 # Journal of Statistical Theory and Applications

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, *, 1 Department of Mathematics and Statistics, College of Science Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, 11432, Saudi Arabia
2 The Higher Institute of Commercial Sciences, Al mahalla Al kubra, 31951, Algarbia, Egypt
3 Department 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
https://doi.org/10.2991/jsta.d.210222.002How 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.

Open Access

## 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. ). 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. ), the Marshall–Olkin - G family by Marshal and Olkin , beta - G family by Eugene et al. , the generalized odd log logistic - G by Cordeiro et al. , the generalized transmuted- G by Nofal et al. , the odd Lindley- G family by Gomes et al. , a new extended alpha power transformed (APT)- G by Ahmad et al. , a new APT- G by Elbatal et al. , a new power Topp-Leone- G by Bantan et al. , type II general inverse exponential- G by Jamal et al. , Truncated Inverted Kumaraswamy- G by Bantan et al. , the exponentiated truncated inverse Weibull-G by Almarashi et al. , truncated Cauchy power- G by Aldahlan et al.  and type II power Topp-Leone- G by Bantan et al. , among others.

For an arbitrary baseline cumulative distribution function (cdf) Gx, Mahadavi and Kundu  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. ), etc.

Proposed the Weibull- GWG 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 (APTWG) 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 APTWG family and its submodels in Section 2. We also demonstrate that the (APTWG) density is given by a linear combination of exponentiated - G (exp - G) densities. Some useful statistical properties of APTWG 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 APTWG 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 APTWG family by

FAPTWGx;θ,α,δ=α1etiθ1α1if α>0,α11etiθif α=1(1)

The pdf of Equation (1) is given by

fAPTWG(x;θ,α,δ)=log(α)θg(x;δ)G(x;δ)θ1(α1)G¯(x;δ)θ+1etiθα1etiθif α>0,α1θg(x;δ)G(x;δ)θ1G¯(x;δ)θ+1etiθif 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,FAPTWG(x;θ,α,δ)=FAPTWG(x) and fAPWG(x;θ,α,δ)=fAPWG(x). A random variable X with pdf (2) is denoted by X~APTWGθ,α,δ.

The survival function RAPWG(x) hazard rate function hAPWG(x) and cumulative hazard rate function HAPWG(x) are given respectively as

RAPTWGx;θ,α,δ=αα1etiθα1if α>0,α1etiθif α=1
hAPTWG(x;θ,α,δ)=log(α)θg(x;δ)G(x;δ)θ1αα1etiθG¯(x;δ)θ+1etiθα1etiθif α>0,α1θg(x;δ)G(x;δ)θ1G¯(x;δ)θ+1if α=1
and
HAPTWGx;θ,α,δ=lnαα1etiθα1if α>0,α1tiθif α=1.

## 2.1. Some Special of the APTW–G Models

We consider three different cases of the APTWG 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)=α1eeβx1θ1α1if α>0,α11eeβx1θif α=1

The corresponding pdf is obtained as follows:

fAPWE(x)=log(α)θβeβx1eβxθ1(α1)eβxθ+1eeβx1θα1eeβx1θif α>0,α1θβeβx1eβxθ1eβxθ+1eeβx1θif α=1

Alpha power transformed Weibull Rayleigh (APTWR) distribution

FAPTWR(x)=α1eeρ2x21θ1α1if α>0,α11eeρ2x21θif α=1

The corresponding pdf is given by

fAPTWR(x)=log(α)θρxeρ2x21eρ2x2θ1(α1)eρ2x2θ+1eeρ2x21θα1eeρ2x21θif α>0,α1θρxeρ2x21eρ2x2θ1eρ2x2θ+1eeρ2x21θif α=1

Alpha-power transformed Weibull Lindley (APTWL) distribution

FAPTWL(x)=α1eeax1+ax1+a1θ1α1if α>0,α11eeax1+ax1+a1θif α=1

The corresponding pdf is obtained as follows:

fAPTWL(x)=log(α)θa21+a(1+x)eax11+ax1+aeaxθ1(α1)1+ax1+aeaxθ+1etiθα1eeax1+ax1+a1θif α>0,α1θa21+a(1+x)eax11+ax1+aeaxθ11+ax1+aeaxθ+1eeax1+ax1+a1θif 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 13 respectively.

From Figures 13, 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(logα)kk!Zk,α>0,α1(3)
the APTWG family density can be given as
fAPTWGx=i=0(logα)i+1θgx;δG(x;δ)θ1i!α1G¯(x;δ)θ+1eGx;δG¯x;δθ1eGx;δG¯x;δθi(4)
but
1eGx;δG¯x;δθi=j=0(1)jijexpjGx;δG¯x;δθ,(5)
applying (5) to the last term in (4) gives
fAPTWGx=i,j=0(1)j(logα)i+1ijθgx;δG(x;δ)θ1i!α1G¯(x;δ)θ+1e(j+1)Gx;δG¯x;δθ(6)

Using the power series of the exponential function, we obtain

expj+1Gx;δG¯x;δθ=k=0(1)kj+1kk!Gx;δG¯x;δθk
inserting this expansion in (4) we get
fAPTWGx=i,j,k=0(1)j+k(logα)i+1j+1kijθi!k!α1×gx;δG(x;δ)θk+11G¯(x;δ)θk+1+1,(7)

Furthermore, using the general binomial expansion

(1z)a=m=0(a)mm!zm(8)
where (a)m=Γa+mΓa=aa+1a+m1 is the ascending factorial, |z|<1 and Γ. is a gamma function, we can write
1Gx;δθk+1+1=d=0θk+1+1dd!G(x;δ)d
so, the pdf of APTWG can be expressed as an infinite linear combination of exp-G density functions as follows:
fAPTWGx=k,d=0ϖk,dπk+1θ+dx(9)
where
ϖk,d=i,j=0(1)j+klogαi+1θλj+1kijθk+1+1di!k!d!α1d+θk+1,
and
πξx=ξgx;δG(x;δ)ξ1
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 APTWG density can be considered as a linear combination of the exp-G densities. Thus, a lot of interesting statistical and mathematical properties of the APTWG family of distributions come directly from those of exp-G distribution. Likewise, the cdf of the APTWG family of distributions can be written as a linear combination of exp-G cdfs which is given by
FAPTWGx=k,d=0ϖk,dk+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 APTWG quantile function (QF), say x=Qu can be obtained by inverting (1), we have

x=Qu=G1log1log(1+uα1log(α)1θ1+log1log(1+uα1log(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=0xrfAPTWGxdx=k,d=0ϖk,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=0ϖk,dEZk+1θ+dr
where EZξr=ξxrgx;δG(x;δ)ξ1,ξ>0 can be obtained using numerical computation in terms of the baseline QF, i.e., QGu=G1u as EZξr=ξ01uξ1QG(u)rdu. Also the nth central moment of X, say Λn is given by
Λn=EXμ1/n=r=0nnrμ1/nrEXr=r=0nk,d=0nrμ1/nrϖk,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=0ϖk,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=0ϖk,dεt,k+1θ+d
where
εpt=p01up1etQGudu
which will be computed numerically from the baseline QF, i.e., QGu=G1u.

## 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=0ϖk,dtxsπk+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|Xμ1/|=2μ1/Fμ1/2ϑ1μ1/(14)
and
δ2x=E|XM|=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=0ϖk,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=0ϖk,dΔk+1θ+dt
where
Δk+1θ+dt=k+1θ+d0GtQGuuk+1θ+ddu
can be calculated numerically.

## 4. MAXIMUM LIKELIHOOD ESTIMATION

Let x1,,xn be a random sample of size n from the APTWG 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=nloglogαnlogα1+nlogθ+i=1nloggxi;δ+θ1i=1nlogGxi;δ(θ+1)i=1nlogG¯xi;δi=1ntiθ+i=1n1etiθlogα(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=Lnθ,Lnα,LnδT are

Lnθ=nθ+i=1nlogGxi;δi=1nlogG¯xi;δi=1ntiθlogti+logαi=1ntiθlogtietiθ(17)
Lnα=nαlogαnα1+1αi=1n1etiθ(18)
and
Lnδk=i=1ngxi;δgxi;δ+θ1i=1nGxi;δGxi;δθ+1i=1n1G¯xi;δ×G¯x;δδkθi=1ntiθ1tiδk+logαθi=1netiθtiθ1tiδk(19)
where gxi;δ=gxi;δδk, Gxi;δ=Gxi;δδk, 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 24.

• 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 24 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 , which signifies the failure times of air-conditioned system of an airplane. The second data set was consider by Aarset , 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. ), alpha power transformed Lindley (APTL) (Dey et al. ), alpha power transformed exponential (APTE) (Mahadavi and Kundu ) distributions. The probability densities of these distributions is given by:

fAPTEx=γlogαeγxα1eγxα1,x,α,γ>0,α1,
fAPILx=ablogαx21+bxa1α1+bxaα1,x,α,a,b>0,α1,
and
fAPTLx=logαα1βγ21+xeγxα1eγx1+γxγ+1γ+1,x,α,γ>0,α1.

 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^=0.0510.107β^=4.941×1030.00251θ^=0.850.13 176.788 359.576 358.007 360.499 360.92 0.1497
APTIL α^=39.286163.336a^=1.7052.581b^=3.54.478 177.573 361.146 359.578 361.668 362.491 0.2007
APTL α^=0.10.104γ^=0.0245.127×103 183.415 370.83 369.784 371.274 371.727 0.2803
APTE α^=8.688×10105.698×108γ^=8.536×1042.844×103 177.388 358.775 357.729 359.22 359.672 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^=99.036100.712β^=0.0780.03θ^=0.1930.069 269.911 545.823 544.92 546.344 548.007 0.14042
APTIL α^=8.8115.855a^=0.5510.281b^=23.45310.978 291.608 589.217 588.314 589.739 591.401 0.2677
APTL α^=4.359×1066.762×105γ^=8.754×1036.886×103 296.747 597.495 596.892 597.939 598.951 0.2186
APTE α^=4.822×1083.528×106γ^=1.361×1036.19×103 283.583 571.167 570.565 571.611 572.623 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;δ1α1if α>0,α1Gx;δif α=1(A1)
and
fAPTx;α,δ=log(α)αGx;δgx;δα1if α>0,α1gx;δif α=1(A2)
where δ is the parameters vector for the basline cdf Gx;δ.
FWGx;θ,δ=0Gx;δG¯x;δθtθ1etθdt=1expGx;δG¯x;δθ(A3)
and
fWGx;θ,δ=θgx;δG(x;δ)θ1G¯(x;δ)θ+1expGx;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.