# Type I Half Logistic BurrX-G Family: Properties, Bayesian, and Non-Bayesian Estimation under Censored Samples and Applications to COVID-19 Data

Authors: Ali Algarni; Abdullah M. Almarashi; I. Elbatal; Amal S. Hassan; Ehab M. Almetwally; Abdulkader M. Daghistani; Mohammed Elgarhy
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we present a new family of continuous distributions known as the type I half logistic BurrX-G. The proposed family’s essential mathematical properties, such as quantile function (QuFu), moments (Mo), incomplete moments (InMo), mean deviation (MeD), Lorenz (Lo) and Bonferroni (Bo) curves, and entropy (En), are provided. Special models of the family are presented, including type I half logistic Burr X-Lomax, type I half logistic Burr X-Rayleigh, and type I half logistic Burr X-exponential. The maximum likelihood (MLL) and Bayesian techniques are utilized to produce parameter estimators for the recommended family using type II censored data. Monte Carlo simulation is used to evaluate the accuracy of estimates for one of the family’s special models. The COVID-19 real datasets from Italy, Canada, and Belgium are analysed to demonstrate the significance and flexibility of some new distributions from the family.


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## Body

## 1. Introduction

Statistical researchers have been encouraged in recent years to propose new broad families of continuous univariate distributions and to focus their efforts on improving their desired characteristics. For the time being, there is still a need for providing wider classes of distributions in order to provide them with greater flexibility and precision when fitting data. Some of the more recent generators sounding in the literature are the beta–G [1], type I half logistic [2], odd exponentiated half logistic G [3], Marshall–Olkin Burr X-G [4], generalized odd log-logistic-G [5], beta Burr type X−G [6], new generalized odd log-logistic-G [7], generalized Burr X–G [8], type II half logistic [9], the transmuted odd Fréchet–G family in [10], Kumaraswamy-type I half logistic [11], and Burr X-exponential-G [12], among others.Reference [13] proposed a new simple family of distributions with cumulative distribution function (CDFu) and probability density function (PDFu) using the Burr X as generator; the so-called Burr X−G family is as follows:(1)HBXx;θ=1−e−Gx;δ/G¯x;δ2θ,(2)hBXx;θ=2θgx;δG¯x;δ3Gx;δe−Gx;δ/G¯x;δ21−e−Gx;δ/G¯x;δ2θ−1,where gx;δ and Gx;δ are the PDFu and CDFu of any baseline distribution based on a parameter δ. The type I half-logistic- GTIHL−G family is [2] a represented family with an additional positive parameter lambda>0. The CDFu of the TIHL−G distribution family is(3)Fx=∫0−log1−Hx2λe−λx1+e−λx2dx=1−1−Hxλ1+1−Hxλ.The corresponding PDFu is(4)fx=2λhx1−Hxλ−11+1−Hxλ2.The failure (hazard) rate function is(5)τx=λhx1−Hx1+1−Hxλ.We intend to benefit from the combined features of the BurrX-G and the TIHL-G in this work by introducing a new generated family of distributions known as the type I half-logistic Burr X-G (HLBX -G). We hope that the new family will provide more flexibility and attract a broader range of applications in reliability, engineering, and other research areas. The CDFu and PDFu of the HLBX−G family of distributions are provided by(6)Fx;λ,θ,δ=1−1−1−e−Gx;δ/G¯x;δ2θλ1+1−1−e−Gx;δ/G¯x;δ2θλ,(7)fx;λ,θ,δ=4λθgx;δG¯x;δ3Gx;δe−Gx;δ/G¯x;δ21−e−Gx;δ/G¯x;δ2θ−11−1−e−Gx;δ/G¯x;δ2θλ−11+1−1−e−Gx;δ/G¯x;δ2θλ−2.Henceforward, a random variableX having PDFu (7) will be defined as X∼HLBXλ,θ,δ. The hazard rate function of HLBX−G family is given by(8)τx;λ,θ,δ=2λθgx;δGx;δe−Gx;δ/G¯x;δ21−e−Gx;δ/G¯x;δ2θ−1G¯x;δ31−1−e−Gx;δ/G¯x;δ2θ×11+1−1−e−Gx;δ/G¯x;δ2θλ.TheHLBX−G quantile function, say x=Qu, can be obtained by inverting (6) as follows:(9)F−1u=QGu=G−1−log1−1−1−u/1+u1/λ1/θ1/21+−log1−1−1−u/1+u1/λ1/θ1/2,where QGu denotes the QuFu. The quantile measurements are crucial in determining the impact of form parameters on skewness and kurtosis. For information, see [14, 15]. The new suggested family is extremely adaptable and includes several additional distributions. This study will present three new models of the family: HLBX Lomax, HLBX exponential, and HLBX Rayleigh. The pdfs of these models can be symmetric, right-skewed, unimodal, and up-side-down shaped; they are novel and extremely adaptable. The HRF of these models can also be increasing, decreasing, J-shaped, or U-shaped. The remainder of the paper is structured as follows: a useful expansion for the HLBX−G density and some special models are investigated in Section 2. Several mathematical properties including Mos, InMos, MeD, Lo, and Bo curves; residual life (ReL) and reversed ReL (RReL) functions; probability weighted moments (PrWMo); and the En of proposed family are derived in Section 3. MLL and Bayesian estimation methods are provided based on censored sample (CS) in Section 4. Applications of COVID-19 dataset to illustrate the flexibility and potentiality of the proposed family are analysed in Section 5. Section 6 discusses simulation analysis. Section 7 concludes with closing comments.


## 2. Useful Expansion

The following results are useful for expansions offx and Fx. If z<1 and b>0 is a real noninteger, then the following power series holds:(10)1+z−b=∑k=0∞−bkzk,(11)1−zb−1=∑k=0∞−1kΓbk!Γb−kzk.When we apply (10) to the final word in (7), we obtain(12)fx;λ,θ,δ=4λθgx;δG¯x;δ3Gx;δe−Gx;δ/G¯x;δ21−e−Gx;δ/G¯x;δ2θ−1∑i=0∞−2i1−1−e−Gx;δ/G¯x;δ2θλi+1−1.Using (11) in (12), we get(13)fx;λ,θ,δ=4λθ∑i,j,k=0∞−1j+k−2iΓλi+1Γθj+1j!k!Γλi+1−jΓθj+1−k×gx;δG¯x;δ3Gx;δe−k+1Gx;δ/G¯x;δ2.The power series expansion ofe−k+1Gx;δ/G¯x;δ2 is(14)e−k+1Gx;δ/G¯x;δ2=∑m=0∞−1mk+1mm!Gx;δ2mG¯x;δ2m.By adding (14) to (13), we get(15)fx;λ,θ,δ=4λθ∑i,j,k=0∞−1j+k−2iΓλi+1Γθj+1j!k!Γλi+1−jΓθj+1−k×∑m=0∞−1mk+1mm!gx;δG¯x;δ2m+3Gx;δ2m+1.Making use of the generalized binomial expansion to1−Gx;δ−2m+3, we can write(16)1−Gx;δ−2m+3=∑d=0∞Γ2m+d+3d!Γ2m+3Gx;δd.TheHLBX−G density function may be represented as an endless combination of Expo-G density functions by substituting (16) into (15)(17)fHLBX−Gx;λ,θ,δ=∑m,d=0∞ϖm,dπ2m+1+dx,where(18)ϖm,d=∑i,j,k=0∞−1j+k+m−2i4λθΓλi+1Γθj+1j!k!m!d!Γλi+1−jΓθj+1−k×k+1mΓ2m+d+3Γ2m+32m+1+d,and π2m+1+dx=2m+1+dgxG2m+d+1x is the expo-G PDFu with power parameter 2m+1+d. As a result, numerous mathematical and statistical features of the HLBX−G distribution are evident from those of the exp-G distribution. Similarly, the HLBX−G family CDFu may be represented as a combination of exp-G CDFus where(19)FHLBX−Gx;λ,θ,δ=∑m,d=0∞ϖm,dΠ2m+1+dx,where Π2m+1+dx is the exp-G cdf with power parameter 2m+1+d.


### 2.1. Some HLBX-G Family Special Models

We present three submodels of this family based on the baseline distributions: Lomax, exponential, and Rayleigh. These models’ CDFu and PDFu files are given in Table1.Table 1
Three examples of baseline lifetime distributions.
ModelCDFu:Gx;δPDFu:gx;δGx;δ/G¯x;δLomax1−1+x/β−αα/β1+x/β−α−11+x/βα−1Exponential1−e−μxμe−μxeμx−1Rayleigh1−e−μ/2x2μxe−μ/2x2eμ/2x2−1


#### 2.1.1. Half-Logistic BurrX Lomax (HLBXL) Distribution

The CDFu and PDFu ofHLBXL distribution are(20)Fx=1−1−1−e−1+x/βα−12θλ1+1−1−e−1+x/βα−12θλ,fx=4λθα/β1+x/β−α−11+x/β−3α1−1+xβ−αe−1+x/βα−121−e−1+x/βα−12θ−11−1−e−1+x/βα−12θλ−11+1−1−e−1+x/βα−12θλ−2.Plots of the HLBXLo densities are represented in Figure1.Figure 1
HLBXL density function and hazard rate for different values of parameters.


#### 2.1.2. Half-Logistic BurrX Exponential (HLBXE) Distribution

The CDFu and PDFu of theHLBXE model (for x>0) are(21)Fx=1−1−1−e−eμx−12θλ1+1−1−e−eμx−12θλ,fx=4λθμe−μxe−3μx1−e−μxe−eμx−121−e−eμx−12θ−11−1−e−eμx−12θλ−11+1−1−e−eμx−12θλ−2.Plots of the HLBXE densities are represented in Figure2.Figure 2
HLBXE density function and hazard rate.


#### 2.1.3. Half-Logistic BurrX Rayleigh (HLBXR) Distribution

The CDFu and PDFu of theHLBXR model (for x>0) are(22)Fx=1−1−1−e−eμ/2x2−12θλ1+1−1−e−eμ/2x2−12θλ,fx=4λθμxe−μ/2x2e−3μ/2x231−e−μ/2x2e−eμ/2x2−121−e−eμ/2x2−12θ−11−1−e−eμ/2x2−12θλ−11+1−1−e−eμ/2x2−12θλ−2.Plots of the HLBXR densities are represented in Figure3.Figure 3
HLBXR density function.


## 2.1. Some HLBX-G Family Special Models

We present three submodels of this family based on the baseline distributions: Lomax, exponential, and Rayleigh. These models’ CDFu and PDFu files are given in Table1.Table 1
Three examples of baseline lifetime distributions.
ModelCDFu:Gx;δPDFu:gx;δGx;δ/G¯x;δLomax1−1+x/β−αα/β1+x/β−α−11+x/βα−1Exponential1−e−μxμe−μxeμx−1Rayleigh1−e−μ/2x2μxe−μ/2x2eμ/2x2−1


### 2.1.1. Half-Logistic BurrX Lomax (HLBXL) Distribution

The CDFu and PDFu ofHLBXL distribution are(20)Fx=1−1−1−e−1+x/βα−12θλ1+1−1−e−1+x/βα−12θλ,fx=4λθα/β1+x/β−α−11+x/β−3α1−1+xβ−αe−1+x/βα−121−e−1+x/βα−12θ−11−1−e−1+x/βα−12θλ−11+1−1−e−1+x/βα−12θλ−2.Plots of the HLBXLo densities are represented in Figure1.Figure 1
HLBXL density function and hazard rate for different values of parameters.


### 2.1.2. Half-Logistic BurrX Exponential (HLBXE) Distribution

The CDFu and PDFu of theHLBXE model (for x>0) are(21)Fx=1−1−1−e−eμx−12θλ1+1−1−e−eμx−12θλ,fx=4λθμe−μxe−3μx1−e−μxe−eμx−121−e−eμx−12θ−11−1−e−eμx−12θλ−11+1−1−e−eμx−12θλ−2.Plots of the HLBXE densities are represented in Figure2.Figure 2
HLBXE density function and hazard rate.


### 2.1.3. Half-Logistic BurrX Rayleigh (HLBXR) Distribution

The CDFu and PDFu of theHLBXR model (for x>0) are(22)Fx=1−1−1−e−eμ/2x2−12θλ1+1−1−e−eμ/2x2−12θλ,fx=4λθμxe−μ/2x2e−3μ/2x231−e−μ/2x2e−eμ/2x2−121−e−eμ/2x2−12θ−11−1−e−eμ/2x2−12θλ−11+1−1−e−eμ/2x2−12θλ−2.Plots of the HLBXR densities are represented in Figure3.Figure 3
HLBXR density function.


## 2.1.1. Half-Logistic BurrX Lomax (HLBXL) Distribution

The CDFu and PDFu ofHLBXL distribution are(20)Fx=1−1−1−e−1+x/βα−12θλ1+1−1−e−1+x/βα−12θλ,fx=4λθα/β1+x/β−α−11+x/β−3α1−1+xβ−αe−1+x/βα−121−e−1+x/βα−12θ−11−1−e−1+x/βα−12θλ−11+1−1−e−1+x/βα−12θλ−2.Plots of the HLBXLo densities are represented in Figure1.Figure 1
HLBXL density function and hazard rate for different values of parameters.


## 2.1.2. Half-Logistic BurrX Exponential (HLBXE) Distribution

The CDFu and PDFu of theHLBXE model (for x>0) are(21)Fx=1−1−1−e−eμx−12θλ1+1−1−e−eμx−12θλ,fx=4λθμe−μxe−3μx1−e−μxe−eμx−121−e−eμx−12θ−11−1−e−eμx−12θλ−11+1−1−e−eμx−12θλ−2.Plots of the HLBXE densities are represented in Figure2.Figure 2
HLBXE density function and hazard rate.


## 2.1.3. Half-Logistic BurrX Rayleigh (HLBXR) Distribution

The CDFu and PDFu of theHLBXR model (for x>0) are(22)Fx=1−1−1−e−eμ/2x2−12θλ1+1−1−e−eμ/2x2−12θλ,fx=4λθμxe−μ/2x2e−3μ/2x231−e−μ/2x2e−eμ/2x2−121−e−eμ/2x2−12θ−11−1−e−eμ/2x2−12θλ−11+1−1−e−eμ/2x2−12θλ−2.Plots of the HLBXR densities are represented in Figure3.Figure 3
HLBXR density function.


## 3. Fundamental Properties

We looked at the statistical properties of theHLBX−G distribution; Mos, InMos, MeD, Lo, and Bo curves; ReL and RReL functions; and PrWMs in this section.


### 3.1. Moments and Moment Generating Functions

The ordinary Mos and Mo generating function (MoGFu) of theHLBX−G family are computed. The different orders for the Mos are very useful in reliability applications to compute the expected life time of a device, skewness, and kurtosis in a given set of observations.


#### 3.1.1. Moments

Therth ordinary Mo of X can be obtained from (17) as(23)μr/=EXr=∑m,d=0∞ϖm,dEZ2m+1+dr,where Z2m+1+d. The exp-G random variable with the power parameter 2m+1+d is denoted. For ξ>0, the second formula for the rth moment follows from (17) as EZξr=ξ∫−∞∞xrgxGxξ−1dx, which is numerically calculable in terms of the baseline QuFu, i.e., QGu=G−1u as EZξn=ξ∫01uξ−1QGundu. For most parent distributions, this integration can be calculated numerically. Skewness and kurtosis can be calculated using the nth central Mo, say Mnx of X, where(24)Mnx=EX−μ1/n=∑r=0∞nr−μ1/n−rEXr=∑r=0∞∑m,d=0∞nr−μ1/n−rϖm,dEZ2m+1+dr.Remark 1.
IfX have the ordinary Mo in (23), the MoGFu of X can be investigated by using two formulae. The first formula can be computed from equation (17) as(25)MXt=EetX=∑m,d=0∞ϖm,dM2m+1+dt,where M2m+1+dt is the MoGFu of Z2m+1+d. As a result, MXt may be simply calculated from the exp-G generating function. The following is a second alternative formula that may be obtained from (17):(26)MXt=∑m,d=0∞ϖm,dφt,2m+1+d,where φt,ε=ε∫01uε−1etQGudu can be computed numerically from the baseline quantile function, i.e., QGu=G−1u.
Figure4 show the mean, variance, skewness, and kurtosis for HLBXE model.Figure 4
3D plots of mean, variance, skewness, and kurtosis of the HLBXE distribution forμ=2.


### 3.2. Incomplete Moments

The MeDs, Bo, and Lo curves, and other applications rely heavily on the first InMos. These curves have a wide range of uses, including economics, demography, and medicine. This is obvious not only in econometrics research, but also in other disciplines. For every reals>0, the sth InMos of X specified by ηst may be calculated from (17) as(27)ηst=∫−∞txsfxdx=∑m,d=0∞ϖm,d∫−∞txsπ2m+1+dxdx.Equation (27) denotes the sth InMos of Z2m+1+d. The MeDs give important information about characteristic of population and also have been applied of income fields. If X has the HLBX−G family of distributions, the MeDs about the mean μ=EX and the MeDs about the median M are defined by(28)δμx=EX−μ1′=2μ1′Fμ1′−2η1μ1′,δMx=EX−M=μ1′−2η1M,respectively, where μ1′=EX, M= median X=Q1/2, Fμ1′ is evaluated from (6), and η1t is the first InMo given by (27) with s=1. We can determine δμx and δMx by two techniques; the first can be obtained from (17) as η1t=∑m,d=0∞ϖm,dZ2m+1+dt, where Z2m+1+dt=∫−∞txπ2m+1+dxdx is the first InMo of the exp-G distribution. The second technique is given by η1t=∑m,d=0∞ϖm,dϑ2m+1+dt where(29)ϑ2m+1+dt=2m+1+d∫0Gtu2m+1+dQGudu,which can be computed numerically and QGu=G−1u;δ. For a positive random variable X, the Lo and Bo curves, for a given probability p, are given by Lp=1/μ1′η1q and Bp=1/pμ1′η1q, respectively, where μ1′=EX, and q=Qp is the QuFu of X at p.


### 3.3. Residual Lives

Therth order Mo of the ReL is given by(30)ηrt=EX−tr|X>t=1F¯t∫t∞x−trfxdx,r≥1=1F¯t∑k,m=0∞ηk,m⊛∫t∞xrπθk+1+mxdx,where ηk,m⊛=∑k,m=0∞ηk,m∑m=0rrm−tr−m. The mean ReL (MReL) of HLBX−G family s can be obtained by setting r=1 in equation (30), defined as(31)η1t=EXt=EX|X>t.The well-known formula can be used to calculate therth order Mo of the RReL (or inactivity time):(32)mrt=Et−Xr|X≤t=1Ft∫0tt−xrfxdx,r≥1=1Ft∑k,m=0∞ηk,m⊛∫0txrπθk+1+mxdx.The proposed family’sMPT can be calculated by setting r=1 in (32), where(33)mt=EXt=Et−X|X<t.


### 3.4. Probability Weighted Moments

Ther,sth PrWMos of the HLBX−G family is given by(34)ωr,s=EXrFxs=∫−∞∞xrFxsfxdx,using equations (6) and (7), and with a little math, we can get(35)fxFxs=∑m,d=0∞ϰm,dsπ2m+1+dx,where(36)ϰm,ds=∑i,j,k,h=0∞−1i+j+k+h+m+dλjh+1mΓs+i+2Γs+1i!j!m!Γs+2Γs−j+12m+1+dλi+j+1−1kθk+1−1h−2m−3d.Therefore, ther,sth PWMs of the HLBX−G family can be expressed as(37)ωr,s=∑m,d=0∞ϰm,ds∫−∞∞xrπ2m+1+dxdx.


### 3.5. Entropy

The Rényi En is defined byρ>0,ρ≠1(38)IRρ=11−ρlog∫−∞∞fρxdx.Using (7), applying the same procedure of the useful expansion (17) and after some simplifications, we get(39)fρx=∑m,d=0∞Ψm,dgxρGx2m+d+ρ,where(40)Ψm,d=4λθρ∑i,j,k=0∞−1j+k+m+dλjk+ρmm!−2ρiλρ+i−ρjθρ+j−ρk−2m−3ρd.Thus, Rényi entropy ofHLBX−G family is defined as(41)IRρ=11−ρlog∑m,d=0∞ψm,d∫−∞∞gxρGx2m+d+ρdx.


## 3.1. Moments and Moment Generating Functions

The ordinary Mos and Mo generating function (MoGFu) of theHLBX−G family are computed. The different orders for the Mos are very useful in reliability applications to compute the expected life time of a device, skewness, and kurtosis in a given set of observations.


### 3.1.1. Moments

Therth ordinary Mo of X can be obtained from (17) as(23)μr/=EXr=∑m,d=0∞ϖm,dEZ2m+1+dr,where Z2m+1+d. The exp-G random variable with the power parameter 2m+1+d is denoted. For ξ>0, the second formula for the rth moment follows from (17) as EZξr=ξ∫−∞∞xrgxGxξ−1dx, which is numerically calculable in terms of the baseline QuFu, i.e., QGu=G−1u as EZξn=ξ∫01uξ−1QGundu. For most parent distributions, this integration can be calculated numerically. Skewness and kurtosis can be calculated using the nth central Mo, say Mnx of X, where(24)Mnx=EX−μ1/n=∑r=0∞nr−μ1/n−rEXr=∑r=0∞∑m,d=0∞nr−μ1/n−rϖm,dEZ2m+1+dr.Remark 1.
IfX have the ordinary Mo in (23), the MoGFu of X can be investigated by using two formulae. The first formula can be computed from equation (17) as(25)MXt=EetX=∑m,d=0∞ϖm,dM2m+1+dt,where M2m+1+dt is the MoGFu of Z2m+1+d. As a result, MXt may be simply calculated from the exp-G generating function. The following is a second alternative formula that may be obtained from (17):(26)MXt=∑m,d=0∞ϖm,dφt,2m+1+d,where φt,ε=ε∫01uε−1etQGudu can be computed numerically from the baseline quantile function, i.e., QGu=G−1u.
Figure4 show the mean, variance, skewness, and kurtosis for HLBXE model.Figure 4
3D plots of mean, variance, skewness, and kurtosis of the HLBXE distribution forμ=2.


## 3.1.1. Moments

Therth ordinary Mo of X can be obtained from (17) as(23)μr/=EXr=∑m,d=0∞ϖm,dEZ2m+1+dr,where Z2m+1+d. The exp-G random variable with the power parameter 2m+1+d is denoted. For ξ>0, the second formula for the rth moment follows from (17) as EZξr=ξ∫−∞∞xrgxGxξ−1dx, which is numerically calculable in terms of the baseline QuFu, i.e., QGu=G−1u as EZξn=ξ∫01uξ−1QGundu. For most parent distributions, this integration can be calculated numerically. Skewness and kurtosis can be calculated using the nth central Mo, say Mnx of X, where(24)Mnx=EX−μ1/n=∑r=0∞nr−μ1/n−rEXr=∑r=0∞∑m,d=0∞nr−μ1/n−rϖm,dEZ2m+1+dr.Remark 1.
IfX have the ordinary Mo in (23), the MoGFu of X can be investigated by using two formulae. The first formula can be computed from equation (17) as(25)MXt=EetX=∑m,d=0∞ϖm,dM2m+1+dt,where M2m+1+dt is the MoGFu of Z2m+1+d. As a result, MXt may be simply calculated from the exp-G generating function. The following is a second alternative formula that may be obtained from (17):(26)MXt=∑m,d=0∞ϖm,dφt,2m+1+d,where φt,ε=ε∫01uε−1etQGudu can be computed numerically from the baseline quantile function, i.e., QGu=G−1u.
Figure4 show the mean, variance, skewness, and kurtosis for HLBXE model.Figure 4
3D plots of mean, variance, skewness, and kurtosis of the HLBXE distribution forμ=2.


## 3.2. Incomplete Moments

The MeDs, Bo, and Lo curves, and other applications rely heavily on the first InMos. These curves have a wide range of uses, including economics, demography, and medicine. This is obvious not only in econometrics research, but also in other disciplines. For every reals>0, the sth InMos of X specified by ηst may be calculated from (17) as(27)ηst=∫−∞txsfxdx=∑m,d=0∞ϖm,d∫−∞txsπ2m+1+dxdx.Equation (27) denotes the sth InMos of Z2m+1+d. The MeDs give important information about characteristic of population and also have been applied of income fields. If X has the HLBX−G family of distributions, the MeDs about the mean μ=EX and the MeDs about the median M are defined by(28)δμx=EX−μ1′=2μ1′Fμ1′−2η1μ1′,δMx=EX−M=μ1′−2η1M,respectively, where μ1′=EX, M= median X=Q1/2, Fμ1′ is evaluated from (6), and η1t is the first InMo given by (27) with s=1. We can determine δμx and δMx by two techniques; the first can be obtained from (17) as η1t=∑m,d=0∞ϖm,dZ2m+1+dt, where Z2m+1+dt=∫−∞txπ2m+1+dxdx is the first InMo of the exp-G distribution. The second technique is given by η1t=∑m,d=0∞ϖm,dϑ2m+1+dt where(29)ϑ2m+1+dt=2m+1+d∫0Gtu2m+1+dQGudu,which can be computed numerically and QGu=G−1u;δ. For a positive random variable X, the Lo and Bo curves, for a given probability p, are given by Lp=1/μ1′η1q and Bp=1/pμ1′η1q, respectively, where μ1′=EX, and q=Qp is the QuFu of X at p.


## 3.3. Residual Lives

Therth order Mo of the ReL is given by(30)ηrt=EX−tr|X>t=1F¯t∫t∞x−trfxdx,r≥1=1F¯t∑k,m=0∞ηk,m⊛∫t∞xrπθk+1+mxdx,where ηk,m⊛=∑k,m=0∞ηk,m∑m=0rrm−tr−m. The mean ReL (MReL) of HLBX−G family s can be obtained by setting r=1 in equation (30), defined as(31)η1t=EXt=EX|X>t.The well-known formula can be used to calculate therth order Mo of the RReL (or inactivity time):(32)mrt=Et−Xr|X≤t=1Ft∫0tt−xrfxdx,r≥1=1Ft∑k,m=0∞ηk,m⊛∫0txrπθk+1+mxdx.The proposed family’sMPT can be calculated by setting r=1 in (32), where(33)mt=EXt=Et−X|X<t.


## 3.4. Probability Weighted Moments

Ther,sth PrWMos of the HLBX−G family is given by(34)ωr,s=EXrFxs=∫−∞∞xrFxsfxdx,using equations (6) and (7), and with a little math, we can get(35)fxFxs=∑m,d=0∞ϰm,dsπ2m+1+dx,where(36)ϰm,ds=∑i,j,k,h=0∞−1i+j+k+h+m+dλjh+1mΓs+i+2Γs+1i!j!m!Γs+2Γs−j+12m+1+dλi+j+1−1kθk+1−1h−2m−3d.Therefore, ther,sth PWMs of the HLBX−G family can be expressed as(37)ωr,s=∑m,d=0∞ϰm,ds∫−∞∞xrπ2m+1+dxdx.


## 3.5. Entropy

The Rényi En is defined byρ>0,ρ≠1(38)IRρ=11−ρlog∫−∞∞fρxdx.Using (7), applying the same procedure of the useful expansion (17) and after some simplifications, we get(39)fρx=∑m,d=0∞Ψm,dgxρGx2m+d+ρ,where(40)Ψm,d=4λθρ∑i,j,k=0∞−1j+k+m+dλjk+ρmm!−2ρiλρ+i−ρjθρ+j−ρk−2m−3ρd.Thus, Rényi entropy ofHLBX−G family is defined as(41)IRρ=11−ρlog∑m,d=0∞ψm,d∫−∞∞gxρGx2m+d+ρdx.


## 4. Statistical Inference under Type II Censored Sample

Reference [16] examined the two most prevalent censoring systems, known as Type I and Type II censoring schemes. In Type II censoring, a life test is stopped after a specific number of failures. n and r are fixed and predefined in this case, while T=xr is a random variable. See [17] for further details.


### 4.1. Maximum Likelihood Estimation

The MLL has desirable features and may be used to calculate confidence intervals and test statistics. In both the Type II CS and the special case (full sample ifr=n), we compute the MLL estimates (MLE) of the parameters of the HLBX−G family. Let x1,x2,…xr,…xn be a n-sample random sample from the HLBX−G distribution provided by (7). We spoke about n−r observations, where r is the number of the uncensored items. Let ψ=λ,θ,δT be q×1 vector of parameters. The likelihood function of HLBX-G family under Type II CS can be written as(42)Lr=rlog4λ+rlogθ+∑i=1rloggxi;δ+∑i=1rlogGxi;δ−3∑i=1rlogG¯xi;δ−∑i=1rti2+θ−1∑i=1rlog1−e−ti2+λ−1∑i=1rlog1−1−e−ti2θ−2∑i=1rlog1+1−1−e−ti2θλ+n−rλlog1−1−e−Gxr;δ/G¯xr;δ2θ−n−rlog1+1−1−e−Gxr;δ/G¯xr;δ2θλ,where ti=Gxi;δ/G¯xi;δ. The components of score function Uψ=Uλ,Uθ,UδT are(43)Uλ=∂Lr∂λ=rλ+∑i=1rlog1−1−e−ti2θ+n−rlog1−1−e−tr2θ−2∑i=1r1−1−e−ti2θλlog1−1−e−ti2θ1+1−1−e−ti2θλ+n−r1−1−e−tr2θλlog1−1−e−tr2θ1+1−1−e−tr2θλ,Uθ=∂Lr∂θ=rθ+∑i=1rlog1−e−ti2−n−rλ1−e−tr2θlog1−e−tr21−1−e−tr2θ−λ−1∑i=1r1−e−ti2θlog1−e−ti21−1−e−ti2θ+2∑i=1rλ1−1−e−ti2θλ−11−e−ti2θlog1−e−ti21+1−1−e−ti2θλ+n−rλ1−1−e−tr2θλ−11−e−tr2θlog1−e−tr21+1−1−e−tr2θλ,Uδk=∂Lr∂δk=∑i=1rg′xi;δgxi;δ+∑i=1rG′xi;δGxi;δ−3∑i=1rG¯′xi;φG¯xi;φ−2∑i=1rti∂ti∂δk+θ−1∑i=1r2tie−ti21−e−ti2∂ti∂δk−λ−1∑i=1r2θtie−ti21−e−ti2θ−11−1−e−ti2θ∂ti∂δk+4λ∑i=1rθtie−ti21−e−ti2θ−11−1−e−ti2θλ−11+1−1−e−ti2θλ∂ti∂δk−2n−rλθtre−tr21−e−tr2θ−11−1−e−tr2θ∂tr∂δk+2n−rλθtre−tr21−e−tr2θ−11−1−e−tr2θλ−11+1−1−e−tr2θλ∂tr∂δk,where g′xi;δ=∂gxi;δ/∂δk,G′xi;δ=∂Gxi;δ/∂δk,G¯′xi;δ=∂G¯xi;δ/∂δk and δ¯k is the kth element of the vector of parameters ψ¯. The MLE of parameters λ,θ, and δ is obtained by setting Uλ=Uθ=Uδ=0 and simultaneously solving these equations to produce the MLL estimators. These equations cannot be solved analytically; however, they can be solved numerically using iterative approaches with statistical software.


### 4.2. Bayesian Estimation

The prior distribution and the loss function (LoFu) are both used in the Bayesian estimation technique. All parameters are regarded as random variables with a particular distribution known as the prior distribution in this technique. We must select one if no prior knowledge is provided, which is typically the case. We picked independent gamma distributions as our priors since the prior distribution is crucial in parameter estimation. The LoFu, on the other hand, plays an important role in the Bayesian approach. Most Bayesian inference techniques are based on symmetric and asymmetric LoFus. Two of the most frequent symmetric LoFus are the squared error and the linear exponential (Linex) LoFus. The independent joint prior density function ofψ can be written as follows:(44)πψ=b1a1Γa1b2a2Γa2b3a3Γa3λa1−1θa2−1δa3−1e−b1λ+b2θ+b3δ.Reference [18] discussed how to elicit the hyperparameters of the informative priors. From the MLEs λ^B,θ^B,δ^B, we will get these beneficial priors by multiplying the estimate and variance by the inverse of the Fisher information matrix (FIM_ij) of ψ, say λ^B,θ^B,δ^B. By equating mean and variance of gamma priors, the estimated hyperparameters can be written as ai=ψi^2/FIMii and bi=ψi^/FIMii, where FIMii is a variance.The joint posterior PDFu ofψ is obtained from LL function and joint prior function:(45)πψ|x¯=ℓx¯|ψ⋅πψ∫ψℓx¯|ψ⋅πψdψ.Then the joint posterior of HLBX-G family under Type II CS can be written as(46)πψ|x¯∝λa1+r−1θa2+r−1δa3−1e−b1λ+b2θ+b3δ21−1−e−Gxr;δ/G¯xr;δ2θλ1+1−1−e−Gxr;δ/G¯xr;δ2θλn−r∏i=1rgxi;δG¯xi;δ3Gxi;δ1−e−Gxi;δ/G¯xi;δ2θ−1e−∑i=1rGxi;δ/G¯xi;δ2∏i=1r1−1−e−Gxi;δ/G¯xi;δ2θλ−11+1−1−e−Gxi;δ/G¯xi;δ2θλ−2.The Bayes estimators ofψ, say λ^B,θ^B,δ^B based on squared error LoFu, is given by(47)p^B−SELλ,θ,δ=Eλ,θ,δ|x¯pλ,θ,δ=∫0∞∫0∞∫0∞pλ,θ,δ×πω|x¯dλdθdδ.The Bayes estimates of the unknown parameterspsi under the Linex LoFu may be calculated as follows: ψ^Linex=−1/ulogEe−uψ|x. See, for example, [16, 18] for more information on Bayesian estimation. It is worth noting that the integrals (47) cannot be obtained explicitly. As a consequence, we estimate the value of integrals using the Markov Chain Monte Carlo (MCMC) approach.Gibbs sampling and, more generally, Metropolis within Gibbs samplers are significant MCMC subclasses. Two popular MCMC techniques are the Metropolis-Hastings (MH) algorithm and Gibbs sampling. The MH algorithm, like acceptance-rejection sampling, evaluates whether a candidate value can be created from a proposal distribution throughout each iteration of the algorithm. The following are the MH inside Gibbs sampling stages that we used to produce random samples from conditional posterior densities of the HLBX-G family in a Type II CS:(48)πλ|θ,δ,x¯∝λa1+r−1e−b1λ∏i=1r1−1−e−ti2θλ−11+1−1−e−ti2θλ−21−1−e−tr2θλ1+1−1−e−tr2θλn−r,πθ|λ,δ,x¯∝θa2+n−1e−b2θ∏i=1r1−e−Gxi;δ/G¯xi;δ2θ−11−1−e−Gxi;δ/G¯xi;δ2θλ−11−1−e−tr2θλ1+1−1−e−tr2θλn−r∏i=1r1+1−1−e−Gxi;δ/G¯xi;δ2θλ−2,πδ|λ,θ,x¯∝δa3−1e−b3δe−∑i=1Rti2∏i=1rgxi;δG¯xi;δ3Gxi;δ1−e−ti2θ−1∏i=1r1−1−e−ti2θλ−11+1−1−e−ti2θλ−21−1−e−tr2θλ1+1−1−e−tr2θλn−r,where ti=Gxi;δ/G¯xi;δ. The MH algorithm (Algorithm 1) generates a sequence of draws from this distribution.Algorithm 1: Algorithm of MCMC.
(1)
Start with any initial valueψli as a length of ψ satisfying πψ>0.(2)
Using the initial value, sample a candidate pointψ∗ from proposal qψ∗.(3)
Given the candidate pointψ∗, calculate the acceptance probability Aj=min1,Lψ∗|xπψ∗qψ/Lx|ψπψqψ∗.(4)
Draw a value ofu from the uniform (0, 1) distribution; if u<Ai, accept ψl∗ as ψli.(5)
Otherwise, rejectψl∗ and do ψli=ψli−1.(6)
Repeat steps 2–5 many times until we geti draws.(7)
Use loss function.


## 4.1. Maximum Likelihood Estimation

The MLL has desirable features and may be used to calculate confidence intervals and test statistics. In both the Type II CS and the special case (full sample ifr=n), we compute the MLL estimates (MLE) of the parameters of the HLBX−G family. Let x1,x2,…xr,…xn be a n-sample random sample from the HLBX−G distribution provided by (7). We spoke about n−r observations, where r is the number of the uncensored items. Let ψ=λ,θ,δT be q×1 vector of parameters. The likelihood function of HLBX-G family under Type II CS can be written as(42)Lr=rlog4λ+rlogθ+∑i=1rloggxi;δ+∑i=1rlogGxi;δ−3∑i=1rlogG¯xi;δ−∑i=1rti2+θ−1∑i=1rlog1−e−ti2+λ−1∑i=1rlog1−1−e−ti2θ−2∑i=1rlog1+1−1−e−ti2θλ+n−rλlog1−1−e−Gxr;δ/G¯xr;δ2θ−n−rlog1+1−1−e−Gxr;δ/G¯xr;δ2θλ,where ti=Gxi;δ/G¯xi;δ. The components of score function Uψ=Uλ,Uθ,UδT are(43)Uλ=∂Lr∂λ=rλ+∑i=1rlog1−1−e−ti2θ+n−rlog1−1−e−tr2θ−2∑i=1r1−1−e−ti2θλlog1−1−e−ti2θ1+1−1−e−ti2θλ+n−r1−1−e−tr2θλlog1−1−e−tr2θ1+1−1−e−tr2θλ,Uθ=∂Lr∂θ=rθ+∑i=1rlog1−e−ti2−n−rλ1−e−tr2θlog1−e−tr21−1−e−tr2θ−λ−1∑i=1r1−e−ti2θlog1−e−ti21−1−e−ti2θ+2∑i=1rλ1−1−e−ti2θλ−11−e−ti2θlog1−e−ti21+1−1−e−ti2θλ+n−rλ1−1−e−tr2θλ−11−e−tr2θlog1−e−tr21+1−1−e−tr2θλ,Uδk=∂Lr∂δk=∑i=1rg′xi;δgxi;δ+∑i=1rG′xi;δGxi;δ−3∑i=1rG¯′xi;φG¯xi;φ−2∑i=1rti∂ti∂δk+θ−1∑i=1r2tie−ti21−e−ti2∂ti∂δk−λ−1∑i=1r2θtie−ti21−e−ti2θ−11−1−e−ti2θ∂ti∂δk+4λ∑i=1rθtie−ti21−e−ti2θ−11−1−e−ti2θλ−11+1−1−e−ti2θλ∂ti∂δk−2n−rλθtre−tr21−e−tr2θ−11−1−e−tr2θ∂tr∂δk+2n−rλθtre−tr21−e−tr2θ−11−1−e−tr2θλ−11+1−1−e−tr2θλ∂tr∂δk,where g′xi;δ=∂gxi;δ/∂δk,G′xi;δ=∂Gxi;δ/∂δk,G¯′xi;δ=∂G¯xi;δ/∂δk and δ¯k is the kth element of the vector of parameters ψ¯. The MLE of parameters λ,θ, and δ is obtained by setting Uλ=Uθ=Uδ=0 and simultaneously solving these equations to produce the MLL estimators. These equations cannot be solved analytically; however, they can be solved numerically using iterative approaches with statistical software.


## 4.2. Bayesian Estimation

The prior distribution and the loss function (LoFu) are both used in the Bayesian estimation technique. All parameters are regarded as random variables with a particular distribution known as the prior distribution in this technique. We must select one if no prior knowledge is provided, which is typically the case. We picked independent gamma distributions as our priors since the prior distribution is crucial in parameter estimation. The LoFu, on the other hand, plays an important role in the Bayesian approach. Most Bayesian inference techniques are based on symmetric and asymmetric LoFus. Two of the most frequent symmetric LoFus are the squared error and the linear exponential (Linex) LoFus. The independent joint prior density function ofψ can be written as follows:(44)πψ=b1a1Γa1b2a2Γa2b3a3Γa3λa1−1θa2−1δa3−1e−b1λ+b2θ+b3δ.Reference [18] discussed how to elicit the hyperparameters of the informative priors. From the MLEs λ^B,θ^B,δ^B, we will get these beneficial priors by multiplying the estimate and variance by the inverse of the Fisher information matrix (FIM_ij) of ψ, say λ^B,θ^B,δ^B. By equating mean and variance of gamma priors, the estimated hyperparameters can be written as ai=ψi^2/FIMii and bi=ψi^/FIMii, where FIMii is a variance.The joint posterior PDFu ofψ is obtained from LL function and joint prior function:(45)πψ|x¯=ℓx¯|ψ⋅πψ∫ψℓx¯|ψ⋅πψdψ.Then the joint posterior of HLBX-G family under Type II CS can be written as(46)πψ|x¯∝λa1+r−1θa2+r−1δa3−1e−b1λ+b2θ+b3δ21−1−e−Gxr;δ/G¯xr;δ2θλ1+1−1−e−Gxr;δ/G¯xr;δ2θλn−r∏i=1rgxi;δG¯xi;δ3Gxi;δ1−e−Gxi;δ/G¯xi;δ2θ−1e−∑i=1rGxi;δ/G¯xi;δ2∏i=1r1−1−e−Gxi;δ/G¯xi;δ2θλ−11+1−1−e−Gxi;δ/G¯xi;δ2θλ−2.The Bayes estimators ofψ, say λ^B,θ^B,δ^B based on squared error LoFu, is given by(47)p^B−SELλ,θ,δ=Eλ,θ,δ|x¯pλ,θ,δ=∫0∞∫0∞∫0∞pλ,θ,δ×πω|x¯dλdθdδ.The Bayes estimates of the unknown parameterspsi under the Linex LoFu may be calculated as follows: ψ^Linex=−1/ulogEe−uψ|x. See, for example, [16, 18] for more information on Bayesian estimation. It is worth noting that the integrals (47) cannot be obtained explicitly. As a consequence, we estimate the value of integrals using the Markov Chain Monte Carlo (MCMC) approach.Gibbs sampling and, more generally, Metropolis within Gibbs samplers are significant MCMC subclasses. Two popular MCMC techniques are the Metropolis-Hastings (MH) algorithm and Gibbs sampling. The MH algorithm, like acceptance-rejection sampling, evaluates whether a candidate value can be created from a proposal distribution throughout each iteration of the algorithm. The following are the MH inside Gibbs sampling stages that we used to produce random samples from conditional posterior densities of the HLBX-G family in a Type II CS:(48)πλ|θ,δ,x¯∝λa1+r−1e−b1λ∏i=1r1−1−e−ti2θλ−11+1−1−e−ti2θλ−21−1−e−tr2θλ1+1−1−e−tr2θλn−r,πθ|λ,δ,x¯∝θa2+n−1e−b2θ∏i=1r1−e−Gxi;δ/G¯xi;δ2θ−11−1−e−Gxi;δ/G¯xi;δ2θλ−11−1−e−tr2θλ1+1−1−e−tr2θλn−r∏i=1r1+1−1−e−Gxi;δ/G¯xi;δ2θλ−2,πδ|λ,θ,x¯∝δa3−1e−b3δe−∑i=1Rti2∏i=1rgxi;δG¯xi;δ3Gxi;δ1−e−ti2θ−1∏i=1r1−1−e−ti2θλ−11+1−1−e−ti2θλ−21−1−e−tr2θλ1+1−1−e−tr2θλn−r,where ti=Gxi;δ/G¯xi;δ. The MH algorithm (Algorithm 1) generates a sequence of draws from this distribution.Algorithm 1: Algorithm of MCMC.
(1)
Start with any initial valueψli as a length of ψ satisfying πψ>0.(2)
Using the initial value, sample a candidate pointψ∗ from proposal qψ∗.(3)
Given the candidate pointψ∗, calculate the acceptance probability Aj=min1,Lψ∗|xπψ∗qψ/Lx|ψπψqψ∗.(4)
Draw a value ofu from the uniform (0, 1) distribution; if u<Ai, accept ψl∗ as ψli.(5)
Otherwise, rejectψl∗ and do ψli=ψli−1.(6)
Repeat steps 2–5 many times until we geti draws.(7)
Use loss function.


## 5. Applications

Three real-world COVID-19 data applications from different countries are presented in this section to test the goodness of the HLBX-G family distributions. The HLBXE, HLBXL, and HLBXR models are compared with other related models such as Weibull-Lomax (WL) [19], Gompertz Lomax (GL) [20], exponentiated power Lomax (EPL) [21], Kumaraswamy exponentiated Rayleigh (KER) [17], Lomax, exponential and Rayleigh distributions. Tables 2–4 show MLE and standard errors (StEr) for all parameter of models. Also, these tables provide Kolmogorov–Smirnov (D1) statistic along with its P value (D2), Cramér–von Mises (D3), and Anderson–Darling (D4) for all models fitted based on three real datasets of COVID-19 data with different countries as Italy, Canada, and Belgium, where these data are formed of drought mortality rate. Furthermore, the histograms of the three datasets are shown in Figures 5–7.Table 2
MLE, StEr,D1, D2, D3, and D4 for COVID-19 data of Italy.
ItalyEstimationStErD1D2D3D4HLBXLλ0.62900.67420.04940.79620.10630.6529θ0.45770.1024β0.08210.0940α0.78460.2262EPLλ3.19861.63240.06150.53230.12090.6981θ3.62420.6122β0.29520.0646α0.01360.0157GLλ0.34291.24560.05940.57670.13720.8042θ4.78095.3358β2.17198.0276α8.528730.7545HLBXRλ9.38235.72070.06490.46450.13270.7778θ0.28460.0286β3.00002.0868HLBXEλ6.42224.83310.05830.60360.13880.8064θ0.52710.0665μ0.74040.3146WLα0.23270.26350.05890.58860.13190.7794λ0.97660.1965θ2.96331.3249β0.19660.2406KERα157.837738.98950.06500.45560.17670.9965λ0.40980.0978θ1.36101.3386β0.36600.3542Table 3
MLE, StEr,D1, D2, D3, and D4 for COVID-19 data of Canada.
CanadaEstimationStErD1D2D3D4HLBXEλ319.217710.15650.08950.20850.25221.6555θ0.60400.0419μ6.73621.4647HLBXLλ9.242213.79790.07780.36020.19201.3741θ0.53690.0756β2.846521.9187α5.673638.9188EPLλ13.007914.79950.08560.25250.21451.4492θ2.17210.6155β0.52980.2089α0.09420.1799HLBXRλ9.36387.10120.07810.35570.19991.3809θ0.27480.0312β9.36588.6373WLα2.08193.09540.08600.24760.20391.3920λ1.18760.1723θ7.863824.5294β1.40065.3603KERα31.373649.52050.09800.13350.24651.6135λ4.27605.3112θ1.06500.7046β0.60840.4025Table 4
MLE, StEr,D1, D2, D3, and D4 for COVID-19 data of Belgium.
BelgiumEstimationStErD1D2D3D4HLBXEλ202.842065.01870.05880.64850.08900.7121θ0.37050.0239μ95.847315.1840HLBXLλ1.06590.61190.05220.78580.10830.7143θ0.53570.1172β0.01090.0060α0.26650.0499GLλ1.93332.57390.06240.57240.11420.7603θ2.22142.1659β2.04962.1158α0.21730.5041EPLλ44.5068104.02390.05300.76970.11410.7518θ0.81410.1925β1.18530.4791α6.833316.5980HLBXRλ17.55025.66560.06250.57150.08930.7528θ0.18070.0116β0.11700.0587WLα0.20080.15120.05490.73150.11760.7574λ1.02660.2274θ0.78240.2850β0.01410.0104KERα2.374913.01340.07190.39200.09240.7604λ4.110212.6465θ0.72500.5077β0.54000.3782Figure 5
Histogram and PDFu of models for COVID-19 data of Italy.Figure 6
Histogram and pdf of models for COVID-19 data of Canada.Figure 7
Histogram and PDFu of models for COVID-19 data of Belgium.The three datasets were obtained from the following electronic address:https://github.com/CSSEGISandData/COVID-19/.The first set of data represents COVID-19 data belonging to Italy of 172 days, from 1 March to 21 August 2020. The data are as follows: 0.0490 0.0601 0.0460 0.0533 0.0630 0.0297 0.0885 0.0540 0.1720 0.0847 0.0713 0.0989 0.0495 0.1025 0.1079 0.0984 0.1124 0.0807 0.1044 0.1212 0.1167 0.1255 0.1416 0.1315 0.1073 0.1629 0.1485 0.1453 0.2000 0.2070 0.1520 0.1628 0.1666 0.1417 0.1221 0.1767 0.1987 0.1408 0.1456 0.1443 0.1319 0.1053 0.1789 0.2032 0.2167 0.1387 0.1646 0.1375 0.1421 0.2012 0.1957 0.1297 0.1754 0.1390 0.1761 0.1119 0.1915 0.1827 0.1548 0.1522 0.1369 0.2495 0.1253 0.1597 0.2195 0.2555 0.1956 0.1831 0.1791 0.2057 0.2406 0.1227 0.2196 0.2641 0.3067 0.1749 0.2148 0.2195 0.1993 0.2421 0.2430 0.1994 0.1779 0.0942 0.3067 0.1965 0.2003 0.1180 0.1686 0.2668 0.2113 0.3371 0.1730 0.2212 0.4972 0.1641 0.2667 0.2690 0.2321 0.2792 0.3515 0.1398 0.3436 0.2254 0.1302 0.0864 0.1619 0.1311 0.1994 0.3176 0.1856 0.1071 0.1041 0.1593 0.0537 0.1149 0.1176 0.0457 0.1264 0.0476 0.1620 0.1154 0.1493 0.0673 0.0894 0.0365 0.0385 0.2190 0.0777 0.0561 0.0435 0.0372 0.0385 0.0769 0.1491 0.0802 0.0870 0.0476 0.0562 0.0138 0.0684 0.1172 0.0321 0.0327 0.0198 0.0182 0.0197 0.0298 0.0545 0.0208 0.0079 0.0237 0.0169 0.0336 0.0755 0.0263 0.0260 0.0150 0.0054 0.0375 0.0043 0.0154 0.0146 0.0210 0.0115 0.0052 0.2512 0.0084 0.0125 0.0125 0.0109 0.0071.The second set of data represents COVID-19 data belonging to Canada of 142 days, from 1 April to 21 August 2020. These data are formed of rough mortality rate. The data are as follows: 0.0122 0.0198 0.0155 0.0514 0.0176 0.0326 0.0418 0.0405 0.0452 0.0477 0.0524 0.0639 0.0554 0.0654 0.0940 0.0699 0.1138 0.0551 0.1060 0.0712 0.0588 0.0923 0.0831 0.0877 0.0948 0.0975 0.0832 0.0878 0.1023 0.1051 0.0881 0.1164 0.0620 0.1775 0.1133 0.1509 0.1176 0.1270 0.1137 0.0822 0.1534 0.1206 0.1399 0.1219 0.1253 0.0825 0.0884 0.0935 0.0436 0.0977 0.1016 0.0975 0.0861 0.0866 0.0714 0.1263 0.1116 0.1329 0.1247 0.0881 0.1239 0.2551 0.0488 0.1173 0.1861 0.1733 0.1925 0.0970 0.0421 0.0642 0.1516 0.1335 0.0840 0.1332 0.1242 0.1034 0.0806 0.1187 0.1062 0.1253 0.1125 0.1641 0.0629 0.0200 0.0552 0.1075 0.0526 0.0233 0.0336 0.0275 0.0659 0.0874 0.0898 0.0658 0.0487 0.0457 0.0226 0.0776 0.0974 0.0323 0.0312 0.0633 0.0412 0.0124 0.0242 0.0350 0.0391 0.0296 0.0273 0.0118 0.0076 0.0070 0.0147 0.0093 0.0131 0.0114 0.0141 0.0160 0.0277 0.0105 0.0365 0.0117 0.0209 0.0140 0.0136 0.0145 0.0101 0.0107 0.0094 0.0254 0.0217 0.0088 0.0138 0.0355 0.0231 0.0120 0.0169 0.0101 0.0076 0.0461 0.0119.The third set of data represents COVID-19 data belonging to Belgium of 157 days, from 15 March to 20 August 2020. The data are as follows: 0.0279 0.0280 0.0284 0.0284 0.0412 0.0393 0.0452 0.0830 0.0844 0.0587 0.0619 0.0801 0.0748 0.0810 0.1576 0.2035 0.0972 0.1112 0.1670 0.1459 0.1379 0.2962 0.3534 0.1406 0.2000 0.1984 0.1206 0.1366 0.2799 0.6527 0.5214 0.1763 0.1684 0.1499 0.1471 0.2881 0.4752 0.1552 0.1589 0.2471 0.1803 0.1825 0.3995 0.8585 0.1682 0.2018 0.1877 0.1433 0.4135 0.2552 0.6857 0.1463 0.1484 0.1359 0.1765 0.1449 0.2833 0.6186 0.1303 0.1215 0.1682 0.1508 0.1098 0.1905 0.4321 0.1104 0.1027 0.0831 0.4143 0.1532 0.1951 0.4533 0.0764 0.1980 0.1319 0.1287 0.0788 0.4098 0.3878 0.3729 0.1105 0.1258 0.0814 0.0779 0.1594 0.2264 0.0592 0.0822 0.1008 0.0737 0.1124 0.1020 0.1364 0.0544 0.0707 0.0569 0.0463 0.0331 0.0600 0.4500 0.0839 0.0364 0.0300 0.0658 0.0177 0.1951 0.2083 0.0236 0.0800 0.0148 0.0538 0.0213 0.0469 0.0833 0.0088 0.0303 0.0073 0.0161 0.0323 0.0930 0.0145 0.0192 0.0221 0.0073 0.0233 0.0154 0.0045 0.0069 0.0036 0.0046 0.0101 0.0044 0.0107 0.0067 0.0015 0.0043 0.0031 0.0044 0.0066 0.0338 0.0038 0.0062 0.0066 0.0052 0.0077 0.0066 0.0331 0.0127 0.0181 0.0180 0.0179 0.0221 0.0429 0.0522 0.0091 0.0237 0.0349.From Tables2–4, when compared to other distributions, the HLBXE, HLBXR, and HLBXL distributions have the lowest values for all information criterion. D2 has the highest value as well. This leads us to conclude that HLBX family is better fitting the three real sets of data from Italy, Belgium, and Canada. Estimated PDFus of models plots shown in Figures 5–7 indicate that our distribution is a good choice for modeling the above COVID-19 data.


## 6. Simulation Results

In this section, the Monte Carlo simulation procedure is performed for comparison between the MLEs and Bayesian estimation method under square error and Linex LoFus based on MCMC, for estimating parameters of HLBXL distribution as an example of HLBX family distribution, and this is the best distribution according to the above section of the application. We can use a different program to generate these analyses as Mathcad, Mathematica, Maple, andR packages. Algorithm 2 is used for the Monte Carlo simulation experiments.Algorithm 2: Algorithm of Monte Carlo simulation experiments.
(1)
Determine the 10000 iteration number.(2)
Different sample sizesn are considered to be 25, 50, and 100.(3)
We use different case of actual values as follows:Case I:λ=0.5,θ=0.5,β=0.5 and α=0.5.Case II:λ=0.5,θ=0.5,β=2 and α=1.5.Case III:λ=0.5,θ=2,β=2 and α=1.5.Case IV:λ=2,θ=2,β=2 and α=1.5.(4)
Generate a randomX sample for the HLBXL distribution.(5)
Use ML and Bayesian estimation methods to estimate the unknown parameters of HLBXL distribution.(6)
Repeat Steps 4 and 5, to obtain the estimates for 10000 iteration.(7)
Use different measures as follows:Bias=Θ^−Θ and MSE=MeanΘ^−Θ2.Algorithm3 is used for the Monte Carlo simulation of Type II censored sample experiments.Algorithm 3: Algorithm of Monte Carlo simulation of type II censored sample experiments.
(1)
Repeat steps 1–4 in Algorithm2.(2)
Determine the length of censored sample asr=nρ, where ρ is 80% and 92%.(3)
Sort sample asx1<x2<⋯<xr<⋯<xn.(4)
Select the firstr sample.(5)
Repeat steps 5 and 7 under Type II censored sample, to obtain the estimators for 10000 iteration in Algorithm2.We could define the best estimation methods as those that minimise estimate bias and mean squared error (MSE). Tables5–7 reveal the following observations:(1)
As sample size increases, the bias and MSE decrease(2)
When the number of failures increases in a Type II CS, the values of the bias and MSE for HLBXL distribution parameters decrease(3)
We find that the Bayesian estimates under Linex (2) LoFu perform better than other estimates of HLBXL distribution with respect to MSE and bias(4)
We find that the Bayesian method under Linex (2) loss function performs better than other estimations for estimating the parameters of HLBXL distribution with respect to MSE and bias(5)
Asθ increases and the others are fixed, then the bias and MSE are increasing for θ,β,α, and the bias and MSE are decreasing for estimates(6)
Asλ increases and the others are fixed, then the bias and MSE are increasing for β,α, and the bias and MSE are decreasing for estimatesTable 5
Bias and MSE of HLBXL distribution under complete sample for MLE and Bayesian with different loss functions.
CaseMLEBayesianSELinex 2Linex −2nBiasMSEBiasMSEBiasMSEBiasMSEI25λ0.00130.03550.04940.01440.01870.00990.08340.0226θ0.11230.10840.04710.02130.00840.01380.08990.0342β0.07420.14910.11410.04790.05040.02680.18820.0870α0.05500.03920.02680.00530.01640.00440.03770.006450λ0.00340.02410.04690.01320.01300.00870.08520.0220θ0.07520.06900.03730.01760.00980.01310.06700.0250β0.00380.04560.09530.04020.04030.02390.15840.0697α0.01580.01470.02010.00390.01220.00340.02840.0046100λ−0.00410.01750.04540.01090.00880.00680.08740.0198θ0.02410.02370.02390.01050.00540.00850.04380.0138β−0.00190.02880.08040.02740.03480.02360.13150.0581α0.00910.00780.01370.00320.00790.00290.01970.0036II25λ0.18740.30250.03030.00820.00800.00610.06590.0129θ0.10970.11640.05880.02310.02170.01560.09940.0356β−0.02530.26090.04100.0098−0.05640.01020.15210.0345α0.03370.22000.01020.0025−0.01660.00260.03840.004150λ0.11990.17140.03720.00800.01130.00540.06090.0122θ0.05160.04640.04460.01480.01920.01120.07190.0207β0.00170.16650.04150.0101−0.05120.00990.14620.0324α0.02590.12350.00950.0023−0.01540.00240.03580.0040100λ0.08470.10080.03010.00740.00260.00510.05500.0120θ0.02220.02270.02390.00930.00830.00800.04020.0114β−0.00830.05050.03570.0108−0.05360.01130.13570.0311α−0.00090.06760.00890.0023−0.01360.00230.03250.0039III25λ0.08230.22370.03350.01180.00770.00870.06080.0171θ0.36750.81640.01420.0116−0.05680.01340.11590.0213β0.08240.44580.03230.0112−0.04400.01120.13730.0294α0.13110.15750.01250.0029−0.00920.00270.03950.003550λ0.01400.16190.03410.01050.00610.00740.07090.0165θ0.15740.31260.01780.0073−0.06220.00990.09150.0223β0.03740.37860.03520.0104−0.04520.01030.12440.0270α0.11110.12870.01530.0025−0.00790.00220.03530.0042100λ−0.01390.05850.03770.00990.00750.00660.06460.0162θ0.09570.17510.02080.0062−0.06420.00910.10650.0205β−0.03090.17440.04010.0098−0.04700.00940.11610.0261α0.05300.07820.01340.0020−0.01110.00190.03500.0040IV25λ−0.00120.13430.06000.0348−0.17110.05290.34780.1746θ0.28600.60690.05830.0737−0.14940.06540.31960.2334β0.09610.60780.16620.1408−0.09770.09260.57280.5429α0.08110.16080.07970.04100.00280.02700.20810.086550λ0.02410.12560.05060.0324−0.16820.04850.32430.1737θ0.14380.21830.04250.0582−0.11060.05370.22850.1391β0.11030.57400.18250.1382−0.11150.08150.56150.5035α0.05880.12870.09280.0376−0.00610.02250.19050.0815100λ0.02130.02280.03540.0268−0.16800.04450.28170.1488θ0.04550.08210.03920.0478−0.06940.04070.16330.0911β0.06690.12690.16290.1079−0.13950.07430.48880.4523α0.03670.02990.09030.0283−0.02420.01820.20240.0736Table 6
Bias and MSE of the MLE and Bayesian estimate for HLBXL distribution based on Type II censored sample at 80%.
80%MLEBayesianSELinex 2Linex −2CaserBiasMSEBiasMSEBiasMSEBiasMSEI20λ−0.01970.04430.04400.02200.00420.01520.08840.0350θ0.14460.14370.08060.04980.02530.03000.14170.0827β0.01380.09790.14350.09180.05610.04790.24240.1688α0.06210.05920.04170.01150.02270.00880.06240.015740λ−0.00830.03300.04470.01960.00080.01270.09510.0341θ0.08220.07020.05360.02770.01660.01880.09360.0418β−0.01260.05580.12510.08010.04790.04400.21290.1409α0.02670.02680.03680.00950.02220.00770.05250.012280λ0.00230.02210.04480.0169−0.00120.01080.09790.0311θ0.04030.03130.03970.01930.01530.01460.06540.0261β0.01660.05480.10880.07910.04300.04380.18140.1317α0.02080.01790.02940.00890.01840.00760.04110.0105II20λ0.19090.35400.10380.08840.03140.04990.18270.1471θ0.12080.14880.07300.04150.02340.02740.12830.0653β−0.12400.14830.05300.0128−0.07400.01400.20190.0576α0.05290.27480.08160.0596−0.03530.04380.22740.125740λ0.21280.31290.08650.08220.00660.04600.17710.1443θ0.05610.05260.04520.02320.01210.01750.08090.0330β−0.08240.10210.04810.0105−0.07990.01320.19850.0526α−0.03180.17640.04210.0533−0.04940.04450.15460.093480λ0.11740.13880.07280.05860.01900.03830.13280.0907θ0.02650.02340.02420.01300.00410.01120.04550.0161β−0.02540.01850.05010.0091−0.07970.01160.20370.0524α−0.01530.08630.04730.0420−0.02750.03410.13530.0699III20λ0.14090.26520.05870.04120.00020.02680.12450.0687θ0.39400.86930.22930.4344−0.28200.23180.88371.6150β0.12310.45150.08230.0438−0.12260.04260.34010.1844α0.10720.10870.05260.0111−0.00920.00730.12320.025840λ0.04690.20240.04280.0411−0.00790.02860.09760.0620θ0.25600.49940.16780.3301−0.21910.17910.64381.0029β0.03390.44870.04170.0516−0.13390.05720.25250.1431α0.08310.10000.05290.01240.00330.01000.12210.029680λ0.03260.05760.02370.0373−0.01850.02880.06820.0511θ0.10430.18300.13210.2010−0.13660.12410.45420.5393β0.03570.15530.06220.0449−0.13030.04820.30300.1619α0.03320.02830.05290.0150−0.00630.00850.12030.0266IV20λ0.08230.19430.02030.0100−0.10000.01800.15740.0402θ0.33140.66580.10380.1253−0.17720.09740.47050.4576β0.15890.46060.09070.0291−0.08200.02280.30100.1298α0.11800.13300.04070.0106−0.00990.00810.09620.020240λ−0.01570.11930.02250.0079−0.09790.01570.16040.0383θ0.20550.30950.08250.1040−0.12070.07640.33230.2847β−0.01510.27850.07710.0288−0.08500.02570.27190.1149α0.01270.06720.03810.0083−0.01700.00630.09870.018480λ−0.00500.03480.02760.0066−0.09800.01440.17500.0417θ0.11020.11910.08290.0797−0.05560.05510.24840.1721β−0.01080.13350.07050.0342−0.08070.03170.24760.1058α0.00600.03420.03620.0080−0.02260.00640.10160.0190Table 7
Bias and MSE of the MLE and Bayesian estimate for HLBXL distribution based on Type II censored sample at 92%.
92%MLEBayesianSELinex 2Linex −2CasesRBiasMSEBiasMSEBiasMSEBiasMSEI23λ−0.00200.03920.05960.0448−0.00530.02610.13410.0816θ0.11970.11380.09140.07440.02730.04480.16050.1201β0.03520.10780.12780.10990.03740.05900.23270.2015α0.05010.04160.03670.01290.01910.01050.05590.016646λ−0.01660.02530.04940.0420−0.00740.02580.11550.0713θ0.05470.05540.06160.04250.01960.02940.10640.0617β0.02870.05220.12560.10720.04230.05900.21910.1840α0.03780.02180.03800.01230.02430.01020.05260.015192λ0.00660.01740.06410.04480.01510.02430.11880.0688θ0.03710.02780.04100.02440.01470.01860.06850.0326β−0.00790.02870.09730.09290.03000.05410.17280.1566α0.00660.01060.01990.00810.01040.00720.02990.0093II23λ0.22890.38310.06750.03920.01080.02360.13290.0688θ0.13390.14400.10190.07030.04000.04220.16830.1114β−0.10480.25680.08010.0409−0.13740.04390.36000.1983α0.00920.26840.03680.0132−0.03220.01140.11660.028446λ0.12100.19630.06370.03150.01230.01910.12170.0541θ0.03970.04870.05160.03010.01660.02210.08870.0420β−0.02590.13860.07760.0403−0.13320.04300.34600.1902α0.03260.14590.03300.0124−0.03030.01080.10440.025092λ0.08090.10580.06520.03360.02260.02240.11230.0520θ0.01840.02300.02960.01500.00960.01290.05030.0181β−0.01650.02420.08740.0506−0.11250.04680.33770.1904α0.00570.07190.02610.0128−0.02910.01180.08800.0223III23λ0.08340.21980.06070.04260.00410.02660.12300.0694θ0.36530.80350.13690.1856−0.24730.14290.64050.7974β0.03460.46530.08120.0375−0.11790.03690.33080.1679α0.08960.13780.04720.0119−0.01570.00850.11950.026846λ0.06460.19070.04960.0380−0.00110.02560.10450.0597θ0.18640.40140.10220.1539−0.20240.12200.49290.5436β0.08230.42320.06750.0385−0.11460.03600.29570.1487α0.09790.12920.05060.0141−0.00850.00840.11770.025892λ0.01510.06270.03170.0396−0.01110.02490.07740.0566θ0.08200.16730.09940.1374−0.12780.09190.37520.3757β0.02820.16450.05960.0458−0.10370.03450.26070.1371α0.05640.06100.06140.01720.00750.00830.12250.0243IV23λ0.02500.11780.02330.0131−0.10500.02120.17010.0492θ0.28890.62770.10910.1516−0.17800.10820.48260.5279β0.11950.36310.08900.0309−0.09000.02610.31230.1414α0.09490.12850.03840.0092−0.02240.00720.10630.021346λ−0.00540.11660.02560.0098−0.10560.01850.17800.0482θ0.19350.27250.08920.1154−0.11040.08390.33280.2887β0.00350.32570.08370.0295−0.08410.02490.28940.1267α0.01450.09520.04380.0093−0.01230.00660.10620.020992λ0.02420.04230.02890.0075−0.10380.01600.18460.0465θ0.07480.09160.07010.0742−0.05790.05480.21960.1484β0.05030.16240.07360.0358−0.08460.03260.26000.1161α0.03060.03660.04240.0112−0.01010.00840.09970.0217


## 7. Conclusion

A new generalized generator of the half-logistic BurrX-G family was proposed and studied in this paper. Several statistical properties, including QuFu, Mos, InMos, MeD, Lo and Bo curves, and En were derived. The HLBX Lomax, HLBX exponential, and HLBX Rayleigh distributions are discussed. MLL and Bayesian estimation methods were used to estimate the unknown parameters. The HLBXL distribution fits better than the other submodels. To distinguish the performance of estimation methods, a simulation analysis was performed using the R package. For Bayesian estimation, the MCMC method was used. Three real COVID-19 datasets from different countries, including Italy, Canada, and Belgium, were considered. Finally, we plan to use this family to generate new models from the proposed generating family and investigate their statistical properties, as well as investigating the statistical inference of the new models using various methods and demonstrating the importance of the new models using new real datasets [19–22].



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*Source: 5461130-2021-10-29.xml*