Alpha Power Transformed Weibull-G Family of Distributions: Theory and Applications

I. Elbatal; M. Elgarhy; B. M. Golam Kibria

doi:10.2991/jsta.d.210222.002

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Volume 20, Issue 2, June 2021, Pages 340 - 354

Alpha Power Transformed Weibull-G Family of Distributions: Theory and Applications

Authors

I. Elbatal¹, M. Elgarhy², B. M. Golam Kibria³^{, *}^,

¹Department of Mathematics and Statistics, College of Science Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, 11432, Saudi Arabia

²The Higher Institute of Commercial Sciences, Al mahalla Al kubra, 31951, Algarbia, Egypt

³Department of Mathematics and Statistics, Florida International University, Miami, FL, 33199, USA

^*Corresponding author. Email: kibriag@fiu.edu

Corresponding Author

B. M. Golam Kibria

Received 1 July 2020, Accepted 19 February 2021, Available Online 1 March 2021.

DOI: 10.2991/jsta.d.210222.002 How to use a DOI?
Keywords: Application; Exponential distribution; Family of distribution; Generating function; Lindley distribution; Maximum likelihood estimation; Weibull distribution; Alpha power transformed; Distributions; Moments; Weibull -G family
Abstract: This paper considers three special cases: Exponential, Rayleigh and Lindley of a family of generalized distributions, called alpha power Weibull G (APW-G) family. Some essential and valuable statistical properties of the family of distributions are obtained. The proposed distributions are very flexible and can be used to model data with decreasing, increasing or bathtub-shaped hazard rates. Model parameters estimation and a simulation study is carried out to evaluate the behaviors of the parameters. We consider two worthwhile data sets to illustrate the finding of the paper.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

There are many classical distributions that have been developed and discussed their statistical properties and applications for various discipline in past several years. However, most of them are not suitable to model the heavy-tailed data sets (Ahmad et al. [1]). As a results, for application purposes, it is required to have the extended forms of these distributions for various fields. So, it becomes great interest among researchers to generate new univariate distributions, by adding more shape parameters to a standard distribution. The generated new distributions are more flexible in modelling data in practice. Some famous generators families are, Exponentiated -Weibull–generator (Elgarhy et al. [2]), the Marshall–Olkin - G family by Marshal and Olkin [3], beta - G family by Eugene et al. [4], the generalized odd log logistic - G by Cordeiro et al. [5], the generalized transmuted- G by Nofal et al. [6], the odd Lindley- G family by Gomes et al. [7], a new extended alpha power transformed (APT)- G by Ahmad et al. [8], a new APT- G by Elbatal et al. [9], a new power Topp-Leone- G by Bantan et al. [10], type II general inverse exponential- G by Jamal et al. [11], Truncated Inverted Kumaraswamy- G by Bantan et al. [12], the exponentiated truncated inverse Weibull-G by Almarashi et al. [13], truncated Cauchy power- G by Aldahlan et al. [14] and type II power Topp-Leone- G by Bantan et al. [15], among others.

For an arbitrary baseline cumulative distribution function (cdf) Gx, Mahadavi and Kundu [16] suggested APT family of distributions by introducing an extra parameter α to a distribution G to get a more flexible family. The cdf and probability density function (pdf) of APT family are provided, respectively, by (A1) and (A2) in Appendix A.

Several properties of the APT family have been investigated by many authors. See (Dey et al. [17,18], Hassan et al. [19,20] and Ihtisham et al. [21]), etc.

Proposed the Weibull- GW−G family of distribution and defined the cdf and pdf of their Weibull- G family mentioned in (A3) and (A4) in Appendix.

The Weibull distribution is a well-known lifetime distribution and it has extensive applications in reliability theory, specifically, it is has been used for analyzing biological, medical, quality control and engineering data sets. However, when the hazard rates are bathtub, upside down bathtub or bimodal shapes, it does not work that well. As a result, in recent years, researchers are motivated to develop numerous generalizations and extensions of the Weibull distribution to model different kind of data. Therefore, the objective of this paper is to introduce a new extended generator called alpha power transformed Weibull-G (APTW−G) family based on combined the alpha power transformed generator with the Weibull-G family of distributions. It is expected that the proposed distribution will be more flexible and will perform better than some existing probability distributions to model life testing data. We will provide three submodels of this family by taking the baseline distributions as exponential, Rayleigh and Lindley, in the hope that they will perform better than some existing distributions. We will provide the maximum likelihood estimation (MLE) technique for the estimations of the parameters.

The organization of this paper as follows. We provide the new family called APTW−G family and its submodels in Section 2. We also demonstrate that the (APTW−G) density is given by a linear combination of exponentiated - G (exp - G) densities. Some useful statistical properties of APTW−G family are obtained in Section 3. The MLE technique will be discussed in Section 4. For illustration purposes, two real-life data are analyzed in Section 5. Finally, some concluding remarks are stated in Section 6.

2. PROPOSED FAMILY OF DISTRIBUTIONS

Here, will construct the new generator of APTW−G family of distributions based on the APT family and Weibull- G (W- G) family, by inserting (A3) into (A1). We have the cdf of new APTW−G family by

FAPTW−Gx;θ,α,δ=α1−e−tiθ−1α−1if α>0,α≠11−e−tiθif α=1(1)

The pdf of Equation (1) is given by

fAPTW−G(x;θ,α,δ)=log (α)θ g(x;δ)G(x;δ)θ−1(α−1)G¯(x;δ)θ+1e−tiθα1−e−tiθif α>0,α≠1θg(x;δ)G(x;δ)θ−1G¯(x;δ)θ+1e−tiθif α=1(2)

where ti=G(x;δ)G¯(x;δ). From now and onward, we will write simply Gx;δ=Gx,gx;δ=gx,FAPTW−G(x;θ,α,δ)=FAPTW−G(x) and fAPW−G(x;θ,α,δ)=fAPW−G(x). A random variable X with pdf (2) is denoted by X~APTW−Gθ,α,δ.

The survival function RAPW−G(x) hazard rate function hAPW−G(x) and cumulative hazard rate function HAPW−G(x) are given respectively as

RAPTW−Gx;θ,α,δ=α−α1−e−tiθα−1if α>0,α≠1e−tiθif α=1

hAPTW−G(x;θ,α,δ)=log (α)θg(x;δ)G(x;δ)θ−1α−α1−e−tiθG¯(x;δ)θ+1e−tiθα1−e−tiθif α>0,α≠1θg(x;δ)G(x;δ)θ−1G¯(x;δ)θ+1if α=1

and

HAPTW−Gx;θ,α,δ=−lnα−α1−e−tiθα−1if α>0,α≠1tiθif α=1.

2.1. Some Special of the APTW–G Models

We consider three different cases of the APTW−G family of distributions by taking the baseline distributions as exponential, Rayleigh and Lindley in this section. The cdf and pdf of the baseline models are given in Table 1.

Model	Cdf : G(x;δ)	pdf : g(x;δ)	G(x; δ)G¯(x; δ)
Exponential	1−e−βx	βe−βx	eβx−1
Rayleigh	1−e−β2x2	βxe−β2x2	eβ2x2−1
Lindley	1−1+ax1+ae−ax	a21+a(1+x)e−ax	eax1+ax1+a−1

Table 1

List of few special members of the alpha power Weibull-G (APTW-G) family distribution.

Alpha power transformed Weibull exponential (APTWE) distribution

FAPTWE(x)=α1−e−eβx−1θ−1α−1if α>0,α≠11−e−eβx−1θif α=1

The corresponding pdf is obtained as follows:

fAPWE(x)=log (α)θβe−βx1−e−βxθ−1(α−1)e−βxθ+1e−eβx−1θα1−e−eβx−1θif α>0,α≠1θβe−βx1−e−βxθ−1e−βxθ+1e−eβx−1θif α=1

Alpha power transformed Weibull Rayleigh (APTWR) distribution

FAPTWR(x)=α1−e−eρ2x2−1θ−1α−1if α>0,α≠11−e−eρ2x2−1θif α=1

The corresponding pdf is given by

fAPTWR(x)=log (α)θρxe−ρ2x21−e−ρ2x2θ−1(α−1)e−ρ2x2θ+1e−eρ2x2−1θα1−e−eρ2x2−1θif α>0,α≠1θρxe−ρ2x21−e−ρ2x2θ−1e−ρ2x2θ+1e−eρ2x2−1θif α=1

Alpha-power transformed Weibull Lindley (APTWL) distribution

FAPTWL(x)=α1−e−eax1+ax1+a−1θ−1α−1if α>0,α≠11−e−eax1+ax1+a−1θif α=1

The corresponding pdf is obtained as follows:

fAPTWL(x)=log (α)θa21+a(1+x)e−ax1−1+ax1+ae−axθ−1(α−1)1+ax1+ae−axθ+1e−tiθα1−e−eax1+ax1+a−1θif α>0,α≠1θa21+a(1+x)e−ax1−1+ax1+ae−axθ−11+ax1+ae−axθ+1e−eax1+ax1+a−1θif α=1

The pdf (left panel) and hazard rate function (right panel) for different set of parameters, of APTWE, APTWR and APTWL are presented in Figures 1–3 respectively.

From Figures 1–3, we observed that the pdf of APTWE, APTWR and APTWL distributions can be decreasing, symmetry, unimodal and skewed to the right, depend on the values of the parameters. Also, the hazard rate function (HRF) of APTWE, APTWR and APTWL distributions can be increasing, j- shaped, U- shaped and decreasing depend on the values of the parameters.

2.2. Expansion of the pdf of APTW–G Family

We will derive the density expansion of APTW-G family of distributions in this section. Using the power series expansion

αZ=∑k=0∞(log α)kk!Zk,α>0,α≠1(3)

the APTW−G family density can be given as

fAPTW−Gx=∑i=0∞(log α)i+1θgx;δG(x;δ)θ−1i !α−1G¯(x;δ)θ+1e−Gx;δG¯x;δθ1−e−Gx;δG¯x;δθi(4)

but

1−e−Gx;δG¯x;δθi=∑j=0∞(−1)jijexp−jGx;δG¯x;δθ,(5)

applying (5) to the last term in (4) gives

fAPTW−Gx=∑i,j=0∞(−1)j(log α)i+1ijθgx;δG(x;δ)θ−1i !α−1G¯(x;δ)θ+1e−(j+1)Gx;δG¯x;δθ(6)

Using the power series of the exponential function, we obtain

exp−j+1Gx;δG¯x;δθ=∑k=0∞(−1)kj+1kk!Gx;δG¯x;δθk

inserting this expansion in (4) we get

fAPTW−Gx=∑i,j,k=0∞(−1)j+k(log α)i+1j+1kijθi ! k !α−1×gx;δG(x;δ)θk+1−1G¯(x;δ)θk+1+1,(7)

Furthermore, using the general binomial expansion

(1−z)−a=∑m=0∞(a)mm!zm(8)

where (a)m=Γa+mΓa=aa+1…a+m−1 is the ascending factorial, | z |<1 and Γ. is a gamma function, we can write

1−Gx;δ−θk+1+1=∑d=0∞θk+1+1dd!G(x;δ)d

so, the pdf of APTW−G can be expressed as an infinite linear combination of exp-G density functions as follows:

fAPTW−Gx=∑k,d=0∞ϖk,d πk+1θ+dx(9)

where

ϖk,d=∑i,j=0∞(−1)j+klog αi+1θλj+1kijθk+1+1di ! k ! d !α−1d+θk+1,

and

πξx=ξgx;δG(x;δ)ξ−1

is the exp-G pdf with power parameter ξ. Equation (9) indicates that the density function of random variable X can be expressed as an infinite mixture of the exp-G densities with parameter k+1θ+d, therefore the APTW−G density can be considered as a linear combination of the exp-G densities. Thus, a lot of interesting statistical and mathematical properties of the APTW−G family of distributions come directly from those of exp-G distribution. Likewise, the cdf of the APTW−G family of distributions can be written as a linear combination of exp-G cdfs which is given by

FAPTW−Gx=∑k,d=0∞ϖk,d∏k+1θ+dx

where Πk+1θ+dx is the exp-G cdf with power parameter k+1θ+d.

3. SOME STATISTICAL PROPERTIES

Most of the formulae or mathematical expressions of this paper can easily be derived using Mathematica and Maple because of their ability to deal with complex expressions.

3.1. Quantile Function

The APTW−G quantile function (QF), say x=Qu can be obtained by inverting (1), we have

x=Qu=G−1−log 1−log (1 + uα − 1log (α)1θ1+−log 1−log (1 + uα − 1log (α)1θ.(10)

One can generates X easily by taking u as a uniform random variable in 0,1.

3.2. Moments

The rth moment of X can be obtained from Equation (9) as follows:

μr/=EXr=∫0∞xrfAPTW−Gxdx=∑k,d=0∞ϖk,dEZk+1θ+dr(11)

where EZk+1θ+dr denotes the rth moment of Zk+1θ+d which follows the exp - G random variable with power parameter k+1θ+d. Another formula for the rth moment follows from (9) as:

μr/=EXr=∑k,d=0∞ϖk,dEZk+1θ+dr

where EZξr=ξ∫−∞∞xrgx;δG(x;δ)ξ−1,ξ>0 can be obtained using numerical computation in terms of the baseline QF, i.e., QGu=G−1u as EZξr=ξ∫01uξ−1QG(u)rdu. Also the nth central moment of X, say Λn is given by

Λn=EX−μ1/n=∑r=0nnr−μ1/n−rEXr =∑r=0n∑k,d=0∞nr−μ1/n−rϖk,dEZk+1θ+dr.

3.3. Moment Generating Function

We display two different expressions for the moment generating function. The first one can be calculated using (9) as follows:

MXt=EetX=∑k,d=0∞ϖk,dMk+1θ+dt,(12)

where Mk+1θ+dt is the moment generating function of Zk+1θ+d. Consequently, we can be easily determined MXt from the exp- G generating function. The second expression for the MXt follows from (9) as:

MXt=EetX=∑k,d=0∞ϖk,dεt,k+1θ+d

where

εpt=p∫01up−1etQGudu

which will be computed numerically from the baseline QF, i.e., QGu=G−1u.

3.4. Conditional Moments

Here, we will discuss about the conditional moments, which are important for prediction in lifetime models. The sth incomplete moments of X defined by ϑst for any real s>0 can be expressed from (9) as

ϑst=∫−∞txsfxdx=∑k,d=0∞ϖk,d∫−∞txsπk+1θ+dxdx(13)

where the integral in (13) represents the sth incomplete moments of Zk+1θ+d.

3.5. Mean Deviations, Bonferroni and Lorenz Curves

The mean deviations about the mean μ=EX and the mean deviations about the median M are defined respectively by

δ1x=E|X−μ1/|=2μ1/Fμ1/−2ϑ1μ1/(14)

and

δ2x=E|X−M|=μ1/−2ϑ1M(15)

where μ1/=EX,M=medianX=Q12, Fμ1/ is easily computed from (1) and ϑ1t is the first complete moment given by (19) with s=1. Now we provide two ways to determine δ1x and δ2x. First, using Equation (9), a general equation for ϑ1t can be obtained as follows:

ϑ1t=∑k,d=0∞ϖk,dΩk+1θ+dt

where Ωk+1θ+dt=∫−∞txπk+1θ+dxdx is the first complete moment of the exp-G distribution. A second formula for ϑ1t is given by

ϑ1t=∑k,d=0∞ϖk,dΔk+1θ+dt

where

Δk+1θ+dt=k+1θ+d∫0GtQGuuk+1θ+ddu

can be calculated numerically.

4. MAXIMUM LIKELIHOOD ESTIMATION

Let x1,…,xn be a random sample of size n from the APTW−G distribution given by (2). Let ψ=(θ,α,δ))T be p×1 vector of parameters. The total log-likelihood function for ψ is given by

Ln=nloglog α−nlogα−1+nlogθ+∑i=1nlog gxi;δ+θ−1∑i=1nlog Gxi;δ −(θ+1)∑i=1nlog G¯xi;δ−∑i=1ntiθ+∑i=1n1−e−tiθlog α(16)

Using R-language, the log-likelihood can be by solving the nonlinear likelihood equations obtained by differentiating (16). The associated components of the score function Unψ=∂Ln∂θ,∂Ln∂α,∂Ln∂δT are

∂Ln∂θ=nθ+∑i=1nlog Gxi;δ−∑i=1nlog G¯xi;δ −∑i=1ntiθlog ti+log α∑i=1ntiθlog tie−tiθ(17)

∂Ln∂α=nαlog α−nα−1+1α∑i=1n1−e−tiθ(18)

and

∂Ln∂δk=∑i=1ng′xi;δgxi;δ+θ−1∑i=1nG′xi;δGxi;δ−θ+1∑i=1n1G¯xi;δ×∂G¯x;δ∂δk −θ∑i=1ntiθ−1∂ti∂δk+logαθ∑i=1ne−tiθtiθ−1∂ti∂δk(19)

where g′xi;δ=∂gxi;δ∂δk, G′xi;δ=∂Gxi;δ∂δk, and δk is the kth element of the vector of parameters δ. The MLE of ψ, say ψ^, is obtained by solving the nonlinear system Unψ=0. Statistical software can be used to solve numerically these equations via iterative methods.

To evaluate the performance of the Maximum Likelihood (ML) of APTWE, a Monte Carlo simulation is consider. The process is arranged as follows:

We generate random samples from APTWE model.
The number of Monte Carlo replications was made 1000 times each with sample sizes 30, 50 and 100.
Selected values for the parameters are choice as reported in Tables 2–4.
Formulas used for calculating mean square error (MSE), lower bound (LB), average length (AL) and upper bound (UB) of 90% and 95% are calculated.
Step (iii) is also repeated for the other parameters.

All calculations in this section getting by using via Mathematica 9.

The simulation results for different sample sizes (30, 50 and 100) and different values of parameters (α=0.5,β=0.5,θ=0.5), α=0.8,β=0.5,θ=0.5, and α=0.8,β=0.8,θ=0.5 are presented in Tables 2–4 respectively. These tables evidence the estimation consistency of the estimators.

			90%			95%
n	MLE	MSE	LB	UB	AL	LB	UB	AL
30	0.591	0.031	0.194	0.989	0.795	0.118	1.065	0.947
	0.627	0.222	−0.361	1.615	1.977	−0.551	1.805	2.355
	0.614	0.073	0.219	1.008	0.79	0.143	1.084	0.941
50	0.476	0.007	0.293	0.659	0.366	0.258	0.694	0.436
	0.434	0.057	−0.002	0.871	0.873	−0.086	0.954	1.04
	0.541	0.009	0.389	0.693	0.304	0.36	0.723	0.362
100	0.506	0.004	0.385	0.628	0.242	0.362	0.651	0.289
	0.572	0.043	0.232	0.911	0.679	0.167	0.976	0.809
	0.488	0.003	0.412	0.564	0.151	0.398	0.578	0.18

Table 2

Numerical results for APTWE model at (α=0.5,β=0.5,θ=0.5).

			90%			95%
n	MLE	MSE	LB	UB	AL	LB	UB	AL
30	1.2770	1.3630	−0.1010	2.6550	2.7570	−0.3650	2.9190	3.2850
	0.9560	1.1530	−0.5940	2.5070	3.1010	−0.8910	2.8040	3.6950
	0.4950	0.0050	0.3760	0.6150	0.2390	0.3530	0.6380	0.2850
50	0.8890	0.0770	0.5180	1.2610	0.7420	0.4470	1.3320	0.8850
	0.6290	0.1920	0.1060	1.1530	1.0470	0.0060	1.2530	1.2470
	0.5140	0.0040	0.4200	0.6090	0.1890	0.4010	0.6270	0.2250
100	0.7950	0.0060	0.5680	1.0230	0.4550	0.5240	1.0660	0.5420
	0.4970	0.0130	0.2070	0.7880	0.5810	0.1510	0.8440	0.6920
	0.5020	0.0010	0.4300	0.5740	0.1430	0.4170	0.5870	0.1710

Table 3

Numerical results for APTWE model at (α=0.8,β=0.5,θ=0.5).

			90%			95%
n	MLE	MSE	LB	UB	AL	LB	UB	AL
30	0.929	0.115	0.365	1.492	1.128	0.257	1.6	1.343
	1.012	0.298	−0.157	2.181	2.338	−0.381	2.405	2.786
	0.493	0.004	0.367	0.619	0.251	0.343	0.643	0.3
50	0.894	0.057	0.539	1.249	0.71	0.471	1.317	0.846
	0.933	0.135	0.243	1.823	1.58	0.092	1.974	1.882
	0.486	0.002	0.402	0.571	0.17	0.385	0.588	0.202
100	0.849	0.026	0.603	1.096	0.493	0.555	1.143	0.588
	0.759	0.085	0.321	1.197	0.877	0.237	1.281	1.044
	0.509	0.001	0.452	0.605	0.153	0.438	0.619	0.182

Table 4

Numerical results for APTWE model at (α=0.8,β=0.8,θ=0.5).

5. APPLICATIONS

For illustration purposes of the APTWE model, two real data sets are analyzed in this section. The first data was consider by Linhart and Zucchini [22], which signifies the failure times of air-conditioned system of an airplane. The second data set was consider by Aarset [23], which represents the failure times of 50 devices. All data sets are presented in Table 5. We fit the APTWE distribution and other four competing models namely; alpha power transformed Weibull inverse Lomax (APTIL) (ZeinEldein et al. [24]), alpha power transformed Lindley (APTL) (Dey et al. [17]), alpha power transformed exponential (APTE) (Mahadavi and Kundu [16]) distributions. The probability densities of these distributions is given by:

fAPTEx=γlog αe−γxα1−e−γxα−1,x,α,γ>0,α≠1,

fAPILx=ablog αx−21+bx−a−1α1+bx−aα−1,x,α,a,b>0,α≠1,

and

fAPTLx=log αα−1βγ21+xe−γxα1−e−γx1+γxγ+1γ+1,x,α,γ>0,α≠1.

Data 1	23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95
Data 2	0.1, 0.2, 1, 1, 1, 1, 1, 2, 3, 6, 7, 11, 12, 18, 18, 18, 18, 18, 21, 32, 36, 40, 45, 46, 47, 50, 55, 60, 63, 63, 67, 67, 67, 67, 72, 75, 79, 82, 82, 83, 84, 84, 84, 85, 85, 85, 85, 85, 86, 86

Table 5

Failures time data.

Figures 4 and 5 demonstrate the box and TTT plots for the two datasets respectively. Aarset (1987) proposed that the hrf is decreasing or increasing if the graph of TTT plot is a convex or concave. The hrf is U-shaped (bathtub) if the TTT plot is firstly convex and then concave.

The ML estimates along with their standard error (SE) of the model parameters for the first and second sets are provided in Tables 6 and 7 respectively. In the same tables, the analytical measures including; minus log-likelihood (-log L) Kolmogorov–Smirnov (KS) test statistic, Akaike information criterion (AIC), corrected Akaike information criterion (CAIC), Bayesian information criterion (BIC) and Hannan–Quinn information criterion (HQIC) are presented.

Model	ML Estimates (SE)	−Log L	AIC	BIC	CAIC	HQIC	KS
APTWE	α^=0.0510.107 β^=4.941×10−30.00251 θ^=0.850.13	176.788	359.576	358.007	360.499	360.92	0.1497
APTIL	α^=39.286163.336 a^=1.7052.581 b^=3.54.478	177.573	361.146	359.578	361.668	362.491	0.2007
APTL	α^=0.10.104 γ^=0.0245.127×10−3	183.415	370.83	369.784	371.274	371.727	0.2803
APTE	α^=8.688×10−105.698×10−8 γ^=8.536×10−42.844×10−3	177.388	358.775	357.729	359.22	359.672	0.19989

APTWE, Alpha power transformed Weibull exponential; SE, standard error; −Log L, minus log-likelihood; APTIL, alpha power transformed Weibull inverse Lomax; APTL, alpha power transformed Lindley; APTE. alpha power transformed exponential; AIC, Akaike information Criterion; BIC, Bayesian information criterion; CAIC, corrected Akaike information criterion; HQIC, Hannan–Quinn information criterion; KS, Kolmogorov–Smirnov test statistic

Table 6

Analytical results of the APTWE model and other competing models for the first data set.

Model	ML Estimates (SE)	−Log L	AIC	BIC	CAIC	HQIC	KS
APTWE	α^=99.036100.712 β^=0.0780.03 θ^=0.1930.069	269.911	545.823	544.92	546.344	548.007	0.14042
APTIL	α^=8.8115.855 a^=0.5510.281 b^=23.45310.978	291.608	589.217	588.314	589.739	591.401	0.2677
APTL	α^=4.359×10−66.762×10−5 γ^=8.754×10−36.886×10−3	296.747	597.495	596.892	597.939	598.951	0.2186
APTE	α^=4.822×10−83.528×10−6 γ^=1.361×10−36.19×10−3	283.583	571.167	570.565	571.611	572.623	0.19311

APTWE, Alpha power transformed Weibull exponential; SE, standard error; −Log L, minus log-likelihood; APTIL, alpha power transformed Weibull inverse Lomax; APTL, alpha power transformed Lindley; APTE. alpha power transformed exponential; AIC, Akaike information Criterion; BIC, Bayesian information criterion; CAIC, corrected Akaike information criterion; HQIC, Hannan–Quinn information criterion; KS, Kolmogorov–Smirnov test statistic

Table 7

Analytical results of the APTWE model and other competing models for the second data set.

From Tables 6 and 7, it is evident that the proposed APWE distribution provides the overall best fit and therefore could be chosen as the more adequate model among APTWE, APTIL, APTL and APTE models. We also plotted epdf, ecdf, esf and PP plots in Figures 6 and 7 for first and second data sets respectively.

6. CONCLUSIONS

In this paper, we developed three special cases (Exponential, Rayleigh and Lindley) of the family of generalized distributions, called APW-G family. The density function of the proposed models can have numerous forms: increasing, decreasing, negatively-skewed, positively skewed or symmetrical depend on the values of parameters. Some of useful statistical properties, such as expansion of density function, moments, generating function, incomplete moments, mean deviation, Bonferroni and Lorenz curves are provided. We also discuss the method of maximum likelihood to estimate the model parameters. The performance of the new family of distributions is illustrated by means of a four real data sets. It is observed that the proposed distribution perform well than some competitive distributions. It is expected that this paper will be useful for the users or researchers, those are listed in Section 1.

CONFLICTS OF INTEREST

The authors declar no conflict of interest.

AUTHORS' CONTRIBUTIONS

All authors have equally contributed for the manuscript.

ACKNOWLEDGMENTS

Authors are grateful to anonymous referees for their valuable comments and suggestions, which improved the quality and presentation of the paper greatly. Author, B. M. Golam Kibria wants to dedicate this paper to his very favorite teacher, Late Prof. M Kabir, Department of Statistics, Jahangirnagar University, Bangladesh for his wisdom, constant inspiration during student life and affection that motivated him to achieve this present position.

APPENDIX

FAPTx;α,δ=αGx;δ−1α−1if α>0,α≠1Gx;δif α=1(A1)

and

fAPTx;α,δ=log(α)αGx;δgx;δα−1if α>0,α≠1gx;δif α=1(A2)

where δ is the parameters vector for the basline cdf Gx;δ.

FWGx;θ,δ=∫0Gx;δG¯x;δθtθ−1e−tθdt=1−exp−Gx;δG¯x;δθ(A3)

and

fWGx;θ,δ=θgx;δG(x;δ)θ−1G¯(x;δ)θ+1exp−Gx;δG¯x;δθ,(A4)

respectively, where gx;δ and Gx;δ are pdf and cdf of any baseline distribution depending on a vector of parameter δ.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 20 - 2
Pages: 340 - 354
Publication Date: 2021/03/01
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.210222.002 How to use a DOI?
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TY  - JOUR
AU  - I. Elbatal
AU  - M. Elgarhy
AU  - B. M. Golam Kibria
PY  - 2021
DA  - 2021/03/01
TI  - Alpha Power Transformed Weibull-G Family of Distributions: Theory and Applications
JO  - Journal of Statistical Theory and Applications
SP  - 340
EP  - 354
VL  - 20
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.210222.002
DO  - 10.2991/jsta.d.210222.002
ID  - Elbatal2021
ER  -

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