Normal Poisson distribution as a lifetime distribution of a series system

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Artigo Origial DOI:10.5902/2179460X29285 Ciêcia e Natura, Sata Maria v.

1 Artigo Origial DOI: / X29285 Ciêcia e Natura, Sata Maria v.40, e23, 2018 Revista do Cetro de Ciêcias Naturais e Exatas - UFSM ISSN impressa: ISSN o-lie: X Recebido: 27/11/2017 Aceito: 17/01/2018 Normal Poisso distributio as a lifetime distributio of a series system Eisa Mahmoudi 1, Hamed Mahmoodia 2 e Fatemeh Esfadiari 3 1,2,3 Departmet of Statistics, Yazd Uiversity, Yazd, Ira Abstract I this paper, we itroduce a ew three parameter skewed distributio. This ew class which is obtaied by compoudig the ormal ad Poisso distributios, is preseted as a alterative to the class of skew-ormal ad ormal distributios, amog others. Differet properties of this ew distributio have bee ivestigated. The desity ad distributio fuctios of proposed distributio, are give by a closed expressio which allows us to easily compute probabilities, momets ad related measuremets. Estimatio of the parameters of this ew model usig maximum likelihood method via a EM-algorithm is give. Fially, some applicatios of this ew distributio to real data are give. Keywords: Normal distributio, Poisso distributio, EM-algorithm, Maximum likelihood estimatio. Correspodig Author: emahmoudi@yazd.ac.ir Recebido: 27/09/2017 Revisado: 28/11/2017 Aceito: 17/01/2018

2 Ciêcia e Natura v.40, e23, Itroductio The ormal distributio is probably the most well kow statistical distributio ad widely used to model may pheomea. Sice the ormal distributio is symmetric, skew-ormal distributios have bee proposed, studied ad geeralized by may authors. Azzalii 1985) proposed the skew ormal SN) distributio with the pdf φz; λ) =2φz)Φλz), for z,λ R. This distributio ad its variatios have bee discussed by several authors icludig Azzalii 1986), Heze 2008), Braco ad Dey 2011), Loperfido 2010), Arold ad Beaver 2002), Balakrisha 2002), Azzalii ad Chioga 2004), Sharafi ad Behboodia 2008), Elal-Olivero 2010). Azzalii ad Valle 1996), Azzalii ad Capitaio 1999) ad Azzalii ad Chioga 2004) have all discussed various multivariate forms of skew-ormal distributios. The umber of compoets i a system ca be fixed umber. recetly, some researcher itroduced a system whe the umber of compoets is a radom variable N with support 1,2,. I such systems, compoets may be parallel or series. If the lifetime of ith compoet is the cotiuous radom variable X i, the the lifetime of such a system is defied by Y = max 1 i N X i or Y = mi 1 i N X i, based o whether the compoets are parallel or series. By takig a system with series compoets i which the radom variable N has Poisso distributio trucated as zero ad the radom variable X i follows the ormal distributio, we itroduce a ew geeralizatio of the ormal distributio. Recetly, some authors focused o these ew compoudig distributio; Mahmoudi ad Mahmoodia 2017) itroduced the ormal power series class of distributios i a system with parallel compoets. This itroduced class of distributios cotais ormal-geometric, ormal-poisso, ormal-logarithmic ad ormal-biomial models as special case. Roozegar ad Nadarajah 2017) also itroduced the power series skew ormal class of distributios usig this fact that the ith compoet X i has SN distributio ad cosider Y = N X i. To begi with, we shall use the followig otatio throughout this paper : φ ) for the stadard ormal probability desity fuctio pdf), φ ; µ,σ) for the pdf of N µ,σ) -variate ormal distributio with mea vector µ ad covariace matrix Σ), Φ ; µ,σ) for the cdf of N µ,σ), simply Φ ; Σ) for the case whe µ = 0. Furthermore, for r N, let 1 r ad I r deote the vector of oes ad the idetity matrix of dimesio r, respectively. The rest of this paper is orgaized as follows: I Sectio 2, we defie the class of ormal Poisso NP) distributio. The desity, hazard rate, survival fuctios ad some of their properties are give i this Sectio. We derive momets of NP distributio i Sectio 3. I Sectio 4, we derive a expasio for the desity of the order statistics. I Sectio 5, the Shao etropy is derived. I Sectio 6, estimates of the parameters based o a radom sample comig from this family of distributios are derived via a EM algorithm. Applicatios to two real data sets are give i Sectio 7. Fially, Sectio 8 cocludes the paper. Desity =1.8, θ= 5 µ= 3.5 µ=2.15 µ=5 Desity µ=1.5, θ= 5.7 =1.2 =3.5 = x x Desity µ=0.9, =3.3 θ= 1.5 θ=0.85 θ= Figure 1: Plots of the NP desity fuctio for selected parameter values. x 2 The Normal Poisso Distributio ad Some Properties

3 Mahmoudi et al. : Normal Poisso distributio as a lifetime of a series system 2 The Normal Poisso distributio ad Some Properties 2 The Normal Poisso Distributio ad Some Properties Let X 1,X 2, be a sample from a ormal distributio with mea µ ad variace 2. Let N be distributed accordig to a Poisso distributio trucated at zero, with the probability mass fuctio where θ>0. Moreover, N is idepedet of X i s. Remark 2.1. Eve whe θ<0, Equatio 2.3) is also a desity fuctio. We ca the defie the NP distributio by Equatio 2.3) for ay θ R {0}. Plots of the NP desity fuctio for selected parameter values are give i Figure 1. A importat characteristic of the NP distributio is that its desity fuctio ca be uimodal that makes this distributio may advatages i modellig lifetime data. The ormal distributio with mea µ ad variace 2 is a special case of the NP distributio whe θ 0. Whe µ =0ad =1, we say that X has stadard NP distributio with pdf Propositio 2.1. If X NP0,1,θ), the we have Proof. From 2.2), it ca be foud that P N = ) = θ!e θ 1), Defiitio 2.1. A radom variable X is said to have a ormal Poisso distributio, deoted by X NPµ,,θ), if From the defiitio i 2.1), we have, for x R Hece the pdf of X is give by F x; µ,,θ) = P X x) = = X = mix 1,,X N ). 2.1) P X x N = )P N = ) =1 P mix 1,,X N ) x N = )P N = ) =1 = 1 =1 1 Φ )) θ!e θ 1) fx; µ,,θ) = θφ fx; θ) = θφx)eθ1 Φx)) e θ. 1) = eθ e θ1 Φ )) e θ = 1 )eθ1 Φ )) e θ 1) F x;0,1,θ) =1 F x;0,1, θ). 1 F x;0,1, θ) =1 1 eθ1 Φx)) 1 e θ = 1 e θφx) 1 e θ = F x;0,1,θ) 1 e θφ ) 1 e θ. 2.2), 2.3) where µ R is the locatio parameter, >0 is the scale parameter ad θ>0 is the shape parameter, which characterize the skewess, kurtosis, ad uimodality of the distributio. Propositio 2.2. If X 1 NP0,1,θ 1 ) ad X 2 NP0,1,θ 2 ) are idepedet radom variables, the the stress-stregth parameter, R = P X 1 <X 2 ), is give by e θ1 θ 2 e θ1 R = e θ1 1) e θ2 1) e θ1 1) + eθ2 e θ 1+θ 2 1 ) ]. θ 1 + θ 2 Proof. The stress-stregth parameter is give by

4 Ciêcia e Natura v.40, e23, 2018 ] R = P X 1 <X 2 )= = F y;0, 1,θ 1 )fy;0,1,θ 2 )dy 1 e θ1φy) θ 2φ y) e θ21 Φy)) dy. 1 e θ1 e θ2 1) By chage of variable Φy) =t ad some simple calculatio, we have e θ1 θ 2 e θ1 R = e θ1 1) e θ2 1) e θ1 1) + eθ2 e θ 1+θ 2 1 ) ]. θ 1 + θ 2 Propositio 2.3. The desities of NP distributio ca be writte as ifiite umber of liear combiatio of desity of order statistics. We kow that e θ = =1 θ 1 1)!, therefore fx; µ,,θ) = where g X1) x; ) deotes the desity fuctio of X 1) = mi X 1,...,X ). ad The survival ad failure rate fuctios of the NP distributio are give respectively by =1 Sy; µ,,θ) = hy; µ,,θ) = θ!e θ 1) g X 1) x; ), 2.4) eθ1 Φ y µ )) 1 e θ, 2.5) 1 θφ )eθ1 Φ )) y µ θ1 Φ e )) 1). 2.6) Plots of the NP failure rate fuctio for selected parameter values are give i Figure 2. A importat characteristic of the NP distributio is that its failure rate fuctio ca be icreasig, uimodal-bathtub shaped that makes this distributio flexible i modelig differet types of lifetime data. Hazard =1.8, θ= 5 µ= 3.5 µ=2.15 µ=5 Hazard µ=2.5, θ=6 =1.3 =3 = x x Hazard µ=0.8, =2.5 θ= 4 θ=2.5 θ= Figure 2: Plots of the NP failure rate fuctio for selected parameter values. x

5 Mahmoudi et al. : Normal Poisso distributio as a lifetime of a series system 3 Quatiles ad Momets 3 Quatiles ad Momets The pth quatile of the NP distributios is give by ] x p = µ + Φ 1 1 logpeθ 1) + 1). θ Oe ca use this expressio for geeratig a radom sample from NP distributios with geeratig data from uiform distributio. Now, by usig equatio 2.4), we derive momet geeratig fuctio of X NPµ,,θ). The momet geeratig fuctio is give by M X t) = =1 θ!e θ 1) M X 1) t). 3.1) ad based o Jamalizadeh ad Balakrisha 2010) 1 M X1) t) = exp 2 2 t 2 + µt Φ t; I T 1). Therefore 1 M X t) = exp 2 2 t 2 θ + µt!e θ 1) Φ t; I T 1). 3.2) =1 Usig 3.2), k-th momet of a radom variable X ca be obtaied. But from the direct ad simple calculatio, we have µ k = By chagig of variable to t = Φx; µ,), we have Thus, first two momets are as follows: µ k = 1 0 x k θφx; µ)eθ1 Φx;µ,)) e θ dx. 1) µ + Φ 1 t)) k θeθ1 t) e θ 1 dt. µ 1 = EX) = µe θ 1 + ) e θ 1 e θ ]+ θe θ e θ 1 0 erf 1 2t 1)e θt dt, 3.3) µ 2 = EX 2 )= θe θ µ )+2µ e θ e θ ]+2 2µ ) 1 θ 1 1 ] erf 1 2t 1)e θt dt +2 4 erf 1 2t 1)] 2 e θt dt, 3.4) 0 Table 1 gives the first four momets, variace, skewess ad kurtosis of the NP0,1,θ) for differet values θ. Figure 3 shows the skewess ad kurtosis plot of the NP0,1,θ) for differet values θ. Table 1: The first four momets, variace, skewess ad kurtosis of NP distributio for µ =0,=1. θ = 3 θ = 1 θ = 0.3 θ =0.01 θ =0.3 θ =1 θ =3 θ = 10 EX) EX 2 ) EX 3 ) EX 4 ) V AR SK KUR

6 Ciêcia e Natura v.40, e23, Skewess Kurtosis θ Figure 3: Plots of skewess ad kurtosis of NP distributio for selected parameter values θ. 4 Order statistics 4 Order statistics Let X 1,..,X be a radom sample from NP distributio. The pdf of the ith order statistic, X i), is give by f i) x; µ,,θ) = By usig the biomial series expasio we have f i) x) =!fx; µ,,θ) i)!i 1)! 1 F x; µ,,θ)] i F x; µ,,θ)] i )!fx; µ,,θ) i)!i 1)! i 1) j F x; µ,,θ)] j+i 1. j j=0 The momets of the probability desity fuctio 4.1) caot be obtaied i a closed form but the values are derived by usig umerical computatio. Figure 4 shows the expected values of the order statistics plot from NP0,1,θ) for = 40, i =1,10,20 ad differet values θ. The expected value i= θ Figure 4: Plots of the expected values of the order statistics from NP0,1,θ) for = 40, i =1,10,20. 5 Shao etropy

7 Mahmoudi et al. : Normal Poisso distributio as a lifetime of a series system 5 Shao etropy 5 Shao etropy Let X be a radom variable from NPµ,,θ). The, the Shao etropy of X is give by Hx) = E logfx; µ,,θ))) = = logθ)+loge θ 1)) θ We ca write the first itegral as fx; µ,,θ) log φ fx; µ,,θ) 1 Φ x µ fx; µ,,θ) log fx; µ,,θ) log φ x µ )] dx. θφ ) e θ1 Φ )) e θ 1) x µ )] dx )] dx = log 2π) EX 2 ) 2µEX)+µ 2], Where EX 2 ) ad EX) follow from 3.3) ad 3.4). We ca write the secod itegral as Fially, we obtai 6 Estimatio ad iferece 6 Estimatio ad iferece x µ fx; µ,,θ) 1 Φ )) Hx) = logθ)+loge θ 1)) + log 2π) dx = eθ θ 1) 1 θe θ. 1) + 1 EX ) 2µEX)+µ 2] eθ θ 1) 1 e θ. 1) Let x 1,,x be observatios from NPµ,,θ) ad Ψ =µ,,θ) T be the parameter vector. The log-likelihood fuctio is give by l Ψ) = logθ) log e θ 1) ) + log φ )) + θ 1 Φ The maximum likelihood estimatio MLE) of Ψ, say Ψ, T l is obtaied by solvig the oliear system of equatios µ, l, l θ = 0, where l Ψ) µ l Ψ) l Ψ) θ = 1 2 x i µ)+ θ φ = x i µ) 2 + θ 2 x i µ)φ = θ 1 e θ ) Φ ] )). dx ), 6.1) ), 6.2) )). 6.3) Clearly, MLEs caot be obtaied i closed forms. The observed iformatio matrix is obtaied for approximate cofidece itervals ad hypothesis tests of o the model parameters. The 3 3 observed iformatio matrix is give by I Ψ) = I µµ I µ I µθ I µ I I θ I µθ I θ I θθ, where

8 Ciêcia e Natura v.40, e23, 2018 where I µµ = 2 + θ 3 x i µ)φ, I µ = 2 3 x i µ) θ 2 φ + θ 4 x i µ) 2 φ, I µθ = 1 φ, I = x i µ) 2 2θ 3 x i µ)φ + θ 5 x i µ) 3 φ, I θ = 1 2 x i µ)φ, I θθ = θ 2 + e θ 1 e θ ) 2. It is well-kow that uder regularity coditios, the asymptotic distributio of Ψ Ψ is N 3 0,J Ψ) 1 ), where J Ψ) = lim = 1 I Ψ). Therefore, a 1001 γ) asymptotic cofidece iterval for each parameter Ψ r is give by Îrr ACI r = Ψr Z γ/2, Ψ Îrr ) r + Z γ/2, where Îrr is the r,r) diagoal elemet of I Ψ) 1 for r =1,2,3 ad Z γ/2 is the quatile 1 γ/2 of the stadard ormal distributio. The solutio of the three o-liear ormal equatios i 6.1)-6.3) is eeded usig a umerical method. We may use the stadard methods such as Newto-Raphso, Nelder-Mead ad BFGS, but they have their usual problem of covergece. If the iitial guesses are ot close to the optimal value, the iteratio may ot coverge. Due to this reaso, we propose to use EM-algorithm to compute the MLEs. Suppose, {x 1,z 1 ),...,x,z )} is a radom sample of size from X,N). We defie a hypothetical complete-data distributio with a joit probability desity fuctio i the form gz,x; Ψ) = ad sice 1 + θ)e θ = z=1 z2 θ z 1 z!, the expected value of Z X = x, is give by θ z 1 z 2 1 Φ EZ X = x) = z=1 z!e θ1 Φ )) 1 = e θ1 Φ )) z=1 x µ = 1+θ 1 Φ )) z 1 z 2 θ 1 Φ z! )). ))) z 1 By usig the maximum likelihood estimatio over Ψ, with the missig Z s replaced by their coditioal expectatios give above, the M-step of EM cycle is completed. The log-likelihood of the model parameters for the complete data set is l x,z; µ,,θ) θ z z!e θ 1) zφ z i log θ log x i µ) 2 ) + z i 1) log 1 Φ log e θ 1 ). x µ ) 1 Φ ) z 1 x µ, where µ R, >0, θ R {0}, x R ad z N. The probability desity fuctio of Z give X = x is give by gz x) = gz,x; Ψ) fx) = θz 1 z 1 Φ z!e θ1 Φ )) )) z 1,

9 Mahmoudi et al. : Normal Poisso distributio as a lifetime of a series system ) The compoets of the score fuctio l µ l l θ l µ, l, l θ are give by = 1 2 x i µ)+ 1 φ x i µ ) z i 1) 1 Φ x i µ ), = x i µ) z i 1) x i µ) φ x i µ ) 1 Φ x i µ ), = 1 z i eθ θ e θ 1. The maximum likelihood estimates ca be obtaied from the iterative algorithm give by 1 h) x i µ h+1)) + 1 h+1) ) ) 2 θ h+1) = e θ h+1) 1 e θ h+1) φ xi µ h+1) ẑ h) i 1 =0, 1 Φ xi µ h+1) h) x i µ h)) 2 1 h+1) ẑ h) i, h) ) ẑ h) i 1 ) x i µ h)) φ where µ h), h) ad θ h) are foud umerically. Here, for i =1,...,, we have that ẑ h) i =1+ θ 1 h) xi µ h) )) Φ. 1 Φ xi µ h) h+1) xi µ h) h+1) ) =0, I the rest of this sectio, we verify the performace of the proposed estimator of, µ, ad θ of the proposed EM method for NP distributio. We simulate 1000 times uder the NP distributio with differet sets of parameters ad sample sizes = 50, 100, 300 ad 500. For each sample size, we compute the MLEs by EM-method. We also compute the root of mea square errors RMSE), stadard errors SE) ad covariaces of the MLEs of the EM-algorithm. The results for the NP distributio are reported i Tables 2. Some of the poits are quite clear from the simulatio results: i) Covergece has bee achieved i all cases ad this emphasizes the umerical stability of the EM-algorithm. ii) The differeces betwee the average estimates ad the true values are almost small. iii) These results suggest that the EM estimates have performed cosistetly. iv) As the sample size icreases, the root of mea square errors ad the stadard errors of the MLEs decrease. h) Table 2: The averages of the 1000 MLE s, mea of the simulated root of mea square errors, mea of simulated stadard errors ad mea of the simulated covariaces of EM estimators for NP distributio. Average estimators RMSE SE Cov µ,,θ) µ θ µ θ µ θ µ, )) µ, θ), θ) 0.0, 1.0, -1.0) , 1.0, 0.5) , 1.0, 1.0) , 1.0,2.0) , 1.0, 5.0) , 1.0, -1.0) , 1.0,0.5) , 1.0, 1.0) , 1.0, 2.0) , 1.0, 5.0) , 1.0, -1.0) , 1.0, 0.5) , 1.0, 1.0) , 1.0, 2.0) , 1.0, 5.0) , 1.0, -1.0) , 1.0, 0.5) , 1.0, 1.0) , 1.0, 2.0) , 1.0, 5.0)

10 Ciêcia e Natura v.40, e23, Applicatios 7 Applicatios I this sectio, the NP distributio is fitted to three real data sets ad also compared the fitted NP with two relative models, ormal N) ad skew-ormal SN) distributios with pdf 2 φ Φ α, to show the superiority of the NP distributio. The first data set cocerig the plasma ferriti cocetratio 102 male ad 100 female athletes collected at the Australia Istitute of Sport. We estimate parameters by umerically maximizig the likelihood fuctio. The variace covariace matrix of the MLEs uder the NP distributio is computed as The MLEs of the parameters, -2log-likelihood, AIC Akaike Iformatio Criterio), the Kolmogorov-Smirov test statistic K-S) ad the associated p-value are displayed i Table 3 for this data set. The results for these data set show that the NP distributio provides a better fit to this data set tha the N ad SN distributios. Also this coclusio is cofirmed from the plots of the fitted desities i Figure 5. Table 3: MLEs, -2 Log L, K-S, p-value ad AIC for plasma ferriti cocetratio. Dist MLE -2 Log L K-S p-value AIC NP ˆµ = , ˆ = ,ˆθ = N ˆµ = , ˆ = SN ˆµ = , ˆ = , ˆα = Desity NP N SN plasma ferriti cocetratio Figure 5: Plots of fitted NP, N ad SN for plasma ferriti cocetratio data. The secod data set is give by Birbaum ad Sauders 1969) that refers to fatigue life of 6061-T6 alumiium coupos cut parallel to the directio of rollig ad oscillated at 18 cycles per secod. The data set cosists of 100 observatios. The variace covariace matrix of the MLEs uder the NP distributio is computed as Table 4 gives the MLEs of the parameters, -2log-likelihood, AIC, the K-S test statistic ad the associated P-value for the secod data set. The fitted desities fuctios of NP, N ad SN models is displayed i Figure 6. The results for this data set show that the NP distributio is a good competitor for the ormal ad SN distributios. Also the plots of the desities i Figure 6 cofirmed this coclusio..

11 Mahmoudi et al. : Normal Poisso distributio as a lifetime of a series system cofirmed this coclusio. Table 4: MLEs, 2LogL, K-S, p-value ad AIC for time betwee failures data. Dist MLE -2 Log L K-S p-value AIC NP ˆµ =9.6705, ˆ =2.7745,ˆθ = N ˆµ =6.8780, ˆ = SN ˆµ = , ˆ = , ˆα = Desity NP N SN Fatigue life Figure 6: Plots of fitted NP, N ad SN for fatigue life data. The third data set from Bjerkedal 1960), represets the survival time i days of 72 guiea pigs ifected with virulet tubercle bacilli. The variace covariace matrix of the MLEs uder the NP distributio is computed as Table 5 gives the MLEs of the parameters, -2log-likelihood, AIC, the K-S test statistic ad the associated P-value for the third data set. The fitted desities fuctios of NP, N ad SN models is displayed i Figure 7. The results for this data set show that the NP distributios yield the best fit amog the N ad SN distributios. Also the plots of the desities i Figure 6 cofirmed this coclusio.. 8 Cocludig remarks I this paper based o compoudig approach, a ew three-parameter ormal-poisso was developed. The proposed NP is a alterative to the Azzalii skew-ormal distributio for fittig skewed data. We obtai expressios for the momets. The estimatio of the ukow parameters of the proposed distributio is approached by the EM-algorithm. Fially, we fitted NP distributio to three real data sets to show the potetial of the ew proposed distributio. Table 5: MLEs, 2LogL, K-S, P-value ad AIC for guiea pigs data. Dist MLE -2 Log L K-S P-value AIC NP ˆµ = , ˆ = ,ˆθ = N ˆµ = , ˆ = SN ˆµ = , ˆ = , ˆα =

12 Ciêcia e Natura v.40, e23, 2018 Desity NP N SN Guiea pigs Figure 7: Plots of fitted NP, N ad SN for air coditioig system data. Ackowledgemet The authors would like to thak the Referees ad the Editor for their valuable commets ad suggestios which have cotributed to substatially improvig the mauscript. The authors are also idebted to Yazd Uiversity for supportig this research. Referêcias Refereces B.C. Arold, R.J. Beaver, Skewed multivariate models related to hidde trucatio ad/or selective reportig with discussio), Test 11 1) 2002) 7-5 A. Azzalii, A class of distributios which icludes the ormal oes, Scadiavia Joural of Statistics ) A. Azzalii, Further results o a class of distributios which icludes the ormal oes, Statistica XLVI 2) 1986) A. Azzalii, A. Capitaio, Statistical applicatios of the multivariate skew-ormal distributio, Joural of the Royal Statistical Society Series B 61 3) 1999) A. Azzalii, M. Chioga, Some results o the stress-stregth model for skew-ormal variates, Metro ) A. Azzalii, A.D. Valle, The multivariate skew-ormal distributio, Biometrika 83 4) 1996) N. Balakrisha, Discussioo Skew multivariate ormal models related to hidde trucatio ad / or selective reportig by B.C. Arold ad R. J. Beaver, Test ) T. Bjerkedal, Acquisitio of resistace i guiea pigs ifected with differet doses of virulet tubercle bacilli.america Joural of Epidemiology ) Z.W. Birbaum, S.C. Sauders, Estimatio for a family of life distributios with applicatios to fatigue, Joural of Applied Probability ) M. Braco, D.K. Dey, A geeral class of multivariate elliptical distributio, Joural of Multivariate Aalysis ) D. DElal-Olivero, Alpha-skew-ormal distributio, Proyeccioes ) N.A. Heze, A probabilistic represetatio of the skew-ormal distributio, Scadiavia Joural of Statistics ) A. Jamalizadeh, N. Balakrisha, Distributios of order statistics ad liear combiatios of order statistics from a elliptical distributio as mixtures of uified skew - elliptical distributios, Joural of Multivariate Aalysis ) N. Loperfido, Quadratic forms of skew-ormal ad radom vectors, Statistics ad Probability Letters ) E. Mahmoudi ad H. Mahmoodia, Normal power series class of distributios: Model, properties ad applicatios, Electroic Joural of Applied Statistical Aalysis 102) 2017) R. Roozegar, S. Nadarajah, The power series skew ormal class of distributios, Commuicatio i Statistics-Theory ad Methods

13 Mahmoudi et al. : Normal Poisso distributio as a lifetime of a series system R. Roozegar, S. Nadarajah, The power series skew ormal class of distributios, Commuicatio i Statistics-Theory ad Methods 4622) 2017) S. Nadarajah, V. Nassiri, A. Mohammadpour, Trucated-expoetial skew-symmetric distributios, Statistics ) A. Noack, A class of radom variables with discrete distributios, Aals of Mathematical Statistics ) M. Sharafi, J. Behboodia, The Balakrisha skew-ormal desity, Statistical Papers ) Eisa Mahmoudi Departmet of Statistics, Yazd Uiversity, Yazd, Ira emahmoudi@yazd.ac.ir author s cotributio: Hamed Mahmoodia Departmet of Statistics, Yazd Uiversity, Yazd, Ira hamed_mahmoodia@yahoo.com author s cotributio: Fatemeh Esfadiari Departmet of Statistics, Yazd Uiversity, Yazd, Ira f.esfadiyari89@gmail.com author s cotributio::

Introducing a Novel Bivariate Generalized Skew-Symmetric Normal Distribution

Introducing a Novel Bivariate Generalized Skew-Symmetric Normal Distribution Joural of mathematics ad computer Sciece 7 (03) 66-7 Article history: Received April 03 Accepted May 03 Available olie Jue 03 Itroducig a Novel Bivariate Geeralized Skew-Symmetric Normal Distributio Behrouz