# Alpha Power Transformed Inverse Lomax Distribution with Different Methods of Estimation and Applications

Authors: Ramadan A. ZeinEldin; Muhammad Ahsan ul Haq; Sharqa Hashmi; Mahmoud Elsehety
Journal: Complexity
 (2020)

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## Abstract

In this paper, a new three-parameter lifetime distribution, alpha power transformed inverse Lomax (APTIL) distribution, is proposed. The APTIL distribution is more flexible than inverse Lomax distribution. We derived some mathematical properties including moments, moment generating function, quantile function, mode, stress strength reliability, and order statistics. Characterization related to hazard rate function is also derived. The model parameters are estimated using eight estimation methods including maximum likelihood, least squares, weighted least squares, percentile, Cramer–von Mises, maximum product of spacing, Anderson–Darling, and right-tail Anderson–Darling. Numerical results are calculated to compare the performance of these estimation methods. Finally, we used three real-life datasets to show the flexibility of the APTIL distribution.


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## Body

## 1. Introduction

The inverse Lomax (IL) is originally developed as a lifetime distribution. The inverse Lomax is member of family of generalized beta distribution. Kleiber and Kotz [1] showed that IL distribution can be used in economics. The IL distribution has many applications in modeling of different trends of hazard rate function (hrf), i.e., decreasing or upside-down bathtub failure rate of life testing of components. The IL distribution is used in [2] to get Lorenz ordering relationship among ordered statistics. McKenzie et al. [3] used the IL model and applied it to geophysical databases. Singh et al. [4] investigated the reliability estimators of IL distribution under Type II censoring. Bayesian estimation of mixture of the IL model under the censoring scheme was studied in [5]. Inverse power Lomax distribution was studied in [6] and Weibull IL distribution in [7].The probability density (pdf) and cumulative distribution function (cdf) of the IL distribution are as follows:(1)gx;a,b=abx21+bx−a−1,x,a,b>0,(2)Gx;a,b=1+bx−α,x,a,b>0.Mahdavi and Kundu [8] introduced the alpha power transformation (APT) method to add an additional parameter to a family of distributions to increase flexibility in given family. The cdf and pdf of APT-G family are(3)Fx;α=αGx−1α−1,ifα>0,α≠1,Gx,ifα=1,and the corresponding pdf is(4)fx;α=logαα−1gxαGx,ifα>0,α≠1,gx,ifα=1.In the literature, many probability distributions are generalized using this approach; for example, alpha power transformed Weibull (APTW) distribution in [9], APT generalized exponential distribution in [10], APT Lindley distribution in [11], APT extended exponential distribution in [12], alpha power inverted exponential distribution in [13], alpha power Inverse-Weibull distribution in [14], APT inverse-Lindley distribution in [15], APT power Lindley studied in [16], and APT Pareto distribution proposed in [17].The main goal of this research article is to introduce a simpler and more flexible model called APT inverse Lomax (APTIL) distribution. Furthermore, the key motivations of using APTIL distribution in the practice are follows:(i)
To improve the flexibility of the existing distributions by using APT-G(ii)
To introduce the extended version of the IL distribution whose closed form of cdf exist.(iii)
To provide better fits than the competing modified modelsThe rest of the paper is arranged as follows. InSection 2, we define the new model called AP inverse Lomax (APTIL) distribution. Various statistical properties of the APTIL distribution are derived in Section 3 along with more attractive expressions for quantile function, median, mode, moments, order statistics, and stress strength parameter. Lemmas 1 and 2 contain expressions for stochastic ordering and characterization related to hazard function, and the asymptotic behavior of density is derived. In Section 4, we estimate the parameters using eight methods of parameters estimation as follows: maximum likelihood (ML), least squares (LS), weighted least squares (WLS), percentile (PC), Cramer–von Mises (CV), maximum product of spacing (MPS), Anderson–Darling (AD), and right-tail Anderson–Darling (RTAD). Section 5 deals with three applications to show the efficiency of the proposed model. Final remarks are mentioned in Section 6.


## 2. APTIL Distribution

The random variable (rv)X is said to have APTIL distribution denoted by APTIL(α, a, and b) with two shape parameters and one scale as α, a, and b, respectively. The pdf of X for x≥0is(5)fx=logαα−1abx21+bx−a−1α1+b/x−a,ifα≠1,a,b,α>0,abx21+bx−a−1,ifα=1,a,b,α>0,(6)Fx=α1+b/x−a−1α−1,ifα≠1,α>0,1+bx−a,ifα=1,α>0.The survival function (sf) and the hrf for APTIL distribution forx > 0 are in the following forms:(7)Sx=α−α1+b/x−aα−1,(8)hx=logαab1+b/x−a−1α1+b/x−ax2α−11−α1+b/x−a−1/α−1.Figure1 demonstrates the graphs of pdf and hazard function of APTIL distribution for different values of α, a, and b. Clearly, the pdf of APTIL distribution is the function for α ≠ 1 and a < 1 and unimodal for α ≠ 1 and a > 1. The hrf of the APTIL model can be decreasing or upside-down bathtub for α ≠ 1 and a < 1 and a > 1, respectively.Figure 1
Plots of the pdf and hrf of the APTIL distribution for different values of parameters.
(a)
(b)


### 2.1. Useful Expansions

Here, an explicit expression for APTIL pdf is given. By using the series representation of exponential function, equation (5) can be written as(9)fx;a,b,α=logαα−1abx21+bx−a−1α1+b/x−a=logαα−1abx21+bx−a−1e1+b/x−alna=logαα−1abx21+bx−a−1∑i=0∞1+b/x−alnaii!=∑i=0∞logαi+1i!α−1abx21+bx−ai+1−1,(10)fx;a,b,α=∑i=0∞wigai+1,bx,wi=∑i=0∞logαi+1i!α−1,where gai+1,bx denotes the pdf of IL distribution given in equation (1) with parameters ai+1and b.


### 2.2. Quantile Function and Median

The generation from APTIL distribution is obtained by inverting equation (6):(11)x=blnα−1u+1lnα−1/a−1−1,α≠1,α>0.Ifq uniform (0, 1), then X APTIL(α, a, b), the qth quantile function of APTIL(α, a, b) is given by(12)xq=blnα−1q+1lnα−1/a−1−1,and the median can be obtained as(13)x0.5=blnα+1/2lnα−1/a−1−1,and when α = 1, the median of the APTIL distribution is equal to the median of the IL distribution. The analysis of shape of distribution can be performed by study of skewness and kurtosis. Using Bowley’s coefficient of skewness as follows:(14)Sk=x0.75−2x0.5+x0.25x0.75−x0.5.Moor’s coefficient of kurtosis is given as(15)K=x0.875−x0.625−x0.375+x0.125x0.75−x0.5,where x. is the quantile function.


## 2.1. Useful Expansions

Here, an explicit expression for APTIL pdf is given. By using the series representation of exponential function, equation (5) can be written as(9)fx;a,b,α=logαα−1abx21+bx−a−1α1+b/x−a=logαα−1abx21+bx−a−1e1+b/x−alna=logαα−1abx21+bx−a−1∑i=0∞1+b/x−alnaii!=∑i=0∞logαi+1i!α−1abx21+bx−ai+1−1,(10)fx;a,b,α=∑i=0∞wigai+1,bx,wi=∑i=0∞logαi+1i!α−1,where gai+1,bx denotes the pdf of IL distribution given in equation (1) with parameters ai+1and b.


## 2.2. Quantile Function and Median

The generation from APTIL distribution is obtained by inverting equation (6):(11)x=blnα−1u+1lnα−1/a−1−1,α≠1,α>0.Ifq uniform (0, 1), then X APTIL(α, a, b), the qth quantile function of APTIL(α, a, b) is given by(12)xq=blnα−1q+1lnα−1/a−1−1,and the median can be obtained as(13)x0.5=blnα+1/2lnα−1/a−1−1,and when α = 1, the median of the APTIL distribution is equal to the median of the IL distribution. The analysis of shape of distribution can be performed by study of skewness and kurtosis. Using Bowley’s coefficient of skewness as follows:(14)Sk=x0.75−2x0.5+x0.25x0.75−x0.5.Moor’s coefficient of kurtosis is given as(15)K=x0.875−x0.625−x0.375+x0.125x0.75−x0.5,where x. is the quantile function.


## 3. Properties of APTIL Distribution

This section deals with some statistical properties of APTIL distribution.


### 3.1. Mode

The mode of APTIL distribution is derived byf′x=0:(16)f′x=abb+x/x−2aαb+x/x−alogαa−1b−2xb+x/xa+ablogαx2b+x2α−1,ifα≠1,aba−1b−2xb+x/x−ax2b+x2,ifα=1.The mode of APTIL distribution cannot be expressed in the closed form. Computer software, e.g., Mathematica orR, can be used to compute the mode of APTIL distribution for specific values of parameters.Forα = 1, the mode of IL distribution can be easily calculated from the following equation:(17)aba−1b−2xb+x/x−ax2b+x2=0.


### 3.2. Asymptotic Behavior

The behavior of APTIL distribution is investigated here asx⟶0andx⟶∞:(18)limx⟶0fx=limx⟶0logαα−1abx21+bx−a−1α1+b/x−a=∞ifa<10ifa>1,limx⟶∞fx=limx⟶∞logαα−1abx21+bx−a−1α1+b/x−a=0.Lemma 1.
It can be shown that, for APTIL,limx⟶0Fx=0 and limx⟶∞Fx=1Proof.
(19)
lim
x
⟶
0
F
x
=
lim
x
⟶
0
α
1
+
b
/
x
−
a
−
1
α
−
1
=
α
∞
−
a
−
1
α
−
1
=
1
−
1
α
−
1
=
0
,
lim
x
⟶
∞
F
x
=
lim
x
⟶
∞
α
1
+
b
/
x
−
a
−
1
α
−
1
=
α
1
+
0
−
a
−
1
α
−
1
=
1.


### 3.3. Moments

Theorem 1.
LetX be a rv from APTIL distribution, then its rth moment is(20)μr′=abr+1∑i=0∞logαi+1α−1i!Γr+ai+1Γ1−rΓai+1+1.Proof.
LetX be an r.v. with pdf given in equation (5). For any real number a>0,b>0,α>0,andr≥0, the rth moments of APTIL distribution are obtained as(21)μr′=∫0∞xrfx;α,a,bdx.
Using expression from equation (10), we get(22)μr′=∫0∞xr∑i=0∞wigai+1,bxdx,μr′=∑i=0∞wiabr+1Γr+ai+1Γ1−rΓai+1+1,r=1,2,3,…,where Γ0=−γ,Γ−r=−1r/r!φr−−1r/r!γforr=1,2,…,γ denotes Euler’s constant and φr=∑k=1r1/k [18]. The mean of X can be obtained using equation (20) by putting r=1:(23)μ=μ1′=EX=ab2∑i=0∞logαi+1α−1i!Γ1+ai+1γΓai+1+1,where γ denotes Euler’s constant. The rth central moment μr of X is derived as(24)μr=EX−μ1′r=∑k=0rrk−1r−kμr−k′μ1′k.
The variance of APTIL distribution is easily obtained as(25)σ2=μ2′−μ1′2=ab3α−1∑i=0∞logαi+1i!Γ2+ai+1Γ1−rΓai+1+1−μ1′2.Lemma 2.
LetX be a r. v. with pdf (equation (5)). For any real number a>0,b>0,α>0,r≥0andδ≥0, the integral,(26)τa,b,α,r,δ=∫0∞xr−21+bx−a−1α1+b/x−aeδxdx,is calculated as(27)τa,b,α,r,δ=∑i=0∞∑j=0∞δjj!logαii!br+j−1Γr+ai+1+jΓ1−r−jΓai+1+1.
Proof of Lemma2 is given in Appendix.


### 3.4. Moment Generating Function

The m.g.f. ofX is obtained as(28)Mxt=abα−1∑i=0∞∑j=0∞btjj!logαi+1i!Γai+1+jΓ1−jΓai+1+1.Proof.
The moment generating function can be derived by(29)MXt=EetX=∫0∞etxfx;a,b,α,Mxt=ablogαα−1∫0∞etxx−21+bx−a−1α1+b/x−adx;
Using Lemma2, the moments, moment generating function, characteristic generating function, and raw moment can be easily obtained by(30)MXt=ablogαα−1τa,b,α,0,t,ΦXt=ablogαα−1τa,b,α,0,it,EXr=ablogαα−1τa,b,α,r,0.


### 3.5. Order Statistics

LetX1,X2,…,Xn be r.sample from the APTIL(α a, b) distribution with order statistics X1,X2,…,Xn. The pdf of rv X(r) (r = 1, 2, …, n) is obtained as(31)fXrx=n!r−1!n−r!Fr−1xfx1−Fr−1xn−r,fXrx=n!ablogαr−1!n−r!x2α−11+b/x−1−aα1+b/x−aα1+b/x−a−1α−1r−11−α1+b/x−a−1α−1n−r.The pdf ofXr can be expressed as(32)fXrx=n!ablogαr−1!n−r!x2α−1ξ1−1/aαξα−αξα−1n−rαξ−1α−1r−1,where 1+b/x−a=ξ. Particularly, pdf of the first and nth order statistics can be easily derived from equation (32) as(33)fX1x=nablogαx2α−1ξ1−1/aαξα−αξα−1n−1,(34)fXnx=nablogαx2α−1ξ1−1/aαξαξ−1α−1n−1,respectively.


### 3.6. Stress-Strength Model

LetX1 and X2 be two independent random variables with APTILα1,a1,b1 and APTILα2,a2,b2distributions, respectively. If X1 represents “stress” and X2 represents “strength,” the reliability is defined by R=PX2>X1=∫0∞f2x;α2,a2,b2F1x;α1,a1,b1dx. Then, we can write(35)R=a2b2logα2α1−1α2−1∫0∞x−21+b2x−1−a2α21+b2/x−a2α11+b1/x−a1−1dx=a2b2logα2α1−1α2−1∫0∞x−21+b2x−1−a2α21+b2/x−a2α11+b1/x−a1−x−21+b2x−1−a2α21+b2/x−a2dx.Using Lemma2 from equation (27), we have(36)=a2b2logα2α1−1α2−1∫0∞α11+b1/x−a1x−21+b2x−1−a2α21+b2/x−a2dx−a2b2logα2α1−1α2−1τa2,b2,α2,0,0=1α1−1a2b2logα2α2−1∫0∞α11+b1/x−a1x−21+b2x−1−a2α21+b2/x−a2α11+b1/x−a1dx−a2b2logα2α1−1α2−1τa2,b2,α2,0,0R=1α1−1Eα11+b1/x−a1−a2b2logα2α1−1α2−1τa2,b2,α2,0,0.The effect of parametersa, b, and α on mean, variance, skewness, and kurtosis is displayed in Figures 2 and 3, respectively.Figure 2
Plots of the mean and variance for the APTIL distribution.
(a)
(b)Figure 3
Plots of the skewness and kurtosis for the APTIL distribution.
(a)
(b)Remark 1.
(i)
The mean and variance of APTIL distribution increase as “a” or “b” increase for fixed value of α
(ii)
For increasingα, the mean of distribution decreases as “b” decreases for higher value of “a” and increases for lower value of “a”
(iii)
For increasingα, the variance of distribution increases as “b” increases for lower value of “a”
(iv)
The skewness and kurtosis of APTIL distribution decrease as “a” increases or “b” decreases for fixed value of α


### 3.7. Characterization Based on Hazard Function

In this section, characterizations of APTIL distribution based on the hazard function are presented. It is known that hazard functionh(x) satisfies the following differential equation:(37)f′xfx=h′xhx−hx.Theorem 2.
LetX: (0, ∞) be a continuous r. v. with pdf (5) iff its hrf h(x) satisfies the differential equation:(38)h′x−a+1x−21+bx−1hx=abb+x/x−2aαb+x/x−alogα2b+xb+x/xa−α+αb+x/x−a+abαlnαx2b+x2α−αb+x/x−a2,under the boundary conditions h(0) ≥ 0.Proof.
If rvX has the hrf given in (8), then(39)h′x=abb+x/x−2aαb+x/x−alogα−a−1b−2xb+x/xa−α+αb+x/x−a+abαlnαx2b+x2α−αb+x/x−a2,h′x−a+1x−21+bx−1hx=−a1+ab21+b/x−a−2α1+b/x−alogαx4α−11−α1+b/x−a−1/α−1+abb+x/x−2aαb+x/x−alogα−a−1b−2xb+x/xa−α+αb+x/x−a+abαlogαx2b+x2α−αb+x/x−a2.
Now the result follows.
Conversely if equation (38) holds, then(40)ddx1+b/xa+1hx=ddxab1+b/x1+ab+x/x−1−aαb+x/x−alogαx2α−αb+x/x−a,hx=abb+x/x−1−aαb+x/x−alogαx2α−αb+x/x−a+C,which implies C = 0.


## 3.1. Mode

The mode of APTIL distribution is derived byf′x=0:(16)f′x=abb+x/x−2aαb+x/x−alogαa−1b−2xb+x/xa+ablogαx2b+x2α−1,ifα≠1,aba−1b−2xb+x/x−ax2b+x2,ifα=1.The mode of APTIL distribution cannot be expressed in the closed form. Computer software, e.g., Mathematica orR, can be used to compute the mode of APTIL distribution for specific values of parameters.Forα = 1, the mode of IL distribution can be easily calculated from the following equation:(17)aba−1b−2xb+x/x−ax2b+x2=0.


## 3.2. Asymptotic Behavior

The behavior of APTIL distribution is investigated here asx⟶0andx⟶∞:(18)limx⟶0fx=limx⟶0logαα−1abx21+bx−a−1α1+b/x−a=∞ifa<10ifa>1,limx⟶∞fx=limx⟶∞logαα−1abx21+bx−a−1α1+b/x−a=0.Lemma 1.
It can be shown that, for APTIL,limx⟶0Fx=0 and limx⟶∞Fx=1Proof.
(19)
lim
x
⟶
0
F
x
=
lim
x
⟶
0
α
1
+
b
/
x
−
a
−
1
α
−
1
=
α
∞
−
a
−
1
α
−
1
=
1
−
1
α
−
1
=
0
,
lim
x
⟶
∞
F
x
=
lim
x
⟶
∞
α
1
+
b
/
x
−
a
−
1
α
−
1
=
α
1
+
0
−
a
−
1
α
−
1
=
1.


## 3.3. Moments

Theorem 1.
LetX be a rv from APTIL distribution, then its rth moment is(20)μr′=abr+1∑i=0∞logαi+1α−1i!Γr+ai+1Γ1−rΓai+1+1.Proof.
LetX be an r.v. with pdf given in equation (5). For any real number a>0,b>0,α>0,andr≥0, the rth moments of APTIL distribution are obtained as(21)μr′=∫0∞xrfx;α,a,bdx.
Using expression from equation (10), we get(22)μr′=∫0∞xr∑i=0∞wigai+1,bxdx,μr′=∑i=0∞wiabr+1Γr+ai+1Γ1−rΓai+1+1,r=1,2,3,…,where Γ0=−γ,Γ−r=−1r/r!φr−−1r/r!γforr=1,2,…,γ denotes Euler’s constant and φr=∑k=1r1/k [18]. The mean of X can be obtained using equation (20) by putting r=1:(23)μ=μ1′=EX=ab2∑i=0∞logαi+1α−1i!Γ1+ai+1γΓai+1+1,where γ denotes Euler’s constant. The rth central moment μr of X is derived as(24)μr=EX−μ1′r=∑k=0rrk−1r−kμr−k′μ1′k.
The variance of APTIL distribution is easily obtained as(25)σ2=μ2′−μ1′2=ab3α−1∑i=0∞logαi+1i!Γ2+ai+1Γ1−rΓai+1+1−μ1′2.Lemma 2.
LetX be a r. v. with pdf (equation (5)). For any real number a>0,b>0,α>0,r≥0andδ≥0, the integral,(26)τa,b,α,r,δ=∫0∞xr−21+bx−a−1α1+b/x−aeδxdx,is calculated as(27)τa,b,α,r,δ=∑i=0∞∑j=0∞δjj!logαii!br+j−1Γr+ai+1+jΓ1−r−jΓai+1+1.
Proof of Lemma2 is given in Appendix.


## 3.4. Moment Generating Function

The m.g.f. ofX is obtained as(28)Mxt=abα−1∑i=0∞∑j=0∞btjj!logαi+1i!Γai+1+jΓ1−jΓai+1+1.Proof.
The moment generating function can be derived by(29)MXt=EetX=∫0∞etxfx;a,b,α,Mxt=ablogαα−1∫0∞etxx−21+bx−a−1α1+b/x−adx;
Using Lemma2, the moments, moment generating function, characteristic generating function, and raw moment can be easily obtained by(30)MXt=ablogαα−1τa,b,α,0,t,ΦXt=ablogαα−1τa,b,α,0,it,EXr=ablogαα−1τa,b,α,r,0.


## 3.5. Order Statistics

LetX1,X2,…,Xn be r.sample from the APTIL(α a, b) distribution with order statistics X1,X2,…,Xn. The pdf of rv X(r) (r = 1, 2, …, n) is obtained as(31)fXrx=n!r−1!n−r!Fr−1xfx1−Fr−1xn−r,fXrx=n!ablogαr−1!n−r!x2α−11+b/x−1−aα1+b/x−aα1+b/x−a−1α−1r−11−α1+b/x−a−1α−1n−r.The pdf ofXr can be expressed as(32)fXrx=n!ablogαr−1!n−r!x2α−1ξ1−1/aαξα−αξα−1n−rαξ−1α−1r−1,where 1+b/x−a=ξ. Particularly, pdf of the first and nth order statistics can be easily derived from equation (32) as(33)fX1x=nablogαx2α−1ξ1−1/aαξα−αξα−1n−1,(34)fXnx=nablogαx2α−1ξ1−1/aαξαξ−1α−1n−1,respectively.


## 3.6. Stress-Strength Model

LetX1 and X2 be two independent random variables with APTILα1,a1,b1 and APTILα2,a2,b2distributions, respectively. If X1 represents “stress” and X2 represents “strength,” the reliability is defined by R=PX2>X1=∫0∞f2x;α2,a2,b2F1x;α1,a1,b1dx. Then, we can write(35)R=a2b2logα2α1−1α2−1∫0∞x−21+b2x−1−a2α21+b2/x−a2α11+b1/x−a1−1dx=a2b2logα2α1−1α2−1∫0∞x−21+b2x−1−a2α21+b2/x−a2α11+b1/x−a1−x−21+b2x−1−a2α21+b2/x−a2dx.Using Lemma2 from equation (27), we have(36)=a2b2logα2α1−1α2−1∫0∞α11+b1/x−a1x−21+b2x−1−a2α21+b2/x−a2dx−a2b2logα2α1−1α2−1τa2,b2,α2,0,0=1α1−1a2b2logα2α2−1∫0∞α11+b1/x−a1x−21+b2x−1−a2α21+b2/x−a2α11+b1/x−a1dx−a2b2logα2α1−1α2−1τa2,b2,α2,0,0R=1α1−1Eα11+b1/x−a1−a2b2logα2α1−1α2−1τa2,b2,α2,0,0.The effect of parametersa, b, and α on mean, variance, skewness, and kurtosis is displayed in Figures 2 and 3, respectively.Figure 2
Plots of the mean and variance for the APTIL distribution.
(a)
(b)Figure 3
Plots of the skewness and kurtosis for the APTIL distribution.
(a)
(b)Remark 1.
(i)
The mean and variance of APTIL distribution increase as “a” or “b” increase for fixed value of α
(ii)
For increasingα, the mean of distribution decreases as “b” decreases for higher value of “a” and increases for lower value of “a”
(iii)
For increasingα, the variance of distribution increases as “b” increases for lower value of “a”
(iv)
The skewness and kurtosis of APTIL distribution decrease as “a” increases or “b” decreases for fixed value of α


## 3.7. Characterization Based on Hazard Function

In this section, characterizations of APTIL distribution based on the hazard function are presented. It is known that hazard functionh(x) satisfies the following differential equation:(37)f′xfx=h′xhx−hx.Theorem 2.
LetX: (0, ∞) be a continuous r. v. with pdf (5) iff its hrf h(x) satisfies the differential equation:(38)h′x−a+1x−21+bx−1hx=abb+x/x−2aαb+x/x−alogα2b+xb+x/xa−α+αb+x/x−a+abαlnαx2b+x2α−αb+x/x−a2,under the boundary conditions h(0) ≥ 0.Proof.
If rvX has the hrf given in (8), then(39)h′x=abb+x/x−2aαb+x/x−alogα−a−1b−2xb+x/xa−α+αb+x/x−a+abαlnαx2b+x2α−αb+x/x−a2,h′x−a+1x−21+bx−1hx=−a1+ab21+b/x−a−2α1+b/x−alogαx4α−11−α1+b/x−a−1/α−1+abb+x/x−2aαb+x/x−alogα−a−1b−2xb+x/xa−α+αb+x/x−a+abαlogαx2b+x2α−αb+x/x−a2.
Now the result follows.
Conversely if equation (38) holds, then(40)ddx1+b/xa+1hx=ddxab1+b/x1+ab+x/x−1−aαb+x/x−alogαx2α−αb+x/x−a,hx=abb+x/x−1−aαb+x/x−alogαx2α−αb+x/x−a+C,which implies C = 0.


## 4. Estimation of Parameters

The parameters of the APTIL distribution are estimated using various methods including maximum likelihood (ML), least squares (LS), weighted least squares (WLS), percentile (PC), Cramer–von Mises (CV), maximum product of spacing (MPS), Anderson–Darling (AD), and right-tail Anderson–Darling (RTAD) methods of estimation.


### 4.1. ML Estimation

LetX1, …, Xn have the observed values from APTIL distribution. The MLEs of the proposed model parameters α, a, and b are derived using the log-likelihood function sayℓ which is given by(41)ℓ=nlnlogα−nlnα−1+nlna+nlnb−2∑i=1nlogxi−a+1∑i=1nln1+bxi+lnα∑i=1n1+bxi−a.The ML equations of the APTIL distribution are given by(42)∂ℓ∂α=nαlogα−nα−1+1α∑i=1n1+bxi−a,∂ℓ∂a=na−∑i=1nln1+bxi−lnα∑i=1n1+bxi−aln1+bxi,∂ℓ∂b=nb−a+1∑i=1n1xi+b+alnα1+bxi−a−11xi.Equating∂ℓ/∂α, ∂ℓ/∂a, and ∂ℓ/∂b with zeros and solving simultaneously, we obtain the ML estimators of α, a, and b.


### 4.2. Ordinary and Weighted LS Estimators

SupposeX1,X2,…,Xn is a random sample from APTIL distribution with corresponding ordered sample of X1,X2,…,Xn. The mean and variance of APTIL are independent of unknown parameter and are as follows:(43)EFXi=in+1,varFXi=in−i+1n+12n+2,where FXi is the cdf of APTIL distribution withXi being the ith order statistic. Then, LS estimators ([19]) are obtained by minimizing the SSE:(44)∑i=1nFix−in+12,with respect to α, a, and b. So, the LS estimators (LSEs) of the parameters α, a, and b of the APTIL are obtained by minimizing the following:(45)∑i=1nα1+b/xi−a−1α−1−in+12,with respect to α, a, and b.The WLS estimators [19] of α, a, and b can be obtained by minimizing the following expression:(46)∑i=1nn+12n+2in−i+1α1+b/xi−a−1α−1−in+12,with respect to α, a, and b.


### 4.3. PC Estimator (PCE)

LetX1<X2<⋯<Xn be order statistics. Using on PC method of estimation ([20, 21]), the estimators of α, a, and b are derived by minimizing the following:(47)∑i=1nlnin+1−lnα1+b/xi−a−1α−12,with respect to α, a, and b.


### 4.4. The Cramer-von Mises Minimum Distance Estimators

The CV method is based on the difference between the estimated cdf and the empirical cdf ([22, 23]). The CV estimators are obtained by minimizing(48)CV=112n+∑i=1nα1+b/xi−a−1α−1−2i−12n2.Macdonald [24] stated about the CV method that it depends on minimum distance estimators providing empirical evidence that the bias of the estimator is smaller than the other minimum distance estimators.


### 4.5. Maximum Product of Spacing Estimators

The MPS method is a powerful alternative to the ML method for estimating the population parameters of continuous distributions ([25]). Let(49)Diα,a,b=Fxiα,a,b−Fxi−1α,a,b,i=1,2,…,n+1,be the uniform spacings of a random sample from the APTL distribution, where(50)Fx0α,a,b=0,Fxn+1α,a,b=1,∑i=1n+1Diα,a,b=1.The MPS estimator is obtained by maximizing thegeometric mean (GM) of the spacings:(51)GMα,a,b=∏i=1n+1Diα,a,b1/n+1,w.r.t. α, a, and b. The MPS estimator of α, a, and b can be obtained by maximizing the logarithm of the GM of sample spacing’s equation (51). No closed solution exists, so the numerical method is used to find the estimates.


### 4.6. Anderson-Darling and Right-Tail Anderson-Darling Estimators

The method of Anderson–Darling estimation was introduced by [26] in the context of statistical tests. By adapting it to the APTIL model, the Anderson–Darling estimates (ADEs) of α, a, and b can be obtained by minimizing, with respect to α, a, and b, the function given by(52)AD=−n−1n∑i=1n2i−1logFxi;α,a,b+log1−Fxn+1−i;α,a,b.Similarly, the right-tail Anderson–Darling estimates (RTADEs) ofα, a, and b can be obtained by minimizing, with respect to α, a, and b, the function given by(53)RTAD=n2−2∑i=1nlogFxi;α,a,b−1n∑i=1n2i−1log1−Fxn+1−i;α,a,b.


## 4.1. ML Estimation

LetX1, …, Xn have the observed values from APTIL distribution. The MLEs of the proposed model parameters α, a, and b are derived using the log-likelihood function sayℓ which is given by(41)ℓ=nlnlogα−nlnα−1+nlna+nlnb−2∑i=1nlogxi−a+1∑i=1nln1+bxi+lnα∑i=1n1+bxi−a.The ML equations of the APTIL distribution are given by(42)∂ℓ∂α=nαlogα−nα−1+1α∑i=1n1+bxi−a,∂ℓ∂a=na−∑i=1nln1+bxi−lnα∑i=1n1+bxi−aln1+bxi,∂ℓ∂b=nb−a+1∑i=1n1xi+b+alnα1+bxi−a−11xi.Equating∂ℓ/∂α, ∂ℓ/∂a, and ∂ℓ/∂b with zeros and solving simultaneously, we obtain the ML estimators of α, a, and b.


## 4.2. Ordinary and Weighted LS Estimators

SupposeX1,X2,…,Xn is a random sample from APTIL distribution with corresponding ordered sample of X1,X2,…,Xn. The mean and variance of APTIL are independent of unknown parameter and are as follows:(43)EFXi=in+1,varFXi=in−i+1n+12n+2,where FXi is the cdf of APTIL distribution withXi being the ith order statistic. Then, LS estimators ([19]) are obtained by minimizing the SSE:(44)∑i=1nFix−in+12,with respect to α, a, and b. So, the LS estimators (LSEs) of the parameters α, a, and b of the APTIL are obtained by minimizing the following:(45)∑i=1nα1+b/xi−a−1α−1−in+12,with respect to α, a, and b.The WLS estimators [19] of α, a, and b can be obtained by minimizing the following expression:(46)∑i=1nn+12n+2in−i+1α1+b/xi−a−1α−1−in+12,with respect to α, a, and b.


## 4.3. PC Estimator (PCE)

LetX1<X2<⋯<Xn be order statistics. Using on PC method of estimation ([20, 21]), the estimators of α, a, and b are derived by minimizing the following:(47)∑i=1nlnin+1−lnα1+b/xi−a−1α−12,with respect to α, a, and b.


## 4.4. The Cramer-von Mises Minimum Distance Estimators

The CV method is based on the difference between the estimated cdf and the empirical cdf ([22, 23]). The CV estimators are obtained by minimizing(48)CV=112n+∑i=1nα1+b/xi−a−1α−1−2i−12n2.Macdonald [24] stated about the CV method that it depends on minimum distance estimators providing empirical evidence that the bias of the estimator is smaller than the other minimum distance estimators.


## 4.5. Maximum Product of Spacing Estimators

The MPS method is a powerful alternative to the ML method for estimating the population parameters of continuous distributions ([25]). Let(49)Diα,a,b=Fxiα,a,b−Fxi−1α,a,b,i=1,2,…,n+1,be the uniform spacings of a random sample from the APTL distribution, where(50)Fx0α,a,b=0,Fxn+1α,a,b=1,∑i=1n+1Diα,a,b=1.The MPS estimator is obtained by maximizing thegeometric mean (GM) of the spacings:(51)GMα,a,b=∏i=1n+1Diα,a,b1/n+1,w.r.t. α, a, and b. The MPS estimator of α, a, and b can be obtained by maximizing the logarithm of the GM of sample spacing’s equation (51). No closed solution exists, so the numerical method is used to find the estimates.


## 4.6. Anderson-Darling and Right-Tail Anderson-Darling Estimators

The method of Anderson–Darling estimation was introduced by [26] in the context of statistical tests. By adapting it to the APTIL model, the Anderson–Darling estimates (ADEs) of α, a, and b can be obtained by minimizing, with respect to α, a, and b, the function given by(52)AD=−n−1n∑i=1n2i−1logFxi;α,a,b+log1−Fxn+1−i;α,a,b.Similarly, the right-tail Anderson–Darling estimates (RTADEs) ofα, a, and b can be obtained by minimizing, with respect to α, a, and b, the function given by(53)RTAD=n2−2∑i=1nlogFxi;α,a,b−1n∑i=1n2i−1log1−Fxn+1−i;α,a,b.


## 5. Simulation Study

Here, we come up with a numerical study to compare the behavior of different estimates. We generate 1000 random samples of sizen = 50, 100, and 200 from the APTIL distribution. Four sets of the parameters are assigned as follows: set1=α=0.5,b=0.5,a=2, set2=α=0.5,b=0.5,a=3, set3=α=0.3,b=0.5,a=2, and set4=α=0.3,b=0.5,a=3. The MLE, LSE, WLSE, CVE, PCE, MPSE, ADE, and RTADE of α,a, and b are determined. Then, the estimates of all methods and their mean square errors (MSEs) are documented in Tables 1–4.Table 1
Estimates and MSEs of APTIL distribution for ML, LS, WLS, CV, PC, MPS, AD, and RTAD estimates for set 1.
n
Set 1:α=0.5,b=0.5,a=2
MLEs
LSEs
WLSEs
CVEs
PCEs
MPSEs
ADEs
RTADEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
50
0.451
0.16
0.491
0.181
0.542
0.276
0.468
0.246
0.587
0.496
0.709
0.489
0.432
0.089
0.429
0.184
0.697
0.202
0.910
0.261
0.827
0.197
0.848
0.199
0.836
0.300
0.882
0.600
0.806
0.173
1.024
0.399
2.151
0.499
1.561
0.346
1.695
0.345
1.693
0.243
1.789
0.536
1.998
1.069
1.725
0.223
1.530
0.368
100
0.401
0.084
0.394
0.123
0.447
0.197
0.401
0.120
0.386
0.065
0.577
0.232
0.395
0.066
0.411
0.139
0.642
0.076
0.825
0.147
0.772
0.138
0.813
0.154
0.845
0.309
0.767
0.372
0.711
0.083
0.905
0.235
1.983
0.135
1.684
0.235
1.803
0.274
1.737
0.278
1.798
0.298
1.93
0.318
1.828
0.135
1.628
0.317
200
0.394
0.068
0.374
0.104
0.409
0.136
0.352
0.099
0.342
0.050
0.496
0.121
0.383
0.044
0.362
0.106
0.68
0.069
0.798
0.122
0.735
0.094
0.797
0.117
0.773
0.192
0.687
0.168
0.671
0.055
0.857
0.157
1.861
0.082
1.683
0.154
1.785
0.137
1.706
0.146
1.797
0.142
1.902
0.155
1.833
0.076
1.607
0.191Table 2
Estimates and MSEs of APTIL distribution for ML, LS, WLS, CV, PC, MPS, AD, and RTAD estimates for set 2.
n
Set 2:α=0.5,b=0.5,a=3
MLEs
LSEs
WLSEs
CVEs
PCEs
MPSEs
ADEs
RTADEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
50
0.472
0.186
0.709
1.148
0.64
1.163
0.518
0.803
0.364
0.086
0.56
0.237
0.377
0.110
0.810
1.526
0.752
0.204
1.094
0.623
1.088
0.523
1.175
0.724
1.104
0.651
0.981
0.670
1.057
0.438
1.118
0.717
2.778
0.540
2.227
1.644
2.172
1.227
2.311
1.599
2.169
1.018
2.532
1.325
2.183
0.880
2.252
1.447
100
0.473
0.159
0.525
0.569
0.52
0.970
0.591
0.900
0.311
0.076
0.482
0.187
0.313
0.082
0.721
1.006
0.738
0.182
1.113
0.612
1.035
0.406
1.136
0.622
1.068
0.549
0.971
0.554
1.046
0.415
0.995
0.508
2.728
0.354
2.303
1.481
2.291
1.161
2.236
1.508
2.211
0.855
2.434
0.944
2.148
0.816
2.411
1.425
200
0.454
0.085
0.521
0.595
0.446
0.660
0.292
0.337
0.262
0.082
0.384
0.087
0.261
0.074
0.498
0.463
0.664
0.100
1.078
0.572
0.995
0.308
1.257
0.692
1.007
0.484
0.833
0.263
0.987
0.309
1.032
0.512
2.771
0.287
2.481
1.798
2.281
1.067
2.054
1.221
2.333
0.609
2.517
0.491
2.231
0.655
2.417
1.229Table 3
Estimates and MSEs of APTIL distribution for ML, LS, WLS, CV, PC, MPS, AD, and RTAD estimates for set 3.
n
Set 3:α=0.3,b=0.5,a=2
MLEs
LSEs
WLSEs
CVEs
PCEs
MPSEs
ADEs
RTADEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
50
0.315
0.053
0.26
0.066
0.263
0.085
0.212
0.042
0.401
0.604
0.413
0.234
0.282
0.052
0.234
0.056
0.509
0.032
0.805
0.128
0.854
0.19
0.895
0.205
0.798
0.255
0.849
0.565
0.71
0.126
0.948
0.285
2.243
0.356
1.628
0.211
1.697
0.225
1.678
0.245
1.717
0.413
1.878
0.523
1.838
0.188
1.57
0.26
100
0.325
0.029
0.221
0.025
0.234
0.035
0.181
0.033
0.303
0.06
0.387
0.171
0.27
0.039
0.199
0.033
0.504
0.033
0.789
0.113
0.771
0.111
0.838
0.14
0.737
0.199
0.682
0.198
0.637
0.058
0.872
0.178
2.138
0.217
1.67
0.149
1.724
0.177
1.686
0.135
1.864
0.321
1.96
0.257
1.906
0.095
1.616
0.179
200
0.336
0.016
0.228
0.026
0.218
0.026
0.179
0.023
0.29
0.03
0.409
0.149
0.274
0.021
0.211
0.022
0.48
8.602∗
0.791
0.099
0.74
0.087
0.788
0.096
0.65
0.105
0.596
0.121
0.602
0.037
0.812
0.119
2.081
0.076
1.632
0.156
1.763
0.083
1.702
0.109
1.907
0.145
2.031
0.204
1.906
0.065
1.624
0.163
∗10−3.Table 4
Estimates and MSEs of APTIL distribution for ML, LS, WLS, CV, PC, MPS, AD, and RTAD estimates for set 4.
n
Set 4:α=0.3,b=0.5,a=3
MLEs
LSEs
WLSEs
CVEs
PCEs
MPSEs
ADEs
RTADEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
Est.
MSEs
50
0.306
0.093
0.544
0.711
0.561
0.916
0.339
0.366
0.266
0.079
0.497
0.468
0.259
0.575
0.380
0.260
0.753
0.166
0.968
0.511
0.951
0.410
0.913
0.435
1.032
0.539
0.865
0.513
0.965
0.819
1.047
0.589
2.684
0.410
2.440
1.512
2.425
1.552
2.668
1.441
2.309
0.850
2.695
0.972
2.270
1.689
2.391
1.282
100
0.304
0.086
0.312
0.242
0.399
0.474
0.328
0.304
0.194
0.031
0.325
0.094
0.217
0.427
0.39
0.310
0.751
0.153
0.949
0.397
0.934
0.342
0.986
0.427
0.998
0.501
0.819
0.327
0.864
0.680
0.916
0.375
2.585
0.353
2.511
1.250
2.516
1.833
2.436
1.296
2.323
0.736
2.626
0.807
2.310
1.547
2.478
1.086
200
0.251
0.033
0.228
0.073
0.250
0.157
0.398
0.311
0.164
0.029
0.326
0.076
0.197
0.227
0.397
0.221
0.69
0.081
0.990
0.394
1.026
0.399
0.881
0.380
0.885
0.276
0.696
0.168
0.872
0.482
0.780
0.288
2.667
0.238
2.412
1.160
2.388
1.406
2.766
1.290
2.500
0.441
2.813
0.477
2.332
1.489
2.781
0.908Form Table5, for the parameter combinations, we can conclude that the ML estimation method outperforms all the other estimation methods (overall score of 20.5). Therefore, depending on our study, we can consider the ML estimation method is the best for APTIL distribution.Table 5
Partial and overall ranks of all the methods of estimation for various combination of parameters.
Parameters
n
ML
LS
WLS
CV
PC
MPS
AD
RTAD
α
=
0.5
,
b
=
0.5
,
a
=
2
50
4.5
4.5
3
2
7
8
1
6
100
1.5
3
5
5
5
8
1.5
7
200
2
6
4.5
3
4.5
7
1
8
α
=
0.5
,
b
=
0.5
,
a
=
3
50
1
6
4
7
3
5
2
8
100
1
6
4.5
8
3
4.5
2
6
200
1
8
5
6.5
4
2
3
6.5
α
=
0.3
,
b
=
0.5
,
a
=
2
50
2.5
4
5
2.5
7
8
1
6
100
3
1.5
5
4
8
7
1.5
6
200
1
5
3
4
7
8
2
6
α
=
0.3
,
b
=
0.5
,
a
=
3
50
1
6.5
6.5
3
2
4
8
5
100
1
4.5
7
6
3
2
8
4.5
200
1
4.5
6.5
6.5
2
3
8
4.5
∑
Ranks
20.5
59.5
59
57.5
55.5
66.5
39
73.5
Overall rank
1
6
5
4
3
7
2
8


## 6. Applications

In this section, we utilized three data sets to show that APTIL can be a better life testing distribution compared with some known probability distributions such as APT Weibull (APW) distribution [9], alpha power transformed inverse exponential (APTIE) distribution [10], Marshall Olkin length biased exponential (MOLBE) distribution [27], APT inverted Weibull (APIW) distribution [14], APT Pareto (APTP) distribution [17], and inverse Lomax (IL) distribution [4]. The corresponding probability density functions of competitor models are given below:(54)fAPIEx=λlogαα−1x2e−λ/xαe−λ/x,fAPIWx=logαλβα−1x−β+1e−λx−βαe−λx−β,fAPWx=logαλβα−1xβ−1e−λxβα1−eλxβ,fAPPx=βlogαα−1x−β+1α1−ex−β,fMOLBEx=x/β2xe−x/β1−1−α1+x/βe−x/β2.The first data set was originally reported by [28] which represents the maximum annual flood discharges of the North Saskachevan in unit of 1000 cubic feet per second, of the North Saskachevan River at Edmonton, over a period of 47 years. The data are as follows: 19.885, 20.940, 21.820, 23.700, 24.888, 25.460, 25.760, 26.720, 27.500, 28.100, 28.600, 30.200, 30.380, 31.500, 32.600, 32.680, 34.400, 35.347, 35.700, 38.100, 39.020, 39.200, 40.000, 40.400, 40.400, 42.250, 44.020, 44.730, 44.900, 46.300, 50.330, 51.442, 57.220, 58.700, 58.800, 61.200, 61.740, 65.440, 65.597, 66.000, 74.100, 75.800, 84.100, 106.600, 109.700, 121.970, 121.970, and 185.560.The second data set represents the marks in Mathematics for 48 students in the slow pace programme in the year 2013 [29]. The data are as follows: 29, 25, 50, 15, 13, 27, 15, 18, 7, 7, 8, 19, 12, 18, 5, 21, 15, 86, 21, 15, 14, 39, 15, 14, 70, 44, 6, 23, 58, 19, 50, 23, 11, 6, 34, 18, 28, 34, 12, 37, 4, 60, 20, 23, 40, 65, 19, and 31.The third data consists of a sample of 30 failure times of air-conditioned system of an airplane [30]. The data are as follows: 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, and 95.For the selection of best fit model, we used the following criteria: Akaike information criterion (AIC), Bayesian information criterion (BIC), Anderson–darling (A∗), and Cramer–von Mises (W∗) test. The maximum likelihood estimates are presented in Table6, and the goodness of fit measures are presented in Table 7.Table 6
Estimated values for the data sets.
Model
Estimates
Data I
APTIL(α, a, b)
0.000439
709.021
0.146377
APW(α, β, λ)
11.22380
1.28552
0.010140
APIE(α, λ)
5.10175 ∗ 1010
1.62081
APP(α, β)
198.1017
0.59646
MOLBE(α, β)
2.270760
18.7701
IL(a, b)
902.7830
0.04409
Data II
APTIL(α, a, b)
0.000327
3.28047
22.8655
APIW(α, β, λ)
24.49500
1.72561
37.9214
APIE(α, λ)
0.041375
29.3658
APP(α, β)
257.0863
0.75805
MOLBE(α, β)
0.823351
13.9648
IL(a, b)
2431.360
0.00646
Data III
APTIL(α, a, b)
5.72005 ∗ 10−8
0.90389
1136.77
APIW(α, β, λ)
84.1137
0.91645
3.32642
APIE(α, λ)
101.484
3.93250
APP(α, β)
964.635
0.72033
MOLBE(α, β)
0.06136
74.9680
IL(a, b)
1.87063
12.6355Table 7
Goodness of fit measures for the data sets.
Model
ℓ
AIC
BIC
A
∗
W
∗
Data I
APTIL
−216.492
438.984
444.597
0.27257
0.038885
APW
−225.272
456.544
462.158
1.52944
0.207808
APIE
−236.609
477.217
480.960
6.46851
1.221870
APP
−263.177
530.354
534.097
10.7680
2.139800
MOLBE
−224.746
453.492
457.235
1.55599
0.216626
IL
−235.826
475.652
479.394
6.48532
1.236730
Data II
APTIL
−197.035
400.070
405.684
0.265554
0.0345284
APIW
−198.597
403.195
408.809
0.539549
0.0651926
APIE
−200.564
405.129
408.871
0.857503
0.1104240
APP
−216.577
437.154
440.896
5.353100
0.9553700
MOLBE
−197.201
398.403
402.145
0.372636
0.0614331
IL
−205.056
414.112
417.854
2.530860
0.4221020
Data III
APTIL
−151.910
309.819
314.023
0.430672
0.056662
APIW
−153.147
312.293
316.497
0.491873
0.069053
APIE
−153.372
310.744
313.546
0.596184
0.089069
APP
−156.025
314.169
316.972
0.796196
0.108866
MOLBE
−155.336
314.673
317.475
2.259080
0.323858
IL
−153.639
310.279
315.081
0.458900
0.066987We can use the likelihood ratio (LR) test to compare the fit of the ALTIL distribution with other models for given data sets. The form of the test is suggested its name(55)LRT=2logLsθ^Lgθ^,where the LR is the ratio of two likelihood functions; the simpler model (s) has fewer parameters than the general (g) model. The LR test rejects the null hypothesis if χ2>χd2, where χd2 denotes the upper 100% point of the χ2 distribution.The shape hazard function for modeling can be analyzed using graphical technique called total time in the test (TTT) plot (for more details, see [31]). From Figure 4, for the first and second data, the TTT plot is concave and provides evidence of the monotonic hazard rate. For the third data set, The TTT plot is convex and according to [31], it provides evidence that the hazard rate is decreasing.Figure 4
TTT Plots for all three data sets.
(a)
(b)
(c)The APTIL distribution gives the lowest values of AIC, BIC,A∗, and W∗ tests among all the fitted models to these data sets. So, it could be selected as the best model among them. The fitted pdf and estimated cumulative distribution function of the APTIL are displayed in Figures 5–7 for the three data sets, respectively.Figure 5
Estimated pdf and cdf of the APTIL model and other competing models for first data.
(a)
(b)Figure 6
Estimated pdf and cdf of APTIL model and other competing models for second data.
(a)
(b)Figure 7
Estimated pdf and cdf of the APTIL model and other competing models for third data.
(a)
(b)The empirical data and estimated density plot show closeness which depict that the APTIL model fits all three data sets well. The APTIL model is compared with other competitive models. The estimated cdf curve of APTIL model also confirms the above results.The likelihood ratio test is performed to compare APTIL distribution with other fitted models to testHo against H1 discussed above, and results are shown in Table 8. Low values of the likelihood ratio mean that the observed result was much less likely to occur under the null hypothesis as compared with the alternative. We conclude that APTIL distribution provides better fit than all other competitive models at the level of significance ≤0.05.Table 8
Likelihood ratio test results for data sets.
Model
Chi-square
df
p value
Data set I
APTIL vs APIE
40.234
1
2.252929e − 10
APTIL vs MOLBE
16.508
1
4.844524e − 05
APTIL vs APP
93.370
1
4.337761e − 22
APTIL vs IL
38.668
1
5.023782e − 10
Data set II
APTIL vs APIE
7.0580
1
0.007891205
APTIL vs MOLBE
2.3320
1
0.054483800
APTIL vs APP
39.084
1
4.059573e − 10
APTIL vs IL
16.042
1
6.195283e − 05
Data set III
APTIL vs APIE
2.9240
1
0.05727128
APTIL vs MOLBE
6.8520
1
0.00885421
APTIL vs APP
8.2300
1
0.00412035
APTIL vs IL
3.4580
1
0.05294641


## 7. Conclusions

In this research, we proposed and studied the APTIL distribution. Some structural characteristics of the APTIL distribution are derived. The asymptotic behavior of its density function is studied. Characterization related to hazard rate function is also obtained. Estimation of the population parameters is achieved using eight various procedures. Simulation results are carried out to assess the performance of estimators. Real data sets are used for the applications to show the flexibility of the APTIL model.



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*Source: 1860813-2020-02-27.xml*