x2,…,xn stand for the order statistics derived from these n variables. The pdf of the rth order statistic, say fr:nx, is then calculated as(26)fr:nx=Cr:nFxr−1fx1−Fxn−r,where Cr:n=n!/r−1!n−r! The binomial expansion is used in this case; then, rth order statistic of TGL distribution is(27)fr:nx=n!r−1!n−r!1−eϑ/π1−1+γxδπ1+βeϑ/π1−1+γxδπr−1δϑγ1+γxδπ−1eϑ/π1−1+γxδπ1−β+2βeϑ/π1−1+γxδπeϑ/πn−r1−1+γxδπ1−β+βeϑ/π1−1+γxδπn−r.For the TGL distribution, thekth moments of rth order statistics are(28)μr:nk=δϑγn!r−1!n−r!∑i=0r−1∑j=0n−r∑l=0ir−1in−rjil−1i1−βn−r−jβj+lγ−k∫0∞1+πϑy1/δπ−1ke−n−r+i+j+l+1y1+πϑy1−1/δπ1−β+2βe−y1δϑγ1+πϑy1/δπ−1dy.By using the binomial expansion,(29)μr:nk=n!γ−kr−1!n−r!∑i=0r−1∑j=0n−r∑l=0i∑m=0k∑t=0∞r−1in−rjilkmk−mδπt−1i+mπϑt×1−βn−r−jβj+l∫0∞yte−n−r+i+j+l+1y1−β+2βe−ydy.For the TGL distribution, thekth moments of rth order statistics are(30)μr:nk=n!γ−kr−1!n−r!∑i=0r−1∑j=0n−r∑l=0i∑m=0k∑t=0∞r−1in−rjilkmk−mδπt−1i+mπϑt1−βn−r−jβj+l×1−βn−r+i+j+l+1t+1+2βn−r+i+j+l+2t+1Γt+1.


## 3.1. Moments

Therth instant near 0 is calculated. We can write as follows from (10).(14)μr′=δϑγ∫0∞xr1+γxδπ−1eϑ/π1−1+γxδπ1−β+2βeϑ/π1−1+γxδπdx=I1+I2.First, to obtainI1, as a result, we use binomial expansion.(15)I1=γ−r1−β∫0∞πϑy+11/δπ−1re−ydy.So,I1 is given by(16)I1=γ−r1−β∑i=0r∑j=0∞ri−1iπϑ1/δπr−i−jΓ1δπr−i−j+1.In a similar way, I2 is as follows:(17)I2=βγ−r∑i=0r∑j=0∞ri1δπr−ij−1iπ2ϑ1/δπr−i−jΓ1δπr−i−j+1.Then,μr′ can be expressed as(18)μr′=Ai,jγ−r∑i=0r∑j=0∞ri1δπr−ij−1iΓ1δπr−i−j+1whereAi,j=1−βπϑ1/δπr−i−j+βπ2ϑ1/δπr−i−j.


## 3.2. Mode

The mode of the TGL distribution was obtained in this subsection. Thelnfx is as follows:lnfx=lnδγϑ+δπ−1ln1+γx+ϑ/π1−1+γxδπ+ln1−β+2βeϑ/π1−1+γxδπ. The mode is the solution of the following equation:(19)2βαγ−1eϑ/π1−1+γxδπ+δπ−1−δϑ1+γxδπ1−β=0.


## 3.3. Quantile and Median

By inverting cdf (9) as follows, the TGL distribution can be easily simulated: on (0, 1), if U follows a uniform distribution, then(20)q=1−eϑ/π1−1+γxqδπ1+βeϑ/π1−1+γxqδπ.Special Caseβ=1:(21)x0.5=1γ1+π2ϑln21/δπ−1.


## 3.4. Rényi Entropy

The variance of the uncertainty is measured by the Rényi entropy of a pdffx with random variable X of TGL distribution. The Rényi entropy is defined as for any real parameter ψ>0andψ≠1:(22)IRψ=1ψ−1log∫Rfψxdx,ψ>0andψ≠1.We can extract the integrated component using the density function (10) as follows:(23)∫Rfψx=δϑγψ∫0∞1+γxψδπ−1eϑ/πψ1−1+γxδπ1−β+2βeϑ/π1−1+γxδπψdx.The binomial expansion is then used as follows:(24)∫Rfψxdx=∑i,j=0∞∑l=0jψiψa−1+bijjl−1i+jγl+2ψa1+βψ−iβa+bi×∫0∞xψ+le−γxψ+jdx,As a result, the TGL distribution's Rényi entropy is(25)IRψ=δϑγψψ−1∑i=0∞∑j=0∞ψi1−ψ1δπ−1j1−βψ−i2βiπ/ψϑj1+ij+1Γj+1.


## 3.5. Order Statistics

In this part, we will look at single-order statistics for the TGL distribution. Let us sayx1,…,xn; there are n TGL random variables that are both independent and identically distributed. Let x1,x2,…,xn stand for the order statistics derived from these n variables. The pdf of the rth order statistic, say fr:nx, is then calculated as(26)fr:nx=Cr:nFxr−1fx1−Fxn−r,where Cr:n=n!/r−1!n−r! The binomial expansion is used in this case; then, rth order statistic of TGL distribution is(27)fr:nx=n!r−1!n−r!1−eϑ/π1−1+γxδπ1+βeϑ/π1−1+γxδπr−1δϑγ1+γxδπ−1eϑ/π1−1+γxδπ1−β+2βeϑ/π1−1+γxδπeϑ/πn−r1−1+γxδπ1−β+βeϑ/π1−1+γxδπn−r.For the TGL distribution, thekth moments of rth order statistics are(28)μr:nk=δϑγn!r−1!n−r!∑i=0r−1∑j=0n−r∑l=0ir−1in−rjil−1i1−βn−r−jβj+lγ−k∫0∞1+πϑy1/δπ−1ke−n−r+i+j+l+1y1+πϑy1−1/δπ1−β+2βe−y1δϑγ1+πϑy1/δπ−1dy.By using the binomial expansion,(29)μr:nk=n!γ−kr−1!n−r!∑i=0r−1∑j=0n−r∑l=0i∑m=0k∑t=0∞r−1in−rjilkmk−mδπt−1i+mπϑt×1−βn−r−jβj+l∫0∞yte−n−r+i+j+l+1y1−β+2βe−ydy.For the TGL distribution, thekth moments of rth order statistics are(30)μr:nk=n!γ−kr−1!n−r!∑i=0r−1∑j=0n−r∑l=0i∑m=0k∑t=0∞r−1in−rjilkmk−mδπt−1i+mπϑt1−βn−r−jβj+l×1−βn−r+i+j+l+1t+1+2βn−r+i+j+l+2t+1Γt+1.


## 4. Parameter Estimation

The MLEs of the parametersℤ=δ,γ,ϑ,π,β under complete and Type-II censored samples are investigated in this section. Approximate confidence intervals (ACIs) for unknown values are also calculated using the Fisher information matrix.


### 4.1. MLEs under Complete Sample

In the statistical literature, various approaches for parameter estimation have been given, with the MLEs method being the most extensively employed. We explore applying MLEs to estimate the parameters of the TGL distribution with a complete sample. Ifx1,x2,…,xn is a random sample of this distribution of size n with a set of parameter vectors ℤ=δ,γ,ϑ,π,β, then the log-likelihood function, say ℓ1ℤ, may be stated as(31)ℓ1ℤ=nlnϑ+lnδ+lnγ+δπ−1∑i=1nln1+γxi+ϑπ∑i=1nψi+∑i=1nln1−β+2βeϑ/πψi,where ψi=1−1+γxiδπ. The partial differential equations ℓ1ℤ are calculated as follows:(32)∂ℓ1ℤ∂δ=nδ+π∑i=1nln1+γxi−ϑ∑i=1n1+γxiδπln1+γxi−∑i=1n2βϑ1+γxiδπln1+γxieϑ/πψi1−β+2βeϑ/πψi,∂ℓ1ℤ∂ϑ=nϑ+1π∑i=1nψi+∑i=1kψi/πeϑ/πψi1−β+2βeϑ/πψi,∂ℓ1ℤ∂γ=nγ+δπ−1∑i=1nxi1+γxi−δϑ∑i=1nxi1+γxiδπ−1−∑i=1n2βδϑeϑ/πψixi1+γxiδπ−11−β+2βeϑ/πψi,∂ℓ1ℤ∂π=δ∑i=1nln1+γxi−ϑπ2∑i=1nπδ1+γxiδπln1+γxi+ψi−∑i=1n2βeϑ/πψiϑδπ−11+γxiδπln1+γxi−ψi1−β+2βeϑ/πψi,∂ℓ1ℤ∂β=∑i=1n2eϑ/πψi−11−β+2βeϑ/πψi.The nonlinear equations are numerically solved to determine ML estimators as∂ℓ1ℤ/∂δ=0, ∂ℓ1ℤ/∂γ=0, ∂ℓ1ℤ/∂ϑ=0, and ∂ℓ1ℤ/∂π=0∂ℓ1ℤ/∂β=0 using an iterative technique.


### 4.2. MLEs under Type-II Censored Sample

The MLEs of parameters for TGL distribution based on Type-II censored samples are investigated in this subsection. The Fisher information matrix for Type-II censored model is also used to calculate the approximate confidence intervals for the unknown parametersℤ=δ,γ,ϑ,π,β. Let x1,x2,…,xn be a random sample of size n, and we only look at the first k-th order statistics based on the Type-II censored sample. Likelihood function in this scenario is of the kind(33)L2ℤ=C∏i=1kfxi:k:n1−Fxi:k:nn−k,x1:k:n≤x2:k:n≤⋯≤xk:k:n.where C is a constant and x1:k:n,x2:k:n,…,xk:k:n is the data that has been censored. The log-likelihood function ℓ2ℤ is possibly written as follows without constant term from (15).(34)ℓ2ℤ=klnϑ+lnδ+lnγ+δπ−1∑i=1kln1+γxi+ϑπ∑i=1kψi+∑i=1kln1−β+2βeϑ/πψi+n−k∑i=1kln1−1−eϑ/πψi1+βeϑ/πψi,where xi=xi:k:n,i=1,2,…,k, denotes the time of the k-th failure and xk denotes the time of the k-th failure. The MLEs ℤ=δ,γ,ϑ,π,β are the solutions to the following five equations:(35)∂ℓ2ℤ∂ϑ=kϑ+1π∑i=1kψi+∑i=1kψi/πeϑ/πψi1−β+2βeϑ/πψi+n−k∑i=1k−βψi/πeϑ/πψi+ψi/πeϑ/πψi+2βψi/πe2ϑ/πψi1−1−eϑ/πψi1+βeϑ/πψi=0,∂ℓ2ℤ∂δ=kδ+π∑i=1kln1+γxi−ϑ∑i=1k1+γxiδπln1+γxi−∑i=1k2βϑ1+γxiδπln1+γxieϑ/πψi1−β+2βeϑ/πψi+n−k∑i=1kϑ1+γxiδπln1+γxieϑ/πψiβ−1−2βeϑ/πψi1−1−eϑ/πψi1+βeϑ/πψi=0,(36)∂ℓ2ℤ∂π=δ∑i=1kln1+γxi−ϑπ2∑i=1kπδ1+γxiδπln1+γxi+ψi−∑i=1k2βeϑ/πψiϑδπ−11+γxiδπln1+γxi−ψi1−β+2βeϑ/πψi−n−k∑i=1kδϑπ−11+γxiδπln1+γxieϑ/πψiβ−1−2βeϑ/πψi−ψi1−1−eϑ/πψi1+βeϑ/πψi=0,(37)∂ℓ2ℤ∂γ=kγ+δπ−1∑i=1kxi1+γxi−δϑ∑i=1kxi1+γxiδπ−1−∑i=1k2βδϑeϑ/πψixi1+γxiδπ−11−β+2βeϑ/πψi−n−k∑i=1kδϑeϑ/πψixi1+γxiδπ−1β−1−βeϑ/πψi1−1−eϑ/πψi1+βeϑ/πψi=0,∂ℓ2ℤ∂β=∑i=1k2eϑ/πψi−11−β+2βeϑ/πψi+n−k∑i=1keϑ/πψieϑ/πψi−11−1−eϑ/πψi1+βeϑ/πψi.It is to be noted that equation (16) cannot be solved explicitly. To obtain the MLEs ℤ=δ,γ,ϑ,π,β, a numerical approach is required, and a numerical technique is needed. We obtain the observed Fisher information matrix since its expectation requires numerical integration. The 5×5 observed information or Hessian matrix Hℤ is(38)Hℤ=HδδHδγHδϑHδπHδβHγδHγγHγϑHγπHγβHϑδHϑγHϑϑHϑπHϑβHπδHπγHπϑHππHπβHβδHβγHβϑHβπHββ.The Fisher information matrixHℤ is given by the negative expected of second partial derivatives of (15) for the unknown parameters ℤ=δ,γ,ϑ,π,β locally at δ^,γ^,ϑ^,π^,β^ given in (16). Under some regularity conditions, δ^,γ^,ϑ^,π^,β^ is approximately normal with mean δ,γ,ϑ,π,β and covariance matrix H−1ℤ. Practically, we estimate H−1ℤ byHδ^,γ^,ϑ^,π^,β^−1; then(39)Iδ^,γ^,ϑ^,π^,β^−1=−HδδHδγHδϑHδπHδβHγδHγγHγϑHγπHγβHϑδHϑγHϑϑHϑπHϑβHπδHπγHπϑHππHπβHβδHβγHβϑHβπHββδ^,γ^,ϑ^,π^,β^−1,Hδ^,γ^,ϑ^,π^,β^−1=Varδ^Covδ^,γ^Covδ^,ϑ^Covδ^,π^Covδ^,β^Covγ^,δ^Varγ^Covγ^,ϑ^Covγ^,π^Covγ^,β^Covϑ^,δ^Covϑ^,γ^Varϑ^Covϑ^,π^Covϑ^,β^Covπ^,δ^Covπ^,γ^Covπ^,ϑ^Varπ^Covπ^,β^Covβ^,δ^Covβ^,γ^Covβ^,ϑ^Covβ^,π^Varβ^.The elements of the observed Hessian matrix are computed using an iterative numerical solution method. Now, the ACIsℤ=δ,γ,ϑ,π,β can be obtained as follows:(40)δ^±z1−α/2.Varδ^,γ^±z1−α/2.Varγ^,ϑ^±z1−α/2.Varϑ^,π^±z1−α/2.Varπ^,β^±z1−α/2.Varβ^,where zα is the 100α−th percentile of a standard normal distribution.


## 4.1. MLEs under Complete Sample

In the statistical literature, various approaches for parameter estimation have been given, with the MLEs method being the most extensively employed. We explore applying MLEs to estimate the parameters of the TGL distribution with a complete sample. Ifx1,x2,…,xn is a random sample of this distribution of size n with a set of parameter vectors ℤ=δ,γ,ϑ,π,β, then the log-likelihood function, say ℓ1ℤ, may be stated as(31)ℓ1ℤ=nlnϑ+lnδ+lnγ+δπ−1∑i=1nln1+γxi+ϑπ∑i=1nψi+∑i=1nln1−β+2βeϑ/πψi,where ψi=1−1+γxiδπ. The partial differential equations ℓ1ℤ are calculated as follows:(32)∂ℓ1ℤ∂δ=nδ+π∑i=1nln1+γxi−ϑ∑i=1n1+γxiδπln1+γxi−∑i=1n2βϑ1+γxiδπln1+γxieϑ/πψi1−β+2βeϑ/πψi,∂ℓ1ℤ∂ϑ=nϑ+1π∑i=1nψi+∑i=1kψi/πeϑ/πψi1−β+2βeϑ/πψi,∂ℓ1ℤ∂γ=nγ+δπ−1∑i=1nxi1+γxi−δϑ∑i=1nxi1+γxiδπ−1−∑i=1n2βδϑeϑ/πψixi1+γxiδπ−11−β+2βeϑ/πψi,∂ℓ1ℤ∂π=δ∑i=1nln1+γxi−ϑπ2∑i=1nπδ1+γxiδπln1+γxi+ψi−∑i=1n2βeϑ/πψiϑδπ−11+γxiδπln1+γxi−ψi1−β+2βeϑ/πψi,∂ℓ1ℤ∂β=∑i=1n2eϑ/πψi−11−β+2βeϑ/πψi.The nonlinear equations are numerically solved to determine ML estimators as∂ℓ1ℤ/∂δ=0, ∂ℓ1ℤ/∂γ=0, ∂ℓ1ℤ/∂ϑ=0, and ∂ℓ1ℤ/∂π=0∂ℓ1ℤ/∂β=0 using an iterative technique.


## 4.2. MLEs under Type-II Censored Sample

The MLEs of parameters for TGL distribution based on Type-II censored samples are investigated in this subsection. The Fisher information matrix for Type-II censored model is also used to calculate the approximate confidence intervals for the unknown parametersℤ=δ,γ,ϑ,π,β. Let x1,x2,…,xn be a random sample of size n, and we only look at the first k-th order statistics based on the Type-II censored sample. Likelihood function in this scenario is of the kind(33)L2ℤ=C∏i=1kfxi:k:n1−Fxi:k:nn−k,x1:k:n≤x2:k:n≤⋯≤xk:k:n.where C is a constant and x1:k:n,x2:k:n,…,xk:k:n is the data that has been censored. The log-likelihood function ℓ2ℤ is possibly written as follows without constant term from (15).(34)ℓ2ℤ=klnϑ+lnδ+lnγ+δπ−1∑i=1kln1+γxi+ϑπ∑i=1kψi+∑i=1kln1−β+2βeϑ/πψi+n−k∑i=1kln1−1−eϑ/πψi1+βeϑ/πψi,where xi=xi:k:n,i=1,2,…,k, denotes the time of the k-th failure and xk denotes the time of the k-th failure. The MLEs ℤ=δ,γ,ϑ,π,β are the solutions to the following five equations:(35)∂ℓ2ℤ∂ϑ=kϑ+1π∑i=1kψi+∑i=1kψi/πeϑ/πψi1−β+2βeϑ/πψi+n−k∑i=1k−βψi/πeϑ/πψi+ψi/πeϑ/πψi+2βψi/πe2ϑ/πψi1−1−eϑ/πψi1+βeϑ/πψi=0,∂ℓ2ℤ∂δ=kδ+π∑i=1kln1+γxi−ϑ∑i=1k1+γxiδπln1+γxi−∑i=1k2βϑ1+γxiδπln1+γxieϑ/πψi1−β+2βeϑ/πψi+n−k∑i=1kϑ1+γxiδπln1+γxieϑ/πψiβ−1−2βeϑ/πψi1−1−eϑ/πψi1+βeϑ/πψi=0,(36)∂ℓ2ℤ∂π=δ∑i=1kln1+γxi−ϑπ2∑i=1kπδ1+γxiδπln1+γxi+ψi−∑i=1k2βeϑ/πψiϑδπ−11+γxiδπln1+γxi−ψi1−β+2βeϑ/πψi−n−k∑i=1kδϑπ−11+γxiδπln1+γxieϑ/πψiβ−1−2βeϑ/πψi−ψi1−1−eϑ/πψi1+βeϑ/πψi=0,(37)∂ℓ2ℤ∂γ=kγ+δπ−1∑i=1kxi1+γxi−δϑ∑i=1kxi1+γxiδπ−1−∑i=1k2βδϑeϑ/πψixi1+γxiδπ−11−β+2βeϑ/πψi−n−k∑i=1kδϑeϑ/πψixi1+γxiδπ−1β−1−βeϑ/πψi1−1−eϑ/πψi1+βeϑ/πψi=0,∂ℓ2ℤ∂β=∑i=1k2eϑ/πψi−11−β+2βeϑ/πψi+n−k∑i=1keϑ/πψieϑ/πψi−11−1−eϑ/πψi1+βeϑ/πψi.It is to be noted that equation (16) cannot be solved explicitly. To obtain the MLEs ℤ=δ,γ,ϑ,π,β, a numerical approach is required, and a numerical technique is needed. We obtain the observed Fisher information matrix since its expectation requires numerical integration. The 5×5 observed information or Hessian matrix Hℤ is(38)Hℤ=HδδHδγHδϑHδπHδβHγδHγγHγϑHγπHγβHϑδHϑγHϑϑHϑπHϑβHπδHπγHπϑHππHπβHβδHβγHβϑHβπHββ.The Fisher information matrixHℤ is given by the negative expected of second partial derivatives of (15) for the unknown parameters ℤ=δ,γ,ϑ,π,β locally at δ^,γ^,ϑ^,π^,β^ given in (16). Under some regularity conditions, δ^,γ^,ϑ^,π^,β^ is approximately normal with mean δ,γ,ϑ,π,β and covariance matrix H−1ℤ. Practically, we estimate H−1ℤ byHδ^,γ^,ϑ^,π^,β^−1; then(39)Iδ^,γ^,ϑ^,π^,β^−1=−HδδHδγHδϑHδπHδβHγδHγγHγϑHγπHγβHϑδHϑγHϑϑHϑπHϑβHπδHπγHπϑHππHπβHβδHβγHβϑHβπHββδ^,γ^,ϑ^,π^,β^−1,Hδ^,γ^,ϑ^,π^,β^−1=Varδ^Covδ^,γ^Covδ^,ϑ^Covδ^,π^Covδ^,β^Covγ^,δ^Varγ^Covγ^,ϑ^Covγ^,π^Covγ^,β^Covϑ^,δ^Covϑ^,γ^Varϑ^Covϑ^,π^Covϑ^,β^Covπ^,δ^Covπ^,γ^Covπ^,ϑ^Varπ^Covπ^,β^Covβ^,δ^Covβ^,γ^Covβ^,ϑ^Covβ^,π^Varβ^.The elements of the observed Hessian matrix are computed using an iterative numerical solution method. Now, the ACIsℤ=δ,γ,ϑ,π,β can be obtained as follows:(40)δ^±z1−α/2.Varδ^,γ^±z1−α/2.Varγ^,ϑ^±z1−α/2.Varϑ^,π^±z1−α/2.Varπ^,β^±z1−α/2.Varβ^,where zα is the 100α−th percentile of a standard normal distribution.


## 5. Bootstrap Confidence Interval

We create two parametric bootstrap confidence intervals (CI)ℤ=δ,γ,ϑ,π,β in this section as follows.


### 5.1. Percentile Bootstrap (Boot-P)

(1)
Compute the MLE ofℤ=δ,γ,ϑ,π,β based on complete and censored samples(2)
Generate a bootstrap sample usingℤ to obtain the bootstrap estimate of ℤ say ℤ^b using the bootstrap sample(3)
Repeat step (2)B times to get ℤb1,ℤb2,…,ℤbB.(4)
Arrangeℤb1,ℤb2,…,ℤbB in order of ascending as ℤb1,ℤb2,…,ℤbB.(5)
A two-side1001−γ% Boot-P CI for the unknown parameters ℤ=δ,γ,ϑ,π,β is set by ℤ^bBγ/2,ℤ^bB1−γ/2


### 5.2. Bootstrap-t (Boot-t)

(1)
The same steps as (1-2) in Boot-p(2)
Compute the t-statistic ofℤ=δ,γ,ϑ,π,βT=ℤ^b−ℤ^/Varℤ^b where Varℤ^b asymptotic variances of ℤ^=δ^,γ^,ϑ^,π^,β^ and it can be obtained using the Fisher information matrix(3)
Repeat steps 2-3B times and obtain T1,T2,…,TB(4)
ArrangeT1,T2,…,TB in ascending order as T1,T2,…,TB(5)
A two-side1001−γ% Boot-t CI for the unknown parameters ℤ=δ,γ,ϑ,π,β is given by(41)ℤ^+TBγ/2Varℤ^,ℤ^+TB1−γ/2Varℤ^.


## 5.1. Percentile Bootstrap (Boot-P)

(1)
Compute the MLE ofℤ=δ,γ,ϑ,π,β based on complete and censored samples(2)
Generate a bootstrap sample usingℤ to obtain the bootstrap estimate of ℤ say ℤ^b using the bootstrap sample(3)
Repeat step (2)B times to get ℤb1,ℤb2,…,ℤbB.(4)
Arrangeℤb1,ℤb2,…,ℤbB in order of ascending as ℤb1,ℤb2,…,ℤbB.(5)
A two-side1001−γ% Boot-P CI for the unknown parameters ℤ=δ,γ,ϑ,π,β is set by ℤ^bBγ/2,ℤ^bB1−γ/2


## 5.2. Bootstrap-t (Boot-t)

(1)
The same steps as (1-2) in Boot-p(2)
Compute the t-statistic ofℤ=δ,γ,ϑ,π,βT=ℤ^b−ℤ^/Varℤ^b where Varℤ^b asymptotic variances of ℤ^=δ^,γ^,ϑ^,π^,β^ and it can be obtained using the Fisher information matrix(3)
Repeat steps 2-3B times and obtain T1,T2,…,TB(4)
ArrangeT1,T2,…,TB in ascending order as T1,T2,…,TB(5)
A two-side1001−γ% Boot-t CI for the unknown parameters ℤ=δ,γ,ϑ,π,β is given by(41)ℤ^+TBγ/2Varℤ^,ℤ^+TB1−γ/2Varℤ^.


## 6. Simulation Study

In this section, we discuss the Monte Carlo simulation study to estimate point and interval estimation of parameters of TGL distribution based on complete sample sizes and Type-II censored samples. The simulation results are in Tables2– 5, and concluding remarks of simulation results are obtained in this section. A Monte Carlo simulation is an initial task for studying different properties of parameters of the TGL model based on different sample schemes; we can use these steps:(1)
Generate random sample from a uniform distribution with intervals 0 and 1 and sample sizen as 30, 50, and 100(2)
Determine different actuals values of parameters of TGL distribution asCase 1:π=0.3,β=0.3,δ=0.5,γ=0.6,ϑ=0.5Case 2:π=1.2,β=0.4,δ=0.8,γ=0.75,ϑ=1.9Case 3:π=0.35,β=−0.4,δ=0.45,γ=0.75,ϑ=0.85Case 4:π=1.5,β=−0.6,δ=1.4,γ=2.7,ϑ=1.8(3)
We use the inverse CDF method to transform the CDF in terms ofu and get the sample of TGL distribution(4)
Sort sample and select the firstm failures as 20 and 25 where n = 30, 30, and 40 where n = 50 and 70 and 90 where n = 100(5)
By using different programs as Mathcad, R-software, Mable, and Mathematica, we can obtain the results of the simulation(6)
We use 10000 iterations in the summation generator(7)
In point estimation, we obtain bias and the mean squared error (MSE) of parameters of TGL distribution(8)
In intervals estimation, we obtain a length of CI for ACI denoted as L.ACI, length of percentile bootstrap CI can be denoted as L.BPCI, length of bootstrap-t CI can be denoted as L.BTCI, and coverage probability (CP) of ACITable 2
Various measures for parameters of TGL distribution based on different schemes of samples: Case 1.
NMBiasMSEL.ACIL.BPCIL.BTCICP (%)3030π0.00460.04990.85720.09920.097996.40β0.01900.03920.77310.08150.081294.60δ0.02410.02110.56150.05510.054797.80γ0.31740.59892.76820.25300.249895.80ϑ−0.01380.03920.76190.08300.083496.2025π0.00500.13811.45720.14650.146195.80β−0.07520.05150.83960.08410.084396.30δ0.07770.03470.66410.06200.061996.90γ0.33700.75753.08970.28470.288095.50ϑ-0.02380.05650.92770.09000.091096.7020π0.04450.25801.98460.18820.185293.20β−0.12130.07130.93330.09700.097998.00δ0.14700.07690.92200.09740.097494.80γ0.36730.86583.35310.36720.364095.00ϑ-0.03950.06240.96700.09190.091996.705050π0.01390.03540.72200.06890.067896.70β0.02500.03720.75060.06220.063693.20δ0.01460.01170.42070.02510.025298.10γ0.26880.46562.45980.26800.260495.30ϑ−0.01020.02860.64490.06660.065696.7040π−0.01430.14371.48340.13390.135794.10β−0.07140.04820.81410.07880.078195.10δ0.07630.03710.69410.06030.060495.30γ0.29310.53762.59740.27050.264396.40ϑ−0.01980.05050.87830.07390.071996.4030π−0.01530.27322.04920.22230.219994.30β−0.09800.05870.86950.08670.087598.20δ0.14240.06790.85620.09530.096095.10γ0.37500.59682.83130.29480.305995.60ϑ−0.03470.05480.90840.08010.079897.70100100π0.00310.01900.52750.04440.044796.60β0.00890.03050.58670.05810.057893.20δ0.00540.00230.18510.01630.016295.60γ0.19320.22611.70400.14980.148195.00ϑ−0.01030.01490.46530.04760.048097.0090π0.00440.04900.86790.09840.098295.20β−0.02840.03120.68360.06470.064593.80δ0.02750.00520.26030.02420.024395.00γ0.20840.31742.05280.17660.178295.10ϑ−0.01270.02270.58740.05120.052496.3070π−0.07500.16731.57710.15760.157394.60β−0.04880.04090.76970.08500.084698.70δ0.08690.02770.55620.05720.056294.90γ0.22160.40242.33110.25740.251496.10ϑ−0.01770.03660.74990.06840.068197.70Table 3
Various measures for parameters of TGL distribution based on different schemes of samples: Case 2.
NmBiasMSEL.ACIL.BPCIL.BTCICP (%)3030π0.01090.18831.64720.15850.156794.60β−0.31842.32564.26230.33380.320499.10δ0.15680.16921.49160.13710.137895.20γ0.18090.69193.06930.23520.235397.10ϑ−0.00900.05010.87730.09410.094194.6025π0.03160.51182.80580.22250.225092.90β−0.48214.08567.69860.43470.403398.00δ0.20030.24671.78240.16430.164495.90γ0.28440.72883.15150.27460.274295.00ϑ0.02720.08391.13110.11150.112894.3020π0.06570.65463.17250.32720.328493.50β−0.851013.707714.13181.10321.092696.90δ0.23690.30871.97130.16590.164996.70γ0.30230.77423.24090.28640.282593.70ϑ0.03420.12151.36390.11060.110794.005050π0.01080.08171.07690.10630.105993.07β−0.08700.09451.16180.12270.122493.07δ0.04850.06030.94860.09390.092297.03γ0.14850.25331.89460.19390.192596.04ϑ−0.00400.03890.76070.07540.076894.0640π0.01830.35122.32300.23850.236795.00β−0.26631.06653.91320.28700.268399.20δ0.15120.16731.49050.14750.148895.6030γ0.18490.39702.36230.25720.254094.80ϑ0.01000.05400.91040.08400.081795.10π−0.03440.70913.29990.40860.401893.60β−0.50914.38477.96600.42420.416198.30δ0.19300.27811.92470.18900.189096.70γ0.33120.69452.99910.25590.259294.30ϑ0.07060.09911.23450.11390.113494.10100100π0.00940.03970.76410.07250.072693.53β−0.00810.06230.98120.09830.098695.88δ0.03090.03480.72280.06630.066495.88γ0.05410.10021.22610.11690.116994.71ϑ−0.01060.00820.34950.03380.034094.7190π−0.01020.09111.18290.12550.125895.20β−0.07260.07461.03230.10640.106198.10δ0.04050.05000.86250.09620.097595.00γ0.08760.13271.38660.13140.131495.40ϑ0.01360.01580.48980.05170.050794.6070π−0.01060.20941.79420.18580.180695.10β−0.15000.11021.16170.11160.112096.50δ0.07570.10981.26500.12820.129496.00γ0.13980.17821.56240.15730.157394.50ϑ0.01510.02300.59240.05220.052296.00Table 4
Various measures for parameters of TGL distribution based on different schemes of samples: Case 3.
NMBiasMSEL.ACIL.BPCIL.BTCICP (%)3030π0.09370.18251.63490.15690.157699.89β−0.92931.06974.08821.29771.316693.69δ0.05070.05930.93390.05230.051997.01γ0.74672.35155.25330.76900.759595.90ϑ−0.08860.13891.42010.14670.146498.4525π0.06750.25301.95470.20900.206298.60β−1.08512.87734.39292.05011.961795.10δ0.09220.06190.90600.08490.085094.80γ0.81032.58175.45760.44110.438996.50ϑ−0.06430.17781.63420.15440.155397.2020π0.05630.37452.39000.23320.235898.50β−1.48703.23505.21921.92311.918094.50δ0.12980.08961.05750.10380.103795.30γ0.90292.77325.48790.59500.593695.20ϑ−0.05550.24291.92050.19080.192097.005050π0.07480.15491.51540.14570.147699.60β−0.38420.92353.62100.54580.546894.40δ0.05100.04860.84150.06480.065196.30γ0.43950.99753.51730.30650.298394.70ϑ−0.09250.10531.21980.12390.122898.5040π0.05570.23131.87350.17510.173598.20β−0.97981.19294.06581.30021.294894.80δ0.08550.07691.03480.07230.072895.70γ0.63231.82104.67550.37890.378895.60ϑ−0.10140.13601.39080.14450.145798.1030π0.03070.32672.23840.22490.227598.60β−1.03553.26094.93722.16152.113594.00δ0.12780.07920.92600.09740.097495.50γ0.65361.96764.86750.51970.522794.30ϑ−0.07650.15601.51960.14690.152297.00100100π0.07700.09721.18460.11120.111397.70β−0.01240.51592.81650.33340.333698.50δ0.02370.01520.47420.04700.046894.30γ0.22460.40432.33300.16850.166097.40ϑ−0.07330.05490.87240.08470.085596.1090π0.07880.12031.32480.14110.140997.00β−0.05300.58082.98170.27310.263998.20δ0.03110.01790.51070.05230.051294.90γ0.25870.50892.48340.22170.218797.50ϑ−0.08160.06130.91700.08520.085496.3070π0.04270.18471.67730.17400.174796.70β−0.10070.68383.21910.37180.352297.70δ0.06560.03260.66020.06570.065493.80γ0.29240.53652.63400.24520.244196.60ϑ−0.08400.08551.09830.09930.099198.30Table 5
Various measures for parameters of TGL distribution based on different schemes of samples: Case 4.
NMBiasMSEL.ACIL.BPCIL.BTCICP (%)3030π0.04570.87163.65720.34520.346793.10β−0.73942.91653.11211.05741.049995.00δ0.20400.12191.11130.10420.104793.80γ0.09721.19474.26980.31810.317994.70ϑ−0.06000.57202.94030.24160.241494.0025π−0.05461.47184.75320.45060.450891.50β−0.84973.07803.23311.17831.156193.00δ0.26970.30051.87170.15700.155896.80γ0.26841.57354.80570.40810.414292.60ϑ−0.06570.90533.72280.35390.350195.4020π−0.11831.86185.33130.50210.505295.60β−0.91153.50463.97771.19381.180693.60δ0.31440.37582.06400.29920.286796.30γ0.35362.00955.38390.53600.532893.50ϑ−0.09671.10294.11430.38620.374696.305050π0.07140.54782.84900.30280.301396.90β−0.16031.81942.58700.60000.533898.00δ0.15160.07400.88550.08220.083094.00γ−0.00480.31422.25360.19780.197696.90ϑ−0.12490.34812.23890.24020.241197.4040π0.08000.73233.34150.31600.315996.10β−0.30222.03532.95691.00540.870397.70δ0.15950.07940.91110.09660.097095.20γ−0.00530.35842.34780.23550.231296.30ϑ−0.13450.45802.69400.29570.294697.2030π0.12540.99183.90460.36610.368995.10β−0.60512.17763.08071.14121.051796.70δ0.18170.20141.60940.11550.116797.50γ0.00840.54612.88930.28320.273296.00ϑ−0.14490.56412.89030.30360.307496.00100100π0.07960.22551.83590.17560.177097.10β0.06070.12291.35420.11230.111798.30δ0.06200.01650.44070.04050.040195.20γ−0.05510.06390.96750.10070.100594.90ϑ−0.10310.17541.59180.15550.154396.0090π0.09650.33912.25240.23110.230796.70β0.06680.12731.37430.10920.109599.30δ0.07880.02330.51240.04610.046295.30γ−0.04680.06961.01820.08780.087296.40ϑ−0.13200.23471.82810.17650.176198.1070π0.09490.47622.68060.30190.301797.00β0.00061.36744.58610.48310.414599.30δ0.09680.03890.67390.04680.047096.60γ−0.03580.12821.39720.14660.146196.90ϑ−0.14160.29312.04940.20830.207098.70


## 7. Concluding Remarks of Simulation Results

Tables2–5 show the simulation results of point and interval estimates of TGL distribution parameters using Type-II censored samples and entire sample sizes. Based on these Tables, the following concluding remakes are noticed.(i)
As sample size increases and fixed other values of the model, the various measures for the parameter of TGL distribution estimates decrease(ii)
As the number of units to be failed increases and fixed other values of the model, the various measures for the parameter of TGL distribution Type-II censored samples estimates decrease(iii)
The bootstrap is the shortest length of CI for interval estimation of parameter TGL distribution


## 8. Real Data Analysis

The relevance and potentiality of the TGL distribution are demonstrated in this section through the application of two real data sets.


### 8.1. Data Set (1): COVID-19 of France

The COVID-19 data in question is from France, and it covers a period of 108 days, from March 1 to June 16, 2021. This data was formed by using daily new deaths (ND), daily cumulative cases (CC), and daily cumulative deaths (CD) as follows:(42)xi=NDiCCi−CDi−1×1000.The data are as follows: 0.0024 0.0025 0.0027 0.0045 0.0045 0.0062 0.0109 0.0113 0.0123 0.0123 0.0126 0.0129 0.0130 0.0139 0.0152 0.0160 0.0160 0.0161 0.0164 0.0169 0.0173 0.0174 0.0188 0.0219 0.0225 0.0226 0.0248 0.0260 0.0284 0.0303 0.0315 0.0318 0.0320 0.0323 0.0327 0.0329 0.0332 0.0343 0.0345 0.0346 0.0346 0.0347 0.0347 0.0352 0.0359 0.0365 0.0366 0.0370 0.0371 0.0376 0.0384 0.0392 0.0396 0.0419 0.0421 0.0443 0.0445 0.0462 0.0492 0.0506 0.0525 0.0536 0.0539 0.0540 0.0568 0.0569 0.0592 0.0600 0.0603 0.0606 0.0617 0.0619 0.0621 0.0629 0.0632 0.0657 0.0665 0.0675 0.0678 0.0683 0.0684 0.0691 0.0693 0.0707 0.0708 0.0719 0.0722 0.0727 0.0771 0.0773 0.0773 0.0774 0.0791 0.0832 0.0837 0.0845 0.0865 0.0897 0.0907 0.0910 0.0946 0.0961 0.0972 0.1010 0.1056 0.1123 0.1149 0.2153.In Table6, the TGL distribution is fitted to COVID-19 of France country. The TGL model is compared with other competitive models as Mead and Afify [16] proposed the Burr-XII model (KEBXII) with Kumaraswamy exponentiated, Weibull-Lomax (WL) distribution, Odds Exponential-Pareto IV (OEPIV) distribution proposed by Baharith et al. [17], Marshall–Olkin Alpha power Weibull (MOAPW) by Almetwally et al. [18], Marshall–Olkin Alpha power extended Weibull (MOAPEW) by Almetwally [19], Marshall–Olkin alpha power inverse Weibull (MOAPIW) by Basheer et al. [20], Marshall–Olkin alpha power Lomax (MOAPL) by Almongy et al. [21], and Gompertz Lomax (GOLOM) distribution by Oguntunde et al. [11]. According to this result, we note that the estimate of TGL has the best measure where it has the smallest value of Cramer-von Mises (W∗), Anderson-Darling (A∗), and Kolmogorov- Smirnov (KS) statistic along with its P value. The fitted TGL, pdf, CDF, and PP-plot of the data set are displayed in Figure 3.Table 6
The TGL model's and other competing models' analytical results using COVID-19 data of France.
πβδγϑKSP valueW∗A∗TGLEstimates8.35880.90780.2296106.50580.11310.06600.73480.07950.4964SE3.38530.11022.27301.73971.3473KEBXIIEstimates1.58950.707252.81891.40501.78940.07190.63270.10130.6654SE1.13600.31156.64340.26701.0125WLEstimates2.87071.54661.02130.1104—0.07110.64660.09680.6432SE20.49280.31180.96360.4204OEPIVEstimates3.79140.63100.86630.1134—0.07210.62810.10240.6726SE22.60070.10391.26770.3183MOAPIWEstimates3171.24390.61220.00260.1791—0.11000.14620.44162.7501SE1410.38180.04140.00040.0317MOAPWEstimates7.74931.01682.12760.0244—0.06630.72920.09090.5547SE26.41130.32372.70800.0155MOAPEWEstimates3.27590.85946.250318.884723.65010.06730.71180.09440.5600SE6.45100.32847.53131.541736.6454MOAPLEstimates1.085829.20037.74920.56950.06670.72230.09410.5544SE4.918455.229817.90491.1921GOLOMEstimates0.6174—6.58875.18360.33400.09120.32990.07980.5392SE0.32853.67302.76940.2097Figure 3
TGL model pdf and CDF estimates, as well as a PP-plot using COVID-19 data of France. Data set (2): COVID-19 of the United Kingdom.The COVID-19 data in question is from the United Kingdom and spans 82 days, from May 1 to July 16, 2021. This data is formed by using daily ND, daily CC, and daily CD as follows:0.0023, 0.0023, 0.0023, 0.0046, 0.0065, 0.0067, 0.0069, 0.0069, 0.0091, 0.0093, 0.0093, 0.0093, 0.0111, 0.0115, 0.0116, 0.0116, 0.0119, 0.0133, 0.0136, 0.0138, 0.0138, 0.0159, 0.0161, 0.0162, 0.0162, 0.0162, 0.0163, 0.0180, 0.0187, 0.0202, 0.0207, 0.0208, 0.0225, 0.0230, 0.0230, 0.0239, 0.0245, 0.0251, 0.0255, 0.0255, 0.0271, 0.0275, 0.0295, 0.0297, 0.0300, 0.0302, 0.0312, 0.0314, 0.0326, 0.0346, 0.0349, 0.0350, 0.0355, 0.0379, 0.0384, 0.0394, 0.0394, 0.0412, 0.0419, 0.0425, 0.0461, 0.0464, 0.0468, 0.0471, 0.0495, 0.0501, 0.0521, 0.0571, 0.0588, 0.0597, 0.0628, 0.0679, 0.0685, 0.0715, 0.0766, 0.0780, 0.0942, 0.0960, 0.0988, 0.1223, 0.1343, and 0.1781.In Table7, the TGL distribution is fitted to COVID-19 of The United Kingdom country. The TGL model is compared with other competitive models as, KEBXII, WL, OEPIV, MOAPW, MOAPEW, and GOLOM distributions. According to this result, we note that the estimate of TGL has the best measure where it has the smallest value of W∗, A∗, and KS statistic along with its P value. The fitted TGL, pdf, CDF, and PP-plot of the data set are displayed in Figure 4.Table 7
The TGL model's and other competing models' analytical results using COVID-19 data of The United Kingdom.
πβδγϑKSP valueW∗A∗TGLEstimates0.37590.72693.7995197.67170.01100.05790.93130.05200.3666SE0.10980.28241.076181.63370.0052KEBXIIEstimates9.29690.022345.11905.87851.74420.06860.80930.05740.4058SE4.91360.013313.00120.00240.0329WLEstimates209.12931.238687.0044253.2098—0.05890.92210.05730.4038SE427.50930.0999267.0487985.0169—OEPIVEstimates344.25890.79954.555913.9408—0.06000.91200.05570.3935SE89.54780.05532.56421.9995—MOAPWEstimates8.37280.65444.81520.0067—0.06200.89120.06140.4256SE11.20110.06293.27050.0020—MOAPEWEstimates12.77820.64463.63287.397711.83450.06430.86460.05620.3930SE14.68370.28037.698511.056612.0845GOLOMEstimates1.4935-1.40725.33733.13700.10030.35740.07190.4967SE7.57158.024517.357513.6175Figure 4
TGL model pdf and CDF estimates, as well as a PP-plot using COVID-19 data of the United Kingdom.


## 8.1. Data Set (1): COVID-19 of France

The COVID-19 data in question is from France, and it covers a period of 108 days, from March 1 to June 16, 2021. This data was formed by using daily new deaths (ND), daily cumulative cases (CC), and daily cumulative deaths (CD) as follows:(42)xi=NDiCCi−CDi−1×1000.The data are as follows: 0.0024 0.0025 0.0027 0.0045 0.0045 0.0062 0.0109 0.0113 0.0123 0.0123 0.0126 0.0129 0.0130 0.0139 0.0152 0.0160 0.0160 0.0161 0.0164 0.0169 0.0173 0.0174 0.0188 0.0219 0.0225 0.0226 0.0248 0.0260 0.0284 0.0303 0.0315 0.0318 0.0320 0.0323 0.0327 0.0329 0.0332 0.0343 0.0345 0.0346 0.0346 0.0347 0.0347 0.0352 0.0359 0.0365 0.0366 0.0370 0.0371 0.0376 0.0384 0.0392 0.0396 0.0419 0.0421 0.0443 0.0445 0.0462 0.0492 0.0506 0.0525 0.0536 0.0539 0.0540 0.0568 0.0569 0.0592 0.0600 0.0603 0.0606 0.0617 0.0619 0.0621 0.0629 0.0632 0.0657 0.0665 0.0675 0.0678 0.0683 0.0684 0.0691 0.0693 0.0707 0.0708 0.0719 0.0722 0.0727 0.0771 0.0773 0.0773 0.0774 0.0791 0.0832 0.0837 0.0845 0.0865 0.0897 0.0907 0.0910 0.0946 0.0961 0.0972 0.1010 0.1056 0.1123 0.1149 0.2153.In Table6, the TGL distribution is fitted to COVID-19 of France country. The TGL model is compared with other competitive models as Mead and Afify [16] proposed the Burr-XII model (KEBXII) with Kumaraswamy exponentiated, Weibull-Lomax (WL) distribution, Odds Exponential-Pareto IV (OEPIV) distribution proposed by Baharith et al. [17], Marshall–Olkin Alpha power Weibull (MOAPW) by Almetwally et al. [18], Marshall–Olkin Alpha power extended Weibull (MOAPEW) by Almetwally [19], Marshall–Olkin alpha power inverse Weibull (MOAPIW) by Basheer et al. [20], Marshall–Olkin alpha power Lomax (MOAPL) by Almongy et al. [21], and Gompertz Lomax (GOLOM) distribution by Oguntunde et al. [11]. According to this result, we note that the estimate of TGL has the best measure where it has the smallest value of Cramer-von Mises (W∗), Anderson-Darling (A∗), and Kolmogorov- Smirnov (KS) statistic along with its P value. The fitted TGL, pdf, CDF, and PP-plot of the data set are displayed in Figure 3.Table 6
The TGL model's and other competing models' analytical results using COVID-19 data of France.
πβδγϑKSP valueW∗A∗TGLEstimates8.35880.90780.2296106.50580.11310.06600.73480.07950.4964SE3.38530.11022.27301.73971.3473KEBXIIEstimates1.58950.707252.81891.40501.78940.07190.63270.10130.6654SE1.13600.31156.64340.26701.0125WLEstimates2.87071.54661.02130.1104—0.07110.64660.09680.6432SE20.49280.31180.96360.4204OEPIVEstimates3.79140.63100.86630.1134—0.07210.62810.10240.6726SE22.60070.10391.26770.3183MOAPIWEstimates3171.24390.61220.00260.1791—0.11000.14620.44162.7501SE1410.38180.04140.00040.0317MOAPWEstimates7.74931.01682.12760.0244—0.06630.72920.09090.5547SE26.41130.32372.70800.0155MOAPEWEstimates3.27590.85946.250318.884723.65010.06730.71180.09440.5600SE6.45100.32847.53131.541736.6454MOAPLEstimates1.085829.20037.74920.56950.06670.72230.09410.5544SE4.918455.229817.90491.1921GOLOMEstimates0.6174—6.58875.18360.33400.09120.32990.07980.5392SE0.32853.67302.76940.2097Figure 3
TGL model pdf and CDF estimates, as well as a PP-plot using COVID-19 data of France. Data set (2): COVID-19 of the United Kingdom.The COVID-19 data in question is from the United Kingdom and spans 82 days, from May 1 to July 16, 2021. This data is formed by using daily ND, daily CC, and daily CD as follows:0.0023, 0.0023, 0.0023, 0.0046, 0.0065, 0.0067, 0.0069, 0.0069, 0.0091, 0.0093, 0.0093, 0.0093, 0.0111, 0.0115, 0.0116, 0.0116, 0.0119, 0.0133, 0.0136, 0.0138, 0.0138, 0.0159, 0.0161, 0.0162, 0.0162, 0.0162, 0.0163, 0.0180, 0.0187, 0.0202, 0.0207, 0.0208, 0.0225, 0.0230, 0.0230, 0.0239, 0.0245, 0.0251, 0.0255, 0.0255, 0.0271, 0.0275, 0.0295, 0.0297, 0.0300, 0.0302, 0.0312, 0.0314, 0.0326, 0.0346, 0.0349, 0.0350, 0.0355, 0.0379, 0.0384, 0.0394, 0.0394, 0.0412, 0.0419, 0.0425, 0.0461, 0.0464, 0.0468, 0.0471, 0.0495, 0.0501, 0.0521, 0.0571, 0.0588, 0.0597, 0.0628, 0.0679, 0.0685, 0.0715, 0.0766, 0.0780, 0.0942, 0.0960, 0.0988, 0.1223, 0.1343, and 0.1781.In Table7, the TGL distribution is fitted to COVID-19 of The United Kingdom country. The TGL model is compared with other competitive models as, KEBXII, WL, OEPIV, MOAPW, MOAPEW, and GOLOM distributions. According to this result, we note that the estimate of TGL has the best measure where it has the smallest value of W∗, A∗, and KS statistic along with its P value. The fitted TGL, pdf, CDF, and PP-plot of the data set are displayed in Figure 4.Table 7
The TGL model's and other competing models' analytical results using COVID-19 data of The United Kingdom.
πβδγϑKSP valueW∗A∗TGLEstimates0.37590.72693.7995197.67170.01100.05790.93130.05200.3666SE0.10980.28241.076181.63370.0052KEBXIIEstimates9.29690.022345.11905.87851.74420.06860.80930.05740.4058SE4.91360.013313.00120.00240.0329WLEstimates209.12931.238687.0044253.2098—0.05890.92210.05730.4038SE427.50930.0999267.0487985.0169—OEPIVEstimates344.25890.79954.555913.9408—0.06000.91200.05570.3935SE89.54780.05532.56421.9995—MOAPWEstimates8.37280.65444.81520.0067—0.06200.89120.06140.4256SE11.20110.06293.27050.0020—MOAPEWEstimates12.77820.64463.63287.397711.83450.06430.86460.05620.3930SE14.68370.28037.698511.056612.0845GOLOMEstimates1.4935-1.40725.33733.13700.10030.35740.07190.4967SE7.57158.024517.357513.6175Figure 4
TGL model pdf and CDF estimates, as well as a PP-plot using COVID-19 data of the United Kingdom.


## 9. Conclusion

We investigate the so-called five-parameter transmuted generalized Lomax distribution in this study. Lomax and Gompertz Lomax (GoLom) distributions are included in the TGL model. The TGL distribution's structural properties are deduced. The maximum likelihood approach is used to estimate the population parameters based on complete and Type-II censored samples. We discussed the Monte Carlo simulation study to estimate point and interval estimation of parameters of TGL distribution based on complete sample sizes and Type-II censored samples. The proposed distribution was applied to two COVID-19 real-world data sets from France and United Kingdom. We compared a new transmuted generalization of the Lomax distribution (TGL) with KEBXII, WL, OEPIV, MOAPW, MOAPEW, and GOLOM distributions. It was shown to provide a better fit than several other models. We hope that the presented model will be used in a variety of fields, including engineering, survival and lifetime data, meteorology, biology, hydrology, economics (income disparity), and others.



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*Source: 5918511-2021-10-07.xml*