Normal–Poisson Distribution As a Lifetime Distribution of a Series

Artigo Original DOI:10.5902/2179460X29285

Ciência e Natura, Santa Maria v.40, e23, 2018 Revista do Centro de Ciências Naturais e Exatas - UFSM ISSN impressa: 0100-8307 ArtigoISSN on-line: Original 2179-460X DOI: 10.5902/2179460X29285 Artigo Original DOI: 10.5902/2179460X29285 Ciência e Natura, Santa Maria, v. 40 2018, Revista do Centro de Ciências Naturais e Exatas - UFSM Recebido: 27/11/2017 Aceito: 17/01/2018 ISSNArtigoCiência impressa: Original e Natura, 0100-8307 Santa Maria, ISSN v. 40 on-line: 2018, Revista 2179-460X do Centro de Ciências Naturais e Exatas - UFSM DOI: 10.5902/2179460X29285 ISSN impressa: 0100-8307 ISSN on-line: 2179-460X Ciência e Natura, Santa Maria, v. 40 2018, Revista do Centro de Ciências Naturais e Exatas - UFSM ISSN impressa: 0100-8307 ISSN on-line: 2179-460X

Normal–Poisson distribution as a lifetime distribution of a series ArtigoNormal–Poisson Original distribution as a lifetime distributionDOI: 10.5902/2179460X29285 of a series system CiênciaNormal–Poisson e Natura, Santa Maria, v. 40 2018, distribution Revista do Centro de Ciênciassystem as a Naturais lifetime e Exatas distribution - UFSM of a series 1 2 3 ISSN impressa: 0100-8307Eisa ISSN Mahmoudi on-line: 2179-460X∗ , Hamed Mahmoodian e Fatemeh Esfandiari 1 2 3 Eisa Mahmoudi∗ , Hamedsystem Mahmoodian e Fatemeh Esfandiari 1,2,3 Department1 of Statistics, Yazd University,2 Yazd, Iran 3 Eisa Mahmoudi1,2,3Department∗ , Hamed of Statistics, Mahmoodian Yazd University,e Fatemeh Yazd, Iran Esfandiari

1,2,3Department of Statistics, Yazd University, Yazd, Iran Normal–Poisson distribution asAbstract a lifetime distribution of a series In this paper, we introduce a new three parameter skewedsystemAbstract distribution. This new class which is obtained by compounding the normalIn this paper, and Poisson we introduce distributions, a new threeis presented parameter as an skewed alternative distribution. to the class This of new skew-normal class which and is obtained normal distributions, by compounding among the Abstract others.normal Different and Poisson properties distributions, of this new is presented distribution1 as an have alternative been investigated. to the class2 The of skew-normaldensity and distribution and normal3 functions distributions, of proposed among Indistribution,others. this paper, Different are we given introduce propertiesEisa by a closed Mahmoudi a of new this expression three new parameterdistribution∗ , which Hamedallows skewed have Mahmoodian beenus distribution. to investigated.easily compute Thise The new Fatemehprobabilities, density class which and Esfandiari moments distribution is obtained and relatedfunctions by compounding measurements. of proposed the Estimationnormaldistribution, and of Poisson are the given parameters distributions, by a closed of this expression is new presented model which asusing an allows alternativemaximum us to easily likelihood to the compute class method of probabilities, skew-normal via an EM-algorithm moments and normal and is relateddistributions, given. measurements.Finally, among some others.applicationsEstimation Different of of the this properties parameters new distribution of of this this1,2,3 new to new realDepartment distribution model data usingare of havegiven. Statistics,maximum been investigated. Yazd likelihood University, method The density Yazd, via an Iran and EM-algorithm distribution isfunctions given. Finally, of proposed some Keywords:distribution,applicationsNormal are of this given distribution, new by distribution a closed Poisson expression to real distribution, data which are allows given. EM-algorithm, us to easily Maximum compute probabilities, likelihood estimation. moments and related measurements. EstimationKeywords: ofNormal the parameters distribution, of this Poisson new distribution,model using EM-algorithm,maximum likelihood Maximum method likelihood via an EM-algorithm estimation. is given. Finally, some applications of this new distribution to real data are given. 1 IntroductionKeywords: Normal distribution, Poisson distribution, EM-algorithm,Abstract Maximum likelihood estimation. 1 IntroductionIn this paper, we introduce a new three parameter skewed distribution. This new class which is obtained by compounding the The normalnormal distribution and Poisson distributions,is probably the is presented most well as anknown alternative statistical to the distribution class of skew-normal and widely and used normal to distributions,model many among phenomena. Since1The normalIntroductionothers. the normal Differentdistribution distribution properties is probably is of symmetric, this thenew most distribution skew-normal well known have been distributions statistical investigated. distribution have The been density and proposed, and widely distribution studied used to functions and model generalized many of proposed phenomena. by many authors.Since the Azzalini normal (1985)distribution proposed is symmetric, the skew normalskew-normal (SN) distribution distributions with have the been pdf proposed,φ(z; λ)=2 studiedφ(z)Φ( andλz) generalized, for z,λ byR. many This distribution, are given by a closed expression which allows us to easily compute probabilities, moments and related measurements.∈ Thedistributionauthors. normal Azzalini and distribution its (1985) variations is proposed probably have been the the skew discussed most normal well by known several(SN) statisticaldistribution authors including distribution with the Azzalini pdf andφ( widelyz; (1986),λ)=2 used Henzeφ(z to)Φ( model (2008),λz), manyfor Brancoz,λ phenomena. andR. ThisDey Estimation of the parameters of this new model using maximum likelihood method via an EM-algorithm is given. Finally, some∈ (2011),Sincedistributionapplications the Loperfido normal and its of (2010),distribution thisvariations new Arnold distribution haveis symmetric, and been Beaver to discussedreal skew-normal(2002),data are by Balakrishnan given. several distributions authors (2002), including have Azzalini been Azzalini andproposed, Chiogna (1986), studied (2004), Henze and (2008),Sharafi generalized and Branco Behboodian by and many Dey authors.(2011), Loperfido Azzalini (1985)(2010), proposed Arnold and the Beaver skew normal(2002), Balakrishnan(SN) distribution (2002), with Azzalini the pdf andφ(z Chiogna; λ)=2φ (2004),(z)Φ(λz Sharafi), for andz,λ BehboodianR. This (2008),Keywords: Elal-OliveroNormal (2010). distribution, Azzalini Poisson and Valle distribution, (1996), EM-algorithm, Azzalini and Maximum Capitanio likelihood (1999) and estimation. Azzalini and Chiogna (2004)∈ have all discusseddistribution(2008), Elal-Olivero various and its multivariate variations (2010). Azzalini have forms been of and skew-normal discussed Valle (1996), by distributions. several Azzalini authors and Capitanioincluding (1999)Azzalini and (1986), Azzalini Henze and (2008), Chiogna Branco (2004) and have Dey all (2011),discussedThe Loperfido number various of (2010), multivariatecomponents Arnold formsin anda system ofBeaver skew-normal can (2002), be fixed Balakrishnan distributions. number n. recently, (2002), Azzalini some researcher and Chiogna introduced (2004), a system Sharafi when and Behboodian the number (2008),of componentsThe Elal-Olivero number is of a random components (2010). variable Azzalini in aN system andwith Valle cansupport be (1996), fixed1,2, number Azzalini. In suchn and. recently, systems, Capitanio some components (1999) researcher and may Azzalini introduced be parallel and a Chiogna system or series. when (2004) If the the havelifetime number all 1 Introduction ··· ofdiscussedofi componentsth component various is a ismultivariate random the continuous variable forms randomN ofwith skew-normal support variable1,X distributions.2,i, then. In the such lifetime systems, of such components a system may is defined be parallel by Y or= series. max If1 thei N lifetimeXi or ··· n ≤ ≤ TheYof=iTheth normal min component number1 i distributionN ofX isicomponents, basedthe continuousis probablyon whether in a system random the the most componentscan variable wellbe fixed knownX number arei, then parallelstatistical the. recently, lifetime or distribution series. someof such researcher and a system widely introduced is used defined to amodel by systemY = many maxwhen phenomena.1 thei N numberXi or ≤ ≤ ≤ ≤ ofY components=By min taking1 i aN is systemX a randomi, based with variable on series whether componentsN with the componentssupport in which1,2, are the. parallel random In such or systems,variable series.N componentshas Poisson may distribution be parallel truncated or series. as If zero the lifetime and the Since the normal≤ ≤ distribution is symmetric, skew-normal··· distributions have been proposed, studied and generalized by many of ithBy component taking a system is the with continuous series components random variable in whichXi, the then random the lifetime variable ofN suchhas a Poisson system is distribution defined by truncatedY = max as1 zeroi N andXi theor authors.random variable AzzaliniX (1985)i follows proposed the normal the skew distribution, normal we (SN) introduce distribution a new with generalization the pdf φ(z of; λ the)=2 normalφ(z)Φ( distribution.λz), for z,λ≤ ≤R. This Yrandom= min variable1 i N XXi, basedfollows on the whether normal the distribution, components we are introduce parallel a or new series. generalization of the normal distribution. ∈ distributionRecently,≤ and≤ some its variationsi authors focused have been on these discussed new compounding by several authors distribution; including Mahmoudi Azzalini and (1986), Mahmoodian Henze (2008), (2017) Branco introduced and Dey the (2011),normalByRecently, takingLoperfido power asome series system (2010), authors class with Arnold of focused series distributions and components on Beaver these in (2002), newa systemin whichcompounding Balakrishnan with the parallelrandom distribution; (2002), components.variable AzzaliniN Mahmoudihas This and Poisson introduced Chiogna and distribution Mahmoodian (2004), class truncatedof Sharafi distributions (2017) and as introduced Behboodianzero contains and the the (2008),normal-geometric,randomnormal Elal-Olivero powervariable seriesXi normal-Poisson, (2010).follows class of Azzalinithe distributions normal normal-logarithmic and distribution, Valle in a system(1996), we with Azzalini introduce and parallel normal-binomial and a newcomponents. Capitanio generalization (1999)models This and as introduced of special the Azzalini normal case. class and distribution. Roozegar Chiogna of distributions (2004) and Nadarajah havecontains all discussed(2017)normal-geometric,Recently, also various introduced some multivariate authors normal-Poisson, the focused power forms series on of normal-logarithmic these skew-normal skew new normal compounding distributions. class and of normal-binomial distribution; distributions Mahmoudi using models this as and fact special Mahmoodian that the case.ith Roozegar component (2017) introducedandX Nadarajahi has SNthe N distributionnormal(2017)Thenumberpower also and introduced series of consider components class theY of= power distributions in a system seriesXi. skew can in a be system normal fixed numberwith class parallel ofn. distributions recently, components. some using researcher This this introduced fact introduced that the classi ath systemof component distributions when theXi contains numberhas SN iN=1 ofnormal-geometric,distribution componentsTo begin and with, is considera random we normal-Poisson, shallY variable use= thei=1N following normal-logarithmicXwithi. support notation1,2, throughout and. In normal-binomial suchthis systems, paper components : modelsφ( ) for as the may special standard be parallel case. normal Roozegar or series. probability and If the Nadarajah lifetime density (2017) also introduced the power series skew normal··· class of distributions using· this fact that the ith component X has SN offunctionithTo component begin (pdf), with,φn is( we the; µ, shall continuousΣ) for use the the pdf random following of Nn variable(µ notation,Σ) (nX-variatei, throughout then the normal lifetime this distribution paper of such : φ( a with) systemfor mean the is standard vector definedµ normal byandY covariance= probability max1 i matrixiN densityXiΣor), distribution and consider· Y = N X . · ≤ ≤ YΦfunctionn=( min; µ,Σ1 (pdf),)iforNφ theXn(i, cdf based; µ of,ΣN) onforn( whetherµ the,Σi=1) pdf, simply the ofi N componentsnΦ(µn(,Σ;)Σ(n) for are-variate the parallel case normal whenor series. distributionµ = 0. Furthermore, with mean vector for r µNand, let covariance1r and Ir denote matrix theΣ), · ≤ ≤ · · ∈ vectorΦn(To; ofµ begin,Σ ones) for with, and the thewe cdf identityshall of N usen( matrixµ the,Σ) following, simply of dimensionΦ notationn( ; Σr,) respectively.for throughout the case when this paperµ =N :0.φ Furthermore,( ) for the standard for r normalN, let 1 probabilityr and Ir denote density the By· taking a system with series components in· which the random variable has· Poisson distribution∈ truncated as zero and the functionvector of (pdf), onesφ andn( the; µ, identityΣ) for the matrix pdf of N dimensionn(µ,Σ) (rn,-variate respectively. normal distribution with mean vector µ and covariance matrix Σ), randomThe variable rest of thisXi· paperfollows is the organized normal asdistribution, follows: In we Section introduce 2, we a new define generalization the class of of normal–Poisson the normal distribution. (NP) distribution. The Φn(The; µ, restΣ) for of thisthe cdf paper of N isn organized(µ,Σ), simply as follows:Φn( ; Σ In) Sectionfor the case 2, we when defineµ = the0. class Furthermore, of normal–Poisson for r N, let (NP)1r and distribution.Ir denote The the density,Recently,· hazard some rate, authors survival focused functions on and these some new of· compounding their properties distribution; are given in Mahmoudi this Section. and We Mahmoodian derive∈ moments (2017) of NP introduced distribution the normalvectorindensity, Section of power hazard ones 3. In series and rate, Section the survivalclass identity 4, of we functionsdistributions matrix deriveof an and dimension expansion in some a system of theirr for, withrespectively. theproperties parallel density arecomponents. of given the order in this statistics. This Section. introduced In We Section derive class moments 5, of the distributions Shannon of NP distribution entropy contains is normal-geometric,derived.in SectionThe rest In Section3. of In this Section normal-Poisson, 6, paper estimates 4, is we organized deriveof the normal-logarithmic parameters an as expansion follows: based In for Section on theand a density randomnormal-binomial 2, we of definesample the order the coming models class statistics. from of as normal–Poisson special this In Section family case. of 5, Roozegar distributions the(NP) Shannon distribution. and areNadarajah entropy derived The is density, hazard rate, survival functions and some of their properties are given in this Section. We derive moments of NP distribution (2017)viaderived. an EM also In algorithm. Sectionintroduced 6, Applications estimates the power of series the to two parameters skew real data normal based sets class are on agiven of random distributions in Section sample 7. comingusing Finally, this from Section fact this that family 8 theconcludesith of componentdistributions the paper.X arei has derived SN distributioninvia Section an EM 3. algorithm. and In consider Section Applications 4,Y we= deriveN toX an two. expansion real datasets for the are density given in of Section the order 7. Finally, statistics. Section In Section 8 concludes 5, the Shannonthe paper. entropy is ∗Corresponding Author: [email protected]=1 i derived.Recebido:To begin In 27/09/2017 Section with, we 6, Revisado: estimates shall use 28/11/2017 ofthe the following parameters Aceito: 17/01/2018 notation based throughout on a random this sample paper coming : φ( ) for from the this standard family normal of distributions probability are density derived ∗Corresponding Author: [email protected] · functionviaRecebido: an EM (pdf), algorithm.27/09/2017φ ( ; µ Revisado: Applications,Σ) for 28/11/2017 the pdf to twoof Aceito:N real(µ 17/01/2018 data,Σ) ( setsn -variate are given normal in Section distribution 7. Finally, with mean Section vector 8 concludesµ and covariance the paper. matrix Σ), n · n Φn∗(Corresponding; µ,Σ) for the Author: [email protected] of Nn(µ,Σ), simply Φn( ; Σ) for the case when µ = 0. Furthermore, for r N, let 1r and Ir denote the · · ∈ vectorRecebido: of ones 27/09/2017 and the Revisado: identity 28/11/2017 matrix of Aceito: dimension 17/01/2018r, respectively. The rest of this paper is organized as follows: In Section 2, we define the class of normal–Poisson (NP) distribution. The density, hazard rate, survival functions and some of their properties are given in this Section. We derive moments of NP distribution in Section 3. In Section 4, we derive an expansion for the density of the order statistics. In Section 5, the Shannon entropy is derived. In Section 6, estimates of the parameters based on a random sample coming from this family of distributions are derived via an EM algorithm. Applications to two real data sets are given in Section 7. Finally, Section 8 concludes the paper.

∗Corresponding Author: [email protected] Recebido: 27/09/2017 Revisado: 28/11/2017 Aceito: 17/01/2018 Artigo Original DOI: 10.5902/2179460X29285

Ciência e Natura, Santa Maria, v. 40 2018, Revista do Centro de Ciências Naturais e Exatas - UFSM ISSN impressa: 0100-8307 ISSN on-line: 2179-460X

Normal–Poisson distribution as a lifetime distribution of a series system

1 2 3 Eisa Mahmoudi∗ , Hamed Mahmoodian e Fatemeh Esfandiari

1,2,3Department of Statistics, Yazd University, Yazd, Iran

Abstract In this paper, we introduce a new three parameter skewed distribution. This new class which is obtained by compounding the normal and Poisson distributions, is presented as an alternative to the class of skew-normal and normal distributions, among others. Different properties of this new distribution have been investigated. The density and distribution functions of proposed distribution, are given by a closed expressionCiência which allows e Natura us to easily v.40, compute e23, 2018 probabilities, moments and related measurements. Estimation of the parameters of this new model using maximum likelihood method via an EM-algorithm is given. Finally, some applications of this new distribution to real data are given. Keywords: Normal distribution, Poisson distribution, EM-algorithm, Maximum likelihood estimation.

1 IntroductionIntroduction

The normal distribution is probably the most well known statistical distribution and widely used to model many phenomena. Since the normal distribution is symmetric, skew-normal distributions have been proposed, studied and generalized by many authors. Azzalini (1985) proposed the skew normal (SN) distribution with the pdf φ(z; λ)=2φ(z)Φ(λz), for z,λ R. This ∈ distribution and its variations have been discussed by several authors including Azzalini (1986), Henze (2008), Branco and Dey (2011), Loperfido (2010), Arnold and Beaver (2002), Balakrishnan (2002), Azzalini and Chiogna (2004), Sharafi and Behboodian (2008), Elal-Olivero (2010). Azzalini and Valle (1996), Azzalini and Capitanio (1999) and Azzalini and Chiogna (2004) have all discussed various multivariate forms of skew-normal distributions. The number of components in a system can be fixed number n. recently, some researcher introduced a system when the number of components is a random variable N with support 1,2, . In such systems, components may be parallel or series. If the lifetime ··· of ith component is the continuous random variable Xi, then the lifetime of such a system is defined by Y = max1 i N Xi or ≤ ≤ Y = min1 i N Xi, based on whether the components are parallel or series. ≤ ≤ By taking a system with series components in which the random variable N has Poisson distribution truncated as zero and the random variable Xi follows the normal distribution, we introduce a new generalization of the normal distribution. Recently, some authors focused on these new compounding distribution; Mahmoudi and Mahmoodian (2017) introduced the normal power series class of distributions in a system with parallel components. This introduced class of distributions contains normal-geometric, normal-Poisson, normal-logarithmic and normal-binomial models as special case. Roozegar and Nadarajah (2017) also introduced the power series skew normal class of distributions using this fact that the ith component Xi has SN N distribution and consider Y = i=1 Xi. To begin with, we shall use the following notation throughout this paper : φ( ) for the standard normal probability density · function (pdf), φ ( ; µ,Σ) for the pdf of N (µ,Σ) (n -variate normal distribution with mean vector µ and covariance matrix Σ), n · n Φn( ; µ,Σ) for the cdf of Nn(µ,Σ), simply Φn( ; Σ) for the case when µ = 0. Furthermore, for r N, let 1r and Ir denote the · · ∈ vector of ones and the identity matrix of dimension r, respectively. The rest of this paper is organized as follows: In Section 2, we define the class of normal–Poisson (NP) distribution. The density, hazard rate, survival functions and some of their properties are given in this Section. We derive moments of NP distribution in Section 3. In Section 4, we derive an expansion for the density of the order statistics. In Section 5, the Shannon entropy is derived. In Section 6, estimates of the parameters based on a random sample coming from this family of distributions are derived Ciênciavia an EM e Natura algorithm. Applications to two real data sets are given in Section 7. Finally, Section 8 concludes the paper. 2

∗Corresponding Author: [email protected] Recebido: 27/09/2017 Revisado: 28/11/2017 Aceito: 17/01/2018 0.4 σ=1.8, θ=−5 µ=−3.5 µ=2.15 µ=1.5, θ=−5.7 µ=5 σ=1.2 σ=3.5 0.3 σ=5 0.2 Density Density 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5

−5 0 5 10 0246810

x x 0.15

µ=0.9, σ=3.3 θ=−1.5 θ=0.85 θ=2.5 0.10 Density 0.05 0.00

0246810

x Figure 1: Plots of the NP density function for selected parameter values.

2 The Normal–Poisson Distribution and Some Properties

Let X ,X , be a sample from a normal distribution with mean µ and variance σ2. Let N be distributed according to a Poisson 1 2 ··· distribution truncated at zero, with the probability mass function

θn P (N = n)= , n!(eθ 1) − where θ>0. Moreover, N is independent of Xi’s.

Deﬁnition 2.1. A random variable X is said to have a normal–Poisson distribution, denoted by X NP(µ,σ,θ), if ∼ X = min(X , ,X ). (2.1) 1 ··· N From the deﬁnition in (2.1), we have, for x R ∈

Remark 2.1. Even when θ<0, Equation (2.3) is also a density function. We can then deﬁne the NP distribution by Equation (2.3) for any θ R 0 . ∈ −{ } Ciência e Natura 2 0.4 σ=1.8, θ=−5 µ=−3.5 µ=2.15 µ=1.5, θ=−5.7 µ=5 σ=1.2 σ=3.5 0.3 σ=5 0.2 Density Density 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5

−5 0 5 10 0246810

x x 0.15

µ=0.9, σ=3.3 θ=−1.5 θ=0.85 θ=2.5 0.10 Density 0.05

Mahmoudi et al. : Normal–Poisson distribution as a lifetime of a series system 0.00

0246810

x Figure 1: Plots of the NP density function for selected parameter values.

22 TheThe Normal–Poisson Normal–Poisson distribution Distribution and Some and Properties Some Properties

θn P (N = n)= , n!(eθ 1) − where θ>0. Moreover, N is independent of Xi’s.

∞ F (x; µ,σ,θ)=P (X x)= P (X x N = n)P (N = n) ≤ ≤ | n=1 ∞ = P (min(X , ,X ) x N = n)P (N = n) 1 ··· N ≤ | n=1 n x µ x µ x µ n θ θ 1 Φ − θΦ − ∞ 1 Φ − θ e e ( − ( σ )) 1 e− ( σ ) =1 − σ = − = − . (2.2) − n!(eθ 1) eθ 1 1 e θ n=1 − − − − Hence the pdf of X is given by x µ x µ θ(1 Φ( − )) θφ( − )e − σ f(x; µ,σ,θ)= σ , (2.3) σ(eθ 1) − where µ R is the location parameter, σ>0 is the scale parameter and θ>0 is the shape parameter, which characterize the ∈ skewness, kurtosis, and unimodality of the distribution. 3 Mahmoudi, Mahmoodian and Esfandiari : Normal–Poisson distribution as a lifetime distribution of a series system Remark 2.1. Even when θ<0, Equation (2.3) is also a density function. We can then deﬁne the NP distribution by Equation (2.3) for any θ R 0 . ∈ −{ } Plots of the NP density function for selected parameter values are given in Figure 1. An important characteristic of the NP distribution is that its density function can be unimodal that makes this distribution many advantages in modelling lifetime data. The normal distribution with mean µ and variance σ2 is a special case of the NP distribution when θ 0. When µ =0and σ =1, → we say that X has standard NP distribution with pdf

θφ(x)eθ(1 Φ(x)) f(x; θ)= − . (eθ 1) − Proposition 2.1. If X NP(0,1,θ), then we have ∼ F (x;0,1,θ)=1 F ( x;0,1, θ) . − − − Proof. From (2.2), it can be found that

θ(1 Φ(x)) θΦ(x) 1 e − 1 e− 1 F ( x;0,1, θ)=1 − = − = F (x;0,1,θ) − − − − 1 e θ 1 e θ − − − −

Proposition 2.2. If X1 NP(0,1,θ1) and X2 NP(0,1,θ2) are independent random variables, then the stress-strength ∼ ∼ parameter, R = P (X1

θ1 θ1 θ2 e− θ2 e− e R = + eθ1+θ2 1 . (eθ1 1) (eθ2 1) (eθ1 1) θ + θ − − − − 1 2 Proof. The stress-strength parameter is given by

+ ∞ R = P (X1

θ1 θ1 θ2 e− θ2 e− e R = + eθ1+θ2 1 . (eθ1 1) (eθ2 1) (eθ1 1) θ + θ − − − − 1 2

Proposition 2.3. The densities of NP distribution can be written as inﬁnite number of linear combination of density of order n 1 θ θ − ∞ statistics. We know that e = n=1 (n 1)! , therefore − ∞ θn f(x; µ,σ,θ)= g (x; n), (2.4) n!(eθ 1) X(1) n=1 − where gX(1) (x; n) denotes the density function of X(1) = min (X1,...,Xn) . The survival and failure rate functions of the NP distribution are given respectively by

y µ θ(1 Φ( − )) e − σ 1 S(y; µ,σ,θ)= − , (2.5) eθ 1 − and x µ x µ θ(1 Φ( −σ )) θφ( −σ )e − h(y; µ,σ,θ)= y µ . (2.6) θ(1 Φ( − )) σ(e σ 1) − − Plots of the NP failure rate function for selected parameter values are given in Figure 2. An important characteristic of the NP distribution is that its failure rate function can be increasing, unimodal-bathtub shaped that makes this distribution ﬂexible in modeling different types of lifetime data. 3 Mahmoudi, Mahmoodian and Esfandiari : Normal–Poisson distribution as a lifetime distribution of a series system

Plots of the NP density function for selected parameter values are given in Figure 1. An important characteristic of the NP distribution is that its density function can be unimodal that makes this distribution many advantages in modelling lifetime data. The normal distribution with mean µ and variance σ2 is a special case of the NP distribution when θ 0. When µ =0and σ =1, → we say that X has standard NP distribution with pdf

θφ(x)eθ(1 Φ(x)) f(x; θ)= − . (eθ 1) − Proposition 2.1. If X NP(0,1,θ), then we have ∼ F (x;0,1,θ)=1 F ( x;0,1, θ) . − − − Proof. From (2.2), it can be found that

θ(1 Φ(x)) θΦ(x) 1 e − 1 e− 1 F ( x;0,1, θ)=1 − = − = F (x;0,1,θ) − − − − 1 e θ 1 e θ − − − −

Proposition 2.2. If X1 NP(0,1,θ1) and X2 NP(0,1,θ2) are independent random variables, then the stress-strength ∼ ∼ parameter, R = P (X1

θ1 θ1 θ2 e− θ2 e− e R = + eθ1+θ2 1 . (eθ1 1) (eθ2 1) (eθ1 1) θ + θ − − − − 1 2 Proof. The stress-strength parameter is given by

+ ∞ R = P (X1

θ1 θ1 θ2 e− θ2 e− e R = + eθ1+θ2 1 . (eθ1 1) (eθ2 1) (eθ1 1) θ + θ − − − − 1 2

y µ θ(1 Φ( − )) e − σ 1 S(y; µ,σ,θ)= − , (2.5) eθ 1 − and x µ x µ θ(1 Φ( −σ )) θφ( −σ )e − h(y; µ,σ,θ)= y µ . (2.6) θ(1 Φ( − )) σ(e σ 1) − − Plots of the NP failure rate function for selected parameter values are given in Figure 2. An important characteristic of the NP distribution is that its failure rate function can be increasing, unimodal-bathtub shaped that makes this distribution ﬂexible in Ciência e Natura 4 modeling different types of lifetime data.

σ=1.8, θ=−5 µ=−3.5 µ=2.5, θ=6 µ=2.15 σ=1.3 µ=5 σ=3 σ=5 Hazard Hazard 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

−5 0 5 10 −5 0 5 10

x x

µ=0.8, σ=2.5 θ=−4 θ=2.5 θ=7.5 Hazard 0.0 0.2 0.4 0.6 0.8 1.0 1.2

−5 0 5 10

x Figure 2: Plots of the NP failure rate function for selected parameter values.

3 Quantiles and Moments

The pth quantile of the NP distributions is given by

θ 1 log(p(e 1) + 1) x = µ + σΦ− 1 − . p − θ One can use this expression for generating a random sample from NP distributions with generating data from uniform distribution. Now, by using equation (2.4), we derive moment generating function of X NP(µ,σ,θ). The moment generating function is ∼ given by

∞ θn M (t)= M (t). (3.1) X n!(eθ 1) X(1) n=1 − and based on Jamalizadeh and Balakrishnan (2010)

1 2 2 T MX(1) (t)=n exp σ t + µt Φn 1( 1n 1σt; In 1 + 1n 11n 1). 2 − − − − − − Therefore n 1 2 2 ∞ θ T MX (t)=exp σ t + µt nΦn 1( 1n 1σt; In 1 + 1n 11n 1). (3.2) 2 n!(eθ 1) × − − − − − − n=1 − Using (3.2), k-th moment of a random variable X can be obtained. But from the direct and simple calculation, we have

+ θ(1 Φ(x;µ,σ)) ∞ θφ(x; µσ)e µ = xk − dx. k × (eθ 1) −∞ − By changing of variable to t = Φ(x; µ,σ), we have

1 θ(1 t) 1 k θe − µ = (µ + σΦ− (t)) dt. k eθ 1 0 − Thus, ﬁrst two moments are as follows: θ θ 1 µe− (1 + σ) θ θe− 2 1 θt µ = E(X)= [1 e− ]+ √2σ erf− (2t 1)e− dt, (3.3) 1 eθ 1 − eθ 1 × − − − 0 Ciência e Natura 4

σ=1.8, θ=−5 µ=−3.5 µ=2.5, θ=6 µ=2.15 σ=1.3 µ=5 σ=3 σ=5 Hazard Hazard 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

−5 0 5 10 −5 0 5 10

x x

µ=0.8, σ=2.5 θ=−4 θ=2.5 θ=7.5 Hazard

Mahmoudi et al. : Normal–Poisson distribution as a lifetime of a series system 0.0 0.2 0.4 0.6 0.8 1.0 1.2

−5 0 5 10

x Figure 2: Plots of the NP failure rate function for selected parameter values.

33 QuantilesQuantiles and andMoments Moments

The pth quantile of the NP distributions is given by

∞ θn M (t)= M (t). (3.1) X n!(eθ 1) X(1) n=1 − and based on Jamalizadeh and Balakrishnan (2010)

+ θ(1 Φ(x;µ,σ)) ∞ θφ(x; µσ)e µ = xk − dx. k × (eθ 1) −∞ − By changing of variable to t = Φ(x; µ,σ), we have

1 θ(1 t) 1 k θe − µ = (µ + σΦ− (t)) dt. k eθ 1 0 − Thus, ﬁrst two moments are as follows: θ θ 1 µe− (1 + σ) θ θe− 2 1 θt µ1 = E(X)= [1 e− ]+ √2σ erf− (2t 1)e− dt, (3.3) 5 Mahmoudi, Mahmoodian and Esfandiarieθ 1 : Normal–Poisson− eθ 1 distribution× as a lifetime− distribution of a series system − − 0

θ 2 2 2 θe− µ (1 + σ )+2µσ θ 2 3 µ = E(X )= [ e− ]+2√2µ(σ + σ ) 2 eθ 1 θ − 1 − 1 1 θt 4 1 2 θt erf− (2t 1)e− dt +2σ [erf− (2t 1)] e− dt , (3.4) × − − 0 0 Table 1 gives the ﬁrst four moments, variance, skewness and kurtosis of the NP(0,1,θ) for different values θ. Figure 3 shows the skewness and kurtosis plot of the NP(0,1,θ) for different values θ.

Table 1: The ﬁrst four moments, variance, skewness and kurtosis of NP distribution for µ =0,σ=1.

θ = 3 θ = 1 θ = 0.3 θ =0.01 θ =0.3 θ =1 θ =3 θ = 10 − − − E(X) 0.7541 -0.2781 0.0845 -0.0028 -0.0845 -0.2781 -0.7541 -1.5045 E(X2) 1.3477 1.0450 1.0041 1.0000 1.0041 1.0450 1.3477 2.6533 E(X3) 2.0013 -0.7003 0.2114 -0.0071 -0.2114 -0.7003 -2.0013 -5.2127 E(X4) 4.5372 3.1954 3.0179 3.0000 3.0179 3.1954 4.5372 11.2262 V AR 0.7790 0.9677 0.9970 1.0000 0.9970 0.9677 0.7790 0.3898 SK -0.2764 0.1349 -0.0421 0.0014 0.0421 0.1349 0.2764 -0.1973 KUR 3.5076 3.0792 3.0074 3.0000 3.0074 3.0792 3.5076 3.4236

Skewness Kurtosis 01234

0 5 10 15 20

Figure 3: Plots of skewness and kurtosis of NP distribution for selected parameter values θ.

4 Order statistics

Let X1,..,Xn be a random sample from NP distribution. The pdf of the ith order statistic, X(i) , is given by

n!f(x; µ,σ,θ) n i i 1 f (x; µ,σ,θ)= [1 F (x; µ,σ,θ)] − [F (x; µ,σ,θ)] − . (4.1) (i) (n i)!(i 1)! − − − By using the binomial series expansion we have

n!f(x; µ,σ,θ) ∞ j n i j+i 1 f (x)= ( 1) − [F (x; µ,σ,θ)] − . (i) (n i)!(i 1)! − j j=0 − −

The moments of the probability density function (4.1) cannot be obtained in a closed form but the values are derived by using numerical computation. Figure 4 shows the expected values of the order statistics plot from NP(0,1,θ) for n = 40, i =1,10,20 and different values θ. 5 Mahmoudi, Mahmoodian and Esfandiari : Normal–Poisson distribution as a lifetime distribution of a series system

Table 1: The ﬁrst four moments, variance, skewness and kurtosis of NP distribution for µ =0,σ=1.

θ = 3 θ = 1 θ = 0.3 θ =0.01 θ =0.3 θ =1 θ =3 θ = 10 − − − E(X) 0.7541 -0.2781 0.0845 -0.0028 -0.0845 -0.2781 -0.7541 -1.5045 E(X2) 1.3477 1.0450 1.0041 1.0000 1.0041 1.0450 1.3477 2.6533 E(X3) 2.0013 -0.7003 0.2114 -0.0071 -0.2114 -0.7003 -2.0013 -5.2127 E(X4) 4.5372 3.1954 3.0179 3.0000 3.0179 3.1954 4.5372 11.2262 Ciência e Natura v.40, e23, 2018 V AR 0.7790 0.9677 0.9970 1.0000 0.9970 0.9677 0.7790 0.3898 SK -0.2764 0.1349 -0.0421 0.0014 0.0421 0.1349 0.2764 -0.1973 KUR 3.5076 3.0792 3.0074 3.0000 3.0074 3.0792 3.5076 3.4236

Skewness Kurtosis 01234

0 5 10 15 20

Figure 3: Plots of skewness and kurtosis of NP distribution for selected parameter values θ.

4 OrderOrder statistics statistics

Let X1,..,Xn be a random sample from NP distribution. The pdf of the ith order statistic, X(i) , is given by

n!f(x; µ,σ,θ) n i i 1 f (x; µ,σ,θ)= [1 F (x; µ,σ,θ)] − [F (x; µ,σ,θ)] − . (4.1) (i) (n i)!(i 1)! − − − By using the binomial series expansion we have

n!f(x; µ,σ,θ) ∞ j n i j+i 1 f (x)= ( 1) − [F (x; µ,σ,θ)] − . (i) (n i)!(i 1)! − j j=0 − −

The moments of the probability density function (4.1) cannot be obtained in a closed form but the values are derived by using Ciência e Natura 6 numerical computation. Figure 4 shows the expected values of the order statistics plot from NP(0,1,θ) for n = 40, i =1,10,20 and different values θ.

The expected value 1

i=1 i=10 i=20 −4 −3 −2 −1 0

0 5 10 15 20

Figure 4: Plots of the expected values of the order statistics from NP(0,1,θ) for n = 40, i =1,10,20.

5 Shannon entropy

Let X be a random variable from NP(µ,σ,θ). Then, the Shannon entropy of X is given by

x µ + x µ θ(1 Φ( −σ )) ∞ θφ −σ e − H(x)= E (log(f(X; µ,σ,θ))) = f(x; µ,σ,θ) log θ dx − − × σ(e 1) −∞ − + ∞ x µ = log(θ)+log(σ(eθ 1)) f(x; µ,σ,θ) log φ − dx − − − σ −∞ + ∞ x µ θ f(x; µ,σ,θ) 1 Φ − dx. − × − σ −∞ We can write the ﬁrst integral as

+ ∞ x µ 1 f(x; µ,σ,θ) log φ − dx = log(√2π) E(X2) 2µE(X)+µ2 , σ − − 2σ2 − −∞ Where E(X2) and E(X) follow from (3.3) and (3.4). We can write the second integral as

+ θ ∞ x µ e (θ 1) 1 f(x; µ,σ,θ) 1 Φ − dx = − − . × − σ θ(eθ 1) −∞ − Finally, we obtain

H(x)= log(θ)+log(σ(eθ 1)) + log(√2π) − − 1 eθ (θ 1) 1 + E(X2) 2µE(X)+µ2 − − . 2σ2 − − (eθ 1) − 6 Estimation and inference

Let x , ,x be n observations from NP(µ,σ,θ) and Ψ =(µ,σ,θ)T be the parameter vector. The log-likelihood function is 1 ··· n given by

n x µ n x µ l (Ψ)=n log(θ) n log σ(eθ 1) + log φ i − + θ 1 Φ i − . n − − σ − σ i=1 i=1 Ciência e Natura 6

The expected value 1

i=1 i=10 i=20 −4 −3 −2 −1 0

Mahmoudi et al. : Normal–Poisson0 5 distribution10 as15 a lifetime20 of a series system θ

Figure 4: Plots of the expected values of the order statistics from NP(0,1,θ) for n = 40, i =1,10,20.

55 ShannonShannon entropy entropy

Let X be a random variable from NP(µ,σ,θ). Then, the Shannon entropy of X is given by

+ ∞ x µ 1 f(x; µ,σ,θ) log φ − dx = log(√2π) E(X2) 2µE(X)+µ2 , σ − − 2σ2 − −∞ Where E(X2) and E(X) follow from (3.3) and (3.4). We can write the second integral as

+ θ ∞ x µ e (θ 1) 1 f(x; µ,σ,θ) 1 Φ − dx = − − . × − σ θ(eθ 1) −∞ − Finally, we obtain

H(x)= log(θ)+log(σ(eθ 1)) + log(√2π) − − 1 eθ (θ 1) 1 + E(X2) 2µE(X)+µ2 − − . 2σ2 − − (eθ 1) − 6 Estimation and inference 6 Estimation and inference

Let x , ,x be n observations from NP(µ,σ,θ) and Ψ =(µ,σ,θ)T be the parameter vector. The log-likelihood function is 1 ··· n given by

n n θ xi µ xi µ 7l Mahmoudi,n(Ψ)=n log( Mahmoodianθ) n log andσ(e Esfandiari1) + : Normal–Poissonlog φ − distribution+ θ as a lifetime1 Φ distribution− . of a series system − − σ − σ i=1 i=1

T ∂ln ∂ln ∂ln The maximum likelihood estimation (MLE) of Ψ, say Ψ, is obtained by solving the nonlinear system of equations ∂µ , ∂σ , ∂θ = 0, where ∂l (Ψ) 1 n θ n x µ n = (x µ)+ φ i − , (6.1) ∂µ σ2 i − σ σ i=1 i=1 ∂l (Ψ) n 1 n θ n x µ n = + (x µ)2 + (x µ)φ i − , (6.2) ∂σ −σ σ3 i − σ2 i − σ i=1 i=1 n ∂ln(Ψ) n θ 1 xi µ = n(1 e− )− + 1 Φ − . (6.3) ∂θ θ − − − σ i=1 Clearly, MLEs cannot be obtained in closed forms. The observed information matrix is obtained for approximate conﬁdence intervals and hypothesis tests of on the model parameters. The 3 3 observed information matrix is given by ×

Iµµ Iµσ Iµθ I (Ψ)= I I I , n −  µσ σσ σθ  Iµθ Iσθ Iθθ   where

n θ n x µ I = + (x µ)φ i − , µµ −σ2 σ3 i − σ i=1 2 n θ n x µ θ n x µ I = (x µ) φ i − + (x µ)2φ i − , µσ −σ3 i − − σ2 σ σ4 i − σ i=1 i=1 i=1 1 n x µ I = φ i − , µθ σ σ i=1 n 3 n 2θ n x µ θ n x µ I = (x µ)2 (x µ)φ i − + (x µ)3φ i − , σσ σ2 − σ4 i − − σ3 i − σ σ5 i − σ i=1 i=1 i=1 1 n x µ I = (x µ)φ i − , σθ σ2 i − σ i=1 n θ θ 2 I = + ne− (1 e− )− . θθ −θ2 −

1 It is well-known that under regularity conditions, the asymptotic distribution of √n Ψ Ψ is N (0,J (Ψ)− ), where − 3 n 1 Jn(Ψ) = limn = n− In(Ψ). Therefore, an 100(1 γ) asymptotic conﬁdence interval for each parameter Ψr is given by →∞ − ACI = Ψ Z Irr,Ψ + Z Irr , r r − γ/2 r γ/2 rr 1 where I is the (r,r) diagonal element of I (Ψ)− for r =1,2,3 and Z is the quantile 1 γ/2 of the standard normal n γ/2 − distribution. The solution of the three non-linear normal equations in (6.1)-(6.3) is needed using a numerical method. We may use the standard methods such as Newton-Raphson, Nelder-Mead and BFGS, but they have their usual problem of convergence. If the initial guesses are not close to the optimal value, the iteration may not converge. Due to this reason, we propose to use EM-algorithm to compute the MLEs. Suppose, (x ,z ),...,(x ,z ) is a random sample of size n from (X,N). We deﬁne a { 1 1 n n } hypothetical complete-data distribution with a joint probability density function in the form

z 1 θz x µ x µ − g(z,x; Ψ)= zφ − 1 Φ − , σz!(eθ 1) σ − σ − where µ R, σ>0, θ R 0 , x R and z N. The probability density function of Z given X = x is given by ∈ ∈ −{ } ∈ ∈ z 1 z 1 x µ − g(z,x; Ψ) θ − z 1 Φ −σ g(z x)= = − x µ , | f(x) θ(1 Φ( −σ )) z!e − 7 Mahmoudi, Mahmoodian and Esfandiari : Normal–Poisson distribution as a lifetime distribution of a series system

Iµµ Iµσ Iµθ I (Ψ)= I I I , n −  µσ σσ σθ  Iµθ Iσθ Iθθ   where

z 1 θz x µ x µ − g(z,x; Ψ)= zφ − 1 Φ − , σz!(eθ 1) σ − σ − where µ R, σ>0, θ R 0 , x R and z N. The probability density function of Z given X = x is given by ∈ ∈ −{ } ∈ ∈ z 1 Ciência e Natura z 1 x µ − 8 g(z,x; Ψ) θ − z 1 Φ −σ g(z x)= = − x µ , | f(x) θ(1 Φ( −σ )) z!e − 2 z 1 θ z θ − and since (1 + θ)e = ∞ , the expected value of Z X = x, is given by z=1 z! |

x µ z 1 ∞ z 1 2 − θ − z 1 Φ −σ E(Z X = x)= − x µ | θ(1 Φ( −σ )) z=1 z!e − x µ z 1 ∞ 2 − 1 z θ 1 Φ −σ = x µ − θ(1 Φ( −σ )) z! e − z=1 x µ = 1+θ 1 Φ − . − σ By using the maximum likelihood estimation over Ψ, with the missing Z’s replaced by their conditional expectations given above, the M-step of EM cycle is completed. The log-likelihood of the model parameters for the complete data set is

n n 1 2 l∗ (x,z; µ,σ,θ) z log θ n log σ (x µ) n ∝ i − − 2σ2 i − i=1 i=1 n x µ + (z 1) log 1 Φ i − n log eθ 1 . i − − σ − − i=1

∂ln∗ ∂ln∗ ∂ln∗ The components of the score function ∂µ , ∂σ , ∂θ are given by n n x µ ∂l 1 1 φ i− n∗ = (x µ)+ (z 1) σ , 2 i i xi µ ∂µ σ − σ − 1 Φ − i=1 i=1 σ − n n xi µ ∂l n 1 1 (xi µ) φ − n∗ = + (x µ)2 + (z 1) − σ , 3 i 2 i xi µ ∂σ −σ σ − σ − 1 Φ − i=1 i=1 − σ n ∂l 1 neθ n∗ = z . ∂θ θ i − eθ 1 i=1 − The maximum likelihood estimates can be obtained from the iterative algorithm given by

(h+1) n n xi µ 1 φ −σ(h) x µ(h+1) + z(h) 1 =0, (h) i i (h+1) σ − − xi µ i=1 i=1 1 Φ − (h) − σ (h) n n (h) xi µ 1 2 1 xi µ φ σ(−h+1) n x µ(h) z(h) 1 − =0, 2 i (h+1) i (h) − (h+1) − − σ − xi µ σ ) i=1 i=1 1 Φ (−h+1) − σ (h+1) n eθ 1 (h+1) (h) θ = − zi , θ(h+1) ne i=1 where µ(h), σ(h) and θ(h) are found numerically. Here, for i =1,...,n, we have that

x µ(h) z(h) =1+θ(h) 1 Φ i − . i − σ(h) In the rest of this section, we verify the performance of the proposed estimator of, µ, σ and θ of the proposed EM method for NP distribution. We simulate 1000 times under the NP distribution with different sets of parameters and sample sizes n = 50, 100, 300 and 500. For each sample size, we compute the MLEs by EM-method. We also compute the root of mean square errors (RMSE), standard errors (SE) and covariances of the MLEs of the EM-algorithm. The results for the NP distribution are reported in Tables 2. Some of the points are quite clear from the simulation results: (i) Convergence has been achieved in all cases and this emphasizes the numerical stability of the EM-algorithm. (ii) The differences between the average estimates and the true values are almost small. (iii) These results suggest that the EM estimates have performed consistently. (iv) As the sample size increases, the root of mean square errors and the standard errors of the MLEs decrease. Ciência e Natura 8

2 z 1 θ z θ − and since (1 + θ)e = ∞ , the expected value of Z X = x, is given by z=1 z! |

n n 1 2 Mahmoudil∗ (x,z; µ,σ,θ et al.) : Normal–Poissonz log θ distributionn log σ as a lifetime(x µ of) a series system n ∝ i − − 2σ2 i − i=1 i=1 n x µ + (z 1) log 1 Φ i − n log eθ 1 . i − − σ − − i=1

Table 2: The averages of the 1000 MLE’s, mean of the simulated root of mean square errors, mean of simulated standard errors and mean of the simulated covariances of EM estimators for NP distribution.

Average estimators RMSE SE Cov n (µ,σ,θ) µ σ θ µ σ θ µ σ θ (µ,σ)) (µ,θ)(σ,θ) (0.0, 1.0, -1.0) -0.0741 1.0103 -1.2628 0.5032 0.1301 0.2982 0.2227 0.1235 0.6131 -0.0118 0.1376 -0.02821 (0.0, 1.0, 0.5) -0.0493 0.8693 0.7323 0.1010 0.1430 0.3171 0.0882 0.0580 0.2160 0.0021 0.0104 0.0020 50 (0.0, 1.0, 1.0) 0.0276 0.9979 1.1047 0.2004 0.0747 0.6387 0.1986 0.0747 0.6303 0.0058 0.1080 0.0134 (0.0, 1.0,2.0) -0.0562 0.9922 1.8759 0.4116 0.1322 1.5530 0.4079 0.1321 1.5488 0.0371 0.5852 0.1245 (0.0, 1.0, 5.0) -0.0142 0.9864 5.5179 0.5468 0.1795 3.0369 0.5469 0.1790 2.9940 0.0876 1.5472 0.4250 (0.0, 1.0, -1.0) -0.0667 1.0073 -1.1713 0.3127 0.2112 1.454 0.2014 0.0667 0.6234 -0.0073 0.1285 -0.0214 (0.0, 1.0,0.5) -0.0277 0.9054 0.6971 0.0912 0.1012 0.2878 0.0869 0.0360 0.2099 0.0016 0.0128 0.0015 100 (0.0, 1.0, 1.0) 0.0180 1.0043 1.0677 0.1710 0.0482 0.5787 0.1701 0.0480 0.5750 0.0044 0.0915 0.0131 (0.0, 1.0, 2.0) -0.0283 0.9947 1.9266 0.3463 0.1058 1.3190 0.3453 0.1057 1.3176 0.0287 0.4315 0.1016 (0.0, 1.0, 5.0) 0.0169 0.9946 5.2860 0.5058 -0.0054 0.2860 0.5057 0.1609 2.6908 0.0761 1.3104 0.3766 (0.0, 1.0, -1.0) -0.0561 1.0016 -1.0981 0.2873 0.1897 1.372 0.1871 0.0424 0.5821 -0.0048 0.1201 -0.0166 (0.0, 1.0, 0.5) 0.0007 0.9311 0.6768 0.0662 0.0705 0.2596 0.0662 0.0150 0.1902 0.0008 0.0102 0.0016 300 (0.0, 1.0, 1.0) 0.0197 1.0040 1.0824 0.1568 0.0373 0.5492 0.1556 0.0371 0.5432 0.0032 0.0810 0.0103 (0.0, 1.0, 2.0) 0.0035 1.0046 2.0353 0.2687 0.0739 1.0202 0.2688 0.0738 1.0201 0.0167 0.2663 0.0605 (0.0, 1.0, 5.0) -0.0534 0.9845 4.8726 0.3311 0.1002 1.6897 0.3270 0.0991 1.6857 0.0313 0.5414 0.1563 (0.0, 1.0, -1.0) -0.0424 1.0040 -1.0665 0.2754 0.1765 1.381 0.1769 0.0382 0.5671 -0.0041 0.1128 -0.0102 (0.0, 1.0, 0.5) -0.0068 0.9237 0.6742 0.0554 0.0794 0.2620 0.0550 0.0219 0.1958 0.0006 0.0094 0.0018 500 (0.0, 1.0, 1.0) 0.0001 0.9873 1.0392 0.2196 0.1055 0.6364 0.2197 0.1048 0.6355 0.0088 0.1064 0.0161 (0.0, 1.0, 2.0) 0.0019 1.0033 2.0232 0.2345 0.0632 0.8698 0.2346 0.0631 0.8699 0.0128 0.1996 0.0455 (0.0, 1.0, 5.0) -0.0208 0.9937 5.0709 0.3311 0.1002 1.6897 0.3270 0.0991 1.6857 0.0313 0.5414 0.1563

7 Applications

In this section, the NP distribution is fitted to three real data sets and also compared the fitted NP with two relative models, normal 2 x µ x µ (N) and skew-normal (SN) distributions with pdf σ φ −σ Φ α −σ , to show the superiority of the NP distribution. The first data set concerning the plasma ferritin concentration 102 male and 100 female athletes collected at the Australian Institute of Sport. We estimate parameters by numerically maximizing the likelihood function. The variance covariance matrix of the MLEs under the NP distribution is computed as

0.2844 0.08030 0.01995 0.08030 0.0458 0.0045 .  0.01995 0.0045 0.0018    The MLEs of the parameters, -2log-likelihood, AIC (Akaike Information Criterion), the Kolmogorov-Smirnov test statistic (K-S) and the associated p-value are displayed in Table 3 for this data set. The results for these data set show that the NP distribution provides a better fit to this data set than the N and SN distributions. Also this conclusion is confirmed from the plots of the fitted densities in Figure 5.

Table 3: MLEs, -2 Log L, K-S, p-value and AIC for plasma ferritin concentration.

Dist MLE -2 Log L K-S p-value AIC NP µˆ = 111.7220, σˆ = 49.9121,θˆ =3.0089 2106.9180 0.0824 0.1288 2112.9180 N µˆ = 76.8762, σˆ = 47.3835 2131.994 0.1217 0.0050 2135.994 SN µˆ = 75.9954, σˆ = 47.3919,αˆ =0.0234 2131.994 0.1217 0.0050 2137.994

NP N SN Density 0.000 0.002 0.004 0.006 0.008 0.010 0.012

0 50 100 150 200

plasma ferritin concentration

Figure 5: Plots of ﬁtted NP, N and SN for plasma ferritin concentration data. 9 Mahmoudi, Mahmoodian and Esfandiari : Normal–Poisson distribution as a lifetime distribution of a series system

Average estimators RMSE SE Cov n (µ,σ,θ) µ σ θ µ σ θ µ σ θ (µ,σ)) (µ,θ)(σ,θ) (0.0, 1.0, -1.0) -0.0741 1.0103 -1.2628 0.5032 0.1301 0.2982 0.2227 0.1235 0.6131 -0.0118 0.1376 -0.02821 (0.0, 1.0, 0.5) -0.0493 0.8693 0.7323 0.1010 0.1430 0.3171 0.0882 0.0580 0.2160 0.0021 0.0104 0.0020 50 (0.0, 1.0, 1.0) 0.0276 0.9979 1.1047 0.2004 0.0747 0.6387 0.1986 0.0747 0.6303 0.0058 0.1080 0.0134 (0.0, 1.0,2.0) -0.0562 0.9922 1.8759 0.4116 0.1322 1.5530 0.4079 0.1321 1.5488 0.0371 0.5852 0.1245 (0.0, 1.0, 5.0) -0.0142 0.9864 5.5179 0.5468 0.1795 3.0369 0.5469 0.1790 2.9940 0.0876 1.5472 0.4250 (0.0, 1.0, -1.0) -0.0667 1.0073 -1.1713 0.3127 0.2112 1.454 0.2014 0.0667 0.6234 -0.0073 0.1285 -0.0214 (0.0, 1.0,0.5) -0.0277 0.9054 0.6971 0.0912 0.1012 0.2878 0.0869 0.0360 0.2099 0.0016 0.0128 0.0015 100 (0.0, 1.0, 1.0) 0.0180 1.0043 1.0677 0.1710 0.0482 0.5787 0.1701 0.0480 0.5750 0.0044 0.0915 0.0131 (0.0, 1.0, 2.0) -0.0283 0.9947 1.9266 0.3463 0.1058 1.3190 0.3453 0.1057 1.3176 0.0287 0.4315 0.1016 (0.0, 1.0, 5.0) 0.0169 0.9946 5.2860 0.5058 -0.0054 0.2860 0.5057 0.1609 2.6908 0.0761 1.3104 0.3766 (0.0, 1.0, -1.0) -0.0561 1.0016 -1.0981 0.2873 0.1897 1.372 0.1871 0.0424 0.5821 -0.0048 0.1201 -0.0166 (0.0, 1.0, 0.5) 0.0007 0.9311 0.6768 0.0662 0.0705 0.2596 0.0662 0.0150 0.1902 0.0008 0.0102 0.0016 300 (0.0, 1.0, 1.0) 0.0197 1.0040 1.0824 0.1568 0.0373 0.5492 0.1556 0.0371 0.5432 0.0032 0.0810 0.0103 (0.0, 1.0, 2.0) 0.0035 1.0046 2.0353 0.2687 0.0739 1.0202 0.2688 0.0738 1.0201 0.0167 0.2663 0.0605 (0.0, 1.0, 5.0) -0.0534 0.9845Ciência 4.8726 0.3311 e Natura 0.1002 v.40, 1.6897 e23, 0.3270 2018 0.0991 1.6857 0.0313 0.5414 0.1563 (0.0, 1.0, -1.0) -0.0424 1.0040 -1.0665 0.2754 0.1765 1.381 0.1769 0.0382 0.5671 -0.0041 0.1128 -0.0102 (0.0, 1.0, 0.5) -0.0068 0.9237 0.6742 0.0554 0.0794 0.2620 0.0550 0.0219 0.1958 0.0006 0.0094 0.0018 500 (0.0, 1.0, 1.0) 0.0001 0.9873 1.0392 0.2196 0.1055 0.6364 0.2197 0.1048 0.6355 0.0088 0.1064 0.0161 (0.0, 1.0, 2.0) 0.0019 1.0033 2.0232 0.2345 0.0632 0.8698 0.2346 0.0631 0.8699 0.0128 0.1996 0.0455 (0.0, 1.0, 5.0) -0.0208 0.9937 5.0709 0.3311 0.1002 1.6897 0.3270 0.0991 1.6857 0.0313 0.5414 0.1563

77 ApplicationsApplications

Table 3: MLEs, -2 Log L, K-S, p-value and AIC for plasma ferritin concentration.

NP N SN Density 0.000 0.002 0.004 0.006 0.008 0.010 0.012

0 50 100 150 200

plasma ferritin concentration

Figure 5: Plots of ﬁtted NP, N and SN for plasma ferritin concentration data. Ciência e Natura 10

The second data set is given by Birnbaum and Saunders (1969) that refers to fatigue life of 6061-T6 aluminium coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set consists of 100 observations. The variance covariance matrix of the MLEs under the NP distribution is computed as

0.0134 0.0042 0.0231 0.0042 0.0015 0.0068 .  0.0231 0.0068 0.0427    Table 4 gives the MLEs of the parameters, -2log-likelihood, AIC, the K-S test statistic and the associated P-value for the second data set. The ﬁtted densities functions of NP, N and SN models is displayed in Figure 6. The results for this data set show that the NP distribution is a good competitor for the normal and SN distributions. Also the plots of the densities in Figure 6 conﬁrmed this conclusion. Table 4: MLEs, 2LogL, K-S, p-value and AIC for time between failures data. − Dist MLE -2 Log L K-S p-value AIC NP µˆ =9.6705, σˆ =2.7745,θˆ =4.6120 442.0941 0.0481 0.9749 446.0941 N µˆ =6.8780, σˆ =2.2499 444.1903 0.0687 0.7325 448.1903 SN µˆ =6.8362, σˆ =2.2300,αˆ = 0.0018 444.1903 0.0701 0.7012 450.1903 − 0.20

NP N 0.15 SN 0.10 Density 0.05 0.00

0 5 10 15

Fatigue life

Figure 6: Plots of ﬁtted NP, N and SN for fatigue life data.

The third data set from Bjerkedal (1960), represents the survival time in days of 72 guinea pigs infected with virulent tubercle bacilli. The variance covariance matrix of the MLEs under the NP distribution is computed as

11.4618 3.3416 0.3748 3.3416 1.8285 0.0900 .  0.3748 0.0900 0.0018    Table 5 gives the MLEs of the parameters, -2log-likelihood, AIC, the K-S test statistic and the associated P-value for the third data set. The fitted densities functions of NP, N and SN models is displayed in Figure 7. The results for this data set show that the NP distributions yield the best fit among the N and SN distributions. Also the plots of the densities in Figure 6 confirmed this conclusion.

8 Concluding remarks

In this paper based on compounding approach, a new three-parameter normal-Poisson was developed. The proposed NP is an alternative to the Azzalini skew-normal distribution for ﬁtting skewed data. We obtain expressions for the moments. The estimation of the unknown parameters of the proposed distribution is approached by the EM-algorithm. Finally, we ﬁtted NP distribution to three real data sets to show the potential of the new proposed distribution. Ciência e Natura 10

0.0134 0.0042 0.0231 0.0042 0.0015 0.0068 .   Mahmoudi et al. : Normal–Poisson0.0231 distribution 0.0068 0.0427 as a lifetime of a series system   Table 4 gives the MLEs of the parameters, -2log-likelihood, AIC, the K-S test statistic and the associated P-value for the second data set. The ﬁtted densities functions of NP, N and SN models is displayed in Figure 6. The results for this data set show that the NP distribution is a good competitor for the normal and SN distributions. Also the plots of the densities in Figure 6 conﬁrmed this conclusion. Table 4: MLEs, 2LogL, K-S, p-value and AIC for time between failures data. − Dist MLE -2 Log L K-S p-value AIC NP µˆ =9.6705, σˆ =2.7745,θˆ =4.6120 442.0941 0.0481 0.9749 446.0941 N µˆ =6.8780, σˆ =2.2499 444.1903 0.0687 0.7325 448.1903 SN µˆ =6.8362, σˆ =2.2300,αˆ = 0.0018 444.1903 0.0701 0.7012 450.1903 − 0.20

NP N 0.15 SN 0.10 Density 0.05 0.00

0 5 10 15

Fatigue life

Figure 6: Plots of ﬁtted NP, N and SN for fatigue life data.

8 Concluding remarks

Table 5: MLEs, 2LogL, K-S, P-value and AIC for guinea pigs data. − Dist MLE -2 Log L K-S P-value AIC NP µˆ = 256.3517, σˆ = 110.7858,θˆ =3.1438 431.9072 0.09718 0.5062 869.8144 N µˆ = 176.8334, σˆ = 102.7445 435.6852 0.1299 0.1762 875.3704 SN µˆ = 174.9396, σˆ = 102.7617,αˆ =0.0232 435.6852 0.9478 0 877.3703

NP N SN Density 0.000 0.001 0.002 0.003 0.004 0.005

0 100 200 300 400 500 600

Guinea pigs

Figure 7: Plots of ﬁtted NP, N and SN for air conditioning system data.

Acknowledgement

The authors would like to thank the Referees and the Editor for their valuable comments and suggestions which have contributed to substantially improving the manuscript. The authors are also indebted to Yazd University for supporting this research.

References

B.C. Arnold, R.J. Beaver, Skewed multivariate models related to hidden truncation and/or selective reporting (with discussion), Test 11 (1) (2002) 7-5 A. Azzalini, A class of distributions which includes the normal ones, Scandinavian Journal of Statistics 12 (1985) 171-178. A. Azzalini, Further results on a class of distributions which includes the normal ones, Statistica XLVI (2) (1986) 199-208. A. Azzalini, A. Capitanio, Statistical applications of the multivariate skew-normal distribution, Journal of the Royal Statistical Society Series B 61 (3) (1999) 579-602. A. Azzalini, M. Chiogna, Some results on the stress-strength model for skew-normal variates, Metron 3 (2004) 315-326. A. Azzalini, A.D. Valle, The multivariate skew-normal distribution, Biometrika 83 (4) (1996) 715-726. N. Balakrishnan, Discussionon“Skew multivariate normal models related to hidden truncation and / or selective reporting” by B.C. Arnold and R. J. Beaver, Test 11 (2002) 37-39. T. Bjerkedal, Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli.American Journal of Epidemiology 72 (1960) 130-148. Z.W. Birnbaum, S.C. Saunders, Estimation for a family of life distributions with applications to fatigue, Journal of Applied Probability 6 (1969) 328-347. M. Branco, D.K. Dey, A general class of multivariate elliptical distribution, Journal of Multivariate Analysis 79 (2001) 99-113. D. DElal-Olivero, Alpha-skew-normal distribution, Proyecciones 29 (2010) 224-240. N.A. Henze, A probabilistic representation of the skew-normal distribution, Scandinavian Journal of Statistics 13 (1986) 271-275. 11 Mahmoudi, Mahmoodian and Esfandiari : Normal–Poisson distribution as a lifetime distribution of a series system

Table 5: MLEs, 2LogL, K-S, P-value and AIC for guinea pigs data. − Dist MLE -2 Log L K-S P-value AIC Ciência e Natura v.40, e23, 2018 NP µˆ = 256.3517, σˆ = 110.7858,θˆ =3.1438 431.9072 0.09718 0.5062 869.8144 N µˆ = 176.8334, σˆ = 102.7445 435.6852 0.1299 0.1762 875.3704 SN µˆ = 174.9396, σˆ = 102.7617,αˆ =0.0232 435.6852 0.9478 0 877.3703

NP N SN Density 0.000 0.001 0.002 0.003 0.004 0.005

0 100 200 300 400 500 600

Guinea pigs

Figure 7: Plots of ﬁtted NP, N and SN for air conditioning system data.

Acknowledgement

ReferênciasReferences

B.C. Arnold, R.J. Beaver, Skewed multivariate models related to hidden truncation and/or selective reporting (with discussion), Test 11 (1) (2002) 7-5 A. Azzalini, A class of distributions which includes the normal ones, Scandinavian Journal of Statistics 12 (1985) 171-178. A. Azzalini, Further results on a class of distributions which includes the normal ones, Statistica XLVI (2) (1986) 199-208. A. Azzalini, A. Capitanio, Statistical applications of the multivariate skew-normal distribution, Journal of the Royal Statistical Society Series B 61 (3) (1999) 579-602. A. Azzalini, M. Chiogna, Some results on the stress-strength model for skew-normal variates, Metron 3 (2004) 315-326. A. Azzalini, A.D. Valle, The multivariate skew-normal distribution, Biometrika 83 (4) (1996) 715-726. N. Balakrishnan, Discussionon“Skew multivariate normal models related to hidden truncation and / or selective reporting” by B.C. Arnold and R. J. Beaver, Test 11 (2002) 37-39. T. Bjerkedal, Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli.American Journal of Epidemiology 72 (1960) 130-148. Z.W. Birnbaum, S.C. Saunders, Estimation for a family of life distributions with applications to fatigue, Journal of Applied Probability 6 (1969) 328-347. M. Branco, D.K. Dey, A general class of multivariate elliptical distribution, Journal of Multivariate Analysis 79 (2001) 99-113. D. DElal-Olivero, Alpha-skew-normal distribution, Proyecciones 29 (2010) 224-240. Ciência e Natura 12 N.A. Henze, A probabilistic representation of the skew-normal distribution, Scandinavian Journal of Statistics 13 (1986) 271-275.

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Eisa Mahmoudi Department of Statistics, Yazd University, Yazd, Iran Email: [email protected] author’s contribution:

Hamed Mahmoodian

Department of Statistics, Yazd University, Yazd, Iran Email: [email protected] author’s contribution:

Fatemeh Esfandiari

Department of Statistics, Yazd University, Yazd, Iran Email: [email protected] author’s contribution::