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The Poisson Burr X Inverse Rayleigh Distribution and Its Applications

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The Poisson Burr X Inverse Rayleigh Distribution and Its Applications Journal of Data Science,18(1). P. 56 – 77,2020 DOI:10.6339/JDS.202001_18(1).0003 The Poisson Burr X Inverse Rayleigh Distribution And Its Applications Rania H. M. Abdelkhalek* Department of Statistics, Mathematics and Insurance, Benha University, Egypt. ABSTRACT A new flexible extension of the inverse Rayleigh model is proposed and studied. Some of its fundamental statistical properties are derived. We assessed the performance of the maximum likelihood method via a simulation study. The importance of the new model is shown via three applications to real data sets. The new model is much better than other important competitive models. Keywords: Inverse Rayleigh; Poisson; Simulation; Modeling. * [email protected] Rania H. M. Abdelkhalek 57 1. Introduction and physical motivation The well-known inverse Rayleigh (IR) model is considered as a distribution for a life time random variable (r.v.). The IR distribution has many applications in the area of reliability studies. Voda (1972) proved that the distribution of lifetimes of several types of experimental (Exp) units can be approximated by the IR distribution and studied some properties of the maximum likelihood estimation (MLE) of the its parameter. Mukerjee and Saran (1984) studied the failure rate of an IR distribution. Aslam and Jun (2009) introduced a group acceptance sampling plans for truncated life tests based on the IR. Soliman et al. (2010) studied the Bayesian and non- Bayesian estimation of the parameter of the IR model Dey (2012) mentioned that the IR model has also been used as a failure time distribution although the variance and higher order moments does not exist for it. The probability density function (PDF) and cumulative distribution function (CDF) of the IR distribution are given by (for 푥 ≥ 0) (훼) 2 −3 2 −2 (훼) 2 −2 푔퐼푅 (푥) = 2훼 푥 exp[−훼 푥 ] and 퐺퐼푅 (푥) = exp[−훼 푥 ], respectively, where 훼 > 0 is a scale parameter. It is worth mentioning that the IR model is a special case for the Inverse Weibull model. In the statistical literature, there are many useful extensions of the IR model, for example: beta IR by Barreto-Souza et al. (2011), Marshall-Olkin IR by Krishna et al. (2013), Transmuted IR by Mahmoud and Mandouh (2013), Weibull IR by Afify et al. (2016), odd Lindley IR by Korkmaz et al. (2017), Burr X IR by Yousof et al. (2017a), transmuted Topp- Leone IR by Yousof et al. (2017b), Topp-Leone Generated IR by Yousof et al. (2018b) and odd log-logistic IR Yousof et al. (2018b), among others. In this paper we propose and study a new extension of the IR distribution using the zero truncated Poisson (ZTP) distribution. Suppose that a system has 푁 subsystems functioning independently at a given time where 푁 has ZTP distribution with parameter 휆. It is the conditional probability distribution of a Poisson-distributed r.v., given that the value of the r.v. is not zero. The probability mass function (PMF) of 푁 is given by 휆 푛 푃푀퐹푍푇푃(푁 = 푛) = [exp(−휆) 휆 ]/{푛! [휏휆]}|푛=1,2,… Note that for ZTP r.v., the expected value 퐄(푁|휆) and variance 푉푎푟(푁|휆) are, respectively, given by 퐄(푁|휆) = 휆/휏(휆), where 휏(휆) = −exp(−휆) + 1 and 휆2 휆 + 휆2 푉푎푟(푁|휆) = − 2 + . 휏( ) [휏(휆)] 휆 Suppose that the failure time of each subsystem has the Burr X IR (BX-IR(휃, 훼) for short) defined by the CDF and PDF given by 58 The Poisson Burr X Inverse Rayleigh Distribution And Its Applications 2 휃 exp⁡[−훼2푥−2] 퐻휃,훼 (푥) = [1 − exp⁡(− { } )] 퐵푋−퐼푅 1 − exp⁡[−훼2푥−2] and 2 휃−1 exp[−훼2푥−2] ℎ휃,훼 (푥) = 4휃훼2푥−(2+1){1 − exp[−훼2푥−2]}−3 × [1 − exp (− { } )] 퐵푋−퐼푅 1 − exp[−훼2푥−2] 2 exp[−훼2푥−2] ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡× exp⁡[−2(훼푥−1)2]exp⁡(− { } ) 1 − exp[−훼2푥−2] respectively, where 훼 > 0 is a scale parameter and 휃 > 0 is the shape parameter. Let 푌 denote the failure time of the ith subsystem and let 푋 = min{푌1, 푌2, ⋯ , 푌푁}. Then the conditional CDF of 푋 given 푁 is 휃,훼 푁 퐹(푥|푁) = 1 − Pr(푋 > 푥|푁) = 1 − [1 − 퐻퐵푋−퐼푅(푥)] Therefore, the unconditional CDF of the Poisson BX-IR (PBX-IR) density function, as described in Ristić and Nadarajah (2014), can be expressed as 휃,훼 휆,휃,훼 1 − exp⁡(−휆퐻퐵푋−퐼푅(푥)) 퐹푃퐵푋−퐼푅(푥) = 휏휆 then we have 2 휃 exp[−훼2푥−2] 1 − exp⁡(−휆 [1 − exp (− { } )] ) ( 2 −2) 휆,휃,훼 1 − exp −훼 푥 퐹푃퐵푋−퐼푅(푥) = 휏휆 the corresponding PDF is 2 (휆,휃,훼) 4휃휆2훼 −(2+1) 2 −2 −3 푓푃퐵푟푋퐼푊(푥) = 푥 {1 − exp[−훼 푥 ]} 휏(휆) 2 exp[−훼2푥−2] ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡× exp (−2(훼푥−1)2 − { } ) 1 − exp[−훼2푥−2] 2 휃−1 exp[−훼2푥−2] ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡× [1 − exp (− { } )] 1 − exp[−훼2푥−2] 2 휃 exp[−훼2푥−2] ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡× exp⁡{−휆 [1 − exp (− { } )] } 1 − exp[−훼2푥−2] The transformation of Ristić and Nadarajah (2014) used by many authors such as Ramos et al. (2015), Aryal and Yousof (2017) and Korkmaz et al. (2018), among others. Rania H. M. Abdelkhalek 59 (휆,휃,훼) The hazard rate function (HRF) can be easily calculated via 푓푃퐵푟푋퐼푊(푥)/[1 − (휆,휃,훼) 퐹푃퐵푋−퐼푅(푥)]. The PBX-IR density can be right-skewed, unimodal and symmetric (see Figure 1) whereas the PBX-IR HRF can be unimodal then bathtub, increasing and bathtub (see Figure 2). Figure 1: Plots of the PBX-IR PDF for selected parameter values. 60 The Poisson Burr X Inverse Rayleigh Distribution And Its Applications Figure 2: Plots of the PBX-IR HRF for selected parameter values. This article is organized as follows: In Section 2, we derive some mathematical properties of the new model. Maximum likelihood method for the model parameters is addressed in Section 3. Sections 4 presents the simulation studies. In Section 5, The potentiality of the proposed model is illustrated by means of three real data sets. Section 6 provides some concluding remarks. 2. Mathematical properties 2.1 Useful expansions Upon the power series ∞ 휁푞 exp(휁) = ∑ ( ) , 푞! 푞=0 Rania H. M. Abdelkhalek 61 the PDF in (6) can be written as ∞ 1+ℎ 2 ℎ (휆,휃,훼) 2휃휆 2훼 (−1) ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡푓푃퐵푟푋퐼푊(푥) = ∑ ℎ! [휏(휆)] 푞=0 exp[−훼2푥−2] ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡× exp (푥−(2+1) ) {1 − exp[−훼2푥−2]}3 2 exp[−훼2푥−2] ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡× exp[− ( ) ] 1 − exp[−훼2푥−2] 2 휃(ℎ+1)−1 exp[−훼2푥−2] ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡× [1 − exp (− { } )] 1 − exp[−훼2푥−2] If |휁| < 1 and 휏 > 0 is a real non-integer, the following power series holds ∞ (−1)Γ(1 + 휏) (1 − 휁)휏 = ∑ 휁 Γ(1 + 휏 − ) =0 Applying (8) to (7) we have 2 −(2+1) 2 −2 (휆,휃,훼) 2휃2훼 푥 exp[−2훼 푥 ] ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡푓푃퐵푟−퐼푅(푥) = [휏(휆)] ∞ 휆1+ℎ(−1)ℎ+Γ(휃(⁡ℎ + 1)) ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡× ∑ i! Γ(휃(ℎ + 1) − ) ℎ,=0 2 exp[−훼2푥−2] 푒푥푝[−( + 1) ( ) ] 1 − exp[−훼2푥−2] ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡× [1 − exp[−훼2푥−2]]3 Via applying the power series to the term 2 exp[−훼2푥−2] exp [−( + 1) ( ) ], 1 − exp[−훼2푥−2] equation (9) becomes ∞ 4훼2휃휆1+ℎ(−1)ℎ++휔( + 1)휔푥−3 ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡푓(휆,휃,훼) (푥) = ∑ 푃퐵푋−퐼푅 ! 휔! [휏 ] ℎ,,휔=0 (휆) Γ(θ(h + 1)) exp[−훼2푥−2] {exp[−훼2푥−2]}2휔+1 ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡× Γ(θ(h + 1) − i) {1 − exp[−훼2푥−2]}2휔+3 Then consider the series expansion 62 The Poisson Burr X Inverse Rayleigh Distribution And Its Applications ∞ Γ(푐 + 휅) (1 − 휁)−푐퐼 = ∑ 휁휅 |휁|<1,푐>0 휅! Γ(푐) =0 Applying the expansion in (11) to (10) for the term {1 − exp[−훼2푥−2]}2휔+3, equation (10) becomes 1+ℎ ℎ++휔 휔 (휆,휃,훼) ∞ 2휃휆 (−1) ⁡(+1) 푓푃퐵푋−퐼푅(푥) = ⁡ℎ,,휔,휅=0 ⁡ !⁡휔!휅![휏(휆)][2(1+휔)+휅] Γ(휃(ℎ+1))Γ(3+2휔+휅) × [2(1 + 휔) + 휅] Γ(휃(ℎ+1)−)Γ(2휔+3) × 2훼2푥−(2+1)exp{−[2(1 + 휔) + 휅](훼푥−1)2}. This can be written as ∞ (휆,휃,훼) 푓푃퐵푋−퐼푅(푥) = ∑ 휐휔,휅⁡ℎ[2(1+휔)+휅](푥; 훼), 휔,휅=0 where ∞ 2휃휆1+ℎ(−1)휔Γ(3 + 2휔 + 휅) (−1)ℎ+⁡Γ(휃(ℎ + 1))( + 1)휔 휐 = ⁡ ∑ ,⁡ 휔,휅 휔! 휅! [휏 ]Γ(2휔 + 3)[2(1 + 휔) + 휅] ! ⁡Γ(휃(ℎ + 1) − ) (휆) ℎ,=0 1 and ℎ[2(1+휔)+휅](푥; 훼) is the IR density with scale parameter 훼[2(1 + 휔) + 휅]2. Similarly, the CDF of the PBX-IR model can also be expressed as ∞ (휆,휃,훼) 퐹푃퐵푋−퐼푅(푥) = ∑ 휐휔,휅퐻[2(1+휔)+휅](푥; 훼), 휔,휅=0 1 where 퐻[2(1+휔)+휅](푥; 훼) is the the IR density with scale parameter 훼[2(1 + 휔) + 휅]2. 2.2 Quantile and random number generation The quantile function (QF) of 푋, where 푋 ∼PBX-IR (휆, 휃, 훼), is obtained by inverting (5) as 1 − 1 2 1 2 휃 −ln{1 − 푢[휏(휆)]} 푄(푢) = 훼 −ln 1 + {−ln [1 − ( ) ]} , 휆 { [( )]} so, simulating the PBX-IR r.v. is straightforward (see Section 4). If 푈 is a uniform variate on the unit interval (0,1), then the r.v. 푋 = 푄(푈) follows (6). 2.3 Moments 푡ℎ ′ The 푟 ordinary moment of 푋, say 휇푟, follows from (12) as Rania H. M. Abdelkhalek 63 ∞ 푟 푟 휇′ | = 퐄(푋푟) = ∑ 휐 훼푟[2(1 + 휔) + 휅]2⁡Γ (1 − ), 푟 (푟<2) 휔,휅 2 휔,휅=0 where ∞ 훕−1 훕 Γ(1 + 훕)|(훕∈ℝ+) =푤=0 (훕 − 푤) = ∫ ⁡푥 ⁡exp(−푡)푑푥 = 훕!, 0 upon setting 푟 = 1 in (15) gives the mean of 푋 as ∞ 1 1 퐄(푋) = ∑ 휐 훼[2(1 + 휔) + 휅]2⁡Γ (1 − ).
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