# On the Extended Generalized Inverted Kumaraswamy Distribution

Authors: Qasim Ramzan; Sadia Qamar; Muhammad Amin; Huda M. Alshanbari; Amna Nazeer; Ahmed Elhassanein
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

In this work, we provide a new generated class of models, namely, the extended generalized inverted Kumaraswamy generated (EGIKw-G) family of distributions. Several structural properties (survival functionsf, hazard rate function hrf, reverse hazard rate function rhrf, quantile function qf and median, sth raw moment, generating function, mean deviation md, etc.) are provided. The estimates for parameters of new G class are derived via maximum likelihood estimation MLE method. The special models of the proposed class are discussed, and particular attention is given to one special model, the extended generalized inverted Kumaraswamy Burr XII (EGIKw-Burr XII) model. Estimators are evaluated via a Monte Carlo simulation MCS. The superiority of EGIKw-Burr XII model is proved using a lifetime data applications.


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## Body

## 1. Introduction

Study of data is the most important and fundamental topic in statistics. The probability distributions help in the characterization of the variability and uncertainty prevailing in data by identifying the patterns of variation. The objective of statistical modeling is to develop appropriate probability distributions that adequately explain a data set generated by surveys, observational studies, experiment, etc.In this context, there have been fundamental and significant thriving in probability distribution theory via the introduction of new generalized families of distributions, and several techniques to develop new distributions have been proposed. Some well-known systems of distributions are the beta generalized family of distributions by Eugene et al. [1], gamma generalized family by Zografos and Balakrishnan [2], Kumaraswamy generalized class of distributions by Cordeiro and de Castro [3], McDonald generalized family by Al-Sarabia [2012], gamma generalized family of distributions (type 2) by Ristic and Balakishnan [4], gamma generalized family (type 3) by Torabi and Hedesh [5], transformed-transformer (T-X) family by Alzaatreh et al. [6], logistic generalized family of distributions by Torabi and Montazeri [7], Weibull generalized class by Bourguignon et al. [8], Lomax generalized family of distributions by Cordeiro et al. [9], logistic X by Tahir et al. [10], odd generalized exponential family (OGE-G) by Tahir et al. [11], Garhy generalized class by Elgarhy et al. [12], Kumaraswamy–Weibull generalized family of distributions by Hassan and Elgarhy [13], exponentiated Weibull generalized family by Hassan and Elgarhy [14], additive Weibull generalized family by Hassan and Hemeda [15], type II half logistic generalized class by Hassan et al. [16], Zubair-G family of distributions by Ahmad [17], generalized inverted Kumaraswamy (GIKw) generated class by Jamal et al. [18], exponentiated Kumaraswamy-G class by Silva et al. [19], and type II Kumaraswamy half logistic family by El-Sherpieny and Elsehetry [20].The inverted distributions are applied in various spheres of life including life testing, biology, environmental science, engineering sciences, and econometrics. Al-Fattah et al. [21] proposed the inverted Kumaraswamy (IKw) model via Y = 1/X − 1 transformation, when X has a Kumaraswamy distribution. Iqbal et al. [22] further generalized the model via transformation T =  Xγ to introduce the IKw distribution and proposed the generalized inverted Kumaraswamy (GIKum) distribution with respective cdf and pdf:(1)Fx=1−1+xγ−αβ,fx=αβγxγ−11+x−α−11−1+xγ−αβ−1,where α>0,β>0,γ>0 are the shape parameters, and x>0. Let st denote the expression for pdf of some random variable rv, T∈a,b, where −∞≤a<b<∞, and consider DWx is some function of cdf of another rv, say X; the T-X family can be defined as(2)Fx=∫aDWxstdt,where DWx satisfies the following:(1)
DWx∈a,b.(2)
DWx is differentiable and monotonically nondecreasing function.(3)
DWx⟶a as x⟶−∞, DWx⟶b as x⟶∞.We give a new G class, the extended generalized inverted Kumaraswamy generated (EGIKw-G) family, consideringst to be GIKum and using the generator Wλx,ϑ/1−Wλx,ϑ as DWx in (2) in order to obtain the distributions which show higher flexibility compared with other commonly used standard distributions; see [23, 24]. For Wx some baseline cdf, the expression for the cdf of EGIKw-G class is(3)FEGIKw-Gx=αβγ∫0Wλx,ϑ/1−Wλx,ϑtγ−11+t−α−11−1+tγ−αβ−1dt,or equivalently(4)FEGIKw-Gx=1−1+Wλx,ϑ1−Wλx,ϑγ−αβ,where α>0,β>0,λandγ>0 are extra positive parameters which offer the skewness, hence promoting the tails weight variation, and ϑ denotes baseline parametric space. For the conditions on baseline distributions, a detailed note can be found in Alzaatreh et al. [6]. In the following section, the pdf, reliability measures, and qf are explored. In Section 3, four special submodels of EGIKw-G class are discussed. In Section 4, several useful properties of the suggested class are provided. In Section 5, MCS study and MLEs are considered to verify the convergence properties. In Section 6, the practical importance of considered G class is examined through real-word data.


## 2. Density and Reliability Measures

In this part of paper, we offer a brief discussion on some of the other basic functions related to the EGIKw-G class of models including thepdf, the sf, the hrf, the rhrf, and the cumulative hazard rate function chrf which have an important role in reliability theory. If X follows EGIKw-G class (4), then its pdf is(5)fEGIKw−Gx=αβγλwx,ϑWλ−1x,ϑ1−Wλx,ϑ2Wλx,ϑ1−Wλx,ϑγ−1×1+Wλx,ϑ1−Wλx,ϑγ−α−1×1−1+Wλx,ϑ1−Wλx,ϑγ−αβ−1=αβγλwx,ϑWγλ−1x,ϑ1−Wλx,ϑ−γ−1×1+Wλx,ϑ1−Wλx,ϑγ−α−1×1−1+Wλx,ϑ1−Wλx,ϑγ−αβ−1.The expressions for thesf, the hrf, the rhrf, and the chrf are given by(6)SEGIKw−Gx=1−1−1+Wλx,ϑ1−Wλx,ϑγ−αβ,hEGIKw−Gx=αβγλwx,ϑWγλ−1x,ϑ1−Wλx,ϑ−γ−1×1+Wλx,ϑ1−Wλx,ϑγ−α−1×1−1+Wλx,ϑ1−Wλx,ϑγ−αβ−1×1−1−1+Wλx,ϑ1−Wλx,ϑγ−αβ−1,HEGIKw−Gx=αβγλwx,ϑWγλ−1x,ϑ1−Wλx,ϑ−γ−1×1+Wλx,ϑ1−Wλx,ϑγ−α−1×1−1+Wλx,ϑ1−Wλx,ϑγ−α−1,ΩEGIKw−Gx=−log1−Fx=−log1−1−1+Wλx,ϑ1−Wλx,ϑγ−αβ,respectively. The EGIKw-G class can be easily simulated through inverting (4) as follows: let u be a standard uniform rv, rv; the inverse cdf or qf is given by solving FEGIKw−Gxu=u as(7)Qu=xu=W−11+1−u1/β−1/α−1−1/γ−1/λ.Furthermore, median, three quartiles, and seven octiles can be, respectively, obtained byQ0.5, Q0.5; qi=Qi/4,i∈1,2,3; and Oj=Qj/8,j∈1,2,3,4,5,6,7. The qf is useful for evaluating some crucial properties including skewness, kurtosis, and central probabilistic results. The Bowley skewness is given by(8)Skb=q3+q1−2q2q3−q1.For some baseline distributionWx when the resulting EGIKw-G distribution is symmetric, right skewed, and left skewed, we have Sk = 0, Sk>0, and Sk<0, respectively. A measure of kurtosis, the Moors kurtosis (see, e.g., Moors [25]), is given as(9)Kum=O3−O1+O7−O5O6−O2.The tail of the EGIKw-G distribution becomes heavier asKum increases, provided that Wx,α,β,γ,  andλ remain unchanged.Note that the EGIKw-G class of models outlined above reduces to generalized inverted Kumaraswamy generated (EGIKw-G) class proposed by Jamal et al. [18], for γ=1, and when γ=1,λ=1, the exponentiated-G class given by Cordeiro et al. [26] is obtained. Hence, parameter γ offers more flexibility to the extremes for the density function curves, and therefor new G class becomes more suitable for data sets which exhibit heavy tail. For every generated model, “W” and “w” represent baseline cdf and pdf, respectively.


## 3. Special Models

The EGIKw-G density function (4) offers high flexibility in tails along with promoting variation in tail weights to extremes of specific model. In this section, we provide four of many possible submodels under EGIKw-G class offering a more better fit to the data. For brevity, in the remainder of this paper, we shall comment in detail on only four of the most impotent EGIKw-G distributions, namely, EGIKw-Normal, EGIKw-Fréchet, EGIKw-Uniform, and EGIKw-Burr XII distributions.


### 3.1. EGIKw-Normal Distribution

The EGIKw-Normalpdf is obtained from (5) for Wx=Φx−μ/σ and Wx=ϕx−μ/σ, so(10)fx=αβγλ1σϕx−μσΦx−μσλγ−1×1−Φx−μσλ−γ−1×1+Φx−μ/σλ1−Φx−μ/σλγ−α−1×1−1+Φx−μ/σλ1−Φx−μ/σλγ−αβ−1,and thecdf is(11)Fx=1−1+Φx−μ/σλ1−Φx−μ/σλγ−αβ,where x∈ℝ,μ∈ℝ, and σ>0; ϕ⋅ and Φ⋅, respectively, denote the standard normal pdf and cdf. The rvX in the above expression is EGIKw-N, e.g., X∼EGIKw−Nα,β,γ,λ,μ,σ2.  Forμ=0andσ=1, it reduces to standard EGIKw-N distribution. The pdf and hrf plots of EGIKw-N model are depicted in Figure 1. As given in Figure 1(b), the hrf gives increasing, inverted bathtub, or decreasing (reversed-J) shapes.Figure 1
Selectedfx and hx graphs of EGIKw-Normal model.
(a)(b)


### 3.2. EGIKw-Fréchet Distribution

The Fréchetcdf and pdf for x>rbin 0, δ>0, andφ>0areWx=exp−δx−φandwx=δφx−φ−1exp−δx−φ, respectively. Correspondingly, the EGIKw-Fréchet EGIKw−Fα,β,γ,λ,δ,ϕ is(12)fx=αβγλδφx−φ−1exp−δx−φexp−δx−φλγ−1×1−exp−δx−φλ−γ−1×1+exp−δx−φλ1−exp−δx−φλγ−α−1×1−1+exp−δx−φλ1−exp−δx−φλγ−αβ−1.Thecdf is(13)Fx=1−1+exp−δx−φλ1−exp−δx−φλγ−αβ,where x,α,β,γ,λ,δ,φ>0. For φ=1, we obtain the extended generalized inverted Kumaraswamy inverse exponential distribution. Figure 2(a) indicates that the EGIKw-Fréchet offers various interesting shapes. Figure 2(b) reveals that the model can also offer various hrf shapes including decreasing, increasing, J, revered-J, and bathtub shapes.Figure 2
Selectedfx and hx graphs for EGIKw-Fréchet model.
(a)(b)


### 3.3. EGIKw-Uniform Distribution

The EGIKw-Updf is obtained from (5), taking Wx=x/θandwx=1/θ, where x∈0,θ, as follows:(14)fx=αβγλ1σϕx−μσxθλγ−1×1−xθλ−γ−11+x/θλ1−x/θλγ−α−1×1−1+x/θλ1−x/θλγ−αβ−1.Thecdf is(15)Fx=1−1+x/θλ1−x/θλγ−αβ.Arv, say X, with above model is given as X∼EGIKw−Uα,β,γ,λ,θ. For θ=1, we have standard EGIKw-Uniform model. Figure 3 illustrates shapes of pdf and hrf for the EGIKw-Uniform model. The pdf plot in Figure 3(a) offers a variety of shapes. Moreover, it is obvious from Figure 3(b) that this model can accommodate constant, decreasing, and unimodal hrf.Figure 3
Selectedfx and hx graphs for EGIKw-Uniform model.
(a)(b)


### 3.4. EGIKw-Burr XII Distribution

The Burr XIIpdf and cdf are wx=ψξxξ−11+xξ−ψ−1 and Wx=1−1+xξ−ψ, respectively. Hence, the EGIKw-Burr XII pdf is(16)fx=αβγλψξxξ−11+xξ−ψ−11−1+xξ−ψλγ−1×1−1−1+xξ−ψλ−γ−1×1+1−1+xξ−ψλ1−1−1+xξ−ψλγ−α−1×1−1+1−1+xξ−ψλ1−1−1+xξ−ψλγ−αβ−1.The correspondingcdf takes the following form:(17)Fx=1−1+1−1+xξ−ψλ1−1−1+xξ−ψλγ−αβ.ArvX with the above pdf is denoted as X∼EGIKw−BurrXIIα,β,γ,λ,ψ,ξ. Figure 4 displays some interesting shapes of EGIKw-Burr XII pdf and hrf. It is obvious from these plots that great flexibility is achieved with the proposed models.Figure 4
Selectedfx and hx graphs for EGIKw-Burr XII model.
(a)(b)


## 3.1. EGIKw-Normal Distribution

The EGIKw-Normalpdf is obtained from (5) for Wx=Φx−μ/σ and Wx=ϕx−μ/σ, so(10)fx=αβγλ1σϕx−μσΦx−μσλγ−1×1−Φx−μσλ−γ−1×1+Φx−μ/σλ1−Φx−μ/σλγ−α−1×1−1+Φx−μ/σλ1−Φx−μ/σλγ−αβ−1,and thecdf is(11)Fx=1−1+Φx−μ/σλ1−Φx−μ/σλγ−αβ,where x∈ℝ,μ∈ℝ, and σ>0; ϕ⋅ and Φ⋅, respectively, denote the standard normal pdf and cdf. The rvX in the above expression is EGIKw-N, e.g., X∼EGIKw−Nα,β,γ,λ,μ,σ2.  Forμ=0andσ=1, it reduces to standard EGIKw-N distribution. The pdf and hrf plots of EGIKw-N model are depicted in Figure 1. As given in Figure 1(b), the hrf gives increasing, inverted bathtub, or decreasing (reversed-J) shapes.Figure 1
Selectedfx and hx graphs of EGIKw-Normal model.
(a)(b)


## 3.2. EGIKw-Fréchet Distribution

The Fréchetcdf and pdf for x>rbin 0, δ>0, andφ>0areWx=exp−δx−φandwx=δφx−φ−1exp−δx−φ, respectively. Correspondingly, the EGIKw-Fréchet EGIKw−Fα,β,γ,λ,δ,ϕ is(12)fx=αβγλδφx−φ−1exp−δx−φexp−δx−φλγ−1×1−exp−δx−φλ−γ−1×1+exp−δx−φλ1−exp−δx−φλγ−α−1×1−1+exp−δx−φλ1−exp−δx−φλγ−αβ−1.Thecdf is(13)Fx=1−1+exp−δx−φλ1−exp−δx−φλγ−αβ,where x,α,β,γ,λ,δ,φ>0. For φ=1, we obtain the extended generalized inverted Kumaraswamy inverse exponential distribution. Figure 2(a) indicates that the EGIKw-Fréchet offers various interesting shapes. Figure 2(b) reveals that the model can also offer various hrf shapes including decreasing, increasing, J, revered-J, and bathtub shapes.Figure 2
Selectedfx and hx graphs for EGIKw-Fréchet model.
(a)(b)


## 3.3. EGIKw-Uniform Distribution

The EGIKw-Updf is obtained from (5), taking Wx=x/θandwx=1/θ, where x∈0,θ, as follows:(14)fx=αβγλ1σϕx−μσxθλγ−1×1−xθλ−γ−11+x/θλ1−x/θλγ−α−1×1−1+x/θλ1−x/θλγ−αβ−1.Thecdf is(15)Fx=1−1+x/θλ1−x/θλγ−αβ.Arv, say X, with above model is given as X∼EGIKw−Uα,β,γ,λ,θ. For θ=1, we have standard EGIKw-Uniform model. Figure 3 illustrates shapes of pdf and hrf for the EGIKw-Uniform model. The pdf plot in Figure 3(a) offers a variety of shapes. Moreover, it is obvious from Figure 3(b) that this model can accommodate constant, decreasing, and unimodal hrf.Figure 3
Selectedfx and hx graphs for EGIKw-Uniform model.
(a)(b)


## 3.4. EGIKw-Burr XII Distribution

The Burr XIIpdf and cdf are wx=ψξxξ−11+xξ−ψ−1 and Wx=1−1+xξ−ψ, respectively. Hence, the EGIKw-Burr XII pdf is(16)fx=αβγλψξxξ−11+xξ−ψ−11−1+xξ−ψλγ−1×1−1−1+xξ−ψλ−γ−1×1+1−1+xξ−ψλ1−1−1+xξ−ψλγ−α−1×1−1+1−1+xξ−ψλ1−1−1+xξ−ψλγ−αβ−1.The correspondingcdf takes the following form:(17)Fx=1−1+1−1+xξ−ψλ1−1−1+xξ−ψλγ−αβ.ArvX with the above pdf is denoted as X∼EGIKw−BurrXIIα,β,γ,λ,ψ,ξ. Figure 4 displays some interesting shapes of EGIKw-Burr XII pdf and hrf. It is obvious from these plots that great flexibility is achieved with the proposed models.Figure 4
Selectedfx and hx graphs for EGIKw-Burr XII model.
(a)(b)


## 4. Structural Properties of EGIKw-G Family of Distributions

In this part of article, we provide some useful expressions for EGIKw-G class including explicit expansions of density and cumulative distribution function,rth moment, md, moment generating function mgf, and pdf of order statistics.


### 4.1. Expansions for EGIKw-Gcdf and pdf

We express EGIKw-Gcdf and pdf in terms of finite (or infinite) weighted sums of exponentiated-G cdf and pdf, respectively. Consider the EGIKw-G cdf given by (4)(18)FEGIKw−Gx=1−1+Wλx1−Wλxγ−α−1β.Ford>0 real noninteger and y<1, the power series representations are(19)1−yd=∑i=0∞di−1iyi,(20)1+y−d=∑i=0∞d+i−1i−1iyi.Ford>0 integer value,(21)1−yd=∑i=0∞di−1iyi.Using the series expansions given above, the EGIKw-G distribution function (4) is rewritten as(22)FEGIKw−Gx=∑i=0∞−1iβi1+Wλx1−Wλxγ−αi=∑i,j=0∞−1i+jβiαi+j−1jWλx1−Wλxγj=∑i,j,k=0∞−1i+jβiαi+j−1jγj+k−1kWxλγj+k=∑i,j,k=0∞ιi,j,kWxλγj+k,where ιi,j,k=−1i+jβiαi+j−1jγj+k−1k. For any integer value of β, index i is stopped at β, β; for an integer α, the index j stops at αi+j−1, αi+j−1; and for an integer value of γ, index k is stopped at γj+k−1, γj+k−1. Thus, (22) reveals that EGIKw-G pdf can be written in baseline pdf as a multiple of its cdf’s power series. Otherwise, in case of γ to be a real noninteger, the Wxλγj+k in (22) can have following form(23)Wxλγj+k=1−1−Wxλγj+k=∑l=0∞−1lλγj+kl1−Wxl.Using the binomial expansion for1−Wxl, we obtain(24)1−Wxl=∑r=0l−1rlrWxr.Using (24) into (23), we have(25)Wxλγj+k=∑l=0∞∑r=0l−1l+rλγj+kllrWxr.Further, (4) is rewritten as(26)FEGIKw-Gx=∑i,j,k,l=0∞∑r=0lti,j,k,l,rWxr,where ti,j,k,l,r=−1l+rλγj+kllrιi,j,k. Replacing ∑l=0∞∑r=0l by ∑r=0∞∑l=r∞ in (22), we have(27)FEGIKw-Gx=∑r=0∞zrWxr,where zr=∑i,j,k=0∞∑l=r∞ti,j,k,l,r is sum in constants. The expansion (27) holds for all real noninteger γ values. It should be noted that EGIKw-G cdf can also be provided in the form of exponential-G cdf as(28)FEGIKw-Gx=∑r=0∞zrVrx,where Vrx=Wxr denotes exponential-Gcdf, where r is power parameter. The corresponding results for EGIKw-G pdf are obtained by differentiating (22) for γ>0 integer and by (27) and (28) for γ>0 real noninteger value, respectively, as(29)fEGIKw-Gx=wx∑i,j,k=0∞li,j,k″Wxλγj+k−1,(30)fEGIKw-Gx=wx∑r=0∞z⌣rWxr,(31)fEGIKw−Gx=∑r=0∞zr″vr+1x,where li,j,k″=λγj+kιi,j,k,z⌣r=r+1zr+1,zr″=zr+1 for r=0,1,2,…, r=0,1,2,…; vr+1x=r+1wxWxr is exponential-G density function having parameter r+1. Equation (31) expresses EGIKw-G density in terms of exponential-G densities. Equations (29)–(31) are among main results from this section.


### 4.2. Moments

Moments play a crucial role in studying some important characteristics (tendency, dispersion, skewness, kurtosis, etc.) of a distribution. Thepth EGIKw-G moment can be given as weighted sum in probability weighted moments (PWMs) of order p,q of the parent distribution. Let X and Y, respectively, come from EGIKw-G and baseline G distribution. We can write pth raw moment for X in terms of p,qth PWM (τp,q=EYpGYq=∫xpwxWxqdx, q=0,1,…) of Y. For γ>0 integer, we have(32)μp′=EXp=∑i,j,k=0∞li,j,k″τp,λγj+k−1,where τp,λγj+k−1, is the p,λγj+k−1th PWM of baseline distribution and li,j,k″ is defined in (29). For γ>0 noninteger, we can write(33)μp′=EXp=∑r=0∞z⌣rτp,r,where z⌣r is from (30) and τp,r denotes p,rth PWM of baseline distribution. Hence, moments for any EGIKw-G model can be calculated using baseline PWMs.Furthermore,μp′ can be obtained using baseline qf, Qu=W−1u=x. For γ>0 integer, from (22), and for γ>0 noninteger, from (30), we, respectively, obtain(34)μp′=∑i,j,k=0∞li,j,k″∫xpwxWxλγj+k−1dx,μp′=∑r=0∞z⌣r∫xpwxWxrdx.Usingu=Wxu in the above expressions, we have(35)μp′=∑i,j,k=0∞li,j,k″∫01uλγj+k−1Qupdu,μp′=∑r=0∞z⌣r∫01urQupdu.respectively. Moreover, we can also provide the EGIKw-G moments in the form of exponential-G moments. Let Xr+1 be an exponential-Grv with cdf, Vr+1x=Wxr, and pdf, vr+1x=r+1wxWxr, and (r + 1) be the power parameter, so(36)EXr+1p=∫xpvr+1xdx.Hence, we have(37)μp′=∑r=0∞zr′′∫xpvr+1xdx,where zr″ is defined in (31). Thus, EGIKw-G moments can be written as function of baseline exponential-G moments.


### 4.3. Moment Generating Function

LetX∼ EGIKw-G α,β,γ,λ. We consider various expressions of mgf for X as(38)Mt=EexptX=E∑p=0∞Xpp!tp=∑p=0∞μp′p!tp,where μp′=EXp is the pth EGIKw-G noncentral moment. Another representation of Mt, when γ>0 integer, is derived from (29) as(39)Mt=∑i,j,k=0∞li,j,k″φt,λγj+k−1,where the function φt,λγj+k−1=∫exptxwxWxλγj+k−1dx is obtained using baseline qf as(40)φt,λγj+k−1=∫01uλγj+k−1exptQudu.Forγ>0 noninteger, using (30) we also have(41)Mt=∑r=0∞z⌣rφt,r,and the functionφt,r=∫exptxwxWxrdx is easily deduced from baseline qf as(42)φt,r=∫01urexptQudu.Another representation forMt for γ>0 noninteger is obtained from (31) as(43)Mt=∑r=0∞zr″Mr+1t,where Mr+1t is mgf of X∼ exponential-G(r+1)rv. Hence, Mt of any EGIKw-G model can be determined from the corresponding exponential-G mgf.


### 4.4. Mean Deviations

Themd of a population measures its amount of scattering. For a rvX having pdff(x) and cdfF(x), the md about mean and md about median are, respectively, written as δμX and δMX and are, respectively, given by(44)δμX=EX−μ1′=2μ1′Fμ1′−2Tμ1′,δMX=EX−M=μ1′−2TM,where μ1′ is the first ordinary moment, Fμ1′ is from (4), M is median obtained from (7) for u=1/2, and Tz=∫−∞zxfxdx represents 1st incomplete moment. Using parent qf, two additional expressions for Tx are derived. Firstly, when γ>0 integer,(45)Tz=∑i,j,k=0∞li,j,k″∫0Gzuλγj+k−1Qudu.For γ>0 real noninteger, we have(46)Tz=∑r=0∞z⌣r∫0GzurQudu,where li,j,k″,z⌣r are defined in (29) and (30), respectively. Another useful expression for Tz is obtained from exponential-G distribution as(47)Tz=∑r=0∞zr″∫−∞zxvr+1xdx,where zr″ is defined by (31).


### 4.5. Rényi Entropy

Entropies of anyrv, say X, are measures of diversity of uncertainty. These measures have been used in various fields including engineering, physics, and economics. Rényi entropy is the most popular measure of entropy and is given as (Rényi [27])(48)Iζx=11−ζlog∫−∞∞fζxdx,ζ>0,ζ≠1.Using (19)–(21) the pdffEGIKw−Gζx becomes(49)fEGIKw−Gζx=αβγλζwζx∑i,j,k=0∞l˜i,j,kWxλγj+k+ζγλ−1.Hence,(50)Iζx=11−ζlogαβγλζ∑i,j,k=0∞l˜i,j,k∫−∞∞Wxλγj+k+ζγλ−1wζxdx.Equivalently depending on the parentqf,(51)Iζx=11−ζlogαβγλζ∑i,j,k=0∞l˜i,j,k∫01uλγj+k+ζγλ−1wζ−1Qudx,where l˜i,j,k=−1i+jζβ−1iαi+ζα+1+j−1jγj+ζγ+1+k−1k. In this section, (50) and (51) are main results.


### 4.6. Stress-Strength Reliability

The reliability measure of industrial components has crucial role especially in engineering. The reliability of a product is the probability that it will do its intended job up to a specific time, given that it is operating under normal conditions. The component fails whenX2 (random stress) placed on it exceeds X1 (random strength), and for X1>X2 it will work satisfactorily. Thus, R=PX2<X1 measures the component’s reliability (Kotz et al. [28]). Let X1 and X2 be independent rv, rv; let X1 be an EGIKw-G rv with f1x, (5), and parameters α1,β1,γ1,λ1; and let X2 be a rv with cdfF2x, (4), and parameters α2,β2,γ2,λ2 with common baseline parametric space ϑ. Then, R is obtained as(52)R=∫f1xF2xdx=α1β1γ1λ1∫wx,ϑWγ1λ1−1x,ϑ1−Wλ1x,ϑ−γ1−1×1+Wλ1x,ϑ1−Wλ1x,ϑγ1−α1−11−1+Wλ1x,ϑ1−Wλ1x,ϑγ1−α1β1−1×1−1+Wλ2x,ϑ1−Wλ2x,ϑγ2−α2β2dx.Alternatively, with the change ofrv, X=Q1u,(53)R=∫01F2Q1udu=∫011−1+1+1−u1β1−1α1−1−1γ1λ2λ1−1−γ2−α2β2du,where Q1u is qf from (7) corresponding to f1x. Interestingly, we see that R is independent of W(x), the baseline distribution. Additionally, various different forms will be yielded by using linear expression. One form is derived for γ1,γ2>0 integers by using(54)f1x=wx∑t,u,v=0∞lt,u,v″Wxλ1γ1u+v−1,F2x=∑i,j,k=0∞ιi,j,kWxλ2γ2j+k,where lt,u,v=λ1γ1u+v−1t+uβ1tα1t+u−1uγ1u+v−1v, andιi,j,k=−1i+jβ2iα2i+j−1jγ2j+k−1k. Thus,(55)R=∑i,j,k,t,u,v=0∞ιi,j,klt,u,v″∫−∞∞wxWxλ1γ1u+v+λ2γ2j+k−1dx=∑i,j,k,t,u,v=0∞ιi,j,klt,u,v″λ1γ1u+v+λ2γ2j+k.Similar expressions can be obtained for the caseγ1,γ2>0 nonintegers. As usual, when α1=α2,β1=β2,γ1=γ2,λ1=λ2, i.e., corresponding to the identically distributed case, we have R=1/2.


### 4.7. LorenzLp and Bonferroni Bp Curves

The Lorenz curve forγ>0 integer, is given as follows:(56)Lp=EX≤xEX=1EX∫0xtftdt=1μ∑i,j,k=0∞lt,u,v″∫0xtWtλγj+k−1wtdt.Equivalently based on parentqf and in the form of exponential-G distribution, we have(57)Lp=1μ∑i,j,k=0∞li,j,k″∫0Wxuλγj+k−1Qudu,Lp=1μ∑i,j,k=0∞li,j,k″λγj+k∫0xtvλγj+ktdt,respectively. The corresponding expressions for Bonferroni curve are, respectively, given by (58)–(60) as(58)Bp=EX≤xFXEX=LXFX=1FXEX∫0xtftdt=1μFX∑i,j,k=0∞li,j,k″∫0xtWtλγj+k−1wtdt,(59)Bp=1μFX∑i,j,k=0∞li,j,k″∫0Wxuλγj+k−1Qudu,(60)Bp=1μFX∑i,j,k=0∞li,j,k″λγj+k∫0xtvλγj+ktdt.Similar expressions can be obtained using (30) for the case of γ>0 noninteger.


### 4.8. Moments of Residual Life Function

In reliability theory and life testing problems, residual life has an important role. Thenth moment is provided by(61)mnt=EX−tnX>t=1Rt∫t∞x−tnfxdx=1Rt∑a=0nna−tn−a∫t∞xafxdx.Similarly,nth residual moment of a rv having EGIKw-G distribution for γ>0 integer and for γ>0 noninteger is obtained by inserting pdf of (29) and (30) in the above expression, respectively, as(62)mnt=1Rt∑i,j,k=0∞∑a=0nli,j,k″na−tn−a∫t∞xaWxλγj+k−1wxdx,mnt=1Rt∑r=0∞∑a=0nz⌣rna−tn−a∫t∞xaWxrwxdx.Equivalently depending upon the parentqf, we have(63)mnt=1Rt∑i,j,k=0∞∑a=0nli,j,k″na−tn−a∫Gt1uλγj+k−1Quadu,mnt=1Rt∑r=0∞∑a=0nz⌣rna−tn−a∫Gt1urQuadu.An alternative representation can be derived from exponential-G distribution as(64)mnt=1Rt∑r=0∞∑a=0nlr″na−tn−a∫t∞xavr+1xdx.


### 4.9. Order Statistics

Order statistics are useful in detection of outliers and robust statistical estimation, characterization of probability distributions, reliability analysis, analysis of censored samples, etc. LetX1,X2,…,Xn be nrv from the EGIKw-G distribution. Let X1,X2,…,Xn denote the order statistics. The density of ith ordered value is(65)fi:nx=fxBi,n−i+1Fxi−11−Fxn−i=fxBi,n−i+1∑h=0n−i−1hn−ihFxh+i−1,where B.,. is expression for beta function. We offer the pdf of EGIKw-G order statistics in the form of baseline pdf as multiple of Wx. Replacing (27) in the above expression yields(66)Fxh+i−1=∑t=0∞ztWxth+i−1=∑t=0∞ztuth+i−1.Let us consider(67)∑t=0∞stytz=∑t=0∞ct,zyt,where c0,z=s0z,ct,z=ts0−1∑m=1tmz+1−tsmct−m,z (Gradshteyn and Ryzhik [1]). Hence, we have(68)Fxh+i−1=∑t=0∞ct,h+i−1Wxt=∑t=0∞ct,h+i−1ut,withc0,h+i−1=z0h+i−1,ct,h+i−1=tz0−1∑m=1tmh+i−tzmct−m,h+i−1. Using (68) in (65) with (29) for γ>0 integer and with (30) for γ>0 noninteger, we, respectively, obtain(69)fi:nx=wxBi,n−i+1∑l,j,k,t=0∞∑h=0n−ili,j,k″ct,h+i−1−1hn−ihWxλγj+k+t−1,fi:nx=wxBi,n−i+1∑r,t=0∞∑h=0n−iz⌣rct,h+i−1−1hn−ihWxr+t.Clearly, the above equations can be given in the form of exponential-G densities as(70)fi:nx=∑l,j,k,t=0∞∑h=0n−iwi,j,k″ct,h+i−1−1hn−ihBi,n−i+1λγj+k+tvλγj+k+tx.(71)fi:nx=∑r,t=0∞∑h=0n−iz⌣rct,h+i−1−1hn−ihBi,n−i+1r+t+1vr+t+1x.Equations (70) for γ>0 integer and (71) for γ>0 noninteger immediately yield the pdf of EGIKw-G order statistics as a function of exponential-G pdf,s. Hence, the corresponding moments can be provided in the form of baseline PWMs for γ>0 integer and for γ>0 noninteger, respectively, by(72)Ei:nxs=1Bi,n−i+1∑l,j,k,t=0∞∑h=0n−iwi,j,k″ct,h+i−1−1hn−ihτs,λγj+k+t−1,Ei:nxs=1Bi,n−i+1∑r,t=0∞∑h=0n−iz⌣rct,h+i−1−1hn−ihτs,r+t.Depending upon the parentqf for γ>0 integer and for γ>0 noninteger, we, respectively, obtain(73)Ei:nxs=∑l,j,k,t=0∞∑h=0n−iwi,j,k″ct,h+i−1−1hn−ihBi,n−i+1∫01uλγj+k+t−1Qusdu,Ei:nxs=1Bi,n−i+1∑r,t=0∞∑h=0n−iz⌣rct,h+i−1−1hn−ih∫01ur+tQusdu.Thus, themgf and other properties for EGIKw-G order statistics can also be obtained likewise.


## 4.1. Expansions for EGIKw-Gcdf and pdf

We express EGIKw-Gcdf and pdf in terms of finite (or infinite) weighted sums of exponentiated-G cdf and pdf, respectively. Consider the EGIKw-G cdf given by (4)(18)FEGIKw−Gx=1−1+Wλx1−Wλxγ−α−1β.Ford>0 real noninteger and y<1, the power series representations are(19)1−yd=∑i=0∞di−1iyi,(20)1+y−d=∑i=0∞d+i−1i−1iyi.Ford>0 integer value,(21)1−yd=∑i=0∞di−1iyi.Using the series expansions given above, the EGIKw-G distribution function (4) is rewritten as(22)FEGIKw−Gx=∑i=0∞−1iβi1+Wλx1−Wλxγ−αi=∑i,j=0∞−1i+jβiαi+j−1jWλx1−Wλxγj=∑i,j,k=0∞−1i+jβiαi+j−1jγj+k−1kWxλγj+k=∑i,j,k=0∞ιi,j,kWxλγj+k,where ιi,j,k=−1i+jβiαi+j−1jγj+k−1k. For any integer value of β, index i is stopped at β, β; for an integer α, the index j stops at αi+j−1, αi+j−1; and for an integer value of γ, index k is stopped at γj+k−1, γj+k−1. Thus, (22) reveals that EGIKw-G pdf can be written in baseline pdf as a multiple of its cdf’s power series. Otherwise, in case of γ to be a real noninteger, the Wxλγj+k in (22) can have following form(23)Wxλγj+k=1−1−Wxλγj+k=∑l=0∞−1lλγj+kl1−Wxl.Using the binomial expansion for1−Wxl, we obtain(24)1−Wxl=∑r=0l−1rlrWxr.Using (24) into (23), we have(25)Wxλγj+k=∑l=0∞∑r=0l−1l+rλγj+kllrWxr.Further, (4) is rewritten as(26)FEGIKw-Gx=∑i,j,k,l=0∞∑r=0lti,j,k,l,rWxr,where ti,j,k,l,r=−1l+rλγj+kllrιi,j,k. Replacing ∑l=0∞∑r=0l by ∑r=0∞∑l=r∞ in (22), we have(27)FEGIKw-Gx=∑r=0∞zrWxr,where zr=∑i,j,k=0∞∑l=r∞ti,j,k,l,r is sum in constants. The expansion (27) holds for all real noninteger γ values. It should be noted that EGIKw-G cdf can also be provided in the form of exponential-G cdf as(28)FEGIKw-Gx=∑r=0∞zrVrx,where Vrx=Wxr denotes exponential-Gcdf, where r is power parameter. The corresponding results for EGIKw-G pdf are obtained by differentiating (22) for γ>0 integer and by (27) and (28) for γ>0 real noninteger value, respectively, as(29)fEGIKw-Gx=wx∑i,j,k=0∞li,j,k″Wxλγj+k−1,(30)fEGIKw-Gx=wx∑r=0∞z⌣rWxr,(31)fEGIKw−Gx=∑r=0∞zr″vr+1x,where li,j,k″=λγj+kιi,j,k,z⌣r=r+1zr+1,zr″=zr+1 for r=0,1,2,…, r=0,1,2,…; vr+1x=r+1wxWxr is exponential-G density function having parameter r+1. Equation (31) expresses EGIKw-G density in terms of exponential-G densities. Equations (29)–(31) are among main results from this section.


## 4.2. Moments

Moments play a crucial role in studying some important characteristics (tendency, dispersion, skewness, kurtosis, etc.) of a distribution. Thepth EGIKw-G moment can be given as weighted sum in probability weighted moments (PWMs) of order p,q of the parent distribution. Let X and Y, respectively, come from EGIKw-G and baseline G distribution. We can write pth raw moment for X in terms of p,qth PWM (τp,q=EYpGYq=∫xpwxWxqdx, q=0,1,…) of Y. For γ>0 integer, we have(32)μp′=EXp=∑i,j,k=0∞li,j,k″τp,λγj+k−1,where τp,λγj+k−1, is the p,λγj+k−1th PWM of baseline distribution and li,j,k″ is defined in (29). For γ>0 noninteger, we can write(33)μp′=EXp=∑r=0∞z⌣rτp,r,where z⌣r is from (30) and τp,r denotes p,rth PWM of baseline distribution. Hence, moments for any EGIKw-G model can be calculated using baseline PWMs.Furthermore,μp′ can be obtained using baseline qf, Qu=W−1u=x. For γ>0 integer, from (22), and for γ>0 noninteger, from (30), we, respectively, obtain(34)μp′=∑i,j,k=0∞li,j,k″∫xpwxWxλγj+k−1dx,μp′=∑r=0∞z⌣r∫xpwxWxrdx.Usingu=Wxu in the above expressions, we have(35)μp′=∑i,j,k=0∞li,j,k″∫01uλγj+k−1Qupdu,μp′=∑r=0∞z⌣r∫01urQupdu.respectively. Moreover, we can also provide the EGIKw-G moments in the form of exponential-G moments. Let Xr+1 be an exponential-Grv with cdf, Vr+1x=Wxr, and pdf, vr+1x=r+1wxWxr, and (r + 1) be the power parameter, so(36)EXr+1p=∫xpvr+1xdx.Hence, we have(37)μp′=∑r=0∞zr′′∫xpvr+1xdx,where zr″ is defined in (31). Thus, EGIKw-G moments can be written as function of baseline exponential-G moments.


## 4.3. Moment Generating Function

LetX∼ EGIKw-G α,β,γ,λ. We consider various expressions of mgf for X as(38)Mt=EexptX=E∑p=0∞Xpp!tp=∑p=0∞μp′p!tp,where μp′=EXp is the pth EGIKw-G noncentral moment. Another representation of Mt, when γ>0 integer, is derived from (29) as(39)Mt=∑i,j,k=0∞li,j,k″φt,λγj+k−1,where the function φt,λγj+k−1=∫exptxwxWxλγj+k−1dx is obtained using baseline qf as(40)φt,λγj+k−1=∫01uλγj+k−1exptQudu.Forγ>0 noninteger, using (30) we also have(41)Mt=∑r=0∞z⌣rφt,r,and the functionφt,r=∫exptxwxWxrdx is easily deduced from baseline qf as(42)φt,r=∫01urexptQudu.Another representation forMt for γ>0 noninteger is obtained from (31) as(43)Mt=∑r=0∞zr″Mr+1t,where Mr+1t is mgf of X∼ exponential-G(r+1)rv. Hence, Mt of any EGIKw-G model can be determined from the corresponding exponential-G mgf.


## 4.4. Mean Deviations

Themd of a population measures its amount of scattering. For a rvX having pdff(x) and cdfF(x), the md about mean and md about median are, respectively, written as δμX and δMX and are, respectively, given by(44)δμX=EX−μ1′=2μ1′Fμ1′−2Tμ1′,δMX=EX−M=μ1′−2TM,where μ1′ is the first ordinary moment, Fμ1′ is from (4), M is median obtained from (7) for u=1/2, and Tz=∫−∞zxfxdx represents 1st incomplete moment. Using parent qf, two additional expressions for Tx are derived. Firstly, when γ>0 integer,(45)Tz=∑i,j,k=0∞li,j,k″∫0Gzuλγj+k−1Qudu.For γ>0 real noninteger, we have(46)Tz=∑r=0∞z⌣r∫0GzurQudu,where li,j,k″,z⌣r are defined in (29) and (30), respectively. Another useful expression for Tz is obtained from exponential-G distribution as(47)Tz=∑r=0∞zr″∫−∞zxvr+1xdx,where zr″ is defined by (31).


## 4.5. Rényi Entropy

Entropies of anyrv, say X, are measures of diversity of uncertainty. These measures have been used in various fields including engineering, physics, and economics. Rényi entropy is the most popular measure of entropy and is given as (Rényi [27])(48)Iζx=11−ζlog∫−∞∞fζxdx,ζ>0,ζ≠1.Using (19)–(21) the pdffEGIKw−Gζx becomes(49)fEGIKw−Gζx=αβγλζwζx∑i,j,k=0∞l˜i,j,kWxλγj+k+ζγλ−1.Hence,(50)Iζx=11−ζlogαβγλζ∑i,j,k=0∞l˜i,j,k∫−∞∞Wxλγj+k+ζγλ−1wζxdx.Equivalently depending on the parentqf,(51)Iζx=11−ζlogαβγλζ∑i,j,k=0∞l˜i,j,k∫01uλγj+k+ζγλ−1wζ−1Qudx,where l˜i,j,k=−1i+jζβ−1iαi+ζα+1+j−1jγj+ζγ+1+k−1k. In this section, (50) and (51) are main results.


## 4.6. Stress-Strength Reliability

The reliability measure of industrial components has crucial role especially in engineering. The reliability of a product is the probability that it will do its intended job up to a specific time, given that it is operating under normal conditions. The component fails whenX2 (random stress) placed on it exceeds X1 (random strength), and for X1>X2 it will work satisfactorily. Thus, R=PX2<X1 measures the component’s reliability (Kotz et al. [28]). Let X1 and X2 be independent rv, rv; let X1 be an EGIKw-G rv with f1x, (5), and parameters α1,β1,γ1,λ1; and let X2 be a rv with cdfF2x, (4), and parameters α2,β2,γ2,λ2 with common baseline parametric space ϑ. Then, R is obtained as(52)R=∫f1xF2xdx=α1β1γ1λ1∫wx,ϑWγ1λ1−1x,ϑ1−Wλ1x,ϑ−γ1−1×1+Wλ1x,ϑ1−Wλ1x,ϑγ1−α1−11−1+Wλ1x,ϑ1−Wλ1x,ϑγ1−α1β1−1×1−1+Wλ2x,ϑ1−Wλ2x,ϑγ2−α2β2dx.Alternatively, with the change ofrv, X=Q1u,(53)R=∫01F2Q1udu=∫011−1+1+1−u1β1−1α1−1−1γ1λ2λ1−1−γ2−α2β2du,where Q1u is qf from (7) corresponding to f1x. Interestingly, we see that R is independent of W(x), the baseline distribution. Additionally, various different forms will be yielded by using linear expression. One form is derived for γ1,γ2>0 integers by using(54)f1x=wx∑t,u,v=0∞lt,u,v″Wxλ1γ1u+v−1,F2x=∑i,j,k=0∞ιi,j,kWxλ2γ2j+k,where lt,u,v=λ1γ1u+v−1t+uβ1tα1t+u−1uγ1u+v−1v, andιi,j,k=−1i+jβ2iα2i+j−1jγ2j+k−1k. Thus,(55)R=∑i,j,k,t,u,v=0∞ιi,j,klt,u,v″∫−∞∞wxWxλ1γ1u+v+λ2γ2j+k−1dx=∑i,j,k,t,u,v=0∞ιi,j,klt,u,v″λ1γ1u+v+λ2γ2j+k.Similar expressions can be obtained for the caseγ1,γ2>0 nonintegers. As usual, when α1=α2,β1=β2,γ1=γ2,λ1=λ2, i.e., corresponding to the identically distributed case, we have R=1/2.


## 4.7. LorenzLp and Bonferroni Bp Curves

The Lorenz curve forγ>0 integer, is given as follows:(56)Lp=EX≤xEX=1EX∫0xtftdt=1μ∑i,j,k=0∞lt,u,v″∫0xtWtλγj+k−1wtdt.Equivalently based on parentqf and in the form of exponential-G distribution, we have(57)Lp=1μ∑i,j,k=0∞li,j,k″∫0Wxuλγj+k−1Qudu,Lp=1μ∑i,j,k=0∞li,j,k″λγj+k∫0xtvλγj+ktdt,respectively. The corresponding expressions for Bonferroni curve are, respectively, given by (58)–(60) as(58)Bp=EX≤xFXEX=LXFX=1FXEX∫0xtftdt=1μFX∑i,j,k=0∞li,j,k″∫0xtWtλγj+k−1wtdt,(59)Bp=1μFX∑i,j,k=0∞li,j,k″∫0Wxuλγj+k−1Qudu,(60)Bp=1μFX∑i,j,k=0∞li,j,k″λγj+k∫0xtvλγj+ktdt.Similar expressions can be obtained using (30) for the case of γ>0 noninteger.


## 4.8. Moments of Residual Life Function

In reliability theory and life testing problems, residual life has an important role. Thenth moment is provided by(61)mnt=EX−tnX>t=1Rt∫t∞x−tnfxdx=1Rt∑a=0nna−tn−a∫t∞xafxdx.Similarly,nth residual moment of a rv having EGIKw-G distribution for γ>0 integer and for γ>0 noninteger is obtained by inserting pdf of (29) and (30) in the above expression, respectively, as(62)mnt=1Rt∑i,j,k=0∞∑a=0nli,j,k″na−tn−a∫t∞xaWxλγj+k−1wxdx,mnt=1Rt∑r=0∞∑a=0nz⌣rna−tn−a∫t∞xaWxrwxdx.Equivalently depending upon the parentqf, we have(63)mnt=1Rt∑i,j,k=0∞∑a=0nli,j,k″na−tn−a∫Gt1uλγj+k−1Quadu,mnt=1Rt∑r=0∞∑a=0nz⌣rna−tn−a∫Gt1urQuadu.An alternative representation can be derived from exponential-G distribution as(64)mnt=1Rt∑r=0∞∑a=0nlr″na−tn−a∫t∞xavr+1xdx.


## 4.9. Order Statistics

Order statistics are useful in detection of outliers and robust statistical estimation, characterization of probability distributions, reliability analysis, analysis of censored samples, etc. LetX1,X2,…,Xn be nrv from the EGIKw-G distribution. Let X1,X2,…,Xn denote the order statistics. The density of ith ordered value is(65)fi:nx=fxBi,n−i+1Fxi−11−Fxn−i=fxBi,n−i+1∑h=0n−i−1hn−ihFxh+i−1,where B.,. is expression for beta function. We offer the pdf of EGIKw-G order statistics in the form of baseline pdf as multiple of Wx. Replacing (27) in the above expression yields(66)Fxh+i−1=∑t=0∞ztWxth+i−1=∑t=0∞ztuth+i−1.Let us consider(67)∑t=0∞stytz=∑t=0∞ct,zyt,where c0,z=s0z,ct,z=ts0−1∑m=1tmz+1−tsmct−m,z (Gradshteyn and Ryzhik [1]). Hence, we have(68)Fxh+i−1=∑t=0∞ct,h+i−1Wxt=∑t=0∞ct,h+i−1ut,withc0,h+i−1=z0h+i−1,ct,h+i−1=tz0−1∑m=1tmh+i−tzmct−m,h+i−1. Using (68) in (65) with (29) for γ>0 integer and with (30) for γ>0 noninteger, we, respectively, obtain(69)fi:nx=wxBi,n−i+1∑l,j,k,t=0∞∑h=0n−ili,j,k″ct,h+i−1−1hn−ihWxλγj+k+t−1,fi:nx=wxBi,n−i+1∑r,t=0∞∑h=0n−iz⌣rct,h+i−1−1hn−ihWxr+t.Clearly, the above equations can be given in the form of exponential-G densities as(70)fi:nx=∑l,j,k,t=0∞∑h=0n−iwi,j,k″ct,h+i−1−1hn−ihBi,n−i+1λγj+k+tvλγj+k+tx.(71)fi:nx=∑r,t=0∞∑h=0n−iz⌣rct,h+i−1−1hn−ihBi,n−i+1r+t+1vr+t+1x.Equations (70) for γ>0 integer and (71) for γ>0 noninteger immediately yield the pdf of EGIKw-G order statistics as a function of exponential-G pdf,s. Hence, the corresponding moments can be provided in the form of baseline PWMs for γ>0 integer and for γ>0 noninteger, respectively, by(72)Ei:nxs=1Bi,n−i+1∑l,j,k,t=0∞∑h=0n−iwi,j,k″ct,h+i−1−1hn−ihτs,λγj+k+t−1,Ei:nxs=1Bi,n−i+1∑r,t=0∞∑h=0n−iz⌣rct,h+i−1−1hn−ihτs,r+t.Depending upon the parentqf for γ>0 integer and for γ>0 noninteger, we, respectively, obtain(73)Ei:nxs=∑l,j,k,t=0∞∑h=0n−iwi,j,k″ct,h+i−1−1hn−ihBi,n−i+1∫01uλγj+k+t−1Qusdu,Ei:nxs=1Bi,n−i+1∑r,t=0∞∑h=0n−iz⌣rct,h+i−1−1hn−ih∫01ur+tQusdu.Thus, themgf and other properties for EGIKw-G order statistics can also be obtained likewise.


## 5. Estimation

We employMLE for estimating unknown parameters of EGIKw-G distribution. Let ϑ be p-dimensional baseline parametric vector. Consider rv ’s X1,X2…,Xn, with each Xi coming from a EGIKw-G α,β,γ,λ,ϑ′ model. The log-likelihood l=lΘ is obtained from (5) as follows:(74)lΘ=nlogαβγλ+∑i=1nlogwxi,ϑ+γλ−1∑i=1nlogWxi,ϑ−γ+1∑i=1nlog1−Wλxi,ϑ−α+1∑i=1nlog1+W−λxi,ϑ−1−γ+β−1∑i=1nlog1−1+W−λxi,ϑ−1−γ−α.The components ofU=Uα,Uβ,Uγ,Uλ,Uϑ′, the score vector, are(75)Uα=nα−∑i=1nlog1+W−λxi,ϑ−1−γ+β−1∑i=1nlog1+W−λxi,ϑ−1−γ1−1+W−λxi,ϑ−1−γα,Uβ=nβ+∑i=1nlog1−1+W−λxi,ϑ−1−γ−α,Uγ=nγ+λ∑i=1nlogWxi,ϑ−∑i=1nlog1−Wλxi,ϑ+α+1∑i=1nlogW−λxi,ϑ−11+W−λxi,ϑ−1γ+β−1α∑i=1nlogW−λxi,ϑ−11−1+W−λxi,ϑ−1−γα1+W−λxi,ϑ−1γ,Uλ=nλ+γ∑i=1nlogWxi,ϑ−γ+1∑i=1nlogWxi,ϑW−λxi,ϑ−1+α+1γ∑i=1nlogWxi,ϑW−λxi,ϑ−1Wλxi,ϑ1+W−λxi,ϑ−1γ+β−1αγ∑i=1nlogWxi,ϑW−λxi,ϑ−11+W−λxi,ϑ−1γ−1Wλxi,ϑ1−1+W−λxi,ϑ−1−γα,Uϑ=∑i=1nw′xi,ϑwxi,ϑ+γλ−1∑i=1nW′xi,ϑWxi,ϑ+λγ+1∑i=1nW′xi,ϑWxi,ϑW−λxi,ϑ−1−α+1γλ∑i=1nW′xi,ϑWxi,ϑ1−Wλxi,ϑ1+W−λxi,ϑ−1γ−β−1αγλ∑i=1nW′xi,ϑ1−W−λxi,ϑ−1−γ+1α−1Wxi,ϑ1−Wλxi,ϑ1+W−λxi,ϑ−1γ.By solvingUα=0,Uβ=0,Uγ=0,Uλ=0,andUϑ=0, we obtain the MLEs α^,β^,γ^,λ^,ϑ^.


## 6. Monte Carlo Simulation

In this part, we examined the usefulness of MLEs for EGIKw-Burr XII (a special model from the family) parameters, through an extensive numerical investigation. Average biasAB and root mean square error RMSE are considered to evaluate the performance of estimators for varying n, s. The qf given by (7) with Burr XII as baseline model was considered for generating EGIKw-Burr XII rv. The simulation was repeated 2,000 times for varying samples. Four different parametric values, I:α=1.3,β=0.1,γ=1.5,λ=0.2,ψ=1.25,ξ=5.0,II:α=0.2,β=0.5,γ=0.15,λ=0.25,ψ=3.3,ξ=2.5,III:α=2.0,β=0.6,γ=1.5,λ=1.75,ψ=0.45,ξ=4.0,IV:α=1.75,β=0.25,γ=1.5,λ=0.05,ψ=0.7,ξ=2.25, were considered. The MLEs,AB, and RMSE values for different n, s are presented in Tables 1 and 2. From the results, it is clear that as n increases, the RMSE for estimators on the average decreases. It is also observed that for all four sets, the AB showed decreasing pattern as n increases. Thus, MLE method performs quite well in parameter estimation of proposed G class.Table 1
Mean estimates, AB, and RMSEs of EGIKw-Burr XII distribution for some parameter values.
nParIIIMLERMSEABMLERMSEAB25α2.0313690.782508−0.731372.0500771.869672−1.85008β0.5060180.426535−0.406020.5239790.111397−0.02398γ1.5397140.101712−0.039711.5361921.391208−1.38619λ1.6794591.620032−1.479461.7215631.589122−1.47156ψ0.5139950.7496180.7360050.4954472.808282.804553Ξ5.5367252.076617−0.536725.6087493.802548−3.1087550α2.0032410.732436−0.703242.0275231.835541−1.82752Β0.49530.406724−0.39530.5118690.073648−0.01187γ1.5165710.060813−0.016571.5194741.370756−1.36947λ1.5534861.408353−1.353491.5986421.388082−1.34864ψ0.5146570.7454240.7353430.494162.8059312.80584ξ5.1406931.181488−0.140695.2303542.985746−2.73035100α1.9994470.71545−0.699452.0114971.815691−1.8115β0.4974150.403892−0.397410.5051930.051041−0.00519γ1.5080290.040566−0.008031.5089471.359594−1.35895λ1.5195431.347079−1.319541.5442231.312065−1.29422ψ0.510660.7460590.739340.4972262.802822.802774ξ5.0595940.784378−0.059595.063912.690264−2.56391200α2.0013630.710236−0.701362.0095531.811641−1.80955β0.4971360.400706−0.397140.502960.03563−0.00296γ1.5046340.029312−0.004631.5054971.35581−1.3555λ1.5131011.327808−1.31311.527031.285339−1.27703ψ0.505960.7478740.744040.4984052.8016182.801595ξ5.0001640.562188−0.000165.0256162.588476−2.52562Table 2
Mean estimates, AB, and RMSEs of EGIKw-Burr XII distribution for some parameter values.
nParIIIIVMLERMSEABMLERMSEAB25α2.0521530.253051−0.052152.0554450.391286−0.30544β0.5197550.1328990.0802460.5168160.286654−0.26682γ1.5387970.099868−0.03881.5398880.101188−0.03989λ1.7126940.5696390.0373061.7218581.76664−1.67186ψ0.4886550.051269−0.038650.4881230.2146350.211877ξ5.5425682.588892−1.542575.5122023.89718−3.262250α2.0149920.176476−0.014992.033340.332539−0.28334β0.5069350.1179080.0930650.5111190.27147−0.26112γ1.5155580.061392−0.015561.5209180.065155−0.02092λ1.5777390.3728160.1722611.6124661.599495−1.56247ψ0.4956120.051045−0.045610.4933620.207970.206638ξ5.1338471.675164−1.133855.2501123.252078−3.00011100α2.0176540.121994−0.017652.0098710.287532−0.25987β0.5037990.1087560.0962010.50490.259768−0.2549γ1.5106310.042348−0.010631.508190.042492−0.00819λ1.5556480.2897580.1943521.5421431.507662−1.49214ψ0.4964230.049195−0.046420.4974240.2032260.202576ξ5.0739911.347996−1.073995.0528912.916263−2.80289200α2.006750.085296−0.006752.0069980.27192−0.257β0.50250.1036310.0975010.5013140.25384−0.25131γ1.5049460.028846−0.004951.5048440.029993−0.00484λ1.5241990.2676190.2258011.525281.483044−1.47528ψ0.4981840.049478−0.048180.4983870.2019370.201613ξ5.0002751.141549−1.000284.9920792.797957−2.74208


## 7. Application

In this part of work, we use EGIKw-Burr XII distribution for cancer patients’ data to illustrate the merit of GIKw-Burr XII model compared to the generalized inverted Kumaraswamy (GIKw-Burr XII) by Jamal et al. (2019), the exponentiated Kumaraswamy Burr XII (EKwBXII) distributions by Paranaiba et al. (2013), the inverse Weibull Burr XII (IW-Burr XII) Model by Amal et al. (2018), the exponentiated Kumaraswamy Burr XII (EK-Burr XII) distribution by Silva et al. (2019), the exponentiated Weibull distribution (EWD) by Nassar and Fissa (2003), the generalized inverse Weibull distribution (GIWD) by De Gusmao et al. (2011), the modified extension of exponential (MEXED) distribution by El-Damcese and Ramadan (2015), and the well-known Burr XII distribution.For each considered model, we obtain the estimates using MLE method and adopt the minimum value of -log(likelihood) at MLE denoted by−l, Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion (CAIC), Hannan–Quinn Information Criterion (HQIC), Anderson-Darling A∗ statistics, Cramér–von Mises W∗ statistics, and Kolmogorov–Smirnov (K-S) tests. Data is about remission times of 128 bladder cancer patients in months from Lee and Wang (2003) and is provided as follows:6.94, 8.66, 0.08, 2.09, 3.48, 4.87, 13.11, 23.63, 0.20, 9.02, 13.29, 0.40, 2.23, 3.52, 4.98, 6.97, 2.26, 3.57, 5.06, 7.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 26.31, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 79.05, 4.18, 5.34, 7.59, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 1.26, 2.83, 4.33, 5.49, 7.66, 4.34, 5.71, 7.93, 11.79, 11.25, 17.14, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 1.40, 3.02, 18.10, 1.46, 4.40, 5.85, 8.26, 3.36, 6.93, 8.65, 11.98, 19.13, 3.36, 6.76, 12.07, 21.73, 2.07, 12.63, 22.69.The key statistics of data are offered in Table3. Furthermore, the TTT- transform curve is depicted by Figure 5, which suggests an upside down bathtub or unimodal failure rate and, therefore, indicates that the EGIKw-Burr XII distribution is suitable for fitting this data set.Table 3
Descriptive statistics.
nMin.Max.MeanVar.Sd.CVSkew.Kurt.1270.08079.0509.076100.49610.0251.10512.31921.451Figure 5
TTT-transform plot for cancer patients’ data set.Table4 gives MLEs and standard error (SE) (within parentheses) results. The computed goodness-of-fit gof results are provided in Table 5. Histograms with estimated pdf plot, cdf plot, QQ-plot, and PP-plot of the EGIKw-Burr XII and other distributions are provided in Figures 6–9, respectively. It is clear from these results that EGIKw-Burr XII model with six parameters offers a better fit than other distributions.Table 4
MLE and SE.
ModelEstimatesEGIKwBXII1.43162 (1.4027)0.68899 (0.50048)1.54554 (3.69725)11.05201 (31.96517)1.16295 (2.92883)0.14219 (0.55648)GIKwBII0.16048 (0.01133)11.76763 (2.09854)1.05116 (0.14219)13.84875 (0.01808)0.58321 (0.00246)-EKwBXII0.36633 (0.42895)0.97155 (1.67659)10.15582 (16.18606)44.13510 (61.61306)0.81470 (0.56507)-KwBXII0.31380 (0.35338)0.96498 (1.84057)9.81034 (17.57105)68.76241 (98.86293)--IWBXII0.97875 (0.24963)3.02715 (0.54086)91.44719 (53.15231)0.40185 (0.08382)--EWD0.68382 (0.13962)2.62405 (1.15772)0.27887 (0.14828)---GIWD1.22766 (90.56746)2.07078 (115.34209)0.75502 (0.04268)---MExXD0.33227 (0.04008)0.08451 (0.03658)0.09726 (0.06376)---BurrXII2.33681 (0.35412)0.23558 (0.04028)----Table 5
gof results.
Model−lAICCAICBICHQICW∗A∗K-SEGIKwBXII402.45120811.26850811.46360819.80100814.735200.017310.112230.03420 (0.9984)GIKwBII418.10950846.21900846.71480860.43990851.996700.355952.284010.09996 (0.1580)EKwBXII403.76400817.52800818.02380831.74890823.305800.042180.291850.04705 (0.9413)KwBXII403.69710815.39430815.72210826.77100820.016500.040930.283880.04576 (0.9531)IWBXII415.47480838.94960839.27740850.32630843.571800.274531.786370.10128 (0.1477)EWD403.52050813.04110813.23620821.57360816.507800.039610.266090.04446 (0.9633)GIWD437.75000881.49990881.69500890.03250884.966600.770334.699690.14576 (0.0090)MExXD402.63420816.90240817.60240833.96750823.835700.020900.139830.03537 (0.9981)BurrXII446.98390897.96780898.06460903.65620900.279000.770304.676680.25203 (0.0001)Figure 6
Thepdf of considered models.Figure 7
Thecdf of considered models.Figure 8
QQ-plots of considered distributions.Figure 9
PP-plots of considered distributions.


## 8. Conclusions

In this work, a four-parameter generated class of models, EGIKw-G class, is proposed. Submodels of the proposed class, namely, the EGIKw-Normal, EGIKw-Fréchet, EGIKw-Uniform, and the EGIKw-Burr XII distributions, are discussed. Various properties includingsf, hrf, rhrf, qf and median, sth raw moment, mgf, md, Rényi entropy, reliability parameter, Lorenz and Bonferroni curves, residual lifetime, and distribution of order statistics are presented. Particular attention is given to EGIKw-Burr XII distribution. A MCS is presented to investigate the performance of AB and RMSE of MLEs. A real application is provided to check the usefulness of EGIKw-G class and its performance compared to other well-known distributions. The gof measures used all revealed that the novel model performed better than its counterparts [29,30].



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