Alpha Power Generalized Inverse Rayleigh Distribution: Its Properties and Applications

Department of Statistics, University of Peshawar, Peshawar, Pakistan Department of Statistics, Yazd University, P.O. Box 89175-741, Yazd, Iran Department of Statistics, Shaheed Benazir Bhutto Women University, Peshawar, Pakistan Department of Statistics, Abdul Wali Khan University Mardan, Mardan, Pakistan Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt


Introduction
Rayleigh distribution (RD) is a special model and a modi ed form of Weibull distribution when shape parameter equals 2. e RD has many applications in various disciplines including engineering and medical sciences, astronomy, and Physics. e RD has been well investigated in the literature. Some researchers have examined its signi cant properties [1][2][3]. Ho man 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. ere are various variants of RD recently introduced by researchers that may be used for tting 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. ey also studied its main mathematical features. Leao et al. [14] proposed beta inverse Rayleigh (BIR) distribution. Ahmad et al. [15] o ered 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.
Let X be a random variable having Rayleigh distribution. Symbolically, X ∼ R(θ). en, its CDF and PDF are f(x) 2θ 2 x exp − (θx) 2 x ≥ 0, θ > 0, One important variant of RD is the Inverse Rayleigh Distribution (IRD), an important lifetime distribution. If X follows RD, then (1/X) has the IRD. e PDF and CDF of IRD are provided by 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. ese 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.
ese 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. e 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. e PDF and CDF of GIR distribution are specified by Here, c 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]. e major goal of doing so is to improve the modelling flexibility of lifetime data when compared with existing distributions.
is article is about the development of new probability distribution, known as Alpha Power Generalized Inverse Rayleigh (APGIR) distribution. is new model is obtained using Alpha Power Transformation [37].

Alpha Power Transformation (APT)
e APT was proposed by Mahdavi and Kundu [37]. is technique can be used to develop new distributions by introducing a new parameter into available distributions. e following is CDF and PDF of APT: and 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.

e Proposed Model.
e 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. e proposed model is a result of using the PDF and CDF of GIR distribution given in (3) and (4).
A random variable X is said to have Alpha Power Generalized Inverse Rayleigh distributed with three-parameters α, λ, and β if its PDF is given by 2 Mathematical Problems in Engineering Definition 1. . A variable X follows Alpha Power Generalized Inverse Rayleigh distributed with CDF as follows: e following are APGIR Hazard Rate (HR) Function and Survival Function (SF): Proof. If f(x) is differentiable function and (d/dx)log f (x) < 0, then f(x) is also decreasing function and vice versa.
Taking the first derivative of the following expression, i.e., For non-negative and less than 1 values of α and for λ and β > 0, it is clear that Hence, for α < 1, f APGIR (x) is decreasing function. □ Differentiating (11), we get Mathematical Problems in Engineering When α is non-negative and less than 1 and when λ and

Median.
To obtain median, we have After some calculations, we obtain the following result of median:

Mode.
To obtain mode, we have Equation (19) is satisfied by mode of APGIR distribution. 2.5. R th Moment of APGIR Distribution. Let X ∼APGIR (α, λ, β), then the following is the r th moment: Put in (20) Mathematical Problems in Engineering Using the following series representation in (22), to have e expression of μ / r is incomplete integral; therefore, it can be solved approximately using numerical integration techniques.

Moment Generating Function (MGF)
. Let X ∼APGIR (α, λ, β), then MGF is defined as follows: Using series notation e tx � ∞ r�0 t r x r /r! in (25), we get Utilize (24) in (26), we get e result in equation (27) is incomplete integral, and it may be solved on the basis of numerical integration methods.

Mean Residual Life Function (MRLF).
e MRLF is the average remaining life of a component that has survived till time t. Here, X is lifetime of an object with f(x) and S(x) provided in (7) and (10), respectively. e MRLF is given by Using the following series representation in (32), we e expression in (33) is an integral that is incomplete. is may be solved approximately using numerical integration techniques.
Using the following series representation in (36), we have (logα) n /n!(z) n ,

Mathematical Problems in Engineering
Putting (10), (31), and (35) in (28), we get e result of μ(t) is an incomplete integral. Numerically, it can be approximated utilizing numerical integration techniques.
e corresponding order statistics are e PDF of i th order statistic is specified by Substituting f(x) and F(x) in (39), we get We get PDF of the smallest order statistic by inserting i � 1 in (40), that is, Put i � n in (40), we acquire the PDF of the largest order statistic To get distribution of the median, substitute i � n/2 in (40) as f n 2 : n (x) � n! (n/2 − 1)!(n − (n/2))!(α − 1) n 2β log α 8 Mathematical Problems in Engineering

Mathematical Problems in Engineering
Using series representation α − ]y β � ∞ k�0 (− log α) k / k!(]y β ) k in the above equation, we get Using log(1 − y) � t⇒dy � (y − 1)dt⇒dy � − e t dt in (51) and simplifying, we get (52) e expression of Renyi entropy is an incomplete integral. e solution of (52) is obtained on the basis of numerical integration techniques.

Lemma 4. e Mean Waiting Time (MWT) say μ(t) is given by
Proof. e MWT of APGIR distribution is described as (54) Substituting the following results in (54), and we obtain the required final expression as e expression for μ(t) is an integral that is incomplete. e solution of (57) may be obtained by numerical integration techniques.

Lemma 5.
e Shannon entropy (SE) expression is given as follows: Proof. e Shannon entropy is described by (60) Insert in (61), 1 − y β � z⇒βy β− 1 dy � − dz, and y � (1 − z) 1/β to have Using the following series in (62), We get the Shannon entropy as Mathematical Problems in Engineering e integral in (63) may be solved approximately with the help of numerical integration techniques.

Maximum Likelihood Estimation.
Let X 1 , X 2 , . . . , X n be a random sample drawn from APGIR (α, λ, β), then likelihood function is as follows: Taking logarithm, (64) becomes 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. ese data are a subset of the data taken from the study by Bekker et al. [45]. e data points are 0. PDF of GIR distribution is as follows: PDF of EIR distribution is as follows: PDF of GR distribution is as follows: PDF of TPR distribution is as follows: PDF of MIR distribution is as follows: PDF of TIR distribution is as follows: PDF of WR distribution is as follows:  Tables 2 and 3.
On the basis of several model selection criteria, Tables 2  and 3 show that our recommended distribution outperforms than other forms of Rayleigh distribution.  Figure 3 represents QQ and PP plot for data set 1. Figure 4 shows theoretical densities and CDFs for data set 1. e 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.

Conclusion
In this paper, we have proposed a new distribution referred to as Alpha Power Generalized Inverse Rayleigh (APGIR) distribution.
is 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 e parameter estimates were derived using the MLE technique. e consistency of MLE's was assessed using simulation studies. e performance of the proposed model was evaluated using two real data sets using some goodness of fit criteria. e results clearly reveal that our proposed model performs well as compared with other types of Rayleigh distribution available in the literature.

Data Availability
e data sets are included within the main body of the paper.