# Alpha Power Generalized Inverse Rayleigh Distribution: Its Properties and Applications

Authors: Muhammad Ali; Alamgir Khalil; Zahra Almaspoor; Sundus Hussain; Umair Khalil; M. El-Morshedy
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This manuscript is related with the development of Alpha Power Generalized Inverse Rayleigh (APGIR) Distribution. The suggested model provides fit of life time data more efficiently. Some of the important characteristics of the suggested model are obtained including moments, moment generating function, quantile, mode, order statistics, stress-strength parameter, and entropies. Parameter estimates are obtained by MLE technique. The performance of the suggested model is evaluated using real-world data sets. The findings of the simulation and real data sets suggest that the newly proposed model is superior to other current competitor models.


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## Body

## 1. Introduction

Rayleigh distribution (RD) is a special model and a modified form of Weibull distribution when shape parameter equals 2. The RD has many applications in various disciplines including engineering and medical sciences, astronomy, and Physics. The RD has been well investigated in the literature. Some researchers have examined its significant properties [1–3]. Hoffman and Karst [4] studied characteristics of the RD and demonstrated how it can be used to analyze the responses of marine vehicles to wave excitation. Dyer and Whisenand [5] also demonstrated the use of RD in communication engineering. Polovko [6] showed how it can be applied to electro vacuum devices. There are various variants of RD recently introduced by researchers that may be used for fitting of data more adequately. Voda [7] proposed generalized Rayleigh (GR) distribution. Voda [8, 9] obtained the ML estimates of the RD. Bhattacharya and Tyagi [10] used RD for the analysis of medical data. Gomes et al. [11] suggested Kumaraswamy generalized Rayleigh (KGR) distribution. Merovci [12] presented transmuted Rayleigh (TR) distribution for investigating lifetime data. Cordeiro et al. [13] developed beta generalized Rayleigh (BGR) distribution. They also studied its main mathematical features. Leao et al. [14] proposed beta inverse Rayleigh (BIR) distribution. Ahmad et al. [15] offered transmuted inverse Rayleigh (TIR) distribution. Iriarte et al. [16] proposed slashed generalized Rayleigh (SGR) distribution. Lalitha and Mishra [17], Ariyawansa and Templeton [18], Howlader and Hossain [19], Sinha and Howlader [20], and Abd Elfattah et al. [21] are just few among others who contributed to RD.LetX be a random variable having Rayleigh distribution. Symbolically, X∼Rθ. Then, its CDF and PDF are(1)fx=2θ2xexp−θx2x≥0,θ>0,Fx=1−exp−θx2x≥0,θ>0,where θ represents scale parameter.One important variant of RD is the Inverse Rayleigh Distribution (IRD), an important lifetime distribution. IfX follows RD, then (1/X) has the IRD. The PDF and CDF of IRD are provided by(2)gy;ω=2ω2x3exp−ωx2x,ω>0,Gx;ω=exp−ωx2x,ω>0.It has several uses in different fields including reliability analysis, engineering, and medicine. Voda [22] used the IRD to estimate the lifetime distribution of many experimental units. Trayer [23] proposed the IRD to accommodate survival and reliability data. Voda [22] discussed several properties and derived expression of ML estimator for parameters of IRD. Mukarjee and Maitim [24] also studied some important statistical properties of IRD. Closed form expressions for some descriptive statistics of the IR distribution were developed by Gharraph [25]. Furthermore, Soliman et al. [26] and Gharraph [25] obtained parameter estimates of IRD using classical and Bayesian estimating approaches, respectively. Various extensions of the IRD are available in the literature. These generalized forms have been used in different disciplines comprising survival and reliability analysis and so on. Rehman and Dar [27], Ahmad et al. [15], and Leao et al. [14] developed EIR, TIR, and BIR distributions, respectively. ShuaibKhan [28] developed a modified form of IRD and discussed it in depth. Potdar and Shirke [29] added an additional shape parameter to scale family of distributions, resulting in generalized inverted scale family of distributions. These distributions fit the complex data better, and conclusions made from them appeared to be quite comprehensive. Mudholkar et al. [30], Gupta et al. [31], Nadarajah and Kotz [32], and Mudholkar and Srivastava [33] studied generalization of several distributions in various statistical publications, generally employed in reliability estimation.Reshi et al. [34] analyzed scale parameter of Generalized Inverse Rayleigh (GIR) distribution. The GIR distribution is quite good at fitting lifetime data. Some of the applications of GIR distribution include reliability analysis, operations research, applied statistics, and communication engineering. Bakoban and Abu Baker [35] discussed many important characteristics of GIR distribution.The PDF and CDF of GIR distribution are specified by(3)gx=2γθ2x3exp−θx−21−exp−θx−2γ−1,x,γ,θ>0,(4)Gx=1−1−exp−θx−2γγ,θ>0.Here,γ and θ represent scale and shape parameter, respectively.In statistical theory, new distributions have been developed in the last few decades by incorporating a spare parameter, employing generators, or mixing existing distributions [36]. The major goal of doing so is to improve the modelling flexibility of lifetime data when compared with existing distributions.This article is about the development of new probability distribution, known as Alpha Power Generalized Inverse Rayleigh (APGIR) distribution. This new model is obtained using Alpha Power Transformation [37].


## 2. Alpha Power Transformation (APT)

The APT was proposed by Mahdavi and Kundu [37]. This technique can be used to develop new distributions by introducing a new parameter into available distributions.The following is CDF and PDF of APT:(5)FAPTx=αFx−1α−1,ifα>0,α≠1.Fx,ifα=1,and(6)fAPTx=logαα−1fxαFx,ifα>0,α≠1,fx,ifα=1.Initially, the proposed method of Mahdavi and Kundu [37] was used for the inclusion of additional parameter in exponential distribution. Later on, some other researchers used APT to some other distributions. Hassan et al. [38] used APT and proposed alpha power transformed extended exponential distribution. Nassar et al. [39] proposed Alpha Power Weibull distribution. Dina and Magdy [40] and Ihtisham et al. [41] introduced alpha power inverse Weibull (APIW) and alpha power Pareto (APP) distribution, respectively.


### 2.1. The Proposed Model

The main goal of this article is to develop a novel probability distribution termed as Alpha Power Generalized Inverse Rayleigh (APGIR) Distribution and to evaluate its flexibility in modelling life time data. The proposed model is a result of using the PDF and CDF of GIR distribution given in (3) and (4).A random variableX is said to have Alpha Power Generalized Inverse Rayleigh distributed with three-parameters α,λ, and β if its PDF is given by(7)fAPGIRx=logαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2β,x>0,α>1,β.λ>0,fx,α=1,0,otherwise.Definition 1.
. A variableX follows Alpha Power Generalized Inverse Rayleigh distributed with CDF as follows:(8)FAPGIRx=α1−1−exp−λx−2β−1α−1,α>1,0,α=1.
The following are APGIR Hazard Rate (HR) Function and Survival Function (SF):(9)hAPGIR=pdfsurvivalfunction,hAPGIRx=2βlogαexp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βλ2x3α−α1−1−exp−λx−2β,α>1,(10)SAPGIR=1−CDF,SAPGIRx=α−α1−1−exp−λx−2βα−1,α>1.
The functions PDF, CDF, HF, and SF are plotted in Figures1(a), 1(b), 2(a), and 2(b), respectively.Figure 1
(a) Graph of PDF. (b) Graph of CDF.
(a)(b)Figure 2
(a) Graphs of HF. (b) Graphs of SF.
(a)(b)Lemma 1.
Iffx is a decreasing function for α<1, then fAPGIRx is also decreasing function.Proof.
Iffx is differentiable function and d/dxlogfx<0, then fx is also decreasing function and vice versa.
Taking the first derivative of the following expression, i.e.,(11)ddxlogfAPGIRx=ddxloglogαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2β,ddxlogfAPTPRx=−3x−2exp−λx−2λ2x31+β−1+βlogα1−exp−λx−2β−1.
For non-negative and less than 1 values ofα and for λ and β>0, it is clear that(12)ddxlogfAPGIRx<0.
Hence, forα<1, fAPGIRx is decreasing function.Lemma 2.
Iffx is decreasing function for α<1, fx is log-convex and then hAPGIRx is decreasing function.Proof.
Ifd2/dx2, fx exist and d2/dx2logfx>0, then fx is log-convex.
Differentiating (11), we get(13)d2dx2logfAPGIRx=3x2−2λ2exp−λx−22−3λ2x2λ2x61+β−1+βlogα1−exp−λx−2β−1−exp−2λx−2λ2x62βlogαβ−11−exp−λx−2β−2.
Whenα is non-negative and less than 1 and when λ and β>0, then d2/dx2logfAPGIRx>0.
Thus, when0<α<1, fAPGIRx is log-convex [42].


### 2.2. Quantile Function (QF)

LetX∼APGIR α,λ,β, then the QF is described by(14)x=F−1u,where u∼U0,1. The QF of APGIR distribution is(15)α1−1−exp−λx−2β−1α−1=u.After simplification, we have(16)Xp=1λ−log1−logα−loguα−1+1logα1/β−1/2.


### 2.3. Median

To obtain median, we have(17)α1−1−exp−λx−2β−1α−1=12.After some calculations, we obtain the following result of median:(18)Median=m=1λ−log1−2logα−α+12logα1/β−1/2.


### 2.4. Mode

To obtain mode, we have(19)ddxfAPGIRx=0⇒ddxlogαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2β=0,−3x−4exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2β+x−3exp−λx−22λλx−31−exp−λx−2β−1α1−1−exp−λx−2β−x−3exp−λx−2β−11−exp−λx−2β−2exp−λx−22λλx−3α1−1−exp−λx−2β+x−3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βlogαβ1−exp−λx−2β−1exp−λx−22λλx−3=0,−3x+2λ2x31−β−1exp−λx−211−exp−λx−2+βexp−λx−21−exp−λx−2=0.Equation (19) is satisfied by mode of APGIR distribution.


### 2.5. Rth Moment of APGIR Distribution

LetX ∼APGIR α,λ,β, then the following is the rth moment:(20)μ/r=EXr=∫0∞xrlogαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βdx.Put in (20) 1−exp−λx−2=y⇒2/λ2x3exp−λx−2dx=−dy, x=1/λ−log1−y−1/2, and x⟶0,y⟶1x⟶∞,y⟶0.(21)μ/r=βlogαα−11λr∫01−log1−y−r2yβ−1α1−yβdy.Letyβ=z⇒βyβ−1dy=dz, y⟶0,z⟶0y⟶1,z⟶1, and y=z1/β.(22)21⇒μ/r=αlogαα−11λr∫01−log1−z1/β−r/2α−zdz.Using the following series representation in (22),(23)−log1−z1/β=∑m=1∞−1m−z1/βmmforz1/β<1,α−z=∑k=0∞−logαkk!zk,to have(24)μ/r=αlogαα−11λr∑k=0∞−logαk/k!∫01∑m=1∞−1m−z1/βm/m−r/2zkdz.The expression ofμ/r is incomplete integral; therefore, it can be solved approximately using numerical integration techniques.


### 2.6. Moment Generating Function (MGF)

LetX ∼APGIR α,λ,β, then MGF is defined as follows:(25)Mxt=Eetx=∫01etxlogαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βdx.Using series notationetx=∑r=0∞trxr/r! in (25), we get(26)Mxt=∑r=0∞tr/r!∫0∞xr2βλ2x3logαα−1exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βdx.Utilize (24) in (26), we get(27)Mxt=αlogαα−1∑r=0∞∑k=0∞trr!−logαkλrk!∫01∑m=1∞−1m−z1/βmm−r/2zkdz.The result in equation (27) is incomplete integral, and it may be solved on the basis of numerical integration methods.


### 2.7. Mean Residual Life Function (MRLF)

The MRLF is the average remaining life of a component that has survived till timet. Here, X is lifetime of an object with fx and Sx provided in (7) and (10), respectively. The MRLF is given by(28)μt=1StEt−∫01xfxdx−t,t≥0,where(29)∫01xfxdx=∫01xlogαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βdx.Let1−exp−λx−2=y⇒2/λ2x3exp−λx−2dx=−dy,x=1/λ−log1−y−1/2.Then,(31)30⇒∫0txfxdx=−logαλα−1∫11−exp−λt−2−log1−y−1/2βyβ−1α1−yβdy.Put1−yβ=z⇒βyβ−1dy=−dz, y=1−z1/β to have(32)∫01xfxdx=logαλα−1∫01−1−exp−λt−2β−log1−1−z1/β−1/2αzdz.Using the following series representation in (32), we have −log1−1−z1/β=∑k=1∞−1k−1−z1/βk/k and αz=∑n=0∞logαn/n!zn:(33)∫01xfxdx=logαλα−1∑n=0∞logαnn!∫01−1−exp−λt−2β∑k=1∞−1k−1−z1βk/k−1/2zndz.The expression in (33) is an integral that is incomplete. This may be solved approximately using numerical integration techniques.(34)Et=∫0∞tft=∫0∞tlogαα−12βλ2t3exp−λt−21−exp−λt−2β−1α1−1−exp−λt−2βdt.Insert1−exp−λt−2=y⇒2/λ2t3&ExponentialE;xp−λt−2dt=−dy and t=1/λ−log1−y−1/2 in (34) to have(35)Et=βlogαλα−1∫01−log1−y−1/2yβ−1α1−yβdy.Put1−yβ=z⇒βyβ−1dy=−dz, y=1−z1/β in (35) to have(36)Et=logαλα−1∫01−log1−1−z1/β−1/2αzdz.Using the following series representation in (36), we have(37)−log1−1−z1/β=∑k=1∞−1k−1−z1/βk/k,αz=∑n=0∞logαn/n!zn,Et=logαλα−1∑n=0∞logαnn!∫01∑k=1∞−1k−1−z1/βkk−1/2zndz.Putting (10), (31), and (35) in (28), we get(38)μt=logαλα−α1−1−e−λx−2β∑n=0∞logαnn!∫01∑k=1∞−1k−1−z1/βkk−1/2zndz−∫01−1−e−λt−2β∑k=1∞−1k−1−z1/βkk−1/2zndz−t.The result ofμt is an incomplete integral. Numerically, it can be approximated utilizing numerical integration techniques.


### 2.8. Order Statistics

SupposeX1,X2,X3,...,Xn denote sample of size n. The corresponding order statistics are X1≤X2≤....≤Xn. The PDF of ith order statistic is specified by(39)fi:nx=n!i−1!n−i!fxFxi−11−Fxn−i.Substitutingfx and Fx in (39), we get(40)fi:nx=n!i−1!n−i!α−1n2βlogαλ2x3exp−λx−2α1−1−exp−λx−2β−1i−11−exp−λx−2β−1α1−1−exp−λx−2βα−α1−1−exp−λx−2βn−i.We get PDF of the smallest order statistic by insertingi=1 in (40), that is,(41)f1:nx=nα−1n2βlogαλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βα−α1−1−exp−λx−2βn−1.Puti=n in (40), we acquire the PDF of the largest order statistic(42)fn:nx=nα−1n2βlogαλ2x3exp−λx−2α1−1−exp−λx−2β−1n−11−exp−λx−2β−1α1−1−exp−λx−2β.To get distribution of the median, substitutei=n/2 in (40) as(43)fn2:nx=n!n/2−1!n−n/2!α−1n2βlogαλ2x3exp−λx−2α1−1−exp−λx−2β−1n/2−11−exp−λx−2β−1α1−1−exp−λx−2βα−α1−1−exp−λx−2βn−n/2.


### 2.9. Stress-Strength Parameter (SSP)

LetX1 and X2 be two independent and identically distributed random variables. Suppose X1∼APGIRα1,λ,β1 and X2∼APGIRα2,λ,β2. The SSP is defined as follows:(44)R=∫−∞∞f1xF2xdx.The SSP is calculated, by incorporating (7) and (8) in the above equation:(45)R=∫0∞logα1α1−12β1λ2x3exp−λx−21−exp−λx−2β1−1α11−1−exp−λx−2β1α21−1−exp−λx−2β2−1α2−1dx.R=∫0∞logα1α1−12β1λ2x3exp−λx−21−exp−λx−2β1−1α11−1−exp−λx−2β1α21−1−exp−λx−2β2−1α2−1dx−1α2−1.Substituting1−exp−λx−2=y⇒2/λ2x3exp−λx−2dx=−dy in (45), we have(46)R=βα1α2logα1α1−1α2−1∫01yβ1−1α1−yβ1α2−yβ2dy−1α2−1.Using series representationα1−yβ1=∑k=0∞−logα1k/k!yβ1k and α2−yβ2=∑m=0∞−logα2m/m!yβ2m in (46) and simplifying, we get the following final result for stress-strength parameter:(47)R=βα1α2logα1α1−1α2−1∑m=0∞∑k=0∞−logα1k−logα2mk!m!β11+k+β2m−1α2−1.Lemma 3.
LetX∼APGIRα,λ,β, then final expression for Renyi entropy is given as follows:(48)REXv=11−vνlog2βlogαλ2α−1−logαννk2r−1λ1−3ν∑k=0∞∑r=0∞logαk−1k+lr!k!∫0∞tl+r1−etmdt.Proof.
Renyi entropy is defined as(49)REXv=11−vlog∫−∞+∞fxvdx,REXv=11−vlog∫0∞logαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βvdx,REXv=11−vlog2βlogαλ2α−1ν+log∫0∞x−3νexp−νλx−21−exp−λx−2νβ−1αν−ν1−exp−λx−2βdx.
Substitute1−exp−λx−2=y to have(50)REXv=11−vνlog2βlogαλ2α−1+logαν2λ1−3ν∫01−log1−y3ν−1/21−yν−1yνβ−1α−νyβdy.
Using series representationα−νyβ=∑k=0∞−logαk/k!νyβk in the above equation, we get(51)REXv=11−vνlog2βlogαλ2α−1+logαννk2λ1−3ν∑k=0∞−logαkk!∫01−log1−y3ν−1/21−yν−1yνβ−1+βkdy.
Usinglog1−y=t⇒dy=y−1dt⇒dy=−etdt in (51) and simplifying, we get(52)REXv=11−vνlog2βlogαλ2α−1−logαννk2r−1λ1−3ν∑k=0∞∑r=0∞logαk−1k+lr!k!∫01tl+r1−etmdt.
The expression of Renyi entropy is an incomplete integral. The solution of (52) is obtained on the basis of numerical integration techniques.Lemma 4.
The Mean Waiting Time (MWT) sayμ¯t is given by(53)μ¯t=t−logαλα1−1−e−λt−2β−1∑n=0∞logαnn!∫01−1−e−λt−2β∑k=1∞−1k−1−z1/βkk−1/2zndz.Proof.
The MWT of APGIR distribution is described as(54)μ¯t=t−1Ft∫0txfxdx.
Substituting the following results in (54),(55)∫0txfxdx=logαλα−1∑n=0∞logαnn!∫01−1−e−λt−2β∑k=1∞−1k−1−z1/βkk−1/2zndz,and(56)Ft=α1−1−exp−λt−2β−1α−1,we obtain the required final expression as(57)μ¯t=t−α−1α1−1−exp−λt−2β−1logαλα−1∑n=0∞logαnn!∫01−1−exp−λt−2β∑k=1∞−1k−1−z1/βkk−1/2zndz.
The expression forμ¯t is an integral that is incomplete. The solution of (57) may be obtained by numerical integration techniques.Lemma 5.
The Shannon entropy (SE) expression is given as follows:(58)S.Ex=loglogαα−122λβ∑n=0∞logαn2nn!∫011−z1−z−1/β−1zn∑k=1∞−1k−1−z1/βkk3/2dz.Proof.
The Shannon entropy is described by(59)S.Ex=E−logfx=E−loglogαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2β.(60)S.Ex=−log2βlogαλ2α−1E1x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2β,S.Ex=−log2βlogαλ2α−1∫0∞1x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βlogαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βdx.
Putting1−exp−λx−2=y in (60), we get(61)S.Ex=log2βλ2logαα−12∫011−yβyβ−1α1−yβ11/λ−log1−y−1/23yβ−1α1−yβdx.
Insert in (61), 1−yβ=z⇒βyβ−1dy=−dz, and y=1−z1/β to have(62)S.Ex=log2βλlogαα−12∫011−z1−z−1/β−1α2z−log1−1−z1/β3/2dz.
Using the following series in (62), α2z=∑m=0∞logαm2zm/m! and −log1−1−z1/β=∑k=1∞−1k−1−z1/βk/k,for1−z1/β<1.
We get the Shannon entropy as(63)S.Ex=loglogαα−122λβ∑n=0∞logαn2nn!∫011−z1−z−1/β−1zn∑k=1∞−1k−1−z1/βkk3/2dz.
The integral in (63) may be solved approximately with the help of numerical integration techniques.


## 2.1. The Proposed Model

The main goal of this article is to develop a novel probability distribution termed as Alpha Power Generalized Inverse Rayleigh (APGIR) Distribution and to evaluate its flexibility in modelling life time data. The proposed model is a result of using the PDF and CDF of GIR distribution given in (3) and (4).A random variableX is said to have Alpha Power Generalized Inverse Rayleigh distributed with three-parameters α,λ, and β if its PDF is given by(7)fAPGIRx=logαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2β,x>0,α>1,β.λ>0,fx,α=1,0,otherwise.Definition 1.
. A variableX follows Alpha Power Generalized Inverse Rayleigh distributed with CDF as follows:(8)FAPGIRx=α1−1−exp−λx−2β−1α−1,α>1,0,α=1.
The following are APGIR Hazard Rate (HR) Function and Survival Function (SF):(9)hAPGIR=pdfsurvivalfunction,hAPGIRx=2βlogαexp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βλ2x3α−α1−1−exp−λx−2β,α>1,(10)SAPGIR=1−CDF,SAPGIRx=α−α1−1−exp−λx−2βα−1,α>1.
The functions PDF, CDF, HF, and SF are plotted in Figures1(a), 1(b), 2(a), and 2(b), respectively.Figure 1
(a) Graph of PDF. (b) Graph of CDF.
(a)(b)Figure 2
(a) Graphs of HF. (b) Graphs of SF.
(a)(b)Lemma 1.
Iffx is a decreasing function for α<1, then fAPGIRx is also decreasing function.Proof.
Iffx is differentiable function and d/dxlogfx<0, then fx is also decreasing function and vice versa.
Taking the first derivative of the following expression, i.e.,(11)ddxlogfAPGIRx=ddxloglogαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2β,ddxlogfAPTPRx=−3x−2exp−λx−2λ2x31+β−1+βlogα1−exp−λx−2β−1.
For non-negative and less than 1 values ofα and for λ and β>0, it is clear that(12)ddxlogfAPGIRx<0.
Hence, forα<1, fAPGIRx is decreasing function.Lemma 2.
Iffx is decreasing function for α<1, fx is log-convex and then hAPGIRx is decreasing function.Proof.
Ifd2/dx2, fx exist and d2/dx2logfx>0, then fx is log-convex.
Differentiating (11), we get(13)d2dx2logfAPGIRx=3x2−2λ2exp−λx−22−3λ2x2λ2x61+β−1+βlogα1−exp−λx−2β−1−exp−2λx−2λ2x62βlogαβ−11−exp−λx−2β−2.
Whenα is non-negative and less than 1 and when λ and β>0, then d2/dx2logfAPGIRx>0.
Thus, when0<α<1, fAPGIRx is log-convex [42].


## 2.2. Quantile Function (QF)

LetX∼APGIR α,λ,β, then the QF is described by(14)x=F−1u,where u∼U0,1. The QF of APGIR distribution is(15)α1−1−exp−λx−2β−1α−1=u.After simplification, we have(16)Xp=1λ−log1−logα−loguα−1+1logα1/β−1/2.


## 2.3. Median

To obtain median, we have(17)α1−1−exp−λx−2β−1α−1=12.After some calculations, we obtain the following result of median:(18)Median=m=1λ−log1−2logα−α+12logα1/β−1/2.


## 2.4. Mode

To obtain mode, we have(19)ddxfAPGIRx=0⇒ddxlogαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2β=0,−3x−4exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2β+x−3exp−λx−22λλx−31−exp−λx−2β−1α1−1−exp−λx−2β−x−3exp−λx−2β−11−exp−λx−2β−2exp−λx−22λλx−3α1−1−exp−λx−2β+x−3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βlogαβ1−exp−λx−2β−1exp−λx−22λλx−3=0,−3x+2λ2x31−β−1exp−λx−211−exp−λx−2+βexp−λx−21−exp−λx−2=0.Equation (19) is satisfied by mode of APGIR distribution.


## 2.5. Rth Moment of APGIR Distribution

LetX ∼APGIR α,λ,β, then the following is the rth moment:(20)μ/r=EXr=∫0∞xrlogαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βdx.Put in (20) 1−exp−λx−2=y⇒2/λ2x3exp−λx−2dx=−dy, x=1/λ−log1−y−1/2, and x⟶0,y⟶1x⟶∞,y⟶0.(21)μ/r=βlogαα−11λr∫01−log1−y−r2yβ−1α1−yβdy.Letyβ=z⇒βyβ−1dy=dz, y⟶0,z⟶0y⟶1,z⟶1, and y=z1/β.(22)21⇒μ/r=αlogαα−11λr∫01−log1−z1/β−r/2α−zdz.Using the following series representation in (22),(23)−log1−z1/β=∑m=1∞−1m−z1/βmmforz1/β<1,α−z=∑k=0∞−logαkk!zk,to have(24)μ/r=αlogαα−11λr∑k=0∞−logαk/k!∫01∑m=1∞−1m−z1/βm/m−r/2zkdz.The expression ofμ/r is incomplete integral; therefore, it can be solved approximately using numerical integration techniques.


## 2.6. Moment Generating Function (MGF)

LetX ∼APGIR α,λ,β, then MGF is defined as follows:(25)Mxt=Eetx=∫01etxlogαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βdx.Using series notationetx=∑r=0∞trxr/r! in (25), we get(26)Mxt=∑r=0∞tr/r!∫0∞xr2βλ2x3logαα−1exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βdx.Utilize (24) in (26), we get(27)Mxt=αlogαα−1∑r=0∞∑k=0∞trr!−logαkλrk!∫01∑m=1∞−1m−z1/βmm−r/2zkdz.The result in equation (27) is incomplete integral, and it may be solved on the basis of numerical integration methods.


## 2.7. Mean Residual Life Function (MRLF)

The MRLF is the average remaining life of a component that has survived till timet. Here, X is lifetime of an object with fx and Sx provided in (7) and (10), respectively. The MRLF is given by(28)μt=1StEt−∫01xfxdx−t,t≥0,where(29)∫01xfxdx=∫01xlogαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βdx.Let1−exp−λx−2=y⇒2/λ2x3exp−λx−2dx=−dy,x=1/λ−log1−y−1/2.Then,(31)30⇒∫0txfxdx=−logαλα−1∫11−exp−λt−2−log1−y−1/2βyβ−1α1−yβdy.Put1−yβ=z⇒βyβ−1dy=−dz, y=1−z1/β to have(32)∫01xfxdx=logαλα−1∫01−1−exp−λt−2β−log1−1−z1/β−1/2αzdz.Using the following series representation in (32), we have −log1−1−z1/β=∑k=1∞−1k−1−z1/βk/k and αz=∑n=0∞logαn/n!zn:(33)∫01xfxdx=logαλα−1∑n=0∞logαnn!∫01−1−exp−λt−2β∑k=1∞−1k−1−z1βk/k−1/2zndz.The expression in (33) is an integral that is incomplete. This may be solved approximately using numerical integration techniques.(34)Et=∫0∞tft=∫0∞tlogαα−12βλ2t3exp−λt−21−exp−λt−2β−1α1−1−exp−λt−2βdt.Insert1−exp−λt−2=y⇒2/λ2t3&ExponentialE;xp−λt−2dt=−dy and t=1/λ−log1−y−1/2 in (34) to have(35)Et=βlogαλα−1∫01−log1−y−1/2yβ−1α1−yβdy.Put1−yβ=z⇒βyβ−1dy=−dz, y=1−z1/β in (35) to have(36)Et=logαλα−1∫01−log1−1−z1/β−1/2αzdz.Using the following series representation in (36), we have(37)−log1−1−z1/β=∑k=1∞−1k−1−z1/βk/k,αz=∑n=0∞logαn/n!zn,Et=logαλα−1∑n=0∞logαnn!∫01∑k=1∞−1k−1−z1/βkk−1/2zndz.Putting (10), (31), and (35) in (28), we get(38)μt=logαλα−α1−1−e−λx−2β∑n=0∞logαnn!∫01∑k=1∞−1k−1−z1/βkk−1/2zndz−∫01−1−e−λt−2β∑k=1∞−1k−1−z1/βkk−1/2zndz−t.The result ofμt is an incomplete integral. Numerically, it can be approximated utilizing numerical integration techniques.


## 2.8. Order Statistics

SupposeX1,X2,X3,...,Xn denote sample of size n. The corresponding order statistics are X1≤X2≤....≤Xn. The PDF of ith order statistic is specified by(39)fi:nx=n!i−1!n−i!fxFxi−11−Fxn−i.Substitutingfx and Fx in (39), we get(40)fi:nx=n!i−1!n−i!α−1n2βlogαλ2x3exp−λx−2α1−1−exp−λx−2β−1i−11−exp−λx−2β−1α1−1−exp−λx−2βα−α1−1−exp−λx−2βn−i.We get PDF of the smallest order statistic by insertingi=1 in (40), that is,(41)f1:nx=nα−1n2βlogαλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βα−α1−1−exp−λx−2βn−1.Puti=n in (40), we acquire the PDF of the largest order statistic(42)fn:nx=nα−1n2βlogαλ2x3exp−λx−2α1−1−exp−λx−2β−1n−11−exp−λx−2β−1α1−1−exp−λx−2β.To get distribution of the median, substitutei=n/2 in (40) as(43)fn2:nx=n!n/2−1!n−n/2!α−1n2βlogαλ2x3exp−λx−2α1−1−exp−λx−2β−1n/2−11−exp−λx−2β−1α1−1−exp−λx−2βα−α1−1−exp−λx−2βn−n/2.


## 2.9. Stress-Strength Parameter (SSP)

LetX1 and X2 be two independent and identically distributed random variables. Suppose X1∼APGIRα1,λ,β1 and X2∼APGIRα2,λ,β2. The SSP is defined as follows:(44)R=∫−∞∞f1xF2xdx.The SSP is calculated, by incorporating (7) and (8) in the above equation:(45)R=∫0∞logα1α1−12β1λ2x3exp−λx−21−exp−λx−2β1−1α11−1−exp−λx−2β1α21−1−exp−λx−2β2−1α2−1dx.R=∫0∞logα1α1−12β1λ2x3exp−λx−21−exp−λx−2β1−1α11−1−exp−λx−2β1α21−1−exp−λx−2β2−1α2−1dx−1α2−1.Substituting1−exp−λx−2=y⇒2/λ2x3exp−λx−2dx=−dy in (45), we have(46)R=βα1α2logα1α1−1α2−1∫01yβ1−1α1−yβ1α2−yβ2dy−1α2−1.Using series representationα1−yβ1=∑k=0∞−logα1k/k!yβ1k and α2−yβ2=∑m=0∞−logα2m/m!yβ2m in (46) and simplifying, we get the following final result for stress-strength parameter:(47)R=βα1α2logα1α1−1α2−1∑m=0∞∑k=0∞−logα1k−logα2mk!m!β11+k+β2m−1α2−1.Lemma 3.
LetX∼APGIRα,λ,β, then final expression for Renyi entropy is given as follows:(48)REXv=11−vνlog2βlogαλ2α−1−logαννk2r−1λ1−3ν∑k=0∞∑r=0∞logαk−1k+lr!k!∫0∞tl+r1−etmdt.Proof.
Renyi entropy is defined as(49)REXv=11−vlog∫−∞+∞fxvdx,REXv=11−vlog∫0∞logαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βvdx,REXv=11−vlog2βlogαλ2α−1ν+log∫0∞x−3νexp−νλx−21−exp−λx−2νβ−1αν−ν1−exp−λx−2βdx.
Substitute1−exp−λx−2=y to have(50)REXv=11−vνlog2βlogαλ2α−1+logαν2λ1−3ν∫01−log1−y3ν−1/21−yν−1yνβ−1α−νyβdy.
Using series representationα−νyβ=∑k=0∞−logαk/k!νyβk in the above equation, we get(51)REXv=11−vνlog2βlogαλ2α−1+logαννk2λ1−3ν∑k=0∞−logαkk!∫01−log1−y3ν−1/21−yν−1yνβ−1+βkdy.
Usinglog1−y=t⇒dy=y−1dt⇒dy=−etdt in (51) and simplifying, we get(52)REXv=11−vνlog2βlogαλ2α−1−logαννk2r−1λ1−3ν∑k=0∞∑r=0∞logαk−1k+lr!k!∫01tl+r1−etmdt.
The expression of Renyi entropy is an incomplete integral. The solution of (52) is obtained on the basis of numerical integration techniques.Lemma 4.
The Mean Waiting Time (MWT) sayμ¯t is given by(53)μ¯t=t−logαλα1−1−e−λt−2β−1∑n=0∞logαnn!∫01−1−e−λt−2β∑k=1∞−1k−1−z1/βkk−1/2zndz.Proof.
The MWT of APGIR distribution is described as(54)μ¯t=t−1Ft∫0txfxdx.
Substituting the following results in (54),(55)∫0txfxdx=logαλα−1∑n=0∞logαnn!∫01−1−e−λt−2β∑k=1∞−1k−1−z1/βkk−1/2zndz,and(56)Ft=α1−1−exp−λt−2β−1α−1,we obtain the required final expression as(57)μ¯t=t−α−1α1−1−exp−λt−2β−1logαλα−1∑n=0∞logαnn!∫01−1−exp−λt−2β∑k=1∞−1k−1−z1/βkk−1/2zndz.
The expression forμ¯t is an integral that is incomplete. The solution of (57) may be obtained by numerical integration techniques.Lemma 5.
The Shannon entropy (SE) expression is given as follows:(58)S.Ex=loglogαα−122λβ∑n=0∞logαn2nn!∫011−z1−z−1/β−1zn∑k=1∞−1k−1−z1/βkk3/2dz.Proof.
The Shannon entropy is described by(59)S.Ex=E−logfx=E−loglogαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2β.(60)S.Ex=−log2βlogαλ2α−1E1x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2β,S.Ex=−log2βlogαλ2α−1∫0∞1x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βlogαα−12βλ2x3exp−λx−21−exp−λx−2β−1α1−1−exp−λx−2βdx.
Putting1−exp−λx−2=y in (60), we get(61)S.Ex=log2βλ2logαα−12∫011−yβyβ−1α1−yβ11/λ−log1−y−1/23yβ−1α1−yβdx.
Insert in (61), 1−yβ=z⇒βyβ−1dy=−dz, and y=1−z1/β to have(62)S.Ex=log2βλlogαα−12∫011−z1−z−1/β−1α2z−log1−1−z1/β3/2dz.
Using the following series in (62), α2z=∑m=0∞logαm2zm/m! and −log1−1−z1/β=∑k=1∞−1k−1−z1/βk/k,for1−z1/β<1.
We get the Shannon entropy as(63)S.Ex=loglogαα−122λβ∑n=0∞logαn2nn!∫011−z1−z−1/β−1zn∑k=1∞−1k−1−z1/βkk3/2dz.
The integral in (63) may be solved approximately with the help of numerical integration techniques.


## 3. Parameters Estimation

### 3.1. Maximum Likelihood Estimation

LetX1,X2,…,Xn be a random sample drawn from APGIR α,λ,β, then likelihood function is as follows:(64)lα,λ,β=2βlogαλ2α−1n1∏i=1nxi3e−∑i=1nλxi−2∏i=1n1−exp−λx−2β−1αn−∑i=1n1−exp−λx−2β.Taking logarithm, (64) becomes(65)loglα,λ,β=nlog2βlogα−nlogλ2α−1−∑i−1nlogxi3−∑i=1nλxi−2+β−1∑i=1nlog1−exp−λxi−2+logαn−∑i=1n1−exp−λxi−2β.By differentiating (65) w.r.t α,λ,andβ and equating to 0, we get the following equations:(66)∂loglα,λ,β∂α=nαlogα−nα−1+1αn−∑i=1n1−exp−λxi−2β=0,(67)∂loglα,λ,β∂λ=−2nλ+2∑i=1nxiλxi−3−2β−1∑i=1nxiλxi−3exp−λxi−21−exp−λxi−2+2βlnα∑i=1nxiλxi−31−exp−λxi−2β−1exp−λxi−2=0,(68)∂loglα,λ,β∂β=nβ+∑i=1nlog1−exp−λxi−2−logα∑i=1n1−exp−λxi−2βlog1−exp−λxi−2=0.We can get estimates ofα,λ, and β by solving (64), (65), and (66) together. The Newton–Raphson technique was adopted for the solution of aforementioned equations. The ML estimators are asymptotically normally distributed, that is, nα^−α,λ^−λ,β^−β∼N30,Σ. The matrix ∑ is achieved by inverting the observed Fisher information matrix F as follows:(69)F=∂2logl∂α2∂2logl∂α∂λ∂2logl∂α∂β∂2logl∂λ∂α∂2logl∂λ2∂2logl∂λ∂β∂2logl∂β∂α∂2logl∂β∂λ∂2logl∂β2.When we differentiate (64)–(66) w.r.t α,λ,andβ, we get(70)∂2logl∂α2=−n1+logααlogα2+nα−12−1α2n−∑i=1n1−exp−λxi−2β,∂2logl∂λ2=2nλ2−6∑i=1n1xi2λ4−2β−1∑i=1n1xi2exp−λxi−2xi2λ61−exp−λxi−2−exp−λxi−23λ21−exp−λxi−2−2xi−2exp−λxi−2λ61−exp−λxi−2+2βlogα∑i=1nxi2λxi−41−exp−λxi−2β−1exp−λxi−2−3+2λxi−21−β−11−exp−λxi−2,∂2logl∂β2=−nβ2−logα∑i=1n1−exp−λxi−2βlog1−exp−λxi−22,∂2logl∂α∂λ=2βαλ3∑i=1nexp−λxi−21−exp−λxi−2β−1xi2,∂2logl∂α∂β=−1α∑i=1n1−exp−λxi−2βlog1−exp−λxi−2,∂2logl∂λ∂β=−2∑i=1nxiλxi−3exp−λxi−21−exp−λxi−2+2logα∑i=1nxiλxi−31−exp−λxi−2β−1+β1−exp−λxi−2β−1log1−exp−λxi−2.The asymptotic1−ζ100% confidence intervals for the parameters of suggested model are as follows:(71)α^±Zζ/2Σ11,λ^±Zζ/2Σ22,β^±Zζ/2Σ33.Here,Zζ represents the upper ζth percentile of standard normal distribution.


### 3.2. Simulation Study

Simulation is used to obtain estimates, Mean Square Error (MSE), and Bias of parameters. The following expression of QF was used to developw=100 samples of size n=50,90,and200, respectively:(72)Xp=1λ−log1−logα−loguα−1+1logα1/β−1/2,where u∼U0,1. The following expression is used to calculate bias and MSE:(73)Bias=1W∑1=1wbi^−b.MSE=1W∑1=1wbi^−b2,where b=α,λ,β. For various choices of α,λ,andβ, simulation results were obtained. Table 1 shows the simulated expected values of MSEs and bias. In Table 1, with increase in sample size, the consistency behavior may be easily observed as estimates approach their parametric values. Furthermore, as the sample size grows, the MSEs and bias of the estimates drop for all parameter combinations. As a result, we can infer that the MLE approach performs well when it comes to estimating the parameters of the APGIR distribution.Table 1
MSE and bias of MLE.
ParameterNMSE, α^MSEβ^MSE, λ^Biasα^Biasβ^Biasλ^α= 18.4510 β= 1.2510 λ= 0.65345045.61740.03880.3961−1.05900.0641−0.32219026.80810.03340.2198−0.62990.0584−0.184320023.11930.01370.17350.53170.0341−0.1389α= 17.4510 β= 1.4510 λ= 0.35345031.90160.09760.2236−1.24080.0827−0.32669018.44790.04540.2013−0.74850.0507−0.292020014.33890.01240.1530−0.5075−0.0053−0.2155α= 17.4510 β= 1.2510 λ= 0.65345044.22460.04630.3910−1.82670.0683−0.32089034.99310.02590.3383−1.63350.0210−0.273820017.57820.01330.18530.44440.0204−0.1502α= 17.4510 β= 1.3010 λ= 0.65345050.135030.05200.2882−1.96960.0756−0.25159033.50260.03340.2850−1.76210.0333−0.225520022.24690.01380.2051−0.00780.0326−0.1635α= 17.4510 β= 1.4510 λ= 0.60345039.78370.07530.4067−0.99350.0371−0.34369037.89690.04060.3365−0.27820.0349−0.283620018.05370.01280.17250.17830.0062−0.1470


### 3.3. Applications

To see the performance and goodness of fit of the proposed mode, the suggested distribution has been fitted to two data sets. We found that the suggested model performed better than other Rayleigh distribution variants such as the Two-Parameter Rayleigh (TPR) distribution proposed by Dey et al. [43], the MIR distribution suggested by Khan [28], the EIR distribution developed by Rehman and Dar [27], the GR distribution offered by Raqab and Madi [44], the TIR distribution by Ahmad et al. [15], and the GIR distribution by Potdar and Shirke [29].Data set 1: the first set of data includes the survival times (in years) of 46 individuals who received just chemotherapy. These data are a subset of the data taken from the study by Bekker et al. [45]. The data points are 0.047,0.115, 0.121, 0.132, 0.164, 0.197,0.203, 0.260, 0.282, 0.296, 0.334, 0.395, 0.458, 0.466, 0.501, 0.507, 0.529, 0.534, 0.540, 0.641, 0.644, 0.696, 0.841, 0.863, 1.099, 1.219, 1.271, 1.326, 1.447, 1.485, 1.553, 1.581, 1.589, 2.178,2.343, 2.416, 2.444, 2.825, 2.830, 3.578, 3.658, 3.743, 3.978, 4.003, 4.033.Data set 2: the failure times of 84 aircraft windshields are included in the second real data set. El-Bassiouny et al. [46] provided this information. The following are the data points:3.70, 2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11, 4.42, 2.41, 3.19, 3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.90, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.53, 2.67, 2.93, 3.22, 3.39, 2.81, 4.20, 3.33, 2.55, 3.31, 3.31, 2.85, 2.56, 3.56, 3.15, 2.35, 2.55, 2.59, 2.38, 2.81, 2.77, 2.17, 2.83, 1.92, 1.41, 3.68, 2.97, 1.36, 0.98, 2.76, 4.91, 3.68, 1.84, 1.59, 3.19, 1.57, 0.81, 5.56, 1.73, 1.59, 2.00, 1.22, 1.12, 1.71, 2.17, 1.17, 5.08, 2.48, 1.18, 3.51, 2.17, 1.69, 1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.70, 2.03, 1.80, 1.57, 1.08, 2.03, 1.61, 2.12, 1.89, 2.88, 2.82, 2.05, 3.65.PDF of GIR distribution is as follows:(74)fx=2βλ2x3exp−λx−21−exp−λx−2β−1x,β,λ>0.PDF of EIR distribution is as follows:(75)fx=2βθx3exp−βθx2,θ,β,X>0.PDF of GR distribution is as follows:(76)fx=2αγ2xexp−γx21−exp−γx2α−1,α,γ,X>0.PDF of TPR distribution is as follows:(77)fx=2αx−μexp−αx−μ2,x>μ,α>0.PDF of MIR distribution is as follows:(78)fx=α+2θx1x2exp−αx−θ1x2,α,θ,X>0.PDF of TIR distribution is as follows:(79)fx=2θx3exp−θx21+λ−2λexp−θx2,θ,X>0.PDF of WR distribution is as follows:(80)fx=αβθxexpθx22expθx22−1β−1e−αexpθx22−1β,α,θ,β,X>0.The APGIR model’s results are compared with other Rayleigh distribution versions using well-known model selection criteria such as Akaike’s Information Criteria (AIC), Consistent Akaike’s Information Criteria (CAIC), Bayesian Information Criterion (BIC), Hannan–Quinn Information Criteria (HQIC), and Kolmogorov–Smirnov (K-S) and theirP values via the R programming language’s Adequacy Model. The results are shown in Tables 2 and 3.Table 2
Goodness of fit results for data set 1.
DistributionMLEAICCAICBICHQICK-SP valueAPGIR10.46620.40918.4132137.2571137.8425142.6771139.27760.140510.3073GIR0.27207.4680148.2952148.5809151.9086149.64220.774845.551e − 16TIR0.0414−0.8352211.8820212.1677215.4953213.2290.431563.923e − 08APEIR10.86610.89340.0477196.7339197.3192202.1538198.75440.756775.551e − 16TPR0.1898−0.5973138.6129138.8986142.2262139.95990.216860.02449EIR0.16640.3224234.1737234.4594237.787235.52070.507722.82e − 11Table 3
Goodness of fit results for data set 2.
DistributionMLEAICCAICBICHQICK-SP valueAPGIR17.45101.45100.6534326.7218326.9718334.5373329.88490.151390.02043GIR1.1357−0.5285353.6821353.8058358.8924355.79080.171080.005739TIR1.9939−0.8843339.1298339.2535344.3401341.23850.153640.01781MIR−2.07686.4548338.6148338.7385343.8251340.72350.400022.531e − 14EIR7.99990.4095354.4818354.6055359.6921356.59050.182580.002544APEIR11.86420.28275.7878339.3922339.6422347.2077342.55530.781822.2e − 16On the basis of several model selection criteria, Tables2 and 3 show that our recommended distribution outperforms than other forms of Rayleigh distribution.Figure3 represents QQ and PP plot for data set 1. Figure 4 shows theoretical densities and CDFs for data set 1. The graphs clearly show better fit for data set 1. Figure 5 represents QQ and PP plot for data set 2. Figure 6 shows theoretical densities and CDFs for data set 2. It is clear from the figures that the data set 2 is better fitted by the proposed distribution.Figure 3
Graphs of QQ and PP plot for data set 1.Figure 4
Graphs of theoretical densities and CDFs for data set 1.Figure 5
Graphs of QQ and PP plot for data set 2.Figure 6
Graphs of theoretical densities and CDFs for data set 2.


## 3.1. Maximum Likelihood Estimation

LetX1,X2,…,Xn be a random sample drawn from APGIR α,λ,β, then likelihood function is as follows:(64)lα,λ,β=2βlogαλ2α−1n1∏i=1nxi3e−∑i=1nλxi−2∏i=1n1−exp−λx−2β−1αn−∑i=1n1−exp−λx−2β.Taking logarithm, (64) becomes(65)loglα,λ,β=nlog2βlogα−nlogλ2α−1−∑i−1nlogxi3−∑i=1nλxi−2+β−1∑i=1nlog1−exp−λxi−2+logαn−∑i=1n1−exp−λxi−2β.By differentiating (65) w.r.t α,λ,andβ and equating to 0, we get the following equations:(66)∂loglα,λ,β∂α=nαlogα−nα−1+1αn−∑i=1n1−exp−λxi−2β=0,(67)∂loglα,λ,β∂λ=−2nλ+2∑i=1nxiλxi−3−2β−1∑i=1nxiλxi−3exp−λxi−21−exp−λxi−2+2βlnα∑i=1nxiλxi−31−exp−λxi−2β−1exp−λxi−2=0,(68)∂loglα,λ,β∂β=nβ+∑i=1nlog1−exp−λxi−2−logα∑i=1n1−exp−λxi−2βlog1−exp−λxi−2=0.We can get estimates ofα,λ, and β by solving (64), (65), and (66) together. The Newton–Raphson technique was adopted for the solution of aforementioned equations. The ML estimators are asymptotically normally distributed, that is, nα^−α,λ^−λ,β^−β∼N30,Σ. The matrix ∑ is achieved by inverting the observed Fisher information matrix F as follows:(69)F=∂2logl∂α2∂2logl∂α∂λ∂2logl∂α∂β∂2logl∂λ∂α∂2logl∂λ2∂2logl∂λ∂β∂2logl∂β∂α∂2logl∂β∂λ∂2logl∂β2.When we differentiate (64)–(66) w.r.t α,λ,andβ, we get(70)∂2logl∂α2=−n1+logααlogα2+nα−12−1α2n−∑i=1n1−exp−λxi−2β,∂2logl∂λ2=2nλ2−6∑i=1n1xi2λ4−2β−1∑i=1n1xi2exp−λxi−2xi2λ61−exp−λxi−2−exp−λxi−23λ21−exp−λxi−2−2xi−2exp−λxi−2λ61−exp−λxi−2+2βlogα∑i=1nxi2λxi−41−exp−λxi−2β−1exp−λxi−2−3+2λxi−21−β−11−exp−λxi−2,∂2logl∂β2=−nβ2−logα∑i=1n1−exp−λxi−2βlog1−exp−λxi−22,∂2logl∂α∂λ=2βαλ3∑i=1nexp−λxi−21−exp−λxi−2β−1xi2,∂2logl∂α∂β=−1α∑i=1n1−exp−λxi−2βlog1−exp−λxi−2,∂2logl∂λ∂β=−2∑i=1nxiλxi−3exp−λxi−21−exp−λxi−2+2logα∑i=1nxiλxi−31−exp−λxi−2β−1+β1−exp−λxi−2β−1log1−exp−λxi−2.The asymptotic1−ζ100% confidence intervals for the parameters of suggested model are as follows:(71)α^±Zζ/2Σ11,λ^±Zζ/2Σ22,β^±Zζ/2Σ33.Here,Zζ represents the upper ζth percentile of standard normal distribution.


## 3.2. Simulation Study

Simulation is used to obtain estimates, Mean Square Error (MSE), and Bias of parameters. The following expression of QF was used to developw=100 samples of size n=50,90,and200, respectively:(72)Xp=1λ−log1−logα−loguα−1+1logα1/β−1/2,where u∼U0,1. The following expression is used to calculate bias and MSE:(73)Bias=1W∑1=1wbi^−b.MSE=1W∑1=1wbi^−b2,where b=α,λ,β. For various choices of α,λ,andβ, simulation results were obtained. Table 1 shows the simulated expected values of MSEs and bias. In Table 1, with increase in sample size, the consistency behavior may be easily observed as estimates approach their parametric values. Furthermore, as the sample size grows, the MSEs and bias of the estimates drop for all parameter combinations. As a result, we can infer that the MLE approach performs well when it comes to estimating the parameters of the APGIR distribution.Table 1
MSE and bias of MLE.
ParameterNMSE, α^MSEβ^MSE, λ^Biasα^Biasβ^Biasλ^α= 18.4510 β= 1.2510 λ= 0.65345045.61740.03880.3961−1.05900.0641−0.32219026.80810.03340.2198−0.62990.0584−0.184320023.11930.01370.17350.53170.0341−0.1389α= 17.4510 β= 1.4510 λ= 0.35345031.90160.09760.2236−1.24080.0827−0.32669018.44790.04540.2013−0.74850.0507−0.292020014.33890.01240.1530−0.5075−0.0053−0.2155α= 17.4510 β= 1.2510 λ= 0.65345044.22460.04630.3910−1.82670.0683−0.32089034.99310.02590.3383−1.63350.0210−0.273820017.57820.01330.18530.44440.0204−0.1502α= 17.4510 β= 1.3010 λ= 0.65345050.135030.05200.2882−1.96960.0756−0.25159033.50260.03340.2850−1.76210.0333−0.225520022.24690.01380.2051−0.00780.0326−0.1635α= 17.4510 β= 1.4510 λ= 0.60345039.78370.07530.4067−0.99350.0371−0.34369037.89690.04060.3365−0.27820.0349−0.283620018.05370.01280.17250.17830.0062−0.1470


## 3.3. Applications

To see the performance and goodness of fit of the proposed mode, the suggested distribution has been fitted to two data sets. We found that the suggested model performed better than other Rayleigh distribution variants such as the Two-Parameter Rayleigh (TPR) distribution proposed by Dey et al. [43], the MIR distribution suggested by Khan [28], the EIR distribution developed by Rehman and Dar [27], the GR distribution offered by Raqab and Madi [44], the TIR distribution by Ahmad et al. [15], and the GIR distribution by Potdar and Shirke [29].Data set 1: the first set of data includes the survival times (in years) of 46 individuals who received just chemotherapy. These data are a subset of the data taken from the study by Bekker et al. [45]. The data points are 0.047,0.115, 0.121, 0.132, 0.164, 0.197,0.203, 0.260, 0.282, 0.296, 0.334, 0.395, 0.458, 0.466, 0.501, 0.507, 0.529, 0.534, 0.540, 0.641, 0.644, 0.696, 0.841, 0.863, 1.099, 1.219, 1.271, 1.326, 1.447, 1.485, 1.553, 1.581, 1.589, 2.178,2.343, 2.416, 2.444, 2.825, 2.830, 3.578, 3.658, 3.743, 3.978, 4.003, 4.033.Data set 2: the failure times of 84 aircraft windshields are included in the second real data set. El-Bassiouny et al. [46] provided this information. The following are the data points:3.70, 2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11, 4.42, 2.41, 3.19, 3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.90, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.53, 2.67, 2.93, 3.22, 3.39, 2.81, 4.20, 3.33, 2.55, 3.31, 3.31, 2.85, 2.56, 3.56, 3.15, 2.35, 2.55, 2.59, 2.38, 2.81, 2.77, 2.17, 2.83, 1.92, 1.41, 3.68, 2.97, 1.36, 0.98, 2.76, 4.91, 3.68, 1.84, 1.59, 3.19, 1.57, 0.81, 5.56, 1.73, 1.59, 2.00, 1.22, 1.12, 1.71, 2.17, 1.17, 5.08, 2.48, 1.18, 3.51, 2.17, 1.69, 1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.70, 2.03, 1.80, 1.57, 1.08, 2.03, 1.61, 2.12, 1.89, 2.88, 2.82, 2.05, 3.65.PDF of GIR distribution is as follows:(74)fx=2βλ2x3exp−λx−21−exp−λx−2β−1x,β,λ>0.PDF of EIR distribution is as follows:(75)fx=2βθx3exp−βθx2,θ,β,X>0.PDF of GR distribution is as follows:(76)fx=2αγ2xexp−γx21−exp−γx2α−1,α,γ,X>0.PDF of TPR distribution is as follows:(77)fx=2αx−μexp−αx−μ2,x>μ,α>0.PDF of MIR distribution is as follows:(78)fx=α+2θx1x2exp−αx−θ1x2,α,θ,X>0.PDF of TIR distribution is as follows:(79)fx=2θx3exp−θx21+λ−2λexp−θx2,θ,X>0.PDF of WR distribution is as follows:(80)fx=αβθxexpθx22expθx22−1β−1e−αexpθx22−1β,α,θ,β,X>0.The APGIR model’s results are compared with other Rayleigh distribution versions using well-known model selection criteria such as Akaike’s Information Criteria (AIC), Consistent Akaike’s Information Criteria (CAIC), Bayesian Information Criterion (BIC), Hannan–Quinn Information Criteria (HQIC), and Kolmogorov–Smirnov (K-S) and theirP values via the R programming language’s Adequacy Model. The results are shown in Tables 2 and 3.Table 2
Goodness of fit results for data set 1.
DistributionMLEAICCAICBICHQICK-SP valueAPGIR10.46620.40918.4132137.2571137.8425142.6771139.27760.140510.3073GIR0.27207.4680148.2952148.5809151.9086149.64220.774845.551e − 16TIR0.0414−0.8352211.8820212.1677215.4953213.2290.431563.923e − 08APEIR10.86610.89340.0477196.7339197.3192202.1538198.75440.756775.551e − 16TPR0.1898−0.5973138.6129138.8986142.2262139.95990.216860.02449EIR0.16640.3224234.1737234.4594237.787235.52070.507722.82e − 11Table 3
Goodness of fit results for data set 2.
DistributionMLEAICCAICBICHQICK-SP valueAPGIR17.45101.45100.6534326.7218326.9718334.5373329.88490.151390.02043GIR1.1357−0.5285353.6821353.8058358.8924355.79080.171080.005739TIR1.9939−0.8843339.1298339.2535344.3401341.23850.153640.01781MIR−2.07686.4548338.6148338.7385343.8251340.72350.400022.531e − 14EIR7.99990.4095354.4818354.6055359.6921356.59050.182580.002544APEIR11.86420.28275.7878339.3922339.6422347.2077342.55530.781822.2e − 16On the basis of several model selection criteria, Tables2 and 3 show that our recommended distribution outperforms than other forms of Rayleigh distribution.Figure3 represents QQ and PP plot for data set 1. Figure 4 shows theoretical densities and CDFs for data set 1. The graphs clearly show better fit for data set 1. Figure 5 represents QQ and PP plot for data set 2. Figure 6 shows theoretical densities and CDFs for data set 2. It is clear from the figures that the data set 2 is better fitted by the proposed distribution.Figure 3
Graphs of QQ and PP plot for data set 1.Figure 4
Graphs of theoretical densities and CDFs for data set 1.Figure 5
Graphs of QQ and PP plot for data set 2.Figure 6
Graphs of theoretical densities and CDFs for data set 2.


## 4. Conclusion

In this paper, we have proposed a new distribution referred to as Alpha Power Generalized Inverse Rayleigh (APGIR) distribution. This distribution has been developed using APT with the input as Generalized Inverse Rayleigh. Several important mathematical properties including the moment generating function, order statistics, mean residual life function, mean waiting time, stress-strength parameter, expression for entropies, quantile function, and rth moment have been derived The parameter estimates were derived using the MLE technique. The consistency of MLE’s was assessed using simulation studies. The performance of the proposed model was evaluated using two real data sets using some goodness of fit criteria. The results clearly reveal that our proposed model performs well as compared with other types of Rayleigh distribution available in the literature.



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