# A Novel Generalized-M Family: Heavy-Tailed Characteristics with Applications in the Engineering Sector

Authors: Mahmoud El-Morshedy; Zahra Almaspoor; Nasir Abbas; Zahid Khan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The Weibull distribution has prominent applications in the engineering sector. However, due to its monotonic behavior of the hazard function, the Weibull model does not provide the best fit for data in many cases. This paper introduces a new family of distributions to obtain new flexible distributions. The proposed family is called a novel generalized-M family. Based on this approach, an updated version of the Weibull distribution is introduced. The updated version of the Weibull distribution is called a novel generalized Weibull distribution. The proposed distribution is able to capture four different patterns of the hazard function. Some mathematical properties of the proposed method are obtained. Furthermore, the maximum likelihood estimators of the proposed family are also obtained. Moreover, a simulation study is conducted for evaluating these estimators. For illustrating the proposed model, two data sets from the engineering sector are analyzed. Based on some well-known analytical measures, it is shown that the novel generalized Weibull distribution is the best competing distribution for analyzing the engineering data sets.


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## Body

## 1. Introduction

The two-parameter Weibull distribution is a famous statistical distribution defined onℝ+. It can be used as a suitable alternative choice for predicting, modeling, and analyzing lifetime data sets in healthcare, engineering, and other related areas. For data modeling in the engineering sector, numerous modifications of the Weibull distribution have been introduced and implemented, refer to Pedrosa et al. [1]; Bala and Napiah [2]; Huo et al. [3]; Liao et al. [4]; Shu et al. [5]; Li et al. [6]; and Bilal et al. [7].On one side, the Weibull distribution generalizes the exponential and Rayleigh distributions whereas, on the other side, it can provide the characteristics of the other distributions. For some review studies based on the applications of the Weibull distribution, we refer to Nadarajah et al. [8]; Almalki and Nadarajah [9]; and Wais [10].Let the random variableV has the Weibull distribution with scale parameter σ>0 and shape parameter α>0, if its HF (hazard function) hv;Ξ has the following form:(1)hv;Ξ=ασvα−1,v>0.From the expression ofhv;Ξ in (1), we can see that the Weibull distribution offers data modeling with (i) decreasing HF shape, if α<1, (ii) increasing HF shape, if α>1, or (iii) constant HF shape, if α=1.In many cases, the data sets in the engineering sector follow the unimodal, bathtub, or modified unimodal HF shapes. In such cases, the Weibull distribution is not a suitable choice to implement for modeling the engineering data sets. In order to provide a close fit to such type of engineering data sets, numerous updated versions of the Weibull distribution have been produced and implemented. For example, (i) Ahmad et al. [11] studied a new Weibull–Weibull distribution for analyzing the strength of the alumina material; (ii) Zhao et al. [12] introduced TIHT–Weibull distribution for modeling the failure time of coating machines; (iii) Wang et al. [13] implemented the NG–Weibull distribution for analyzing the failure time of the electronic machine. For more recent modifications of the Weibull distribution, we refer to Klakattawi [14]; Wang et al. [15]; Tianshuai et al. [16]; Al-Sobhi [17]; Gonzalez et al. [18]; Emam and Tashkandy [19]; Prataviera [20]; and Rehman et al. [21].To overcome the above deficiency of the Weibull distribution, we introduce a new updated form of the Weibull distribution. The new updated form of the Weibull distribution is introduced by proposing a new statistical family of distributions. The new family is called a novel generalized-M (NGen-M) family of distributions. The NGen-M family can be used to update/increase the fitting power of the classical/traditional or other modified and generalized distributions.Definition: a random variableV has the NGen-M distributions, if its DF (distribution function) Kv;λ,Ξ is given by(2)Kv;λ,Ξ=1−1−Mv;Ξλλ−Mv;Ξ2,v∈ℝ,where λ≥1,λ≤−1, and Mv;Ξ represents the baseline DF. In order to show that Kv;λ,Ξ is a valid DF, a complete proof is provided in Proposition 1 and Proposition 2.Proposition 1.
Let’s consider the DFKv;λ,Ξ in Eq. (2), first, we have to show that(3)limv⟶−∞Kv;λ,Ξ=0,and(4)limv⟶−∞Kv;λ,Ξ=1.Proof.
(5)limv⟶−∞Kv;λ,Ξ=limv⟶−∞1−1−Mv;Ξλλ−Mv;Ξ2,limv⟶−∞Kv;λ,Ξ=1−1−M−∞;Ξλλ−M−∞;Ξ2.
SinceMv;Ξ is a DF, we have(6)limv⟶−∞Mv;Ξ=0.
Hence, from (5), we have(7)limv⟶−∞Kv;λ,Ξ=1−1−0λλ−0,limv⟶−∞Kv;λ,Ξ=1−1λλ,limv⟶−∞Kv;λ,Ξ=0.
Also, we have(8)limv⟶−∞Kv;λ,Ξ=limv⟶−∞1−1−Mv;Ξλλ−Mv;Ξ2,limv⟶−∞Kv;λ,Ξ=1−1−M∞;Ξλλ−M∞;Ξ2.
As we stated above thatMv;Ξ is a DF, we have(9)limv⟶−∞Mv;Ξ=1.
Therefore, from (8), we have(10)limv⟶−∞Kv;λ,Ξ=1−1−1λλ−1,limv⟶−∞Kv;λ,Ξ=1−0λλ,limv⟶−∞Kv;λ,Ξ=1.Proposition 2.
The DFKv;λ,Ξ presented in Eq. (2) is a differentiable as well as a right continuous function.Proof.
(11)ddvKv;λ,Ξ=kv;λ,Ξ.
The proof of Proposition2 is very straightforward, and hence, omitted. Based on the mathematical derivations in Proposition 1 and Proposition 2, we can see that Kv;λ,Ξ is a valid DF.
Forv∈ℝ, in link to Kv;λ,Ξ, the PDF (probability density function) kv;λ,Ξ is given by(12)kv;λ,Ξ=mv;Ξλλ+2Mv;Ξ−3Mv;Ξ2,where d/dvMv;Ξ=mv;Ξ.
Corresponding to DFKv;λ,Ξ and PDF kv;λ,Ξ, the SF Sv;λ,Ξ=1−Kv;λ,Ξ, HF hv;λ,Ξ=kv;λ,Ξ/1−Kv;λ,Ξ, and cumulative HF (CHF) Hv;λ,Ξ=−log1−Kv;λ,Ξ are given by(13)Sv;λ,Ξ=1−Mv;Ξλλ−Mv;Ξ2,hv;λ,Ξ=mv;Ξλ+2Mv;Ξ−3Mv;Ξ21−Mv;Ξλ−Mv;Ξ2,and(14)Sv;λ,Ξ=−log1−Mv;Ξλλ−Mv;Ξ2,respectively.
By using the DFMv;Ξ in (2) as a base model, we can introduce an updated version of any base model. In this paper, we use Mv;Ξ as a DF of the Weibull model, to introduce a new version of the Weibull distribution. The updated version of the Weibull model is called a novel generalized-Weibull (NGen-Weibull) distribution. Some basic expressions of the NGen-Weibull distribution along with different plots of PDF and HF are obtained in Section 2.


## 2. A NGen-Weibull Distribution

Consider the DFMv;Ξ of the Weibull distribution given by(15)Mv;Ξ=1−e−σvα,v≥0,α>0,σ>0,with PDFmv;Ξ given by(16)mv;Ξ=ασvα−1e−σvα,v>0,α>0,δ>0,where Ξ=α,σ.By incorporating (15) in (1), we obtain the DF of the NGen-Weibull distribution. Let V has the NGen-Weibull distribution, if its DF Kv;λ,Ξ is given by(17)Kv;λ,Ξ=1−e−σvαλλ−1−e−σvα2.ForV>0, the PDF kv;λ,Ξ of the NGen-Weibull distribution is given by(18)kv;λ,Ξ=ασvα−1e−σvαλλ+21−e−σvα−31−e−σvα2.Visual display of different plots ofkv;λ,Ξ and Kv;λ,Ξ is provided in Figure 1. From the plots of kv;λ,Ξ, we can see that the NGen-Weibull has four different patterns of PDF such as (i) reverse-J shaped (blue curve), (ii) right-skewed (red curve), (iii) left-skewed (green curve), and (vi) symmetrical (black curve).Figure 1
Visual display of different plots ofkv;λ,Ξ and Kv;λ,Ξ.The plots ofkv;λ,Ξ in Figure 1 are obtained for (i) α=1.2,σ=1.800,λ=1.2 (blue curve), (ii) α=5.9,σ=0.001,λ=2.4 (red curve), (iii) α=3.2,σ=0.080,λ=5.2 (green curve), and (iv) α=0.5,σ=1.300,λ=3.2 (black curve).Furthermore, forV>0, the SF, HF, and CHF of the NGen-Weibull distribution are presented as follows:(19)Sv;λ,Ξ=e−σvαλλ−1−e−σvα2,hv;λ,Ξ=ασvα−1λ+21−e−σvα−31−e−σvα2λ−1−e−σvα2,and(20)Hv;λ,Ξ=−loge−σvαλλ−1−e−σvα2,respectively.A graphical illustration ofhv;λ,Ξ and Sv;λ,Ξ is presented in Figure 2. From the plots of hv;λ,Ξ, we can see that the NGen-Weibull distribution has the ability to model data with four different shapes of failure function. The HF shapes of NGen-Weibull distribution include (i) increasing shaped (blue curve), (ii) unimodal (green curve), (iii) modified unimodal (red curve), and (vi) decreasing (black curve).Figure 2
Graphical illustration ofhv;λ,Ξ and Sv;λ,Ξ.The plots ofhv;λ,Ξ in Figure 2 are obtained for (i) α=0.9,σ=1.1,λ=1.2 (blue curve), (ii) α=1.0,σ=1.1,λ=1.8 (red curve), (iii) α=0.8,σ=1.5,λ=1.3 (green curve), and (iv) α=1.2,σ=1.2,λ=1.9 (black curve).The proposed model has certain advantages over the other classical and modified distributions, for example.(i)
The proposed NGen-Weibull distribution is a simple extension of the Weibull model by adding one additional parameter. Most of the updated versions of the Weibull distribution have more than one additional parameter.(ii)
The proposed NGen-Weibull distribution has four different shapes of the HF. There are only a few distributions that can model real-life data sets with unimodal and decreasing-increasing-decreasing shapes of HF.(iii)
The proposed NGen-Weibull distribution obeys the HT properties. This fact indicates that the NGen-Weibull distribution can be a suitable candidate distribution for modeling the financial data sets. Due to the HT characteristics, the NGen-Weibull distribution can be used quite effectively for dealing with the extreme value data sets.(iv)
The NGen-Weibull distribution provides the fit best to real-life engineering data sets as compared to other competing models, see Section5.


## 3. Some Distributional Properties

This section offers the derivation of some distributional properties of the NGen-M distributions. These properties include identifiability property, heavy-tailed property, quantile function, and rth moment.


### 3.1. The Identiability Property

In this subsection, we derive the identifiability property of the NGen-M distributions. The parameter λ is called identifiable, if(21)λ1=λ2.Supposeλ1 and λ2 be the two parameters with DFs Kv;λ1,Ξ and Kv;λ2,Ξ, respectively. From the mathematical definition of the identifiability property, we have(22)Kv;λ1,Ξ=Kv;λ2,Ξ.Using (2) in (22), we get(23)1−1−Mv;Ξλ1λ1−Mv;Ξ2=1−1−Mv;Ξλ2λ2−Mv;Ξ2,1−Mv;Ξλ1λ1−Mv;Ξ2=1−Mv;Ξλ2λ2−Mv;Ξ2,λ21−Mv;Ξλ1−Mv;Ξ2=λ11−Mv;Ξλ2−Mv;Ξ2,λ21−Mv;ΞMv;Ξ2=λ11−Mv;ΞMv;Ξ2,λ1=λ2.


### 3.2. The HT Property

A distribution with DFKv;λ,Ξ is called a HT model, if its SF satisfies(24)limv⟶∞epv1−Kv;λ,Ξ=∞,where p>0.The HT distributions obey an important property called the regular variational property. A probability distribution is called regularly varying if it satisfies(25)limv⟶∞1−Ktv;λ,Ξ1−Kv;λ,Ξ=ta,a>0,t>0,where the term a is called the index of regular variation.Here, we mathematical prove the regular variational property of the NGen-M distributions. Based on the findings of Seneta [22], in terms of SF 1−Mv;Ξ, we have the following:Theorem 1.
If a distribution with SF1−Mv;Ξis regular varying distribution, then1−Kv;λ,Ξis a regular varying distribution.Proof.
Supposelimv⟶−∞1−Mtv;Ξ/1−Mv;Ξ=gv is finite but nonzero ∀v>0. Then, incorporating (2), we get(26)limv⟶∞Kav;λ,ΞKv;λ,Ξ=limv⟶∞1−Mav;Ξ/λλ−Mav;Ξ21−Mv;Ξ/λλ−Mv;Ξ2,limv⟶∞Kav;λ,ΞKv;λ,Ξ=limv⟶∞1−Mav;Ξ1−Mv;Ξ×λ−Mav;Ξ2λ−Mv;Ξ2,limv⟶∞Kav;λ,ΞKv;λ,Ξ=gv×limv⟶∞λ−Mav;Ξ2λ−Mv;Ξ2,limv⟶∞Kav;λ,ΞKv;λ,Ξ=gvλ−Ma.∞;Ξ2λ−M∞;Ξ2,limv⟶∞Kav;λ,ΞKv;λ,Ξ=gvλ−−1λ−1,limv⟶∞Kav;λ,ΞKv;λ,Ξ=gv.
Since (26) is nonzero, ∀v>0. Therefore, 1−Kav;λ,Ξ is the SF of the regular varying distribution.


### 3.3. The Quantile Function

The QF (quantile function) of the NGen-M distributions with DF Kv;λ,Ξ is derived as(27)v=Qu=K−1u=M−1t,where u∈0,1 and t is the solution of 1−tλ−t2−λ1−u. The expression in (27) can be implemented to generate random numbers from any special model of the NGen-M family.


### 3.4. Therth Moment

Some basic distributional properties of any probability distribution can be derived and studied with the help of moments. This subsection offers the derivation of therth moment of NGen-M distributions. Let V have the NGen-M distributions with PDF kv;λ,Ξ, then, its rth moment can be obtained as(28)μr′=∫Ωvrkv;λ,Ξdv.Using (12) in (28), we get(29)μr′=∫Ωvrmv;Ξλλ+2Mv;Ξ−3Mv;Ξ2dv,μr′=∫Ωvrmv;Ξdv+2λ∫Ωvrmv;ΞMv;Ξdv−3λ∫Ωvrmv;ΞMv;Ξ2dv,μr′=∫Ωvrmv;Ξdv+1λ∫Ω2vrmv;ΞMv;Ξ2−1dv−1λ∫Ω3vrmv;ΞMv;Ξ3−1dv,μr′=∫Ωvrmv;Ξdv+1λ∫Ωvrf1v;Ξdv−1λ∫Ωvrf2v;Ξdv,where f1v;Ξ=2mv;ΞMv;Ξ2−1 is the PDF of the exponentiated version of the NGen-M distributions with the exponentiated parameter 2 whereas f2v;Ξ=3mv;ΞMv;Ξ3−1 is the PDF of the exponentiated version of the NGen-M distributions with the exponentiated parameter 3.


## 3.1. The Identiability Property

In this subsection, we derive the identifiability property of the NGen-M distributions. The parameter λ is called identifiable, if(21)λ1=λ2.Supposeλ1 and λ2 be the two parameters with DFs Kv;λ1,Ξ and Kv;λ2,Ξ, respectively. From the mathematical definition of the identifiability property, we have(22)Kv;λ1,Ξ=Kv;λ2,Ξ.Using (2) in (22), we get(23)1−1−Mv;Ξλ1λ1−Mv;Ξ2=1−1−Mv;Ξλ2λ2−Mv;Ξ2,1−Mv;Ξλ1λ1−Mv;Ξ2=1−Mv;Ξλ2λ2−Mv;Ξ2,λ21−Mv;Ξλ1−Mv;Ξ2=λ11−Mv;Ξλ2−Mv;Ξ2,λ21−Mv;ΞMv;Ξ2=λ11−Mv;ΞMv;Ξ2,λ1=λ2.


## 3.2. The HT Property

A distribution with DFKv;λ,Ξ is called a HT model, if its SF satisfies(24)limv⟶∞epv1−Kv;λ,Ξ=∞,where p>0.The HT distributions obey an important property called the regular variational property. A probability distribution is called regularly varying if it satisfies(25)limv⟶∞1−Ktv;λ,Ξ1−Kv;λ,Ξ=ta,a>0,t>0,where the term a is called the index of regular variation.Here, we mathematical prove the regular variational property of the NGen-M distributions. Based on the findings of Seneta [22], in terms of SF 1−Mv;Ξ, we have the following:Theorem 1.
If a distribution with SF1−Mv;Ξis regular varying distribution, then1−Kv;λ,Ξis a regular varying distribution.Proof.
Supposelimv⟶−∞1−Mtv;Ξ/1−Mv;Ξ=gv is finite but nonzero ∀v>0. Then, incorporating (2), we get(26)limv⟶∞Kav;λ,ΞKv;λ,Ξ=limv⟶∞1−Mav;Ξ/λλ−Mav;Ξ21−Mv;Ξ/λλ−Mv;Ξ2,limv⟶∞Kav;λ,ΞKv;λ,Ξ=limv⟶∞1−Mav;Ξ1−Mv;Ξ×λ−Mav;Ξ2λ−Mv;Ξ2,limv⟶∞Kav;λ,ΞKv;λ,Ξ=gv×limv⟶∞λ−Mav;Ξ2λ−Mv;Ξ2,limv⟶∞Kav;λ,ΞKv;λ,Ξ=gvλ−Ma.∞;Ξ2λ−M∞;Ξ2,limv⟶∞Kav;λ,ΞKv;λ,Ξ=gvλ−−1λ−1,limv⟶∞Kav;λ,ΞKv;λ,Ξ=gv.
Since (26) is nonzero, ∀v>0. Therefore, 1−Kav;λ,Ξ is the SF of the regular varying distribution.


## 3.3. The Quantile Function

The QF (quantile function) of the NGen-M distributions with DF Kv;λ,Ξ is derived as(27)v=Qu=K−1u=M−1t,where u∈0,1 and t is the solution of 1−tλ−t2−λ1−u. The expression in (27) can be implemented to generate random numbers from any special model of the NGen-M family.


## 3.4. Therth Moment

Some basic distributional properties of any probability distribution can be derived and studied with the help of moments. This subsection offers the derivation of therth moment of NGen-M distributions. Let V have the NGen-M distributions with PDF kv;λ,Ξ, then, its rth moment can be obtained as(28)μr′=∫Ωvrkv;λ,Ξdv.Using (12) in (28), we get(29)μr′=∫Ωvrmv;Ξλλ+2Mv;Ξ−3Mv;Ξ2dv,μr′=∫Ωvrmv;Ξdv+2λ∫Ωvrmv;ΞMv;Ξdv−3λ∫Ωvrmv;ΞMv;Ξ2dv,μr′=∫Ωvrmv;Ξdv+1λ∫Ω2vrmv;ΞMv;Ξ2−1dv−1λ∫Ω3vrmv;ΞMv;Ξ3−1dv,μr′=∫Ωvrmv;Ξdv+1λ∫Ωvrf1v;Ξdv−1λ∫Ωvrf2v;Ξdv,where f1v;Ξ=2mv;ΞMv;Ξ2−1 is the PDF of the exponentiated version of the NGen-M distributions with the exponentiated parameter 2 whereas f2v;Ξ=3mv;ΞMv;Ξ3−1 is the PDF of the exponentiated version of the NGen-M distributions with the exponentiated parameter 3.


## 4. Estimation and Simulation

This part of the paper offers the derivation of the maximum likelihood estimators (MLEs) of the NGen-M distributions with parameters λ and Ξ. Let V1,V2,…,Vn be a set of sample taken from kv;λ,Ξ. The corresponding LF (likelihood function) Υv;λ,Ξ is(30)Υv;λ,Ξ=∏i=1nkvi;λ,Ξ,Υv;λ,Ξ=∏i=1nmvi;Ξλλ+2Mvi;Ξ−3Mvi;Ξ2,Υv;λ,Ξ=1λn∏i=1nmvi;Ξ∏i=1nλ+2Mvi;Ξ−3Mvi;Ξ2.Corresponding to (30), the log LF Δλ,Ξ is(31)Δλ,Ξ=nlog1λ+∑i=1nlogmvi;Ξ+∑i=1nlogλ+2Mvi;Ξ−3Mvi;Ξ2.Corresponding toΔλ,Ξ, the partial derivatives are(32)∂∂λΔλ,Ξ=−nλ+∑i=1n1λ+2Mvi;Ξ−3Mvi;Ξ2,and(33)∂∂ΞΔλ,Ξ=∑i=1n∂/∂Ξmvi;Ξmvi;Ξ+∑i=1n2∂/∂ΞMvi;Ξ−6Mvi;Ξ∂/∂ΞMvi;Ξλ+2Mvi;Ξ−3Mvi;Ξ2.Equating∂/∂λΔλ,Ξ and ∂/∂ΞΔλ,Ξ to zero, and solving, we obtain the MLEs λ^,Ξ^ of λ,Ξ.After obtaining the expressions of the MLEs, now we provide a simulation study to assess performances ofλ^ and Ξ^. To carry out the evaluation of λ^ and Ξ^ through a simulation study, we generate RNs (random numbers) from the PDF gu;λ,Ξ using the inverse DF method.The simulation results are obtained for (i) α=1.4,σ=1,λ=1.5 and (ii) α=1.2,σ=1,λ=1.9. The simulation results are obtained using R-function with ``L-BFGS-B′′ algorithm and AdequacyModel library. To check the performances of λ^ and Ξ^, two statistical evaluation criteria such as(34)MSEλ^=∑i=1nλ^i−λ2,and(35)Biasλ^=∑i=1nλ^i−λ,were chosen. These two evaluation criteria were also calculated forΞ^.In link toα=1.4,σ=1,λ=1.5, the simulation results of the NGen-Weibull distribution are presented in Table 1 whereas the simulation results for α=1.2,σ=1,λ=1.9 are provided in Table 2.Table 1
The results of the simulation study of the NGen-Weibull model forα=1.4,σ=1,λ=1.5.
nParametersMLEsMSEsBiases25α1.572727000.105651450.17272731σ1.154072000.065730260.15407212λ3.230664 006.477324402.0306643050α1.522884000.053602540.12288441σ1.138389000.044146550.13838869λ2.696242004.516387001.4962416075α1.497386000.032608280.09738595σ1.121207000.031705360.12120685λ2.500344003.782307301.30034420100α1.484352000.025873310.08435211σ1.109394000.027527660.10939399λ2.249304002.821263701.04930410150α1.463615000.016553460.06361484σ1.095838000.023074590.09583788λ2.164244002.765360400.96424440200α1.456049000.014086980.05604902σ1.088937000.020164550.08893705λ2.022918002.185587900.82291830300α1.442332000.009481090.04233205σ1.074784000.014691600.07478449λ1.809146001.479983100.60914590400α1.434387000.007104420.03438667σ1.065242000.012516530.06524163λ1.679365001.089319200.47936480500α1.438202000.006480740.03820168σ1.065776000.012415150.06577620λ1.675391001.024791900.47539100600α1.430618000.005280210.03061809σ1.060799000.011330820.06079897λ1.647639001.010684400.44763910700α1.432310000.005165720.03231025σ1.059613000.010921450.05961322λ1.622840000.914562000.42284040750α1.414981000.004000020.02498083σ1.049650000.008706710.04964955λ1.510436000.636275100.32043650Table 2
The results of simulation study of the NGen-Weibull model forα=1.2,σ=1,λ=1.9.
nParametersMLEsMSEsBiases25α1.315931000.060189980.11593107σ1.156097000.069722260.15609747λ3.521391 006.814517002.1213912050α1.277562000.029811950.07756164σ1.117904000.034266200.11790373λ3.093050005.335289001.6930502075α1.256809000.019706350.05680943σ1.110597000.029871760.11059712λ2.827675004.373158001.42767490100α1.248572000.014392790.04857249σ1.097524000.024428310.09752428λ2.704424003.939458001.30442440150α1.240335000.009775990.04033521σ1.087142000.018688920.08714193λ2.548721003.394285001.14872050200α1.232622000.007963380.03262186σ1.080381000.016730550.08038127λ2.371564 002.867188000.97156410300α1.227361000.005301840.02736133σ1.070034000.012886010.07003359λ2.221988002.336190000.82198780400α1.224368000.004054920.02436773σ1.063477000.010881230.06347681λ2.128215001.932337000.72821510500α1.225677000.003681870.02567661σ1.061764000.010147810.06176444λ2.087973001.743413000.68797340600α1.224082000.003133560.02408230σ1.062215000.010171820.06221497λ2.072379001.676317000.67237860700α1.224706000.002981510.02470586σ1.057195000.009264260.05719467λ2.012889001.096058000.61288930750α1.211702000.002691400.02170158σ1.054507000.008550280.05450734λ1.915065000.993430000.57506490From the simulation results obtained in Tables1 and 2, it is obvious that as the size of n increases.(i)
The estimated values ofλ^MLE and Ξ^MLE tend to be stable(ii)
The MSEs ofλ^MLE and Ξ^MLE decrease(iii)
The biases ofλ^MLE and Ξ^MLE tend to zero


## 5. Data Analyses

This section offers the illustration of the NGen-Weibull distribution using two data sets taken from the engineering sector. We apply the NGen-Weibull distribution to these data sets and compare its fitting results with the Weibull distribution, and its other versions include (i) Marshall–Olkin–Weibull (MO-Weibull), (ii) exponentiated Weibull (Exp-Weibull), and (iii) Kumaraswamy–Weibull (Kum-Weibull) distributions.The DFs of the MO-Weibull, Exp-Weibull, and Kum-Weibull distributions are given, respectively, by(36)Mv;β,Ξ=1−e−σvα1−1−βe−σvα,v≥0,β>0,α>0,σ>0,Mv;a,Ξ=1−e−σvαa,v≥0,a>0,α>0,σ>0,and(37)Mv;a,b,Ξ=1−1−1−e−σvαab,v≥0,a>0,b>0,α>0,σ>0.Furthermore, we consider some analytical measures to compare the fitting results of the NGen-Weibull and other competing distributions. These measures include (i) AD (Anderson Darling) test, (ii) CM (Cramér-von Mises) test, and (iii) KS (Kolmogorov-Smirnov) test withp-value. The numerical values of these measures are computed as(38)−n−1n∑i=1n2i−1logMvi+log1−Mvn−i+1,112n+∑i=1n2i−12n−Mvi2,and(39)supvMnv−Mv,respectively.


### 5.1. Data 1

This subsection offers the first illustration of the NGen-Weibull distribution by analyzing a data set representing the breaking stress of carbon fibres (in Gba). Previously, several authors have also studied this data set. For example, (i) Barreto-Souza et al. [23] considered this data set using the Frechet, beta Frechet (B-Frechet), and exponentiated Frechet (Exp-Frechet) distributions, (ii) Ogunde et al. [24] analyzed this data set using the Nadarajah Haghighi Gompertz (NH-Gompertz) distribution, and (iii) Eghwerido et al. [25] studied this data set using the inverse odd Weibull (IO-Weibull) and a different variant of the Weibull distribution.Some key measures of Data 1 are given by minimum =  0.390,1st quartile = 1.840, median = 2.700, mean = 2.621, 3rd quartile = 3.220, maximum = 5.560, skewness = 0.3681541, kurtosis = 3.104939, variance = 1.027964, and range = 5.17. Corresponding to Data 1, some useful plots are obtained in Figure 3.Figure 3
Some basic plots using data 1.Corresponding to Data 1, the values of the MLEsα^MLE,σ^MLE,λ^MLE, β^MLE, and a^MLE,b^MLE of the fitted models are obtained in Table 3. The profiles of the Log-LF of the MLEs of the NGen-Weibull model are plotted in Figure 4. Furthermore, the values of the analytical measures of the fitted models are presented in Table 4. Based on the numerical illustration presented in Table 4, we can see that the NGen-Weibull distribution is the best competing model.Table 3
The values ofα^MLE,σ^MLE,λ^MLE,β^MLE,a^MLE, and b^MLE of the fitted models using data 1.
Modelsα^MLEσ^MLEλ^MLEa^MLEb^MLEβ^MLENGen-Weibull2.742150.043831.96167———Weibull2.796130.04843————Exp-Weibull2.291810.11245—1.41472——MO-Weibull2.417590.10374———1.73356Kum-Weibull2.126950.09729—1.528381.58933—Figure 4
The profiles of the log-likelihood function of the MLEs using data 1.Table 4
The values of the analytical tests andP-value of the fitted models for data 1.
ModelsCMADKSP-valueNGen-Weibull0.058800.364480.060990.85080Weibull0.062300.415840.062710.82640Exp-Weibull0.074540.423560.073320.65530MO-Weibull0.059450.410850.063890.80890Kum-Weibull0.075830.426020.071850.68010Moreover, in support of the best fitting of the NGen-Weibull distribution using Data 1, a visual illustration of the NGen-Weibull distribution is presented in Figure5. The plots in Figure 5 reveal that the NGen-Weibull distribution closely fits the fitted PDF, CDF, SF, HF, CFH, probability-probability (PP), and quantile-quantile (QQ) functions.Figure 5
The estimated PDF, CDF, SF, HF, CHF, PP, and QQ plots of the proposed modeling using data 1.


### 5.2. Data 2

This subsection offers the second illustration of the NGen-Weibull distribution by considering a data set representing the tensile strength, measured in GPa, of 69 carbon fibres. Previously, numerous authors have also analyzed this data set. For example, (i) Aldahlan [26] considered this data set using the Burr-X exponentiated exponential (B X EExp) distribution, (ii) Eyob et al. [27] studied this data set using the weighted quasi Akash (WQA) distribution, and (iii) Shukla and Shanker [28] analyzed this data set using the truncated Akash (TA) distribution.The basic measures of Data 2 are given by minimum =  1.312,1st quartile = 2.150, median = 2.513, mean = 2.477, 3rd quartile = 2.816, maximum = 3.585, skewness = -0.1541874, kurtosis = 2.95123, variance = 0.2378321, and range = 2.273. Corresponding to Data 2, some basic plots are presented in Figure 6.Figure 6
Some basic plots for data 2.Corresponding to data 2, the numerical values of the MLEsα^MLE,σ^MLE,λ^MLE, a^MLE, and β^MLE,b^MLE of the competing distributions are presented in Table 5. The profiles of the Log-LF of α^MLE,σ^MLE, and λ^MLE of the NGen-Weibull distribution are plotted in Figure 7. The values of the analytical measures of the competing distributions are obtained in Table 6. Based on the numerical findings in Table 6, we can see that the NGen-Weibull distribution has the smallest values of the analytical measures. This fact reveals that the NGen-Weibull model is the best model among the competing distributions.Table 5
The values ofα^MLE,σ^MLE,λ^MLE,β^MLE,a^MLE, and b^MLE of the fitted models using data 2.
Modelsα^MLEσ^MLEλ^MLEa^MLEb^MLEβ^MLENGen-Weibull5.676210.003413.33031———Weibull5.499960.00452————Exp-Weibull4.276540.02298—1.86440——MO-Weibull5.106750.00839———1.54105Kum-Weibull4.155660.01751—1.722031.62877—Figure 7
The profiles of the log-likelihood function of the MLEs of the proposed model using data 2.Table 6
The values of the analytical tests andP-value of the fitted models using data 2.
ModelsCMADKSP-valueNGen-Weibull0.024350.207240.059870.95350Weibull0.026190.231180.070030.86110Exp-Weibull0.029070.219120.063760.92430MO-Weibull0.029330.258070.072220.83480Kum-Weibull0.026630.208150.061170.94890After showing the best fitting of the NGen-Weibull distribution using Data 2 (see, Table6), a graphical illustration is also provided in Figure 8. The plots of the fitted PDF, CDF, SF, HF, CFH, PP, and QQ functions are presented in Figure 8. These plots confirm the best fitting of the NGen-Weibull distribution.Figure 8
The estimated PDF, CDF, SF, HF, CHF, PP, and QQ plots of the proposed modeling for data 2.


## 5.1. Data 1

This subsection offers the first illustration of the NGen-Weibull distribution by analyzing a data set representing the breaking stress of carbon fibres (in Gba). Previously, several authors have also studied this data set. For example, (i) Barreto-Souza et al. [23] considered this data set using the Frechet, beta Frechet (B-Frechet), and exponentiated Frechet (Exp-Frechet) distributions, (ii) Ogunde et al. [24] analyzed this data set using the Nadarajah Haghighi Gompertz (NH-Gompertz) distribution, and (iii) Eghwerido et al. [25] studied this data set using the inverse odd Weibull (IO-Weibull) and a different variant of the Weibull distribution.Some key measures of Data 1 are given by minimum =  0.390,1st quartile = 1.840, median = 2.700, mean = 2.621, 3rd quartile = 3.220, maximum = 5.560, skewness = 0.3681541, kurtosis = 3.104939, variance = 1.027964, and range = 5.17. Corresponding to Data 1, some useful plots are obtained in Figure 3.Figure 3
Some basic plots using data 1.Corresponding to Data 1, the values of the MLEsα^MLE,σ^MLE,λ^MLE, β^MLE, and a^MLE,b^MLE of the fitted models are obtained in Table 3. The profiles of the Log-LF of the MLEs of the NGen-Weibull model are plotted in Figure 4. Furthermore, the values of the analytical measures of the fitted models are presented in Table 4. Based on the numerical illustration presented in Table 4, we can see that the NGen-Weibull distribution is the best competing model.Table 3
The values ofα^MLE,σ^MLE,λ^MLE,β^MLE,a^MLE, and b^MLE of the fitted models using data 1.
Modelsα^MLEσ^MLEλ^MLEa^MLEb^MLEβ^MLENGen-Weibull2.742150.043831.96167———Weibull2.796130.04843————Exp-Weibull2.291810.11245—1.41472——MO-Weibull2.417590.10374———1.73356Kum-Weibull2.126950.09729—1.528381.58933—Figure 4
The profiles of the log-likelihood function of the MLEs using data 1.Table 4
The values of the analytical tests andP-value of the fitted models for data 1.
ModelsCMADKSP-valueNGen-Weibull0.058800.364480.060990.85080Weibull0.062300.415840.062710.82640Exp-Weibull0.074540.423560.073320.65530MO-Weibull0.059450.410850.063890.80890Kum-Weibull0.075830.426020.071850.68010Moreover, in support of the best fitting of the NGen-Weibull distribution using Data 1, a visual illustration of the NGen-Weibull distribution is presented in Figure5. The plots in Figure 5 reveal that the NGen-Weibull distribution closely fits the fitted PDF, CDF, SF, HF, CFH, probability-probability (PP), and quantile-quantile (QQ) functions.Figure 5
The estimated PDF, CDF, SF, HF, CHF, PP, and QQ plots of the proposed modeling using data 1.


## 5.2. Data 2

This subsection offers the second illustration of the NGen-Weibull distribution by considering a data set representing the tensile strength, measured in GPa, of 69 carbon fibres. Previously, numerous authors have also analyzed this data set. For example, (i) Aldahlan [26] considered this data set using the Burr-X exponentiated exponential (B X EExp) distribution, (ii) Eyob et al. [27] studied this data set using the weighted quasi Akash (WQA) distribution, and (iii) Shukla and Shanker [28] analyzed this data set using the truncated Akash (TA) distribution.The basic measures of Data 2 are given by minimum =  1.312,1st quartile = 2.150, median = 2.513, mean = 2.477, 3rd quartile = 2.816, maximum = 3.585, skewness = -0.1541874, kurtosis = 2.95123, variance = 0.2378321, and range = 2.273. Corresponding to Data 2, some basic plots are presented in Figure 6.Figure 6
Some basic plots for data 2.Corresponding to data 2, the numerical values of the MLEsα^MLE,σ^MLE,λ^MLE, a^MLE, and β^MLE,b^MLE of the competing distributions are presented in Table 5. The profiles of the Log-LF of α^MLE,σ^MLE, and λ^MLE of the NGen-Weibull distribution are plotted in Figure 7. The values of the analytical measures of the competing distributions are obtained in Table 6. Based on the numerical findings in Table 6, we can see that the NGen-Weibull distribution has the smallest values of the analytical measures. This fact reveals that the NGen-Weibull model is the best model among the competing distributions.Table 5
The values ofα^MLE,σ^MLE,λ^MLE,β^MLE,a^MLE, and b^MLE of the fitted models using data 2.
Modelsα^MLEσ^MLEλ^MLEa^MLEb^MLEβ^MLENGen-Weibull5.676210.003413.33031———Weibull5.499960.00452————Exp-Weibull4.276540.02298—1.86440——MO-Weibull5.106750.00839———1.54105Kum-Weibull4.155660.01751—1.722031.62877—Figure 7
The profiles of the log-likelihood function of the MLEs of the proposed model using data 2.Table 6
The values of the analytical tests andP-value of the fitted models using data 2.
ModelsCMADKSP-valueNGen-Weibull0.024350.207240.059870.95350Weibull0.026190.231180.070030.86110Exp-Weibull0.029070.219120.063760.92430MO-Weibull0.029330.258070.072220.83480Kum-Weibull0.026630.208150.061170.94890After showing the best fitting of the NGen-Weibull distribution using Data 2 (see, Table6), a graphical illustration is also provided in Figure 8. The plots of the fitted PDF, CDF, SF, HF, CFH, PP, and QQ functions are presented in Figure 8. These plots confirm the best fitting of the NGen-Weibull distribution.Figure 8
The estimated PDF, CDF, SF, HF, CHF, PP, and QQ plots of the proposed modeling for data 2.


## 6. Concluding Remarks

In this article, a new statistical approach for generating new probability distributions was introduced. The new statistical approach was named as NGen-M distributions. Based on the proposed NGen-M distributions method, an updated form of the Weibull distribution was introduced. The updated extension of the Weibull model was named as NGen-Weibull distribution. Certain distributional properties of the NGen-M distributions were derived. The parameters of the NGen-M distributions were estimated via implementing the maximum likelihood approach. A simulation study was also conducted to evaluate the estimators of the NGen-M distributions. In the end, the NGen-M distributions approach was illustrated by analyzing two data sets considered from the engineering sector. To both the engineering data sets, the NGen-Weibull distribution was applied in comparison with other competing models. Using certain statistical test (analytical measures), it is shown that the NGen-Weibull distribution was the best model.In the future, we are motivated to implement the NGen-Weibull distribution for modeling real-life data sets in other fields such as the healthcare sector, hydrology, agriculture, metrology, geology, finance, banking, civil engineering, and education. We are also motivated to introduce the bivariate extension of the NGen-Weibull distribution for modeling the bivariate financial data sets such as import and export, among others.



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*Source: 8569332-2022-08-08.xml*