# On the Alpha Power Transformed Power Lindley Distribution

Authors: Amal S. Hassan; M. Elgarhy; Rokaya E. Mohamd; Sharifah Alrajhi
Journal: Journal of Probability and Statistics
 (2019)

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## Abstract

In this paper, we introduce a new generalization of the power Lindley distribution referred to asthe alpha power transformed power Lindley (APTPL). The APTPL model provides a better fit than the power Lindley distribution. It includes the alpha power transformed Lindley, power Lindley, Lindley, and gamma as special submodels. Various properties of the APTPL distribution including moments, incomplete moments, quantiles, entropy, and stochastic ordering are obtained. Maximum likelihood, maximum products of spacings, and ordinary and weighted least squares methods of estimation are utilized to obtain the estimators of the population parameters. Extensive numerical simulation is performed to examine and compare the performance of different estimates. Two important data sets are employed to show how the proposed model works in practice.


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## Body

## 1. Introduction

The one-parameter Lindley distribution was primarily proposed by Lindley [1] as an alternative model for data with a nonmonotone hazard rate shape. Ghitany et al. [2] provided the properties of the Lindley distribution under a careful mathematical treatment. They also displayed in a numerical application that the Lindley distribution provides a better model than the exponential distribution. A random variableY is said to follow the Lindley distribution with scale parameter θ > 0, if itsprobability density function (pdf) is given by (1)gy;θ=θ2θ+11+ye-θy;y,θ>0,which is two-component mixture of exponential (θ) and gamma (2,θ). The Lindley distribution has only increasing failure rate which has been identified as a major difficulty in lifetime analysis. To overcome this situation, many generalizations of the Lindley distribution have been introduced in literature. Nadarajah et al. [3] proposed a generalized Lindley distribution. Bakouch et al. [4] introduced theextended Lindley (EL) distribution. Shanker et al. [5] introduced the two-parameter Lindley distribution. Ghitany et al. [6] proposed thepower Lindley (PL) as an extension of Lindley distribution. Based on pdf (1) and by using the transformationX = Y1/β, the pdf of the PL distribution is given by(2)gx;θ,β=βθ2θ+1xβ-11+xβe-θxβ;x,θ,β>0.Thecumulative distribution function (cdf) corresponding to (2) is given by (3)Gx;θ,β=1-e-θxβ1+θxβθ+1;x,θ,β>0.Many other generalizations of Lindley distribution have been proposed by several authors; see for example, a new generalized Lindley distribution [7], transmuted quasi Lindley distribution [8], exponentiated power Lindley [9], transmuted generalized Lindley distribution [10], transmuted generalized quasi Lindley distribution [11], and transmuted Kumaraswamy quasi Lindley distribution [12].Thealpha power transformation (APT) is one of the procedures that make the distributions richer and flexible to model the real life data. The APT has been proposed by Mahdavi and Kundu [13] with the parameter α to incorporate skewness to the base distribution. The cdf of APT is specified by(4)
F
A
P
T
x
;
α
=
α
G
x
-
1
α
-
1
,
if
α
>
0
,
α
≠
1
.
G
x
,
if
α
=
1
.Mahdavi and Kundu [13] applied the APT method to the exponential distribution and they studied various properties of the alpha power exponential distribution. Based on APT method, some new distributions are considered like APT Weibull [14], APT generalized exponential [15], APT Lindley [16], APT extended exponential [17], and APT inverse Lindley [18].Following the similar idea of Mahdavi and Kundu [13], we introduce a new three-parameter distribution, the so-called APTPL distribution. The main purpose of the new model is that the additional parameter can give several desirable properties and more flexibility in the form of the hazard and density functions. The APTPL distribution contains a number of known lifetime submodels such as the APTLindley (APTL) distribution, PL distribution, Lindley distribution, and gamma distribution. Properties, estimation of the parameters and applications with real data, are considered.This paper is constructed as follows. We introduce and study the properties of APTPL in Sections2 and 3, respectively. In Section 4, we obtain themaximum likelihood (ML) estimator,least square (LS) estimator,weighted least squares (WLS) estimator, andmaximum product of spacing (MPS) estimator of APTPL distribution. In the same section, a simulation study is executed to investigate the effectiveness of the estimates. The analyses of two real data sets are employed in Section 5. The paper ends with concluding remarks.


## 2. The APTPL Model

Here, we now introduce the notion of APTPL distribution.Definition 1.
A random variableXis said to have a three-parameter APTPL distribution with the scale parameter θ>0 and shape parameters α,β >0, if its cdf is of the form(5)
F
A
P
T
P
L
x
;
φ
=
α
1
-
e
-
θ
x
β
1
+
θ
x
β
/
θ
+
1
-
1
α
-
1
,
if
α
>
0
,
α
≠
1
.
1
-
e
-
θ
x
β
1
+
θ
x
β
θ
+
1
,
if
α
=
1
,where φ=(α,θ,β) is a set of parameters. The pdf of the APTPL distribution is given by(6)
f
A
P
T
P
L
x
;
φ
=
log
⁡
α
α
-
1
β
θ
2
x
β
-
1
1
+
x
β
e
-
θ
x
β
α
1
-
e
-
θ
x
β
1
+
θ
x
β
/
θ
+
1
θ
+
1
,
if
α
>
0
,
α
≠
1
.
β
θ
2
θ
+
1
x
β
-
1
1
+
x
β
e
-
θ
x
β
,
if
α
=
1
.Also, the reliability function, say RAPTPL(x;φ) andhazard rate function (hrf), say hAPTPL(x;φ)(x;φ) ofX are given, respectively, as follows:(7)
R
A
P
T
P
L
x
;
φ
=
α
-
α
1
-
e
-
θ
x
β
1
+
θ
x
β
/
θ
+
1
α
-
1
,
if
α
>
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,
α
≠
1
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e
-
θ
x
β
1
+
θ
x
β
θ
+
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if
α
=
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,and(8)
h
A
P
T
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L
x
;
φ
=
β
θ
2
log
⁡
α
x
β
-
1
1
+
x
β
e
-
θ
x
β
α
1
-
e
-
θ
x
β
1
+
x
β
/
θ
+
1
θ
+
1
α
-
α
1
-
e
-
θ
x
β
1
+
x
β
/
θ
+
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α
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θ
2
x
β
-
1
1
+
x
β
θ
+
1
+
θ
x
β
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if
α
=
1
.A random variableXwith distribution (6) is denoted byX~ APTPL (φ). Submodels of the APTPL distribution are as follows:(i)
Forβ =1, the pdf (6) reduces to APTL [16].(ii)
Forα =1, the pdf (6) reduces to PL [6].(iii)
Forβ =1 and α=1, the pdf (6) reduces to Lindley [1].(iv)
Forβ =1 and α=1, the pdf (6) reduces to gamma (2, θ).
Some descriptive pdf and hrf plots ofXare illustrated Figures 1 and 2 for specific parameter choices of φ (see Figures 1 and 2).Figure 1
The density plots of APTPL distribution.Figure 2
The hrf plots of APTPL distribution.We conclude from Figure1 that the pdf of the APTPL distribution can be reversed J-shaped, uni-model, and right and left skewed for some values of φ. Also, we conclude that, for some values of φ, the hrf of APTPL distribution can be increasing, decreasing, and reversed J-shaped.


## 3. Statistical Properties

Here, some statistical properties of the APTPL distribution including quantile, moments, moment generating function, incomplete moments, Bonferroni and Lorenz curves, probability weighed moments, Rényi entropy, and stochastic ordering are derived.


### 3.1. Quantile Function

The quantile function, sayQ(u) =F−1(u),u∈ (0, 1), is obtained by inverting (5) as follows:(9)α1-e-θQuβ1+θQuβ/θ+1-1α-1=u,which yields (10)1+θQuβθ+1e-θQuβ=1-log⁡uα-1+1log⁡α.Multiply both sides by (1+θ)e-(1+θ), then we have the Lambert equation(11)-1+θ+θQuβe-1+θ+θQuβ=-1+θe-1+θ1-log⁡uα-1+1log⁡α.Hence, we have the negative LambertW function of the real argument(12)-1+θe-1+θ1-log⁡uα-1+1log⁡α,i.e.,(13)Qu=-1θ-1-1θW-1-1+θe-1+θ1-log⁡uα-1+1log⁡α1/β,whereu∈ (0, 1) andW−1(.) is the negative LambertW function. The first quantile (25%), median (50%), and third quantile (75%) for some choices values of the parameters are computed numerically (see Table 1).Table 1
First quantile, median, and third quantile for some values ofθ, α, and β.
θ α β = 0.3 β = 1 25% 50% 75% 25% 50% 75% 1 0.5 0.045 0.735 6.662 0.394 0.912 1.766 2 0.279 3.175 19.709 0.682 1.414 2.446 5 0.852 6.908 33.388 0.953 1.786 2.865 7 1.225 8.780 39.174 1.063 1.919 3.005 10 1.743 11.040 45.625 1.181 2.055 3.146  1.5 0.5 0.007 0.134 1.339 0.228 0.547 1.091 2 0.049 0.617 4.151 0.403 0.865 1.533 5 0.156 1.390 7.183 0.572 1.104 2.330 7 0.228 1.786 8.481 0.641 1.190 1.899 10 0.329 2.268 9.935 0.717 1.279 1.991  2 0.5 0.002 0.040 0.428 0.157 0.381 0.775 2 0.014 0.193 1.366 0.279 0.610 1.098 5 0.047 0.444 2.398 0.399 0.784 1.2999 7 0.069 0.575 2.843 0.449 0.847 1.368 10 0.101 0.735 3.343 0.503 0.912 1.436We detect form Table1 that as the value of α increases, for fixed values of β and θ, the value of percentage points increases. As the value of θ increases, for fixed values of α and β, the percentage value points decrease. Also, as the value of β increases, for fixed values of α and θ, the percentage value points decrease.


### 3.2. Moments and Related Measures

Thekth moment of a random variableX having the APTPL distribution is obtained as follows:(14)μk′=βθ2log⁡αα-1θ+1∫0∞xk+β-11+xβe-θxβα1-e-θxβ1+θxβ/θ+1dx.Since the power series can be written as (15)αd=∑i=0∞log⁡αii!di,hence (14) can be expressed as follows:(16)μk′=βθ2α-1θ+1∑i=0∞log⁡αi+1i!∫0∞xk+β-11+xβe-θxβ1-e-θxβ1+θxβθ+1idx.Using the binomial expansion, then(17)μk′=βα-1∑i=0∞∑j=0i∑m=0j-1jijjmlog⁡αi+1θm+2i!θ+1m+1∫0∞xk+β+βm-11+xβe-θj+1xβdx.Hence, the kth moment of the APTPL distribution takes the form(18)μk′=∑i=0∞∑j=0i∑m=0jNi,j,mΓk/β+m+1θj+1k/β+m+1+Γk/β+m+2θj+1k/β+m+2,k=1,2,...where (19)Ni,j,m=-1jijjmlog⁡αi+1θm+2i!θ+1m+1α-1.Particularly, the mean is as follows:(20)μ1′=EX=∑i=0∞∑j=0i∑m=0jNi,j,mΓ1/β+m+1θj+11/β+m+1+Γ1/β+m+2θj+11/β+m+2.Similarly, the kth central moment of a given random variable X can be defined as (21)μk=EX-μ1′k=∑i=0k-1ikiμ1′iμk-i′.Thecoefficient of skewness (CS) andcoefficient of kurtosis (CK) are defined by (22)CS=μ3μ23/2,andCK=μ4μ22.Thus, numerical values of the μ1′,μ2′,μ3′,μ4′,variance (σ2),coefficient of variation (CV), CS, and CK of the APTPL distribution for some certain values of parameters are obtained and recorded in Table 2.Table 2
Summary statistics of moments of the APTPL distribution.
μ k ′ α = 0.5 ,   θ = 1 ,   β = 0.5 α = 1.5 ,   θ = 1 ,   β = 1 α = 3 ,   θ = 2 ,   β = 2 α = 5 ,   θ = 2 ,   β = 2.5 α = 7 ,   θ = 2.5 ,   β = 3 μ 1 ′ 1.344 0.713 0.366 0.391 0.371 μ 2 ′ 22.379 1.994 0.369 0.393 0.342 μ 3 ′ 874.97 7.675 0.418 0.427 0.333 μ 4 ′ 6085 37.375 0.520 0.493 0.340 σ 2 20.574 1.486 0.234 0.240 0.204 CV 3.376 1.709 1.322 1.252 1.219 CS 8.461 2.284 0.980 0.726 0.596 CK 133.188 9.423 2.728 2.011 1.704Furthermore, the moment generating function of the APTPL distribution is given by(23)Mxt=∑k,i=0∞∑j=0i∑m=0jNi,j,mtkk!Γk/β+m+1θj+1k/β+m+1+Γk/β+m+2θj+1k/β+m+2.


### 3.3. Incomplete Moments

Thesth lower incomplete moment, say ηs(t), of the APTPL distribution is given by(24)ηst=βθ2log⁡αα-1θ+1∫0txs+β-11+xβe-θxβα1-e-θxβ1+θxβ/θ+1dx.Using the power expansion (15) and binomial expansion, then (24) will be (25)ηst=βα-1∑i=0∞∑j=0i∑m=0j-1jijjmlog⁡αi+1θm+2i!θ+1m+1∫0txs+β+βm-11+xβe-θj+1xβdx.So, thesth lower incomplete moment of the APTPL distribution is (26)ηst=∑i=0∞∑j=0i∑m=0jNi,j,mγs/β+m+1,θj+1tβθj+1s/β+m+1+γs/β+m+2,θj+1tβθj+1s/β+m+2,where γ(.,.), is the lower incomplete moments. The first incomplete moment of the APTPL distribution can be obtained by settings =1 in (26). The first incomplete moment is related to the Bonferroni and Lorenz curves, the mean residual, and mean waiting times. The Bonferroni and Lorenz curves are important in economics, reliability, demography, insurance, and medicine. The Lorenz curves, say LO(x), and Bonferroni curve, say BO(x), are defined by (27)LOx=η1xEX,andBOx=LOxFAPTPLx;φ.For the proposed model, they can be determined from (27) as (28)LOx=∑i=0∞∑j=0i∑m=0jNi.j.mγ1/β+m+1,θj+1xβθj+11/β+m+1+γ1/β+m+2,θj+1xβθj+11/β+m+2∑i=0∞∑j=0i∑m=0jNi,j,mΓ1/β+m+1θj+11/β+m+1+Γ1/β+m+2θj+11/β+m+2-1,and,(29)BOx=∑i=0∞∑j=0i∑m=0jNi,j,mγ1/β+m+1,θj+1xβθj+11/β+m+1+γ1/β+m+2,θj+1xβθj+11/β+m+2FAPTPLx;φ∑i=0∞∑j=0i∑m=0jNi,j,mΓ1/β+m+1θj+11/β+m+1+Γ1/β+m+2θj+11/β+m+2-1.


### 3.4. The Probability-Weighted Moments

The class ofprobability-weighted moments (PWMs), denoted by πr,q, for a random variableX, is defined as follows:(30)πr,q=EXrFxq=∫-∞∞xrfxFxqdx.Substituting (5) and (6) in (30), then the PWM of the APTPL distribution is (31)πr,q=∫0∞xrlog⁡αα-1βθ2xβ-11+xβe-θxβα1-e-θxβ1+θxβ/θ+1θ+1α1-e-θxβ1+θxβ/θ+1-1α-1qdx.Using the binomial expansion, then (32)πr,q=βθ2log⁡αθ+1α-1q+1∑m=0q-1q-mqm∫0∞xrαq+11-e-θxβ1+θxβ/θ+11+xβe-θxβdx.Using (15) and binomial expansions, then (32) will be (33)πr,q=∑m=0q∑i=0∞∑j=0i∑l=0j-1q-m+jβθ2+llog⁡αi+1α-1q+1θ+1l+1ijjlqmq+1ii!∫0∞xr+β+βl-11+xβe-θj+1xβdx.Hence, the PWM of the APTPL distribution is given by(34)πr,q=∑m=0q∑i=0∞∑j=0i∑l=0jΞi,j,m,lΓr/β+l+1θj+1r/β+l+1+Γr/β+l+2θj+1r/β+l+2,where (35)Ξi,j,m,l=-1q-m+jjlqmijq+1ii!θ2+llog⁡αi+1θ+1l+1α-1q+1.


### 3.5. Rényi Entropy

The Rényi entropy of a random variable represents a measure of variation of the uncertainty. The Rényi entropy is defined by(36)τRx=1-ε-1log∫0∞fxεdx.Substituting (6) in (36), then we have (37)τRx=1-ε-1βθ2log⁡αα-1θ+1εlog⁡∫0∞xεβ-11+xβεe-θεxβαε1-e-θxβ1+θxβ/θ+1dx.Using (15), then τR(x) is converted to(38)τRx=1-ε-1βθ2log⁡αα-1θ+1ε∑i=0∞εlog⁡αii!log⁡∫0∞xεβ-11+xβεe-θεxβ1-e-θxβ1+θxβθ+1idx.Using binomial expansions, then we have(39)τRx=1-ε-1βθ2log⁡αα-1θ+1ε∑i=0∞∑j=0i∑k=0j-1jijjkε+jilog⁡αiθmi!θ+1jlog∫0∞xεβ-11+xβε+je-θε+jxβdx.The above integral is complicated to obtain, so it will be solved numerically. Some of the numerical values of Rényi entropy for some certain values of parameters are given in Table 3.Table 3
Rényi entropy for some values ofθ,α,ε, and β.
θ α β = 0.8   ε = 2 β = 1.5   ε = 3.5 1 0.5 1.116 0.8085 2 1.359 0.8855 5 1.479 0.8996 7 1.512 0.8989 10 1.5407 0.8990  1.5 0.5 0.832 0.623 2 1.090 0.8282 5 1.221 0.9179 7 1.258 0.9387 10 1.2901 0.9547  2 0.5 0.634 0.4637 2 0.899 0.6773 5 1.037 0.7738 7 1.076 0.7967 10 1.1104 0.8146


### 3.6. Stochastic Ordering

Here, we compare the APTPL1(φ1) and the APTPL2 (φ2) with respect to stochastic ordering information. Let X1 and X2 be two random variables with cdfs, reliability functions and pdfs FAPTPL1x;φ1 and FAPTPL2x;φ2;RAPTPL1x;φ1 and RAPTPL2x;φ2; and fAPTPL1x;φ1 and fAPTPL2x;φ2, respectively, where φ1=(α1,β1,θ1) and φ2=(α2,β2,θ2). A random variable X1 is said to be smaller than X2 in the following ordering (see [19]), if the following holds:(i)
Stochastic orderX1≤srX2 if RAPTPL1x;φ1≤RAPTPL2x;φ2 for all x.(ii)
Likelihood ratio orderX1≤lrX2 if fAPTPL1x;φ1/fAPTPL2x;φ2 is decreasing in x.(iii)
Hazard rate orderX1≤hrX2 if RAPTPL1x;φ1/RAPTPL2x;φ2 is decreasing in x.(iv)
Mean residual life orderX1≤mrlX2 if EX1-t∣X1<t≤EX2-t∣X2<t.We have the following chain of implications among the various partial orderings discussed above:(40)X1≤lrX2⇒X1≤hrX2⇒X1≤mrlX2⇒X1≤stX2Theorem 2.
LetX1~ APTPL1 (φ1) andX2 ~ APTPL2(φ2). If α1=α2=α, β1≥β2, and θ1≥θ2, then X1≤lrX2, X1≤hrX2, X1≤mrlX2, and X1≤srX2.Proof.
It is sufficient to showfAPTPL1x;φ1/fAPTPL2x;φ2 is a decreasing function of x; the likelihood ratio is (41)fAPTPL1x;φ1fAPTPL2x;φ2=α2-1θ2+1log⁡α1β1θ12xβ1-11+xβ1e-θ1xβ1α1-e-θ1xβ11+θ1xβ1/θ1+1α1-1θ1+1log⁡α2β2θ22xβ2-11+xβ2e-θ2xβ2α1-e-θ2xβ21+θ2xβ2/θ2+1,therefore,(42)ddxlog⁡fAPTPL1x;φ1fAPTPL2x;φ2=β1-1x-β2-1x+β1xβ1-11+β1xβ1+β2xβ2-11+β2xβ2-θ1β1xβ1-1+θ2β2xβ2-1-θ1β1xβ1θ1+1e-θ1xβ1log⁡α1+θ1β1xβ1e-θ1xβ11+θ1xβ1θ1+1log⁡α1+θ2β2xβ2θ2+1e-θ2xβ2log⁡α2-θ2β2xβ2e-θ2xβ21+θ2xβ2θ2+1log⁡α2Hence, if α1=α2=α, β1≥β2, and θ1≥θ2, then d/dxlog⁡fAPTPL1x;φ1/fAPTPL2x;φ2≤0, which implies that X2 is stochastically greater than X1 with respect to likelihood ratio order. Similarly, we can conclude X1≤hrX2, X1≤mrlX2, and X1≤srX2.


## 3.1. Quantile Function

The quantile function, sayQ(u) =F−1(u),u∈ (0, 1), is obtained by inverting (5) as follows:(9)α1-e-θQuβ1+θQuβ/θ+1-1α-1=u,which yields (10)1+θQuβθ+1e-θQuβ=1-log⁡uα-1+1log⁡α.Multiply both sides by (1+θ)e-(1+θ), then we have the Lambert equation(11)-1+θ+θQuβe-1+θ+θQuβ=-1+θe-1+θ1-log⁡uα-1+1log⁡α.Hence, we have the negative LambertW function of the real argument(12)-1+θe-1+θ1-log⁡uα-1+1log⁡α,i.e.,(13)Qu=-1θ-1-1θW-1-1+θe-1+θ1-log⁡uα-1+1log⁡α1/β,whereu∈ (0, 1) andW−1(.) is the negative LambertW function. The first quantile (25%), median (50%), and third quantile (75%) for some choices values of the parameters are computed numerically (see Table 1).Table 1
First quantile, median, and third quantile for some values ofθ, α, and β.
θ α β = 0.3 β = 1 25% 50% 75% 25% 50% 75% 1 0.5 0.045 0.735 6.662 0.394 0.912 1.766 2 0.279 3.175 19.709 0.682 1.414 2.446 5 0.852 6.908 33.388 0.953 1.786 2.865 7 1.225 8.780 39.174 1.063 1.919 3.005 10 1.743 11.040 45.625 1.181 2.055 3.146  1.5 0.5 0.007 0.134 1.339 0.228 0.547 1.091 2 0.049 0.617 4.151 0.403 0.865 1.533 5 0.156 1.390 7.183 0.572 1.104 2.330 7 0.228 1.786 8.481 0.641 1.190 1.899 10 0.329 2.268 9.935 0.717 1.279 1.991  2 0.5 0.002 0.040 0.428 0.157 0.381 0.775 2 0.014 0.193 1.366 0.279 0.610 1.098 5 0.047 0.444 2.398 0.399 0.784 1.2999 7 0.069 0.575 2.843 0.449 0.847 1.368 10 0.101 0.735 3.343 0.503 0.912 1.436We detect form Table1 that as the value of α increases, for fixed values of β and θ, the value of percentage points increases. As the value of θ increases, for fixed values of α and β, the percentage value points decrease. Also, as the value of β increases, for fixed values of α and θ, the percentage value points decrease.


## 3.2. Moments and Related Measures

Thekth moment of a random variableX having the APTPL distribution is obtained as follows:(14)μk′=βθ2log⁡αα-1θ+1∫0∞xk+β-11+xβe-θxβα1-e-θxβ1+θxβ/θ+1dx.Since the power series can be written as (15)αd=∑i=0∞log⁡αii!di,hence (14) can be expressed as follows:(16)μk′=βθ2α-1θ+1∑i=0∞log⁡αi+1i!∫0∞xk+β-11+xβe-θxβ1-e-θxβ1+θxβθ+1idx.Using the binomial expansion, then(17)μk′=βα-1∑i=0∞∑j=0i∑m=0j-1jijjmlog⁡αi+1θm+2i!θ+1m+1∫0∞xk+β+βm-11+xβe-θj+1xβdx.Hence, the kth moment of the APTPL distribution takes the form(18)μk′=∑i=0∞∑j=0i∑m=0jNi,j,mΓk/β+m+1θj+1k/β+m+1+Γk/β+m+2θj+1k/β+m+2,k=1,2,...where (19)Ni,j,m=-1jijjmlog⁡αi+1θm+2i!θ+1m+1α-1.Particularly, the mean is as follows:(20)μ1′=EX=∑i=0∞∑j=0i∑m=0jNi,j,mΓ1/β+m+1θj+11/β+m+1+Γ1/β+m+2θj+11/β+m+2.Similarly, the kth central moment of a given random variable X can be defined as (21)μk=EX-μ1′k=∑i=0k-1ikiμ1′iμk-i′.Thecoefficient of skewness (CS) andcoefficient of kurtosis (CK) are defined by (22)CS=μ3μ23/2,andCK=μ4μ22.Thus, numerical values of the μ1′,μ2′,μ3′,μ4′,variance (σ2),coefficient of variation (CV), CS, and CK of the APTPL distribution for some certain values of parameters are obtained and recorded in Table 2.Table 2
Summary statistics of moments of the APTPL distribution.
μ k ′ α = 0.5 ,   θ = 1 ,   β = 0.5 α = 1.5 ,   θ = 1 ,   β = 1 α = 3 ,   θ = 2 ,   β = 2 α = 5 ,   θ = 2 ,   β = 2.5 α = 7 ,   θ = 2.5 ,   β = 3 μ 1 ′ 1.344 0.713 0.366 0.391 0.371 μ 2 ′ 22.379 1.994 0.369 0.393 0.342 μ 3 ′ 874.97 7.675 0.418 0.427 0.333 μ 4 ′ 6085 37.375 0.520 0.493 0.340 σ 2 20.574 1.486 0.234 0.240 0.204 CV 3.376 1.709 1.322 1.252 1.219 CS 8.461 2.284 0.980 0.726 0.596 CK 133.188 9.423 2.728 2.011 1.704Furthermore, the moment generating function of the APTPL distribution is given by(23)Mxt=∑k,i=0∞∑j=0i∑m=0jNi,j,mtkk!Γk/β+m+1θj+1k/β+m+1+Γk/β+m+2θj+1k/β+m+2.


## 3.3. Incomplete Moments

Thesth lower incomplete moment, say ηs(t), of the APTPL distribution is given by(24)ηst=βθ2log⁡αα-1θ+1∫0txs+β-11+xβe-θxβα1-e-θxβ1+θxβ/θ+1dx.Using the power expansion (15) and binomial expansion, then (24) will be (25)ηst=βα-1∑i=0∞∑j=0i∑m=0j-1jijjmlog⁡αi+1θm+2i!θ+1m+1∫0txs+β+βm-11+xβe-θj+1xβdx.So, thesth lower incomplete moment of the APTPL distribution is (26)ηst=∑i=0∞∑j=0i∑m=0jNi,j,mγs/β+m+1,θj+1tβθj+1s/β+m+1+γs/β+m+2,θj+1tβθj+1s/β+m+2,where γ(.,.), is the lower incomplete moments. The first incomplete moment of the APTPL distribution can be obtained by settings =1 in (26). The first incomplete moment is related to the Bonferroni and Lorenz curves, the mean residual, and mean waiting times. The Bonferroni and Lorenz curves are important in economics, reliability, demography, insurance, and medicine. The Lorenz curves, say LO(x), and Bonferroni curve, say BO(x), are defined by (27)LOx=η1xEX,andBOx=LOxFAPTPLx;φ.For the proposed model, they can be determined from (27) as (28)LOx=∑i=0∞∑j=0i∑m=0jNi.j.mγ1/β+m+1,θj+1xβθj+11/β+m+1+γ1/β+m+2,θj+1xβθj+11/β+m+2∑i=0∞∑j=0i∑m=0jNi,j,mΓ1/β+m+1θj+11/β+m+1+Γ1/β+m+2θj+11/β+m+2-1,and,(29)BOx=∑i=0∞∑j=0i∑m=0jNi,j,mγ1/β+m+1,θj+1xβθj+11/β+m+1+γ1/β+m+2,θj+1xβθj+11/β+m+2FAPTPLx;φ∑i=0∞∑j=0i∑m=0jNi,j,mΓ1/β+m+1θj+11/β+m+1+Γ1/β+m+2θj+11/β+m+2-1.


## 3.4. The Probability-Weighted Moments

The class ofprobability-weighted moments (PWMs), denoted by πr,q, for a random variableX, is defined as follows:(30)πr,q=EXrFxq=∫-∞∞xrfxFxqdx.Substituting (5) and (6) in (30), then the PWM of the APTPL distribution is (31)πr,q=∫0∞xrlog⁡αα-1βθ2xβ-11+xβe-θxβα1-e-θxβ1+θxβ/θ+1θ+1α1-e-θxβ1+θxβ/θ+1-1α-1qdx.Using the binomial expansion, then (32)πr,q=βθ2log⁡αθ+1α-1q+1∑m=0q-1q-mqm∫0∞xrαq+11-e-θxβ1+θxβ/θ+11+xβe-θxβdx.Using (15) and binomial expansions, then (32) will be (33)πr,q=∑m=0q∑i=0∞∑j=0i∑l=0j-1q-m+jβθ2+llog⁡αi+1α-1q+1θ+1l+1ijjlqmq+1ii!∫0∞xr+β+βl-11+xβe-θj+1xβdx.Hence, the PWM of the APTPL distribution is given by(34)πr,q=∑m=0q∑i=0∞∑j=0i∑l=0jΞi,j,m,lΓr/β+l+1θj+1r/β+l+1+Γr/β+l+2θj+1r/β+l+2,where (35)Ξi,j,m,l=-1q-m+jjlqmijq+1ii!θ2+llog⁡αi+1θ+1l+1α-1q+1.


## 3.5. Rényi Entropy

The Rényi entropy of a random variable represents a measure of variation of the uncertainty. The Rényi entropy is defined by(36)τRx=1-ε-1log∫0∞fxεdx.Substituting (6) in (36), then we have (37)τRx=1-ε-1βθ2log⁡αα-1θ+1εlog⁡∫0∞xεβ-11+xβεe-θεxβαε1-e-θxβ1+θxβ/θ+1dx.Using (15), then τR(x) is converted to(38)τRx=1-ε-1βθ2log⁡αα-1θ+1ε∑i=0∞εlog⁡αii!log⁡∫0∞xεβ-11+xβεe-θεxβ1-e-θxβ1+θxβθ+1idx.Using binomial expansions, then we have(39)τRx=1-ε-1βθ2log⁡αα-1θ+1ε∑i=0∞∑j=0i∑k=0j-1jijjkε+jilog⁡αiθmi!θ+1jlog∫0∞xεβ-11+xβε+je-θε+jxβdx.The above integral is complicated to obtain, so it will be solved numerically. Some of the numerical values of Rényi entropy for some certain values of parameters are given in Table 3.Table 3
Rényi entropy for some values ofθ,α,ε, and β.
θ α β = 0.8   ε = 2 β = 1.5   ε = 3.5 1 0.5 1.116 0.8085 2 1.359 0.8855 5 1.479 0.8996 7 1.512 0.8989 10 1.5407 0.8990  1.5 0.5 0.832 0.623 2 1.090 0.8282 5 1.221 0.9179 7 1.258 0.9387 10 1.2901 0.9547  2 0.5 0.634 0.4637 2 0.899 0.6773 5 1.037 0.7738 7 1.076 0.7967 10 1.1104 0.8146


## 3.6. Stochastic Ordering

Here, we compare the APTPL1(φ1) and the APTPL2 (φ2) with respect to stochastic ordering information. Let X1 and X2 be two random variables with cdfs, reliability functions and pdfs FAPTPL1x;φ1 and FAPTPL2x;φ2;RAPTPL1x;φ1 and RAPTPL2x;φ2; and fAPTPL1x;φ1 and fAPTPL2x;φ2, respectively, where φ1=(α1,β1,θ1) and φ2=(α2,β2,θ2). A random variable X1 is said to be smaller than X2 in the following ordering (see [19]), if the following holds:(i)
Stochastic orderX1≤srX2 if RAPTPL1x;φ1≤RAPTPL2x;φ2 for all x.(ii)
Likelihood ratio orderX1≤lrX2 if fAPTPL1x;φ1/fAPTPL2x;φ2 is decreasing in x.(iii)
Hazard rate orderX1≤hrX2 if RAPTPL1x;φ1/RAPTPL2x;φ2 is decreasing in x.(iv)
Mean residual life orderX1≤mrlX2 if EX1-t∣X1<t≤EX2-t∣X2<t.We have the following chain of implications among the various partial orderings discussed above:(40)X1≤lrX2⇒X1≤hrX2⇒X1≤mrlX2⇒X1≤stX2Theorem 2.
LetX1~ APTPL1 (φ1) andX2 ~ APTPL2(φ2). If α1=α2=α, β1≥β2, and θ1≥θ2, then X1≤lrX2, X1≤hrX2, X1≤mrlX2, and X1≤srX2.Proof.
It is sufficient to showfAPTPL1x;φ1/fAPTPL2x;φ2 is a decreasing function of x; the likelihood ratio is (41)fAPTPL1x;φ1fAPTPL2x;φ2=α2-1θ2+1log⁡α1β1θ12xβ1-11+xβ1e-θ1xβ1α1-e-θ1xβ11+θ1xβ1/θ1+1α1-1θ1+1log⁡α2β2θ22xβ2-11+xβ2e-θ2xβ2α1-e-θ2xβ21+θ2xβ2/θ2+1,therefore,(42)ddxlog⁡fAPTPL1x;φ1fAPTPL2x;φ2=β1-1x-β2-1x+β1xβ1-11+β1xβ1+β2xβ2-11+β2xβ2-θ1β1xβ1-1+θ2β2xβ2-1-θ1β1xβ1θ1+1e-θ1xβ1log⁡α1+θ1β1xβ1e-θ1xβ11+θ1xβ1θ1+1log⁡α1+θ2β2xβ2θ2+1e-θ2xβ2log⁡α2-θ2β2xβ2e-θ2xβ21+θ2xβ2θ2+1log⁡α2Hence, if α1=α2=α, β1≥β2, and θ1≥θ2, then d/dxlog⁡fAPTPL1x;φ1/fAPTPL2x;φ2≤0, which implies that X2 is stochastically greater than X1 with respect to likelihood ratio order. Similarly, we can conclude X1≤hrX2, X1≤mrlX2, and X1≤srX2.


## 4. Estimation

The population parameters of the APTPL distribution can be estimated using ML, MPS, LS, and WLS methods of estimation.


### 4.1. Maximum Likelihood Estimators

To obtain the ML estimators of the APTPL distribution with set of parametersφ={α,β,θ}T, letX1,…,Xn be observed values from this distribution. Hence, the log-likelihood function for the vector of parameters, say l, can be written as(43)l=nlog⁡log⁡αα-1+2nlog⁡θ+nlog⁡β-nlog⁡θ+1-∑i=1nθxiβ+∑i=1nlog⁡1+xiβ+log⁡α∑i=1n1-e-xiβ1+θxiβθ+1.The ML equations of the APTPL distribution are obtained as follows:(44)∂l∂α=nα-1-αlog⁡ααα-1log⁡α+1α∑i=1n1-e-θxiβ1+θxiβθ+1,∂l∂θ=2nθ-nθ+1-∑i=1nxiβ+log⁡α∑i=1nxiβe-θxiθxiβθ+1-xiβ1+θ2,and(45)∂l∂β=nβ-∑i=1nθxiβln⁡xi+∑i=1nxiβln⁡xi1+xiβ+log⁡α∑i=1n1-e-xiβ1+θxiβθ+1.Equating ∂l/∂α∂l/∂β and ∂l/∂θ with zeros and solving simultaneously, we obtain the ML estimators of α, β, and θ.


### 4.2. Maximum Product of Spacings Estimators

The MPS method is a powerful alternative to ML method for estimating the population parameters of continuous distributions (see [20]). Let (46)Diα,β,θ=Fxi∣α,β,θ-Fxi-1∣α,β,θ,i=1,2,...,n+1,be the uniform spacings of a random sample from the APTPL distribution, where (47)Fx0∣α,β,θ=0,Fxn+1∣α,β,θ=1and∑i=1n+1Diα,β,θ=1.The MPS estimator is obtained by maximizing thegeometric mean (GM) of the spacings (48)GMα,β,θ=∏i=1n+1Diα,β,θ1/n+1,with respect to α, β, and θ. Equivalently, the MPS estimator of α, β, and θ can be obtained by maximizing the logarithm of the GM of sample spacings (48). There is no closed solution, so the numerical technique is applied to find the required estimates.


### 4.3. Least Squares and Weighted Least Squares Estimators

SupposeX1,X2,...,Xn is a random sample of sizen from APTPL distribution and suppose X(1),X(2),...,X(n) is the corresponding ordered sample. Then, the LS estimators can be obtained by minimizing the sum of squares errors, (49)∑i=1nFXi-in+12,with respect to the unknown parameters. So, the LS estimators of the population parameters of APTPL distribution are obtained by minimizing the following quantity (50)∑i=1nα1-e-θxβ1+θxiβ/θ+1-1α-1-in+12,with respect to α,θ, and β. Also, the WLS estimators of APTPL distribution can be obtained by minimizing the following with respect to the population parameters α,θ, and β.(51)∑i=1n1varFXiα1-e-θxβ1+θxiβ/θ+1-1α-1-in+12,where (52)varFXi=n+12n+2in-i+1.


### 4.4. Numerical Illustration

As seen from the three previous subsections, that the ML, MPS, LS, and WLS estimators of the APTPL are very hard to obtain. Therefore, a simulation study is implemented to study the behavior of estimates, based on certain measures, which aremean square errors (MSEs) andabsolute biases (ABs). 1000 random sampleX1,…Xn of sizesn = 10, 20, 30, and 100 are generated from the APTPL distribution. Exact values of parameters are considered ass e t 1 ≡ α = 1.5 ,   θ = 0.5 ,   β = 0.5 ,   s e t 2 ≡ ( α = 2 ,   θ = 0.5 ,   β = 0.8 ) ,   s e t 3 ≡ ( α = 0.5 ,   θ = 1.5 ,   β = 1.25 ) and set4≡(α=0.5,θ=1.5,β=1.75). The ML MPS, LS, and WLS estimates of φ and their measures are calculated. Numerical outcomes are listed in Table 4 and we conclude the following.(i)
As the sample sizes increase, the MSEs and ABs of the ML, MPS, LS, and WLS estimates of parameters decrease.(ii)
Based on ML and MPS methods, for fixed values ofθ, α and as the value of β increases, the MSEs and ABs increase for α and θ estimates and decrease for β estimates.(iii)
Based on the ML method, as the values ofβ and θ increase while the value of α decreases, the MSEs and ABs for θ and β estimates increase, whereas the values of α increase for approximately sample size and different sets.(iv)
The MSEs and ABs of the MPS estimates are smaller than the corresponding for the other estimators.(v)
The MSEs and ABs of the WLS estimates forα and β estimates are smaller than the corresponding for LS estimates in set 1 at most sample sizes.(vi)
Based on the LS method, as the values ofβ and θ increase, while the value of α decreases, the MSEs for θ and β estimates decrease for all sample sizes and different set of parameters.(vii)
Generally, the MPS estimates of all parameters have the smallest MSEs followed, respectively, by ML, LS, and WLS estimates. Also, the MSEs of LS estimates are better than the MSEs of WLS estimates for most sample sizes.Table 4
MSEs and ABs of the APTPL distribution.
n Methods Properties set1≡(α=1.5, θ=0.5, β=0.5) set2≡(α=2, θ=0.5, β=0.8) set3≡(α=0.5, θ=1.5, β=1.25) set4≡(α=0.5, θ=1.5, β=1.75) α θ β α θ Β α θ β α θ β 10 ML MSE 0.463 0.250 0.503 0.370 0.508 0.656 0.236 2.036 1.563 0.510 2.250 0.941 AB 0.547 0.500 0.704 0.580 0.707 0.800 0.486 1.427 1.250 0.645 1.500 0.948 MPS MSE 0.230 0.160 0.250 0.202 0.247 0.640 0.118 0.573 1.546 0.250 0.845 0.940 AB 0.466 0.387 0.500 0.375 0.476 0.797 0.318 0.725 1.241 0.500 0.917 0.924 LS MSE 2.250 1.398 0.546 0.402 0.868 0.659 1.195 2.480 2.548 2.095 2.681 1.391 AB 1.000 2.304 1.444 0.308 1.790 0.999 1.695 0.702 0.661 2.115 0.434 0.260 WLS MSE 2.249 1.510 0.521 0.412 0.915 0.660 1.295 2.394 2.282 2.406 2.610 1.666 AB 1.000 2.425 1.413 0.312 1.855 1.000 1.806 0.757 0.673 2.315 0.534 0.254  20 ML MSE 0.326 0.250 0.480 0.361 0.514 0.647 0.131 2.034 1.543 0.455 2.250 0.933 AB 0.510 0.500 0.690 0.591 0.712 0.797 0.362 1.426 1.248 0.629 1.500 0.932 MPS MSE 0.213 0.147 0.250 0.163 0.239 0.637 0.114 0.556 1.562 0.250 0.710 0.740 AB 0.455 0.375 0.500 0.355 0.473 0.473 0.694 1.497 1.240 0.500 0.840 0.828 LS MSE 2.249 1.345 0.523 0.384 0.887 0.658 0.971 2.205 1.013 1.515 2.532 1.235 AB 1.000 2.240 1.414 0.310 1.884 1.000 1.689 0.850 0.533 2.067 0.497 0.275 WLS MSE 2.247 1.400 0.483 0.411 0.901 0.659 1.227 2.186 1.015 2.036 2.495 1.122 AB 1.000 2.336 1.358 0.314 1.722 1.000 1.937 0.928 0.586 2.416 0.754 0.105  30 ML MSE 0.298 0.245 0.250 0.360 0.506 0.640 0.121 2.014 1.562 0.451 2.249 0.920 AB 0.512 0.499 0.693 0.589 0.707 0.798 0.348 1.419 0.284 0.632 1.500 0.949 MPS MSE 0.212 0.128 0.244 0.157 0.235 0.636 0.079 0.345 1.542 0.250 0.696 0.721 AB 0.450 0.348 0.497 0.345 0.473 0.800 0.149 0.461 1.243 0.500 0.831 0.824 LS MSE 2.249 1.258 0.490 0.370 0.789 0.649 0.899 2.133 0.990 1.225 2.444 1.148 AB 1.000 2.138 1.365 0.302 1.777 0.988 1.672 0.848 0.523 1.989 0.489 0.298 WLS MSE 2.249 1.248 0.450 0.410 0.821 0.650 1.195 2.184 0.805 1.857 2.187 1.120 AB 1.000 2.219 1.332 0.312 1.712 0.997 2.011 0.968 0.586 0.020 0.886 2.459  100 ML MSE 0.494 0.249 0.372 0.318 0.339 0.639 0.101 1.971 1.561 0.368 2.248 0.661 AB 0.449 0.499 0.601 0.482 0.567 0.799 0.318 1.404 1.241 0.597 1.499 0.811 MPS MSE 0.214 0.125 0.243 0.156 0.233 0.634 0.028 0.036 1.541 0.245 0.589 0.591 AB 0.453 0.342 0.211 0.344 0.471 0.629 0.119 0.048 1.213 0.486 0.765 0.757 LS MSE 2.248 1.147 1.147 0.368 0.769 0.640 0.774 2.116 0.892 1.130 2.067 0.664 AB 1.000 2.000 1.256 0.308 1.769 0.977 1.663 0.742 0.519 2.058 0.763 0.197 WLS MSE 2.249 1.142 0.430 0.391 0.713 0.642 1.128 2.117 0.643 1.894 2.180 0.747 AB 1.000 2.381 1.351 0.307 1.231 0.642 2.002 0.953 0.518 2.665 0.977 0.734


## 4.1. Maximum Likelihood Estimators

To obtain the ML estimators of the APTPL distribution with set of parametersφ={α,β,θ}T, letX1,…,Xn be observed values from this distribution. Hence, the log-likelihood function for the vector of parameters, say l, can be written as(43)l=nlog⁡log⁡αα-1+2nlog⁡θ+nlog⁡β-nlog⁡θ+1-∑i=1nθxiβ+∑i=1nlog⁡1+xiβ+log⁡α∑i=1n1-e-xiβ1+θxiβθ+1.The ML equations of the APTPL distribution are obtained as follows:(44)∂l∂α=nα-1-αlog⁡ααα-1log⁡α+1α∑i=1n1-e-θxiβ1+θxiβθ+1,∂l∂θ=2nθ-nθ+1-∑i=1nxiβ+log⁡α∑i=1nxiβe-θxiθxiβθ+1-xiβ1+θ2,and(45)∂l∂β=nβ-∑i=1nθxiβln⁡xi+∑i=1nxiβln⁡xi1+xiβ+log⁡α∑i=1n1-e-xiβ1+θxiβθ+1.Equating ∂l/∂α∂l/∂β and ∂l/∂θ with zeros and solving simultaneously, we obtain the ML estimators of α, β, and θ.


## 4.2. Maximum Product of Spacings Estimators

The MPS method is a powerful alternative to ML method for estimating the population parameters of continuous distributions (see [20]). Let (46)Diα,β,θ=Fxi∣α,β,θ-Fxi-1∣α,β,θ,i=1,2,...,n+1,be the uniform spacings of a random sample from the APTPL distribution, where (47)Fx0∣α,β,θ=0,Fxn+1∣α,β,θ=1and∑i=1n+1Diα,β,θ=1.The MPS estimator is obtained by maximizing thegeometric mean (GM) of the spacings (48)GMα,β,θ=∏i=1n+1Diα,β,θ1/n+1,with respect to α, β, and θ. Equivalently, the MPS estimator of α, β, and θ can be obtained by maximizing the logarithm of the GM of sample spacings (48). There is no closed solution, so the numerical technique is applied to find the required estimates.


## 4.3. Least Squares and Weighted Least Squares Estimators

SupposeX1,X2,...,Xn is a random sample of sizen from APTPL distribution and suppose X(1),X(2),...,X(n) is the corresponding ordered sample. Then, the LS estimators can be obtained by minimizing the sum of squares errors, (49)∑i=1nFXi-in+12,with respect to the unknown parameters. So, the LS estimators of the population parameters of APTPL distribution are obtained by minimizing the following quantity (50)∑i=1nα1-e-θxβ1+θxiβ/θ+1-1α-1-in+12,with respect to α,θ, and β. Also, the WLS estimators of APTPL distribution can be obtained by minimizing the following with respect to the population parameters α,θ, and β.(51)∑i=1n1varFXiα1-e-θxβ1+θxiβ/θ+1-1α-1-in+12,where (52)varFXi=n+12n+2in-i+1.


## 4.4. Numerical Illustration

As seen from the three previous subsections, that the ML, MPS, LS, and WLS estimators of the APTPL are very hard to obtain. Therefore, a simulation study is implemented to study the behavior of estimates, based on certain measures, which aremean square errors (MSEs) andabsolute biases (ABs). 1000 random sampleX1,…Xn of sizesn = 10, 20, 30, and 100 are generated from the APTPL distribution. Exact values of parameters are considered ass e t 1 ≡ α = 1.5 ,   θ = 0.5 ,   β = 0.5 ,   s e t 2 ≡ ( α = 2 ,   θ = 0.5 ,   β = 0.8 ) ,   s e t 3 ≡ ( α = 0.5 ,   θ = 1.5 ,   β = 1.25 ) and set4≡(α=0.5,θ=1.5,β=1.75). The ML MPS, LS, and WLS estimates of φ and their measures are calculated. Numerical outcomes are listed in Table 4 and we conclude the following.(i)
As the sample sizes increase, the MSEs and ABs of the ML, MPS, LS, and WLS estimates of parameters decrease.(ii)
Based on ML and MPS methods, for fixed values ofθ, α and as the value of β increases, the MSEs and ABs increase for α and θ estimates and decrease for β estimates.(iii)
Based on the ML method, as the values ofβ and θ increase while the value of α decreases, the MSEs and ABs for θ and β estimates increase, whereas the values of α increase for approximately sample size and different sets.(iv)
The MSEs and ABs of the MPS estimates are smaller than the corresponding for the other estimators.(v)
The MSEs and ABs of the WLS estimates forα and β estimates are smaller than the corresponding for LS estimates in set 1 at most sample sizes.(vi)
Based on the LS method, as the values ofβ and θ increase, while the value of α decreases, the MSEs for θ and β estimates decrease for all sample sizes and different set of parameters.(vii)
Generally, the MPS estimates of all parameters have the smallest MSEs followed, respectively, by ML, LS, and WLS estimates. Also, the MSEs of LS estimates are better than the MSEs of WLS estimates for most sample sizes.Table 4
MSEs and ABs of the APTPL distribution.
n Methods Properties set1≡(α=1.5, θ=0.5, β=0.5) set2≡(α=2, θ=0.5, β=0.8) set3≡(α=0.5, θ=1.5, β=1.25) set4≡(α=0.5, θ=1.5, β=1.75) α θ β α θ Β α θ β α θ β 10 ML MSE 0.463 0.250 0.503 0.370 0.508 0.656 0.236 2.036 1.563 0.510 2.250 0.941 AB 0.547 0.500 0.704 0.580 0.707 0.800 0.486 1.427 1.250 0.645 1.500 0.948 MPS MSE 0.230 0.160 0.250 0.202 0.247 0.640 0.118 0.573 1.546 0.250 0.845 0.940 AB 0.466 0.387 0.500 0.375 0.476 0.797 0.318 0.725 1.241 0.500 0.917 0.924 LS MSE 2.250 1.398 0.546 0.402 0.868 0.659 1.195 2.480 2.548 2.095 2.681 1.391 AB 1.000 2.304 1.444 0.308 1.790 0.999 1.695 0.702 0.661 2.115 0.434 0.260 WLS MSE 2.249 1.510 0.521 0.412 0.915 0.660 1.295 2.394 2.282 2.406 2.610 1.666 AB 1.000 2.425 1.413 0.312 1.855 1.000 1.806 0.757 0.673 2.315 0.534 0.254  20 ML MSE 0.326 0.250 0.480 0.361 0.514 0.647 0.131 2.034 1.543 0.455 2.250 0.933 AB 0.510 0.500 0.690 0.591 0.712 0.797 0.362 1.426 1.248 0.629 1.500 0.932 MPS MSE 0.213 0.147 0.250 0.163 0.239 0.637 0.114 0.556 1.562 0.250 0.710 0.740 AB 0.455 0.375 0.500 0.355 0.473 0.473 0.694 1.497 1.240 0.500 0.840 0.828 LS MSE 2.249 1.345 0.523 0.384 0.887 0.658 0.971 2.205 1.013 1.515 2.532 1.235 AB 1.000 2.240 1.414 0.310 1.884 1.000 1.689 0.850 0.533 2.067 0.497 0.275 WLS MSE 2.247 1.400 0.483 0.411 0.901 0.659 1.227 2.186 1.015 2.036 2.495 1.122 AB 1.000 2.336 1.358 0.314 1.722 1.000 1.937 0.928 0.586 2.416 0.754 0.105  30 ML MSE 0.298 0.245 0.250 0.360 0.506 0.640 0.121 2.014 1.562 0.451 2.249 0.920 AB 0.512 0.499 0.693 0.589 0.707 0.798 0.348 1.419 0.284 0.632 1.500 0.949 MPS MSE 0.212 0.128 0.244 0.157 0.235 0.636 0.079 0.345 1.542 0.250 0.696 0.721 AB 0.450 0.348 0.497 0.345 0.473 0.800 0.149 0.461 1.243 0.500 0.831 0.824 LS MSE 2.249 1.258 0.490 0.370 0.789 0.649 0.899 2.133 0.990 1.225 2.444 1.148 AB 1.000 2.138 1.365 0.302 1.777 0.988 1.672 0.848 0.523 1.989 0.489 0.298 WLS MSE 2.249 1.248 0.450 0.410 0.821 0.650 1.195 2.184 0.805 1.857 2.187 1.120 AB 1.000 2.219 1.332 0.312 1.712 0.997 2.011 0.968 0.586 0.020 0.886 2.459  100 ML MSE 0.494 0.249 0.372 0.318 0.339 0.639 0.101 1.971 1.561 0.368 2.248 0.661 AB 0.449 0.499 0.601 0.482 0.567 0.799 0.318 1.404 1.241 0.597 1.499 0.811 MPS MSE 0.214 0.125 0.243 0.156 0.233 0.634 0.028 0.036 1.541 0.245 0.589 0.591 AB 0.453 0.342 0.211 0.344 0.471 0.629 0.119 0.048 1.213 0.486 0.765 0.757 LS MSE 2.248 1.147 1.147 0.368 0.769 0.640 0.774 2.116 0.892 1.130 2.067 0.664 AB 1.000 2.000 1.256 0.308 1.769 0.977 1.663 0.742 0.519 2.058 0.763 0.197 WLS MSE 2.249 1.142 0.430 0.391 0.713 0.642 1.128 2.117 0.643 1.894 2.180 0.747 AB 1.000 2.381 1.351 0.307 1.231 0.642 2.002 0.953 0.518 2.665 0.977 0.734


## 5. Real Data Analysis

In this section, we analyze two real data sets to demonstrate the performance of the APTPL distribution in practice. One of the data sets is taken from Linhart and Zucchini [21, page 69]. The other data are taken from Aarset [22] representing the failure times of 50 devices. For the two data sets, we compare the results of the fits with the APTL, PL, EL,Lindley (L), andexponential(E) distributions. The pdfs of APTL and EL are given by(53)fELx;α,β,θ=θ1+θ+θxα-1e-θxβθ+1αβ1+θ+θxθxβ-1-α,x,α,β,θ>0,and(54)fAPTLx;α,θ=log⁡αα-1θ21+xe-θxα1-e-θx1+θx/θ+1θ+1,x,α,θ>0,α≠1To test the superiority of the APTPL distribution, measures of goodness-of-fit can be applied in comparison to some other models. Mainly we useminus log-likelihood(-log L)Kolmogorov-Smirnov (KS) test statistic,Akaike information criterion (AIC),corrected Akaike information criterion (CAIC),Bayesian information criterion (BIC), andHannan-Quinn information criterion (HQIC).


### 5.1. Linhart and Zucchini Data

The first data consists of a sample of 30 failure times of air-conditioned system of an airplane. The data are 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. The ML estimates, SEs of the model parameters, and analytical measures are provided in Table5.Table 5
Analytical results of the APTPL model and other competing models for the first data.
Model ML Estimates (SE) -Log L AIC BIC CAIC HQIC KS APTPL α ^ = 0.276 ( 0.527 ) β^=0.673(0.085) θ^=0.108(0.065) 176.725 359.45 357.881 360.373 360.795 0.13989  EL α ^ = 0.634 ( 1.995 ) β^=0.194(0.161) θ^=104.392(149.921) 217.635 441.27 439.701 442.193 442.615 0.86825  PL β ^ = 1.525 ( 0.155 ) θ^=2.63×10-3(2.058×10-3) 195.999 395.999 394.953 396.443 396.895 0.4927  APTL α ^ = 0.1 ( 0.104 ) θ^=0.024(5.127×10-3) 183.415 370.83 369.784 371.274 371.727 0.2803  L θ ^ = 1.889 × 1 0 - 3 ( 0.00024 ) 276.715 555.429 554.906 555.572 555.877 0.9113  E θ ^ = 4.339 × 1 0 - 4 ( 7.922 × 1 0 - 5 ) 233.056 468.111 467.589 468.254 468.56 0.8929According to the criteria mentioned in Table5, we found that the APTPL model is the best fitted model compared to the other competitive models. The histogram of data and the estimated pdfs and cdfs for the fitted models are displayed in Figure 3. It is clear from Figure 3 that the APTPL model provides a better fit to this data.Figure 3
Estimated pdf, cdf, and reliability function of APTPL model and other competing models for first data.


### 5.2. Aarset Data

The second data consists of 50 failure times of devices. The data are 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. The ML estimates, SEs of the model parameters, and analytical measures are provided in Table6.Table 6
Analytical results of the APTPL model and other competing models for the second data.
Model ML Estimates (SE) -LogL AIC BIC CAIC HQIC KS APTPL α ^ = 5.205 ( 4.844 ) β^=0.605(0.08) θ^=0.258(0.101) 282.257 570.515 569.612 571.037 572.699 0.17452  EL α ^ = 1.068 ( 17.205 ) β^=0.217(2.128×10-3) θ^=168.806(6.413×10-3) 348.441 702.883 701.98 703.405 705.067 0.86086  PL β ^ = 1.753 ( 0.196 ) θ^=1.825×10-3(1.535×10-3) 305.67 615.34 614.738 615.595 616.796 0.3249  APTL α ^ = 4.359 × 1 0 - 6 ( 6.762 × 1 0 - 5 ) θ^=8.754×10-3(6.886×10-3) 296.747 597.495 596.892 597.939 598.951 0.2186  L θ ^ = 2.538 × 1 0 - 3 ( 0.00025 ) 440.025 882.05 881.749 882.133 882.778 0.9789  E θ ^ = 4.204 × 1 0 - 4 ( 5.9448 × 1 0 - 5 ) 389.68 781.36 781.059 781.503 782.088 0.9645As we see from Table6 that the APTPL distribution gives better fit than the other models; therefore, it could be more adequate model for explaining the used data set. More information can be provided in Figure 4.Figure 4
Estimated pdf, cdf, and survival function of the APTPL and other competing distributions for second data.Also, from Figure4, we conclude that the APTPL distribution provides better fits then we expect that the proposed model may be an interesting alternative model for a wider range of statistical research.


## 5.1. Linhart and Zucchini Data

The first data consists of a sample of 30 failure times of air-conditioned system of an airplane. The data are 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. The ML estimates, SEs of the model parameters, and analytical measures are provided in Table5.Table 5
Analytical results of the APTPL model and other competing models for the first data.
Model ML Estimates (SE) -Log L AIC BIC CAIC HQIC KS APTPL α ^ = 0.276 ( 0.527 ) β^=0.673(0.085) θ^=0.108(0.065) 176.725 359.45 357.881 360.373 360.795 0.13989  EL α ^ = 0.634 ( 1.995 ) β^=0.194(0.161) θ^=104.392(149.921) 217.635 441.27 439.701 442.193 442.615 0.86825  PL β ^ = 1.525 ( 0.155 ) θ^=2.63×10-3(2.058×10-3) 195.999 395.999 394.953 396.443 396.895 0.4927  APTL α ^ = 0.1 ( 0.104 ) θ^=0.024(5.127×10-3) 183.415 370.83 369.784 371.274 371.727 0.2803  L θ ^ = 1.889 × 1 0 - 3 ( 0.00024 ) 276.715 555.429 554.906 555.572 555.877 0.9113  E θ ^ = 4.339 × 1 0 - 4 ( 7.922 × 1 0 - 5 ) 233.056 468.111 467.589 468.254 468.56 0.8929According to the criteria mentioned in Table5, we found that the APTPL model is the best fitted model compared to the other competitive models. The histogram of data and the estimated pdfs and cdfs for the fitted models are displayed in Figure 3. It is clear from Figure 3 that the APTPL model provides a better fit to this data.Figure 3
Estimated pdf, cdf, and reliability function of APTPL model and other competing models for first data.


## 5.2. Aarset Data

The second data consists of 50 failure times of devices. The data are 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. The ML estimates, SEs of the model parameters, and analytical measures are provided in Table6.Table 6
Analytical results of the APTPL model and other competing models for the second data.
Model ML Estimates (SE) -LogL AIC BIC CAIC HQIC KS APTPL α ^ = 5.205 ( 4.844 ) β^=0.605(0.08) θ^=0.258(0.101) 282.257 570.515 569.612 571.037 572.699 0.17452  EL α ^ = 1.068 ( 17.205 ) β^=0.217(2.128×10-3) θ^=168.806(6.413×10-3) 348.441 702.883 701.98 703.405 705.067 0.86086  PL β ^ = 1.753 ( 0.196 ) θ^=1.825×10-3(1.535×10-3) 305.67 615.34 614.738 615.595 616.796 0.3249  APTL α ^ = 4.359 × 1 0 - 6 ( 6.762 × 1 0 - 5 ) θ^=8.754×10-3(6.886×10-3) 296.747 597.495 596.892 597.939 598.951 0.2186  L θ ^ = 2.538 × 1 0 - 3 ( 0.00025 ) 440.025 882.05 881.749 882.133 882.778 0.9789  E θ ^ = 4.204 × 1 0 - 4 ( 5.9448 × 1 0 - 5 ) 389.68 781.36 781.059 781.503 782.088 0.9645As we see from Table6 that the APTPL distribution gives better fit than the other models; therefore, it could be more adequate model for explaining the used data set. More information can be provided in Figure 4.Figure 4
Estimated pdf, cdf, and survival function of the APTPL and other competing distributions for second data.Also, from Figure4, we conclude that the APTPL distribution provides better fits then we expect that the proposed model may be an interesting alternative model for a wider range of statistical research.


## 6. Concluding Remarks

In this paper a new three-parameter power Lindley distribution, called the APTPL distribution, based on alpha power transformation is proposed. The main objective behind this generalization is to give more flexibility for more data that can be analyzed using the proposed distribution. Various statistical properties are discussed such as moments, moment generating function, incomplete moments, and quantile function. The model parameters are estimated via the maximum likelihood, maximum product of spacing, and ordinary and weighted least squares methods. Simulation study is employed to examine the behavior of different estimates. The usefulness of the proposed model is illustrated motivated by two real data.



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