# A New Extension of the Exponentiated Weibull Model Mathematical Properties and Modelling

Authors: Majdah Mohammed Badr; Amal T. Badawi; Alya S. Alzubidi
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Recently, several writers have extended the exponentiated Weibull distribution. The five-parameter exponentiated Weibull-exponentiated Weibull (EW-EW) distribution is introduced. In terms of fit, the EW-EW distribution outperforms the EW distribution. Some EW-EW distribution features, such as precise formulas for ordinary moments, quantile, median, and order statistics, are found. Model parameters were estimated using the maximum likelihood technique (ML). The behavior of the various estimators was investigated using a simulated exercise. A medical dataset was utilized to evaluate the practical importance of the EW-EW distribution using additional criteria such as the Akaike information criterion (AKINC), the correct AKINC (COAKINC), the Bayesian INC (BINC), and the Hannan-Quinn INC (HQINC). In terms of performance, we show that the EW-EW distribution beats other models.


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## Body

## 1. Introduction

Probabilistic models play an important role in fitting real-world data. Statistical distributions have been widely used to model and analyze data in a variety of areas, including engineering, biology, economics, finance, and medicine. One can achieve satisfactory results by employing several well-known statistical distributions that function well with various types of data. However, in many circumstances, these distributions lack the flexibility required to analyze the complicated behavior displayed by such data. As a result, current attention has been brought to the notion of more flexible distributions, which can fit different types of data with varying degrees of complexity.To achieve a more flexible and accurate distribution, one of the most typical techniques is to add more parameters to any current model. The E method is one of these methods, and it was proposed by [1]. As a result, numerous academics used the E approach to create more flexible distributions.For example, [2] presented the E exponential (generalized exponential) distribution; reference [3] proposes some new E models, like the Gumbel, W, and gamma models; reference [4] proposed the generalized Gompertz distribution; reference [5] proposed the E-modified W extension distribution; reference [6] introduced the E Gompertz distribution, E-truncated inverse Weibull-G by [7], E power-generalized Weibull power series family [8], and E-exponentiated Weibull Rayleigh distribution [9].Another effective strategy is to create new families of distributions. Different sorts of generators may be used in this method to generate new distribution families for any continuous distribution. Because of its versatility, the transformed-transformer T-X family developed by [10] has become widely employed.Attempts have recently been made to build new distributions as reference [11] offered the W-G family.The E W (EW) distribution is regarded as one of the most often-used statistical models for modelling lifespan data. LetX be an R_V from the E W model with the distribution function (CDF) and density function (PDF) determined by:
(1)Rt=1−e−αxβa,rt=aαβxβ−1e−αxβ1−e−αxβa−1.As a result, numerous academics have proposed changing the EW distribution. The EW family was proposed by [12]. The CDF of an EW-generated family (EW-G) is provided by
(2)Fx;a,α,β=1−e−αGx/1−Gxβa,x>0,a,α,β>0.CDF (2) encompasses a broader range of continuous models. The PDF that corresponds to (2) is provided by
(3)fx;a,α,β=aαβgx−gxGx+gxGx1−Gx2Gxβ−11−Gxβ−1e−αGx/1−Gxβ1−e−αGx/1−Gxβa−1.(i)
The following is the major objective of this article(ii)
To present the exponentiated Weibull-exponentiated Weibull (EW-EW) distribution, a novel four-parameter life time model is proposed(iii)
Investigate some of its statistical properties(iv)
Demonstrate the EW-EW model’s statistical inference(v)
Run a simulation study to demonstrate the model parameter behavior(vi)
Give leading fits than some known models with favourable results for the EW-EW modelThis study proposes an EW-EW distribution. The remainder of this work is divided into several parts: the EW-EW distribution is introduced with certain specific examples in the second section. The third section discusses the statistical characteristics of the EW-EW model. The fourth section calculates the order statistics of the EW-EW distribution. The maximum likelihood (ML) estimates (MLEs) of the EW-EW distribution parameters are provided in the fifth section, and the performance of the EW-EW model is proposed in Section6, which includes a Monte Carlo simulation and the use of real-world data to demonstrate the adaptability of the EW-EW distribution in comparison to numerous models in the literature. Section 7 concludes by summarizing the findings.


## 2. Construction of the EW-EW Model

The five-parameter distribution is produced in this section using the EW-G family. Allow theR_VX to follow the EW distribution, with the CDF provided by
(4)Gx;θ,λ=1−e−xθλ,and its PDF is computed with
(5)gx;θ,λ=λθxθ−1e−xθ1−e−xθλ−1.Substituting (5) and (4) into (2) yields the CDF of EW-EW distribution, and EW-EW(a,α,β,λ,θ) can really be described as follows:
(6)Fx;ω=1−e−α1−e−xθλ/1−1−e−xθλβa,x>0,a,α,β,λ,θ>0.Substituting (5) and (4) into (3) yields the PDF of EW-EW(a,α,β,λ,θ) which can really be described as follows:
(7)fx;ω=aαβλθxθ−1e−xθ1−e−xθλ−11−e−xθλβ−11−1−e−xθλβ+1e−α1−e−xθλ/1−1−e−xθλβ×1−e−α1−e−xθλ/1−1−e−xθλβa−1,x>0,a,α,β,λ,θ>0,where a,β,λ,andθ are shape parameters and α is a scale parameter.The plots for CDF and PDF for numerous parameter values are described in Figures1(a) and 1(b), respectively.Figure 1
Plots of (a) CDF and (b) PDF with numerous values ofa,α,β,λ, and θ.
(a)(b)


### 2.1. Special Cases of the EW-EW Distribution

As special examples, the five-parameter EW-EW distribution includes certain well-known distributions. We included several specific situations from the suggested EW-EW distribution in Table1. In (6),Table 1
Subdistributions of the EW-EW distribution.
ModelαaβλθCDFAuthorEW-W√√√1√Fx=1−e−αexθ−1βa[13]EW-E√√√11Fx=1−e−αex−1βa[10]EW-Rayleigh√√√12Fx=1−e−αex2−1βaNewE exponential W√√11√Fx=1−e−αexθ−1a[6]E exponential exponential√√111Fx=1−e−αex−1aNewE exponential Rayleigh√√112Fx=1−e−αex2−1aNewE Rayleigh W√√21√Fx=1−e−αexθ−12aNewE Rayleigh exponential√√211Fx=1−e−αex−12aNewE Rayleigh Rayleigh√√212Fx=1−e−αex2−12aNewW Weibull√1√1√Fx=1−e−αexθ−1β[14]W exponential√1√11Fx=1−e−αex−1β[15]W Rayleigh√1√12Fx=1−e−αex2−1β[16]Exponential W√1√1√Fx=1−e−αexθ−1βNewExponential Rayleigh√1112Fx=1−e−αex2−1[17]Rayleigh W√121√Fx=1−e−αexθ−12NewRayleigh exponential√1211Fx=1−e−αex−12NewTheR_VX’s survival function (SURF) and hazard rate function (HAZRF) are supplied by
(8)Sx=1−1−e−α1−e−xθλ/1−1−e−xθλβahx=aαβλθxθ−1e−xθ1−e−xθλ−11−e−xθλβ−1/1−1−e−xθλβ+1e−α1−e−xθλ/1−1−e−xθλβ1−e−α1−e−xθλ/1−1−e−xθλβa−11−1−e−α1−e−xθλ/1−1−e−xθλβa.The plots for SURF and HAZRF for different parameters values are provided in Figure2.Figure 2
Plots of (a) HAZRF and (b) SURF with different values ofa,α,β,λ, and θ.
(a)(b)


## 2.1. Special Cases of the EW-EW Distribution

As special examples, the five-parameter EW-EW distribution includes certain well-known distributions. We included several specific situations from the suggested EW-EW distribution in Table1. In (6),Table 1
Subdistributions of the EW-EW distribution.
ModelαaβλθCDFAuthorEW-W√√√1√Fx=1−e−αexθ−1βa[13]EW-E√√√11Fx=1−e−αex−1βa[10]EW-Rayleigh√√√12Fx=1−e−αex2−1βaNewE exponential W√√11√Fx=1−e−αexθ−1a[6]E exponential exponential√√111Fx=1−e−αex−1aNewE exponential Rayleigh√√112Fx=1−e−αex2−1aNewE Rayleigh W√√21√Fx=1−e−αexθ−12aNewE Rayleigh exponential√√211Fx=1−e−αex−12aNewE Rayleigh Rayleigh√√212Fx=1−e−αex2−12aNewW Weibull√1√1√Fx=1−e−αexθ−1β[14]W exponential√1√11Fx=1−e−αex−1β[15]W Rayleigh√1√12Fx=1−e−αex2−1β[16]Exponential W√1√1√Fx=1−e−αexθ−1βNewExponential Rayleigh√1112Fx=1−e−αex2−1[17]Rayleigh W√121√Fx=1−e−αexθ−12NewRayleigh exponential√1211Fx=1−e−αex−12NewTheR_VX’s survival function (SURF) and hazard rate function (HAZRF) are supplied by
(8)Sx=1−1−e−α1−e−xθλ/1−1−e−xθλβahx=aαβλθxθ−1e−xθ1−e−xθλ−11−e−xθλβ−1/1−1−e−xθλβ+1e−α1−e−xθλ/1−1−e−xθλβ1−e−α1−e−xθλ/1−1−e−xθλβa−11−1−e−α1−e−xθλ/1−1−e−xθλβa.The plots for SURF and HAZRF for different parameters values are provided in Figure2.Figure 2
Plots of (a) HAZRF and (b) SURF with different values ofa,α,β,λ, and θ.
(a)(b)


## 3. Fundamental Characteristics

This section describes some of the mathematical aspects of the suggested EW-EW model. This section also provides the quantile function, median, and moments.


### 3.1. Expansions

Expansion forms for the PDF and CDF for EW-EW distribution are obtained as follows.The following mathematical properties are used to obtain the expansion forms for the PDF and CDF for EW-EW distribution:Binomial expansion(9)1−bn=∑k=1∞−1knkbk,where0≤k≤n.Common Taylor series,(10)e−x=∑i=0∞−1ixii!,forallx>0,i≠0.Negative binomial expansion,(11)1−b−n=∑k=0∞−1kn+k−1kbk.


#### 3.1.1. Expansion for the PDF

Using (9), the PDF of EW-EW distribution, given in equation (7), can be written as
(12)fx;ω=aαβλθxθ−1e−xθ1−e−xθλ−11−e−xθλβ−11−1−e−xθλβ+1∑i=0∞−1ia−1i×e−αi+11−e−xθλ/1−1−e−xθλβ.Using common Taylor series (10), the PDF of EW-EW distribution can be written as
(13)fx;ω=aβλθxθ−1e−xθ∑i,m=0∞−1i+mαm+1i+1mm!a−1i1−e−xθλβ−λβm−11−1−e−xθλβm+β+1.Using (11) and after some simplifications, the expansion form for the PDF of EW-EW distribution is of the form
(14)fx;ω=I1xθ−1e−l+1xθ,where
(15)I1=aβλθ∑i,m,k,l=0∞−1i+m+k+lαm+1i+1mm!a−1iβm+1+kkλk+β+βm−1l.


#### 3.1.2. Expansion for the CDF

Using binomial expansion (9), the CDF of EW-EW distribution, given in equation (6), can be written as
(16)Fx;ω=∑i=0∞−1iaie−αi1−e−xθλ/1−1−e−xθλβ.Next, using the common Taylor series (10), we get
(17)Fx;ω=∑i,j=0∞−1i+jaiαijj!1−e−xθλ1−1−e−xθλβj,Fx;ω=∑i,j=0∞−1i+jaiαijj!1−e−xθλβj×1−1−e−xθλ−βj.After, using negative binomial expansion (11), we have the expansion form for the CDF of EW-EW distribution which is of the form
(18)Fx;ω=I2xmθ,where
(19)I2=∑i,j,k,l,m=0∞−1i+j+k+l+maiβj+k−1kaλβj+kllmαijj!m!.


### 3.2. Quantile and Median

IfX follows the EW-EW distribution with parameter vector ω, then, the quantile function (QUF) is provided through
(20)Qu=−log1−1−−1αlog1−u1/a1/β+1−11/λ1/θ.And, hence, the median is obtained by takingu=0.5.


### 3.3. Moments

Assume thatX is a R_V from the EW-EW distribution, and the rth moments is provided through
(21)μr′=EXr=∫0∞xrfx,ωdx.Then, therth moment of the EW-EW model by using equation (14) in previous relation is given by
(22)μr′=EXr=I31θl+1r/θ+1Γrθ+1;rθ+1≥0,θ>0,r=1,2,⋯,where
(23)I3=aβλθ∑i,m,k,l=0∞−1i+m+lαm+1i+1mm!a−1iβm+1+kkλk+β+βm−1l.Γ. is the gamma function. Therefore, the mean of EW-EW distribution takes the form
(24)μ1′=EX=I31θl+11/θ+1Γ1θ+1.In addition,EX2 is computed by putting r=2:
(25)μ2′=EX2=I31θl+12/θ+1Γ2θ+1.Thus, from (24) and (25), the variance is expressed as
(26)σ2=I31θl+12/θ1l+1Γ2θ+1−I31θl+12Γ21θ+1.


## 3.1. Expansions

Expansion forms for the PDF and CDF for EW-EW distribution are obtained as follows.The following mathematical properties are used to obtain the expansion forms for the PDF and CDF for EW-EW distribution:Binomial expansion(9)1−bn=∑k=1∞−1knkbk,where0≤k≤n.Common Taylor series,(10)e−x=∑i=0∞−1ixii!,forallx>0,i≠0.Negative binomial expansion,(11)1−b−n=∑k=0∞−1kn+k−1kbk.


### 3.1.1. Expansion for the PDF

Using (9), the PDF of EW-EW distribution, given in equation (7), can be written as
(12)fx;ω=aαβλθxθ−1e−xθ1−e−xθλ−11−e−xθλβ−11−1−e−xθλβ+1∑i=0∞−1ia−1i×e−αi+11−e−xθλ/1−1−e−xθλβ.Using common Taylor series (10), the PDF of EW-EW distribution can be written as
(13)fx;ω=aβλθxθ−1e−xθ∑i,m=0∞−1i+mαm+1i+1mm!a−1i1−e−xθλβ−λβm−11−1−e−xθλβm+β+1.Using (11) and after some simplifications, the expansion form for the PDF of EW-EW distribution is of the form
(14)fx;ω=I1xθ−1e−l+1xθ,where
(15)I1=aβλθ∑i,m,k,l=0∞−1i+m+k+lαm+1i+1mm!a−1iβm+1+kkλk+β+βm−1l.


### 3.1.2. Expansion for the CDF

Using binomial expansion (9), the CDF of EW-EW distribution, given in equation (6), can be written as
(16)Fx;ω=∑i=0∞−1iaie−αi1−e−xθλ/1−1−e−xθλβ.Next, using the common Taylor series (10), we get
(17)Fx;ω=∑i,j=0∞−1i+jaiαijj!1−e−xθλ1−1−e−xθλβj,Fx;ω=∑i,j=0∞−1i+jaiαijj!1−e−xθλβj×1−1−e−xθλ−βj.After, using negative binomial expansion (11), we have the expansion form for the CDF of EW-EW distribution which is of the form
(18)Fx;ω=I2xmθ,where
(19)I2=∑i,j,k,l,m=0∞−1i+j+k+l+maiβj+k−1kaλβj+kllmαijj!m!.


## 3.1.1. Expansion for the PDF

Using (9), the PDF of EW-EW distribution, given in equation (7), can be written as
(12)fx;ω=aαβλθxθ−1e−xθ1−e−xθλ−11−e−xθλβ−11−1−e−xθλβ+1∑i=0∞−1ia−1i×e−αi+11−e−xθλ/1−1−e−xθλβ.Using common Taylor series (10), the PDF of EW-EW distribution can be written as
(13)fx;ω=aβλθxθ−1e−xθ∑i,m=0∞−1i+mαm+1i+1mm!a−1i1−e−xθλβ−λβm−11−1−e−xθλβm+β+1.Using (11) and after some simplifications, the expansion form for the PDF of EW-EW distribution is of the form
(14)fx;ω=I1xθ−1e−l+1xθ,where
(15)I1=aβλθ∑i,m,k,l=0∞−1i+m+k+lαm+1i+1mm!a−1iβm+1+kkλk+β+βm−1l.


## 3.1.2. Expansion for the CDF

Using binomial expansion (9), the CDF of EW-EW distribution, given in equation (6), can be written as
(16)Fx;ω=∑i=0∞−1iaie−αi1−e−xθλ/1−1−e−xθλβ.Next, using the common Taylor series (10), we get
(17)Fx;ω=∑i,j=0∞−1i+jaiαijj!1−e−xθλ1−1−e−xθλβj,Fx;ω=∑i,j=0∞−1i+jaiαijj!1−e−xθλβj×1−1−e−xθλ−βj.After, using negative binomial expansion (11), we have the expansion form for the CDF of EW-EW distribution which is of the form
(18)Fx;ω=I2xmθ,where
(19)I2=∑i,j,k,l,m=0∞−1i+j+k+l+maiβj+k−1kaλβj+kllmαijj!m!.


## 3.2. Quantile and Median

IfX follows the EW-EW distribution with parameter vector ω, then, the quantile function (QUF) is provided through
(20)Qu=−log1−1−−1αlog1−u1/a1/β+1−11/λ1/θ.And, hence, the median is obtained by takingu=0.5.


## 3.3. Moments

Assume thatX is a R_V from the EW-EW distribution, and the rth moments is provided through
(21)μr′=EXr=∫0∞xrfx,ωdx.Then, therth moment of the EW-EW model by using equation (14) in previous relation is given by
(22)μr′=EXr=I31θl+1r/θ+1Γrθ+1;rθ+1≥0,θ>0,r=1,2,⋯,where
(23)I3=aβλθ∑i,m,k,l=0∞−1i+m+lαm+1i+1mm!a−1iβm+1+kkλk+β+βm−1l.Γ. is the gamma function. Therefore, the mean of EW-EW distribution takes the form
(24)μ1′=EX=I31θl+11/θ+1Γ1θ+1.In addition,EX2 is computed by putting r=2:
(25)μ2′=EX2=I31θl+12/θ+1Γ2θ+1.Thus, from (24) and (25), the variance is expressed as
(26)σ2=I31θl+12/θ1l+1Γ2θ+1−I31θl+12Γ21θ+1.


## 4. Order Statistics for EW-EW Distribution

Letx1:n,x2:n,⋯,xn:n be the order statistics (ORS) acquired as a result of a random sampling (RAS) x1,x2,⋯,xn from the with PDF and CDF, given by (7) and (6), respectively. Then, the mth ORS can really be phrased simply:
(27)fm:nx=n!m−1!n−m!fxFxm−11−Fxn−m,where m=1,2,⋯,n.
(28)fm:nx=I51βm,n−m+1xθ−1e−1+wxθ,where
(29)I5=1βm,n−m+1∑v,q,t,p,w=0∞−1v+q+p+t+wαt+11+qtt!av+m−1qn−mvβt+1+p+1pλβ+λβt+λp−1waβλθ.


## 5. Maximum Likelihood Estimation

Within this part, the ML approach will be used to estimate the EW-EW distribution parametersa,α,β,λ, and θ. The likelihood function for a RAS x1,x2,⋯,xnfrom the EW-EW system is expressed by
(30)L=∏i=1nfxi;a,α,β,λ,θ.The log likelihood function may now be represented by inserting (7) into (30) and taking the log
(31)ℓ=nloga+nlogα+nlogβ+nlogλ+nlogθ+θ−1∑i=1nlogxi−∑i=1nxiθ+λ−1∑i=1nlog1−e−xθ+β−1λ∑i=1nlog1−e−xθ−β+1∑i=1nlog1−1−e−xθλ−α∑i=1n1−e−xθλ1−1−e−xθλβ+a−1∑i=1nlog1−e−α1−e−xθλ/1−1−e−xθλβ.All derivatives of (31) with regard to the parameters a,α,β,λ, and θ are calculated by
(32)∂ℓ∂a=na+∑i=1nlog1−e−α1−e−xθλ/1−1−e−xθλβ(33)∂ℓ∂α=nα+a−1∑i=1ne−α1−e−xθλ/1−1−e−xθλβ1−e−xθλ/1−1−e−xθλβ1−e−α1−e−xθλ/1−1−e−xθλβ−∑i=1n1−e−xθλ1−1−e−xθλβ,(34)∂ℓ∂β=nβ+λ∑i=1nlog1−e−xθ−∑i=1nlog1−1−e−xθλ+a−1∑i=1ne−α1−e−xθλ/1−1−e−xθλβα1−e−xθλ/1−1−e−xθλβlog1−e−xθλ/1−1−e−xθλ1−e−α1−e−xθλ/1−1−e−xθλβ−α∑i=1n1−e−xθλ1−1−e−xθλβlog1−e−xθλ1−1−e−xθλ,(35)∂ℓ∂λ=nλ+∑i=1nlog1−e−xθ+β−1∑i=1nlog1−e−xθ+β+1∑i=1n1−e−xθλlog1−e−xθ1−1−e−xθλ−α∑i=1nβ1−e−xθ2λ1−e−xθλ/1−1−e−xθλβ−1log1−e−xθ1−1−e−xθλ2+β1−e−xθλ1−e−xθλ/1−1−e−xθλβ−1log1−e−xθ1−1−e−xθλ+a−1∑i=1nαβe−α1−e−xθλ/1−1−e−xθλβ1−e−xθ2λ1−e−xθλ/1−1−e−xθλβ−1log1−e−xθ/1−1−e−xθλ2+αβe−α1−e−xθλ/1−1−e−xθλβ1−e−xθλ1−e−xθλ/1−1−e−xθλβ−1log1−e−xθ/1−1−e−xθλ1−e−α1−e−xθλ/1−1−e−xθλβ,(36)∂ℓ∂θ=nθ+∑i=1nlogx−∑i=1nxθlogx+λ−1∑i=1ne−xθxθlogx1−e−xθ+β−1λ∑i=1ne−xθxθlogx1−e−xθ+β+1∑i=1nλ1−e−xθλ−1e−xθxθlogx1−1−e−xθλ−α∑i=1nβλe−xθ1−e−xθ2λ−11−e−xθλ/1−1−e−xθλβ−1xθlogx1−1−e−xθλ2+βλe−xθ1−e−xθλ−11−e−xθλ/1−1−e−xθλβ−1xθlogx1−1−e−xθλ+a−1∑i=1nλαβe−α−xθ1−e−xθλ/1−1−e−xθλβ1−e−xθ2λ−11−e−xθλ/1−1−e−xθλβ−1xθlogx/1−1−e−xθλ2+λαβe−α−xθ1−e−xθλ/1−1−e−xθλβ1−e−xθλ−11−e−xθλ/1−1−e−xθλβ−1xθlogx/1−1−e−xθλ1−e−α1−e−xθλ/1−1−e−xθλβ.The MLEs of the parameters can indeed be acquired by adjusting equations (32)–(36) to 0.


## 6. Performance of EW-EW Distribution

The suggested model is evaluated in two ways in this section. To begin, the performance of the MLEs is evaluated utilizing a simulated study. Second, the goodness of fit of the EW-EW model is assessed using real-world data and compared to other current models.


### 6.1. Numerical Outcomes

To examine the behavior of MLEs for the EW-EW model, a thorough numerical examination is performed. To evaluate the performance of ML estimates, the ML estimators (MLE), biases (∇1), and mean square errors (∇2) are determined for different sample sizes (n).In addition, we note that, for eachn, the MLEs are evaluated using two accuracy measures: ∇1 and ∇2. Tables 2–4 present the MLEs together with the ∇1 and the ∇2. Generally, we note that as the n grows, the ∇2s of the estimates for EW-EW distribution with the parameters decrease, which shows consistency of the estimated parameters.Table 2
The values of MLE,∇1 and ∇2 for the unknown parameters a,α,β,λ, and θ at (0.8, 0.3, 0.5, 0.3, and 0.2), respectively.
nThe MLEs for initial values (0.8, 0.3, 0.5, 0.3, 0.2)a^α^β^λ^θ^10MLE1.59901.48100.69550.17970.5673∇10.79901.18100.1955−0.12030.3673∇20.63841.39480.03820.01450.1349MLE1.36700.32230.68500.26560.4673∇10.56700.02230.1850−0.03440.2673∇20.32140.00050.03420.00120.0714100MLE1.29550.32110.32950.26810.4670∇10.49550.0211−0.1705−0.03190.2670∇20.24550.00040.02910.00100.0713Table 3
The values of MLE,∇1, and ∇2 for the unknown parameters a,α,β,λ, and θ at (0.5, 0.2, 0.3, 0.7, and 0.3), respectively.
nThe MLEs for initial values (0.5, 0.2, 0.3, 0.7, 0.3)a^α^β^λ^θ^10MLE0.96711.03190.86900.54470.5517∇10.46710.83190.5690−0.15530.2517∇20.21820.69200.32380.02410.063350MLE0.92500.35640.51020.74250.5301∇10.42500.15640.21020.04250.2301∇20.18060.02450.04420.00180.0530100MLE0.32590.30430.28870.67270.5137∇10.12590.1043−0.0113−0.02730.2137∇20.01580.01090.00010.00070.4566Table 4
The values of MLE,∇1, and ∇2 for the unknown parameters a,α,β,λ, and θ at (0.9, 0.1, 0.5, 0.5, and 0.3).
nThe MLEs for initial values (0.9, 0.1, 0.5, 0.5, and 0.3)a^α^β^λ^θ^10MLE1.74040.34120.17020.06490.8716∇10.84040.2412−0.3298−0.43510.5716∇20.70630.05820. 108750.18930.326750MLE1.41380.84040.21010.14780.8030∇10.51380.2123−0.2899−0.35220.5030∇20.26400.04510.08400.12410.2530100MLE1.16660.04510.30220.57410.5466∇10.2666-0.0549−0.19780.07410.2466∇20.07110.00300.03910.00550.0608


### 6.2. Modelling to Biomedical and Engineering Real Data

For additional illustration, this section compares the productivity of the goodness-of-fit for the EW-EW distribution with specific chosen distributions from the literature. A real dataset is utilized in particular to compare the suggested distribution to the EW exponential distribution (EW-E), the W exponential distribution (WE), the E exponential exponential distribution (EE-E), and the EW W distribution (EW-W). We obtain the MLE of the model parameters.To compare the performance of different distribution models, we considered AKINC, COAKINC, BINC, and HQINC. However, the better distribution corresponds to the smaller values of AKINC, COAKINC, BINC, and HQINC criteria. Furthermore, we plot the empirical CDF of the datasets and estimated CDF of EW-E, WE, EE-E, and EW-W models are displayed.


#### 6.2.1. The First Data

The survival periods (in 100 days) of 72 guinea pigs infected with virulent tubercle bacilli were derived from [13]. The following are the data sets: 1.460, 1.530, 2.530, 2.540, 0.440, 0.560, 0.590, 0.720, 0.740, 0.770, 0.920, 0.930, 0.960, 1.000, 1.050, 1.070, 1.070, 1.080, 1.080, 1.080, 1.090, 1.120, 1.130, 1.150, 1.160, 1.200, 1.210, 1.220, 1.220, 1.240, 1.300, 1.340, 1.360, 1.390, 1.440, 1.590, 1.600, 1.630, 1.630, 1.680, 1.710, 1.720, 1.760, 1.830, 1.950, 1.960, 1.970, 2.020, 2.130, 0.100, 0.330, 2.150, 2.160, 2.220, 2.300, 2.310, 2.400, 2.450, 2.510, 2.540, 2.780, 2.930, 3.270, 3.420, 3.470, 3.610, 4.020, 1.000, 1.020, 4.320, 4.580, and 5.550.Table5 shows that the EW-EW distribution with five parameters gives a better match than their particular submodels, according to our findings. It has the lowest AKINC, COAKINC, BINC, and HQINC values of all of the candidates studied here. Figures 3 and 4 show plots of the fitted densities and the histogram, respectively.Table 5
Parameter estimates and values of AKINC, COAKINC, BINC, and HQINC for the 1st data.
ModelMethoda^α^β^λ^θ^StatisticsAKINCBINCCOAKINCHQINCEW-EWMLE1.58980.61430.45151.62871.0460237.8770249.2610238.7860240.4090EW-EMLE1.32380.40020.0132——271.0240280.1310271.6210274.6500WEMLE—0.77690.4928——294.4660301.2960294.8190297.1850EE-EMLE0.70610.3862———310.3940317.2240310.7470313.1130EW-WMLE0.98000.61760.6642—0.4818356.8300368.2130357.7390361.3620Figure 3
Plot CDF of the EW-EW, EW-E, WE, EE-E, and EW-EW with the empirical distribution for the 1st dataFigure 4
Plot PDF of the EW-EW, EW-E, WE, EE-E, and EW-EW with the empirical distribution for the 1st data


#### 6.2.2. The Second Data

Reference [18] collected the data concerning the tensile strength of 100 carbon fiber measurements, and they are as follows: 3.700, 3.110, 4.420, 3.280, 3.750, 2.960, 3.390, 3.310, 3.150, 2.810, 1.410, 2.760, 3.190, 1.590, 2.170, 3.510, 1.840, 1.610, 1.570, 1.890, 2.740, 3.270, 2.410, 3.090, 2.430, 2.530, 2.810, 3.310, 2.350, 2.770, 3.680, 4.910, 1.570, 2.000, 1.170, 2.170, 0.390, 2.790, 1.080, 2.880, 2.730, 2.870, 3.190, 1.870, 2.950, 2.670, 4.200, 2.850, 2.550, 2.170, 2.970, 3.680, 0.810, 1.220, 5.080, 1.690, 3.680, 4.700, 2.030, 2.820, 2.500, 1.470, 3.220, 3.150, 2.970, 2.930, 3.330, 2.560, 2.590, 2.830, 1.360, 1.840, 5.560, 1.120, 2.480, 1.250, 2.480, 2.030, 1.610, 2.050, 3.600, 3.110, 1.690, 4.900, 3.390, 3.220, 2.550, 3.560, 2.380, 1.920, 0.980, 1.590, 1.730, 1.710, 1.180, 4.380, 0.850, 1.800, 2.120, and 3.650.Table6 shows that the EW-EW distribution has the fewest values of the investigated metrics when compared to other models. As a result, it is a better model for this data than their particular submodels. Figures 5 and 6 show plots of the fitted densities and the histogram.Table 6
Parameter estimates and values of AKINC, COAKINC, BINC, and HQINC for the 2nd data.
ModelMethodaαβλθStatisticsHQINCCOAKINCBINCAKINCEW-EWMLE3.07560.34520.49540.72001.0938299.0530312.0720299.6920298.0520EW-EMLE35.41862.18610.0279——306.5690316.9900306.9900310.7870WEMLE—0.51010.4532——314.2480321.0780314.6010316.9670EW-WMLE0.95620.61790.5802—0.3156538.5710549.9540539.4800543.1020EE-EMLE0.35480.0501———678.8780686.6940679.1280682.0410Figure 5
Plot CDF of the EW-EW, EW-E, WE, EE-E, and EW-EW with the empirical distribution for the 2nd data.Figure 6
Plot PDF of the EW-EW, EW-E, WE, EE-E and EW-EW with the empirical distribution for the 2nd data.


## 6.1. Numerical Outcomes

To examine the behavior of MLEs for the EW-EW model, a thorough numerical examination is performed. To evaluate the performance of ML estimates, the ML estimators (MLE), biases (∇1), and mean square errors (∇2) are determined for different sample sizes (n).In addition, we note that, for eachn, the MLEs are evaluated using two accuracy measures: ∇1 and ∇2. Tables 2–4 present the MLEs together with the ∇1 and the ∇2. Generally, we note that as the n grows, the ∇2s of the estimates for EW-EW distribution with the parameters decrease, which shows consistency of the estimated parameters.Table 2
The values of MLE,∇1 and ∇2 for the unknown parameters a,α,β,λ, and θ at (0.8, 0.3, 0.5, 0.3, and 0.2), respectively.
nThe MLEs for initial values (0.8, 0.3, 0.5, 0.3, 0.2)a^α^β^λ^θ^10MLE1.59901.48100.69550.17970.5673∇10.79901.18100.1955−0.12030.3673∇20.63841.39480.03820.01450.1349MLE1.36700.32230.68500.26560.4673∇10.56700.02230.1850−0.03440.2673∇20.32140.00050.03420.00120.0714100MLE1.29550.32110.32950.26810.4670∇10.49550.0211−0.1705−0.03190.2670∇20.24550.00040.02910.00100.0713Table 3
The values of MLE,∇1, and ∇2 for the unknown parameters a,α,β,λ, and θ at (0.5, 0.2, 0.3, 0.7, and 0.3), respectively.
nThe MLEs for initial values (0.5, 0.2, 0.3, 0.7, 0.3)a^α^β^λ^θ^10MLE0.96711.03190.86900.54470.5517∇10.46710.83190.5690−0.15530.2517∇20.21820.69200.32380.02410.063350MLE0.92500.35640.51020.74250.5301∇10.42500.15640.21020.04250.2301∇20.18060.02450.04420.00180.0530100MLE0.32590.30430.28870.67270.5137∇10.12590.1043−0.0113−0.02730.2137∇20.01580.01090.00010.00070.4566Table 4
The values of MLE,∇1, and ∇2 for the unknown parameters a,α,β,λ, and θ at (0.9, 0.1, 0.5, 0.5, and 0.3).
nThe MLEs for initial values (0.9, 0.1, 0.5, 0.5, and 0.3)a^α^β^λ^θ^10MLE1.74040.34120.17020.06490.8716∇10.84040.2412−0.3298−0.43510.5716∇20.70630.05820. 108750.18930.326750MLE1.41380.84040.21010.14780.8030∇10.51380.2123−0.2899−0.35220.5030∇20.26400.04510.08400.12410.2530100MLE1.16660.04510.30220.57410.5466∇10.2666-0.0549−0.19780.07410.2466∇20.07110.00300.03910.00550.0608


## 6.2. Modelling to Biomedical and Engineering Real Data

For additional illustration, this section compares the productivity of the goodness-of-fit for the EW-EW distribution with specific chosen distributions from the literature. A real dataset is utilized in particular to compare the suggested distribution to the EW exponential distribution (EW-E), the W exponential distribution (WE), the E exponential exponential distribution (EE-E), and the EW W distribution (EW-W). We obtain the MLE of the model parameters.To compare the performance of different distribution models, we considered AKINC, COAKINC, BINC, and HQINC. However, the better distribution corresponds to the smaller values of AKINC, COAKINC, BINC, and HQINC criteria. Furthermore, we plot the empirical CDF of the datasets and estimated CDF of EW-E, WE, EE-E, and EW-W models are displayed.


### 6.2.1. The First Data

The survival periods (in 100 days) of 72 guinea pigs infected with virulent tubercle bacilli were derived from [13]. The following are the data sets: 1.460, 1.530, 2.530, 2.540, 0.440, 0.560, 0.590, 0.720, 0.740, 0.770, 0.920, 0.930, 0.960, 1.000, 1.050, 1.070, 1.070, 1.080, 1.080, 1.080, 1.090, 1.120, 1.130, 1.150, 1.160, 1.200, 1.210, 1.220, 1.220, 1.240, 1.300, 1.340, 1.360, 1.390, 1.440, 1.590, 1.600, 1.630, 1.630, 1.680, 1.710, 1.720, 1.760, 1.830, 1.950, 1.960, 1.970, 2.020, 2.130, 0.100, 0.330, 2.150, 2.160, 2.220, 2.300, 2.310, 2.400, 2.450, 2.510, 2.540, 2.780, 2.930, 3.270, 3.420, 3.470, 3.610, 4.020, 1.000, 1.020, 4.320, 4.580, and 5.550.Table5 shows that the EW-EW distribution with five parameters gives a better match than their particular submodels, according to our findings. It has the lowest AKINC, COAKINC, BINC, and HQINC values of all of the candidates studied here. Figures 3 and 4 show plots of the fitted densities and the histogram, respectively.Table 5
Parameter estimates and values of AKINC, COAKINC, BINC, and HQINC for the 1st data.
ModelMethoda^α^β^λ^θ^StatisticsAKINCBINCCOAKINCHQINCEW-EWMLE1.58980.61430.45151.62871.0460237.8770249.2610238.7860240.4090EW-EMLE1.32380.40020.0132——271.0240280.1310271.6210274.6500WEMLE—0.77690.4928——294.4660301.2960294.8190297.1850EE-EMLE0.70610.3862———310.3940317.2240310.7470313.1130EW-WMLE0.98000.61760.6642—0.4818356.8300368.2130357.7390361.3620Figure 3
Plot CDF of the EW-EW, EW-E, WE, EE-E, and EW-EW with the empirical distribution for the 1st dataFigure 4
Plot PDF of the EW-EW, EW-E, WE, EE-E, and EW-EW with the empirical distribution for the 1st data


### 6.2.2. The Second Data

Reference [18] collected the data concerning the tensile strength of 100 carbon fiber measurements, and they are as follows: 3.700, 3.110, 4.420, 3.280, 3.750, 2.960, 3.390, 3.310, 3.150, 2.810, 1.410, 2.760, 3.190, 1.590, 2.170, 3.510, 1.840, 1.610, 1.570, 1.890, 2.740, 3.270, 2.410, 3.090, 2.430, 2.530, 2.810, 3.310, 2.350, 2.770, 3.680, 4.910, 1.570, 2.000, 1.170, 2.170, 0.390, 2.790, 1.080, 2.880, 2.730, 2.870, 3.190, 1.870, 2.950, 2.670, 4.200, 2.850, 2.550, 2.170, 2.970, 3.680, 0.810, 1.220, 5.080, 1.690, 3.680, 4.700, 2.030, 2.820, 2.500, 1.470, 3.220, 3.150, 2.970, 2.930, 3.330, 2.560, 2.590, 2.830, 1.360, 1.840, 5.560, 1.120, 2.480, 1.250, 2.480, 2.030, 1.610, 2.050, 3.600, 3.110, 1.690, 4.900, 3.390, 3.220, 2.550, 3.560, 2.380, 1.920, 0.980, 1.590, 1.730, 1.710, 1.180, 4.380, 0.850, 1.800, 2.120, and 3.650.Table6 shows that the EW-EW distribution has the fewest values of the investigated metrics when compared to other models. As a result, it is a better model for this data than their particular submodels. Figures 5 and 6 show plots of the fitted densities and the histogram.Table 6
Parameter estimates and values of AKINC, COAKINC, BINC, and HQINC for the 2nd data.
ModelMethodaαβλθStatisticsHQINCCOAKINCBINCAKINCEW-EWMLE3.07560.34520.49540.72001.0938299.0530312.0720299.6920298.0520EW-EMLE35.41862.18610.0279——306.5690316.9900306.9900310.7870WEMLE—0.51010.4532——314.2480321.0780314.6010316.9670EW-WMLE0.95620.61790.5802—0.3156538.5710549.9540539.4800543.1020EE-EMLE0.35480.0501———678.8780686.6940679.1280682.0410Figure 5
Plot CDF of the EW-EW, EW-E, WE, EE-E, and EW-EW with the empirical distribution for the 2nd data.Figure 6
Plot PDF of the EW-EW, EW-E, WE, EE-E and EW-EW with the empirical distribution for the 2nd data.


## 6.2.1. The First Data

The survival periods (in 100 days) of 72 guinea pigs infected with virulent tubercle bacilli were derived from [13]. The following are the data sets: 1.460, 1.530, 2.530, 2.540, 0.440, 0.560, 0.590, 0.720, 0.740, 0.770, 0.920, 0.930, 0.960, 1.000, 1.050, 1.070, 1.070, 1.080, 1.080, 1.080, 1.090, 1.120, 1.130, 1.150, 1.160, 1.200, 1.210, 1.220, 1.220, 1.240, 1.300, 1.340, 1.360, 1.390, 1.440, 1.590, 1.600, 1.630, 1.630, 1.680, 1.710, 1.720, 1.760, 1.830, 1.950, 1.960, 1.970, 2.020, 2.130, 0.100, 0.330, 2.150, 2.160, 2.220, 2.300, 2.310, 2.400, 2.450, 2.510, 2.540, 2.780, 2.930, 3.270, 3.420, 3.470, 3.610, 4.020, 1.000, 1.020, 4.320, 4.580, and 5.550.Table5 shows that the EW-EW distribution with five parameters gives a better match than their particular submodels, according to our findings. It has the lowest AKINC, COAKINC, BINC, and HQINC values of all of the candidates studied here. Figures 3 and 4 show plots of the fitted densities and the histogram, respectively.Table 5
Parameter estimates and values of AKINC, COAKINC, BINC, and HQINC for the 1st data.
ModelMethoda^α^β^λ^θ^StatisticsAKINCBINCCOAKINCHQINCEW-EWMLE1.58980.61430.45151.62871.0460237.8770249.2610238.7860240.4090EW-EMLE1.32380.40020.0132——271.0240280.1310271.6210274.6500WEMLE—0.77690.4928——294.4660301.2960294.8190297.1850EE-EMLE0.70610.3862———310.3940317.2240310.7470313.1130EW-WMLE0.98000.61760.6642—0.4818356.8300368.2130357.7390361.3620Figure 3
Plot CDF of the EW-EW, EW-E, WE, EE-E, and EW-EW with the empirical distribution for the 1st dataFigure 4
Plot PDF of the EW-EW, EW-E, WE, EE-E, and EW-EW with the empirical distribution for the 1st data


## 6.2.2. The Second Data

Reference [18] collected the data concerning the tensile strength of 100 carbon fiber measurements, and they are as follows: 3.700, 3.110, 4.420, 3.280, 3.750, 2.960, 3.390, 3.310, 3.150, 2.810, 1.410, 2.760, 3.190, 1.590, 2.170, 3.510, 1.840, 1.610, 1.570, 1.890, 2.740, 3.270, 2.410, 3.090, 2.430, 2.530, 2.810, 3.310, 2.350, 2.770, 3.680, 4.910, 1.570, 2.000, 1.170, 2.170, 0.390, 2.790, 1.080, 2.880, 2.730, 2.870, 3.190, 1.870, 2.950, 2.670, 4.200, 2.850, 2.550, 2.170, 2.970, 3.680, 0.810, 1.220, 5.080, 1.690, 3.680, 4.700, 2.030, 2.820, 2.500, 1.470, 3.220, 3.150, 2.970, 2.930, 3.330, 2.560, 2.590, 2.830, 1.360, 1.840, 5.560, 1.120, 2.480, 1.250, 2.480, 2.030, 1.610, 2.050, 3.600, 3.110, 1.690, 4.900, 3.390, 3.220, 2.550, 3.560, 2.380, 1.920, 0.980, 1.590, 1.730, 1.710, 1.180, 4.380, 0.850, 1.800, 2.120, and 3.650.Table6 shows that the EW-EW distribution has the fewest values of the investigated metrics when compared to other models. As a result, it is a better model for this data than their particular submodels. Figures 5 and 6 show plots of the fitted densities and the histogram.Table 6
Parameter estimates and values of AKINC, COAKINC, BINC, and HQINC for the 2nd data.
ModelMethodaαβλθStatisticsHQINCCOAKINCBINCAKINCEW-EWMLE3.07560.34520.49540.72001.0938299.0530312.0720299.6920298.0520EW-EMLE35.41862.18610.0279——306.5690316.9900306.9900310.7870WEMLE—0.51010.4532——314.2480321.0780314.6010316.9670EW-WMLE0.95620.61790.5802—0.3156538.5710549.9540539.4800543.1020EE-EMLE0.35480.0501———678.8780686.6940679.1280682.0410Figure 5
Plot CDF of the EW-EW, EW-E, WE, EE-E, and EW-EW with the empirical distribution for the 2nd data.Figure 6
Plot PDF of the EW-EW, EW-E, WE, EE-E and EW-EW with the empirical distribution for the 2nd data.


## 7. Conclusion

In this article, we introduced a novel five-parameter exponentiated Weibull distribution and studied its various statistical properties in this study. It should be noted that the proposed EW-EW distribution has a lot of appealing characteristics. The EW-EW distribution includes numerous existing distributions as well as some new ones. Simulation results are used to assess the estimation accuracy and performance. In terms of fit, the proposed model surpasses certain other rival models when tested on actual data. We expect that the proposed distribution can entice a wider range of applications in domains such as lifetime analysis, dependability, hydrology, and engineering. In the future, we can use the estimate parameters of the new model using censored schemes.



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*Source: 4669412-2022-04-28.xml*