Multivariate and Big Data Modeling and Related IssuesView this Special Issue
Alpha Power Generalized Inverse Rayleigh Distribution: Its Properties and Applications
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.
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  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  also demonstrated the use of RD in communication engineering. Polovko  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  proposed generalized Rayleigh (GR) distribution. Voda [8, 9] obtained the ML estimates of the RD. Bhattacharya and Tyagi  used RD for the analysis of medical data. Gomes et al.  suggested Kumaraswamy generalized Rayleigh (KGR) distribution. Merovci  presented transmuted Rayleigh (TR) distribution for investigating lifetime data. Cordeiro et al.  developed beta generalized Rayleigh (BGR) distribution. They also studied its main mathematical features. Leao et al.  proposed beta inverse Rayleigh (BIR) distribution. Ahmad et al.  offered transmuted inverse Rayleigh (TIR) distribution. Iriarte et al.  proposed slashed generalized Rayleigh (SGR) distribution. Lalitha and Mishra , Ariyawansa and Templeton , Howlader and Hossain , Sinha and Howlader , and Abd Elfattah et al.  are just few among others who contributed to RD.
Let X be a random variable having Rayleigh distribution. Symbolically, . Then, its CDF and PDF arewhere represents scale parameter.
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. The PDF and CDF of IRD are provided by
It has several uses in different fields including reliability analysis, engineering, and medicine. Voda  used the IRD to estimate the lifetime distribution of many experimental units. Trayer  proposed the IRD to accommodate survival and reliability data. Voda  discussed several properties and derived expression of ML estimator for parameters of IRD. Mukarjee and Maitim  also studied some important statistical properties of IRD. Closed form expressions for some descriptive statistics of the IR distribution were developed by Gharraph . Furthermore, Soliman et al.  and Gharraph  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 , Ahmad et al. , and Leao et al.  developed EIR, TIR, and BIR distributions, respectively. ShuaibKhan  developed a modified form of IRD and discussed it in depth. Potdar and Shirke  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. , Gupta et al. , Nadarajah and Kotz , and Mudholkar and Srivastava  studied generalization of several distributions in various statistical publications, generally employed in reliability estimation.
Reshi et al.  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  discussed many important characteristics of GIR distribution.
The PDF and CDF of GIR distribution are specified by
Here, and 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 . 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 .
2. Alpha Power Transformation (APT)
The APT was proposed by Mahdavi and Kundu . 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:and
Initially, the proposed method of Mahdavi and Kundu  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.  used APT and proposed alpha power transformed extended exponential distribution. Nassar et al.  proposed Alpha Power Weibull distribution. Dina and Magdy  and Ihtisham et al.  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 variable X is said to have Alpha Power Generalized Inverse Rayleigh distributed with three-parameters , and if its PDF is given by
Definition 1. . A variable X follows Alpha Power Generalized Inverse Rayleigh distributed with CDF as follows:The following are APGIR Hazard Rate (HR) Function and Survival Function (SF):The functions PDF, CDF, HF, and SF are plotted in Figures 1(a), 1(b), 2(a), and 2(b), respectively.
Lemma 1. If is a decreasing function for , then is also decreasing function.
Proof. If is differentiable function and , then 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 and , it is clear thatHence, for , is decreasing function.
Lemma 2. If is decreasing function for , is log-convex and then is decreasing function.
Proof. If , exist and , then is log-convex.
Differentiating (11), we getWhen α is non-negative and less than 1 and when and , then
Thus, when , is log-convex .
2.2. Quantile Function (QF)
Let X∼APGIR , then the QF is described bywhere . The QF of APGIR distribution is
After simplification, we have
To obtain median, we have
After some calculations, we obtain the following result of median:
To obtain mode, we have
Equation (19) is satisfied by mode of APGIR distribution.
2.5. Rth Moment of APGIR Distribution
Let X ∼APGIR , then the following is the rth moment:
Put in (20) , , and .
Let , , and
Using the following series representation in (22),to have
The expression of is incomplete integral; therefore, it can be solved approximately using numerical integration techniques.
2.6. Moment Generating Function (MGF)
Let X ∼APGIR , then MGF is defined as follows:
Using series notation in (25), we get
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 time t. Here, X is lifetime of an object with and provided in (7) and (10), respectively. The MRLF is given bywhere
Put , to have
Using the following series representation in (32), we have and :
The expression in (33) is an integral that is incomplete. This may be solved approximately using numerical integration techniques.
Insert and in (34) to have
Put , in (35) to have
Using the following series representation in (36), we have
Putting (10), (31), and (35) in (28), we get
The result of is an incomplete integral. Numerically, it can be approximated utilizing numerical integration techniques.
2.8. Order Statistics
Suppose denote sample of size n. The corresponding order statistics are The PDF of ith order statistic is specified by
Substituting and in (39), we get
We get PDF of the smallest order statistic by inserting in (40), that is,
Put in (40), we acquire the PDF of the largest order statistic
To get distribution of the median, substitute in (40) as
2.9. Stress-Strength Parameter (SSP)
Let and be two independent and identically distributed random variables. Suppose and . The SSP is defined as follows:
The SSP is calculated, by incorporating (7) and (8) in the above equation:
Substituting in (45), we have
Using series representation and in (46) and simplifying, we get the following final result for stress-strength parameter:
Lemma 3. Let , then final expression for Renyi entropy is given as follows:
Proof. Renyi entropy is defined asSubstitute to haveUsing series representation in the above equation, we getUsing in (51) and simplifying, we getThe 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) say is given by
Proof. The MWT of APGIR distribution is described asSubstituting the following results in (54),andwe obtain the required final expression asThe expression for 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:
Proof. The Shannon entropy is described byPutting in (60), we getInsert in (61), , and to haveUsing the following series in (62), and .
We get the Shannon entropy asThe integral in (63) may be solved approximately with the help of numerical integration techniques.
3. Parameters Estimation
3.1. Maximum Likelihood Estimation
Let be a random sample drawn from APGIR , then likelihood function is as follows:
Taking logarithm, (64) becomes
By differentiating (65) w.r.t and equating to 0, we get the following equations:
We can get estimates of , and 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, . The matrix is achieved by inverting the observed Fisher information matrix F as follows: