Iteratioal Joural of Advaced Statistics ad Probability, 3 (2 (2015 146-160 www.sciecepubco.com/idex.php/ijasp Sciece Publishig Corporatio doi: 10.14419/ijasp.v3i2.4624 Research Paper The complemetary Poisso-Lidley class of distributios Amal S. Hassa*, Salwa M. Assar, Kareem A. Ali Istitute of Statistical Studies ad Research, Departmet of Mathematical Statistics, Cairo Uiversity, Egypt *Correspodig author E-mail: dr_amal2@hotmail.com Copyright 2015 Amal S. Hassa et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract This paper proposed a ew geeral class of cotiuous lifetime distributios, which is a complemetary to the Poisso- Lidley family proposed by Asgharzadeh et al. [3]. The ew class is derived by compoudig the maximum of a radom umber of idepedet ad idetically cotiuous distributed radom variables, ad Poisso-Lidley distributio. Several properties of the proposed class are discussed, icludig a formal proof of probability desity, cumulative distributio, ad reliability ad hazard rate fuctios. The ukow parameters are estimated by the maximum likelihood method ad the Fisher s iformatio matrix elemets are determied. Some sub-models of this class are ivestigated ad studied i some details. Fially, a real data set is aalyzed to illustrate the performace of ew distributios. Keywords: Poisso-Lidley Distributio; Complemetary; Lifetime Distributios; Distributio of the Maximum. 1. Itroductio Lifetime distributios are of great importace i several applicatios for theoretical research ad applied fields such as; isurace, medical, egieerig, biological, commuicatios ad life testig. Recetly, attempts have bee made to defie ew lifetime probability distributios, that provide great flexibility i modelig data i practice. Oe way to geerate a ew lifetime distributio is compoudig procedure. Firstly; Adamidis ad Loukas [1] itroduced a ew distributio with decreasig failure rate by mixig the distributio of the miimum of a fixed umber of expoetial distributed radom variables with geometric radom variable. Similar procedure for derivig aother lifetime distributios discussed by several authors; such as; Kus [11]; Thamasbi ad Rezaei [17] ad Chahkadi ad Gajali [6]. I the same way; a compoud class of lifetime distributios with Poisso distributio is derived by Alkari ad Oraby [2]. Asgharzadeh et al. [3] obtaied a ew compoud class of Poisso Lidley by compoudig distributio of the miimum of a fixed umber of ay lifetime distributio with Poisso Lidley distributio. Recetly based o reliability studies, some researches have proposed a series of ew distributios for the maximum of a sequece of idetically idepedet distributed radom variables, which represets the risk times of the system compoet. Louzada et al. [15] itroduced a two-parameter lifetime distributio with icreasig failure rate by compoudig the distributio of the maximum of sequece of idepedet ad idetically compoets radom variables from expoetial distributio ad geometric radom variable. Next, Flores et al. [8] treat the distributio of a vector s with maximum compoets that are expoetially distributed i a radom umber of a power series distributio type. This type of distributio is called complemetary expoetial power series distributio. Likewise, Leahu et al. [12] itroduced two ew families of distributios amed as max- Erlag power series distributio ad mi- Erlag power series distributio. They mixed miimum ad maximum of a radom umber of idepedetly; idetically Erlag distributed radom variables with power series distributio. Cordeiro ad Silva [7] itroduced the complemetary class of exteded Weibull power series distributios by usig maximum distributio of exteded Weibull. I this article, a ew compoud class of Poisso-Lidley distributio is suggested by mixig the maximum of a fixed umber of ay cotiuous lifetime radom variables with Poisso- Lidley radom variable. I particular, some submodels of this class are derived ad studied i some details. This paper is orgaized as followig. I Sectio 2, the ew
Iteratioal Joural of Advaced Statistics ad Probability 147 class of Poisso-Lidley lifetime distributios with its probability desity, cumulative distributio, ad reliability ad hazard rate fuctios are itroduced. I Sectio 3, some statistical measuremets of the ew class will be derived. Maximum likelihood estimators for the ukow parameters from the class of Poisso Lidley distributios are discussed i Sectio 4. Four special sub-models of the proposed class are ivestigated i Sectio 5. A applicatio to a real data set is preseted i Sectio 6. Fially, some cocludig remarks are addressed i Sectio 7. 2. The ew class Followig the same idea of Adamidis ad Loukas [1], the ew class of distributios is defied as follows. Let X 1, X 2,, X w are idetically idepedet distributed (iid radom variables from a cotiuous probability desity fuctio h(x; θ o (0,, with some cotiuous ukow parameters (θ = θ 1, θ 2,, θ k. Let W be a zero trucated Poisso-Lidley radom variable idepedet of X s with the followig probability mass fuctio: P(W = w = α 2 2+α+w (1+α Defie, X = max{ X i } W of X W = w is give by: f X W=w (x = wh(x; θh(x; θ w 1 w, w = {1,2.. }, α > 0. (1 as the Poisso-Lidley- H radom variable. The coditioal probability desity fuctio (PDF The joit distributio of X ad W is obtaied as the followig: f X,W (x, w; α, θ = α 2 2+α+w (1+α w wh(x; θh(x; θw 1. So, the ew class of the complemetary Poisso-Lidley (CPL lifetime distributios is derived as the margial PDF of X as follows: f(x; α, θ = α 2 h(x;θ (1+α ((2 + α w=1 w (H(x;θ w 2 ( H(x;θ w=1, 1+α w 1 1+α w 1 After some simplificatios, it reduces to: f(x; α, θ = α2 (1+α 2 h(x;θ(3+α H(x;θ (1+α H(x;θ 3, x > 0. (2 Propositio 1: The PDF of complemetary Poisso-Lidley lifetime distributios ca be writte as a liear combiatio of h i (x, θs Proof The PDF (2 ca be expressed as; f(x; α, θ = α2 (1+α 2 [h 1(x; θ + h 2 (x; θ]; Where; h i (x; θ = ih(x; θ[1 + α H(x; θ] (i+1 ; i = 1,2. So, f(x; α, θ is a liear combiatio of h i (x, θs. The correspodig cumulative distributio fuctio (CDF associated to (2 is give by: F(x; α, θ = ( α 2 1+3α+α 2 H(x;θ (1+α H(x;θ 2 {1 + α + (2 + α (1 + α H(x; θ}, x > 0. (3 Propositio 2: F(x; α, θ H(x; θ as α. So H(x; θ is the limitig case of F(x; α, θ. Proof lim F(x; α, θ = lim α ( α2 α 1+3α+α 2 H(x;θ (1+α H(x;θ 2 {1 + α + (2 + α (1 + α H(x; θ} = H(x; θ.
148 Iteratioal Joural of Advaced Statistics ad Probability So H(x; θ is the limitig case of (3. Furthermore, the reliability ad hazard rate fuctios of complemetary Poisso-Lidley lifetime distributios are obtaied, respectively, as the followig: R(x; α, θ = 1 α2 H(x;θ{1+α+(2+α(1+α H(x;θ} ad (1+α H(x;θ 2, (4 υ(x; α, θ = α 2 (1+α 2 h(x;θ (1+3α+α 2 ( 3+α H(x;θ 3 (1+α H(x;θ 1 α2 H(x;θ{1+α+(2+α(1+α H(x;θ} (1+α H(x;θ 2. (5 Now, let us deote a radom variable X followig the complemetary Poisso-Lidley lifetime distributios with parameters θ ad α by X~CPL(θ, α. This ew class of distributios is preseted as a geeralizatio of several distributios. 3. Momets ad momet geeratig fuctio Momets are commoly used to characterize the probability distributio or observed data set. Some of the most importat features ad characteristics of a distributio ca be studied through momets (e.g. tedecy, dispersio, skewess ad kurtosis. The r th raw momet of X about the origi ca be determied from (2 as follows: E(X r = α2 (1+α 2 0 xr h(x; θ(3 + α H(x; θ (1 + α H(x; θ 3 Usig the biomial expasio the E(X r ca be writte as: 0 xr E(X r = α2 (1+α 1 After some simplificatio, E(X r = α2 (1+α 1 Where, I(i = (i+2 2 0 i=0 ( i+2 2 dx. (1 + α i (H(x; θ i h(x; θ (3 + α H(x; θ dx. i=0 (1 + α i [ (3 + αi(i I(i + 1], (6 x r h(x, θ (H(x; θ i dx. Hece, the momets of this class is obtaied directly by substitutig r=1, 2 i Equatio (6. Additioally, the momet geeratig fuctio ca be writte as: t r M x (t = r=0 E(X r r! Therefore, the momet geeratig fuctio of CPL class of distributios is obtaied from (6 as follows: M x (t = α2 (1+α 1 t r r=0 i=0 ( i+2 (1 + α i [ (3 + αi(i I(i + 1]. (7 r! 2 4. Parameter estimatio I this sectio; the maximum likelihood estimates (MLEs of the model parameters of the complemetary Poisso- Lidley class of distributios are determied from complete samples. I additio, a expressio for the associated Fisher s iformatio matrix is give. Let X 1, X 2,, X is a radom sample from the ew class of Poisso-Lidley with parameters α ad θ. the loglikelihood fuctio based o observed radom sample of size is give by:
Iteratioal Joural of Advaced Statistics ad Probability 149 l = 2lα(1 + α l(1 + 3α + α 2 + lh(x i ; θ + l (3 + α H(x i ; θ 3 l (1 + α H(x i ; θ. The first partial derivatives for the log-likelihood equatio with respect to α ad θ are give respectively as follows: = 2(2α+1 α α(α+1 Ad (3+2α + 1 1 1+3α+α 2 3 3+α H(x i ;θ, (8 1+α H(x i ;θ θ = h(xi;θ θ h(x i ;θ ( H(x i ;θ θ 3+α H(x i ;θ + 3 ( H(x i ;θ θ. (9 1+α H(x i ;θ The MLEs of the parameters; α ad θ; ca be obtaied from (8 ad (9 with respect to α ad θ respectively, by settig l l = 0 ad = 0 ad solvig for the values of α ad θ. α θ For large sample size, the maximum likelihood estimators, uder appropriate regularity coditios, are cosistet ad asymptotically ormally distributed. Tests of hypothesis ad cofidece itervals for the parameters ca be obtaied based o Fisher ' s iformatio matrix. Therefore, the two sided approximate cofidece limits for the maximum likelihood estimates α ad ˆ of populatio parameters α ad θ ca be costructed, such that: ( α θ = (α θ ± Zδ 2 ( diagoal(i 1, Where, zδ is the stadard ormal percetile at δ 2, δ is the sigificat level, I is the asymptotic Fisher iformatio 2 matrix which is obtaied by substitutig ˆ for θ ad α for α as follows: I = ( 2 l α 2 2 l α θ 2 l α θ 2 l θ 2. The elemets of the asymptotic Fisher iformatio matrix I ca be expressed as the followig: 2 l α 2 = 2(2α 2+2α +1 (α (α +1 2 + (7+6α +2α 2 (1+3α +α 2 1 2 (3+α H(x i ;θ 2 + 3 1 (1+α H(x i ;θ 2, 2 θ 2h(xi;θ 2 l = h(x θ 2 i ;θ ( h(x i ;θ θ 2 [ h(x i ;θ 2 ( 2 H(x i ;θ (3+α H(x θ 2 i ;θ +( H(x i ;θ 2 θ (3+α H(x i ;θ 2 ] +3 ( 2 H(x i ;θ (1+α H(x θ 2 i ;θ +( H(x i ;θ 2 θ (1+α H(x i ;θ 2, 2 l α θ = ( H(x i ;θ θ (3+α H(x i ;θ 2 3 ( H(x i ;θ θ (1+α H(x i ;θ 2. 5. Special models I this sectio, some special cases of CPL lifetime distributios will be discussed, icludig the complemetary Burr XII Poisso Lidley (CBXIIPL distributio, complemetary Burr III Poisso Lidley (CBIIIPL distributio, complemetary Weibull Poisso Lidley (CWPL distributio, ad complemetary iverse Weibull Poisso Lidley (CIWPL distributio. To illustrate the flexibility of distributios, plots of PDF ad hazard rate fuctio for some values of the parameter are preseted.
150 Iteratioal Joural of Advaced Statistics ad Probability 5.1. Complemetary Burr XII Poisso Lidley distributio Burr [4] itroduced family of distributios which cocluded twelve distributios. Burr XII distributio was the most popular betwee Burr distributios family. Burr [5] ad Tadikamalla [16] showed that the skewess ad kurtosis of Burr XII distributio have differet shapes ad degrees. Characteristics of Burr XII distributio are ear to several distributios like expoetial family, ormal, logormal etc. To check extra properties of Burr XII distributio (see Headrick et al. [9]. The distributio fuctio of Burr XII with shape parameters γ ad β takes the followig form: H(x; γ, β = 1 (1 + x β γ, x 0, (10 Substitutig the CDF (10 ad its correspodig desity fuctio ito geeral expressios (2 ad (3 to obtai the desity ad distributio fuctios of complemetary Burr XII Poisso Lidley as follows: f(x; α, γ, β = ( α2 (1+α 2 γβ 1+3α+α 2 xβ 1 (1 + x β γ 1 ( 2+α+(1+xβ γ Ad, F(x; α, γ, β = ( α 2 (1 (1+xβ γ 1+3α+α 2 (α+(1+x β γ 3, x > 0, β, γ > 0, (11 (α+(1+x β γ 2 (1 + α + (2 + α(α + (1 + x β γ (12 where α > 0 Is the scale parameter. Note that; from propositio (2; as α, F(x; α, γ, β 1 (1 + x β γ = H(x; γ, β, which is the distributio fuctio of Burr XII. Hece, the Burr XII is obtaied as limitig distributio for CBXIIPL distributio. The probability desity fuctio of CBXIIPL is displayed i Figure (1 for some selected values of parameter to show its flexibility to model lifetime data. (a α = 1, 2, 3, γ = 1, 1.5, 2, β = 0.5 (b α = 0.5, 1,1.5, γ = 1.5, 1, 0.5, β = 1.5 Fig. 1: Plots of the CBXIIPL Desities Fuctio for Some Parameter Values Furthermore, from the geeral expressios (4 ad (5, the reliability ad hazard rate fuctios of CBXIIPL reduce to R(x; α, γ, β = 1 Ad υ(x; α, γ, β = α2 (1 (1+xβ γ 1+3α+α 2 (α+(1+x β γ 2 (1 + α + (2 + α(α + (1 + x β γ, α 2 (1+α 2 γβx β 1 (1+x β γ 1 ( 2+α+(1+xβ γ (α+(1+x β γ 3 α 1 2 (1 (1+xβ γ 1+3α+α 2 (α+(1+x β γ 2 (1+α+(2+α(α+(1+xβ γ.
Iteratioal Joural of Advaced Statistics ad Probability 151 Figure (2 represets the shapes of hazard rate fuctio for CBXIIPL for selected values of α, β ad γ. It is clear from plots that for small values of β ad γ, the hazard rate fuctio takes decreasig form (Figure. 2(a. Also, for large values of β ad γ, the hazard rate fuctio takes covex form (Figure. 2(b. (a α = 10, 20, 30, γ = 0.5, 1, 1.5, β = 0.5 (b α = 20, 25, 30, γ = 3, 3.5, 4, β = 10 Fig. 2: Plots of the CBXIIPL Hazard Rate Fuctios for Some Parameter Values Based o geeral expressio (6, the r th raw momet of the radom variable X about the origi havig the CBXIIPL is determied as follows: (i+2 2 E(X r = α2 (1+α 1 i=0 (1 + α i [(3 + αi(i I(i + 1], Where, I(i is obtaied by substitutig (10 ad its correspodig desity fuctio i (6 as follows: I(i = βγ 0 x r+β 1 (1 + x β γ (1 (1 + x β γ i dx By usig biomial expasio ad after some calculatios, I(i takes the followig form: i I(i = k=0 ( i k ( 1 k 1 (k+1 Also, the momet geeratig fuctio is: 1 Γ(γ(k+1 Γ (1 + r β Γ (γ(k + 1 r β. M x (t = α2 (1+α 1 tr r=0 i=0 ( i+2 2 (1 + α i [(3 + αi(i I(i + 1]. r! Furthermore, the log-likelihood fuctio based o observed radom sample of size is give as follows: l = 2lα(1 + α l(1 + 3α + α 2 + lγβ + (β 1 (γ + 1 l(1 + x β i + l (2 + α + (1 + x β i γ 3 l (α + (1 + x β i γ lx i. The first partial derivatives for the log-likelihood equatio with respect to α, γ ad β are give respectively as follows: = 2(2α+1 α α(α+1 + [ 1 1+3α+α 2 (3+2α 2+α+(1+x i β γ 3 β γ], (13 α+(1+x i = l(1 + x γ γ i β [ (1+x i Ad β γ l(1+xi β 2+α+(1+x i β γ 3(1+x β γ β i l(1+xi β γ ], (14 α+(1+x i = + lx β β i (γ + 1 x i β lx i γ [ (1+x i β γ 1 xi β lx i (1+x β i 2+α+(1+x i β γ 3(1+x i β γ 1 xi β lx i α+(1+x i β γ ]. (15
152 Iteratioal Joural of Advaced Statistics ad Probability MLEs of the ukow parameters are obtaied after settig o-liear Equatios (13-(15 to be zero; l α = 0, l γ = 0 ad l = 0. as it seems, there is o closed solutio, so a extesive umerical solutio will be applied. β 5.2. Complemetary Burr III Poisso Lidley distributio The Burr III distributio properly approximates may familiar distributios such as ormal, logormal, gamma, Weibull, ad expoetial distributios. It plays a importat role i reliability egieerig, statistical quality cotrol, ad risk aalysis models. Burr type III has bee itroduced to forestry by Lidsay et al. [14].This distributio is a importat because this is iheretly more flexible tha the Weibull distributio, which is ofte used i forestry applicatio. Burr type III covers a much larger area of the skewess kurtosis plae tha the Weibull distributio. The cumulative distributio fuctio for Burr III distributio with shape parameters γ ad β is give by: H(x; γ, β = (1 + x β γ, x > 0. (16 The PDF ad CDF of CBIIIPL are obtaied by direct substitutio of CDF (16 ad its PDF i geeral expressio (2 ad (3 as the followig: f(x; α, γ, β = α2 (1+α 2 γβx β 1 (1+x β γ 1 Ad F(x; α, γ, β = α2 (1+x β γ 1+3α+α 2 ( 3+α (1+x β γ (1+α (1+x β γ 3, x > 0, β, γ, α > 0, (17 (1+α (1+x β γ 2 (1 + α + (2 + α(1 + α (1 + x β γ (18 From propositio (2; it is quite clear that as α, the Burr III is the limitig distributio for CBIIIPL. That is, lim F(x; α, γ, β (1 + α x β γ = H(x; γ, β The probability desity fuctio of CBIIIPL is illustrated i Figure (3 for some selected values of parameter. (a α = 2, 3, 4, γ = 0.5, 1.5, 2, β = 0.5 (b α = 2,3, 4, γ = 1, β = 1.5, 2, 2.5 Fig. 3: Plots of CBIIIPL Desities Fuctio for Some Parameter Values Furthermore, the reliability ad hazard rate fuctios of CBIIIPL are obtaied from geeral expressios (4 ad (5 as follows: R(x; α, γ, β = 1 Ad α2 (1+x β γ 1+3α+α 2 (1+α (1+x β γ 2 (1 + α + (2 + α(1 + α (1 + x β γ,
Iteratioal Joural of Advaced Statistics ad Probability 153 υ(x; α, γ, β = α 2 (1+α 2 γβx β 1 (1+x β γ 1 ( 3+α (1+x β γ (1+α (1+x β γ 3 α 1 2 (1+x β γ 1+3α+α 2 (1+α (1+x β γ 2 (1+α+(2+α(1+α (1+x β γ Figure (4 represets the shapes of hazard rate fuctio for CBIIIPL for selected values of α, β ad γ. Clearly, for small values of β ad γ, plots of the hazard rate fuctio take decreasig form (Figure. 4(a. Also, the hazard rate fuctio takes covex form for large values of β ad γ ((Figure. 4(b. (a α = 10, 20, 30, γ = 0.1, β = 0.5, 1, 1.5 (b α = 5, 10, 15, γ = 0.1, 1, 10, β = 10, 15, 20 Fig. 4: Plots of the CBIIIPL Hazard Rate Fuctios for Some Parameter Values To obtai the r th raw momet of CBIIIPL, firstly it must to obtai the itegrated part, I(i, defied i (6 as follows: I(i = 0 βγx r β 1 (1 + x β γ(i+1 dx = Γ(1 r β Γ(γ(i+1+r β (i+1γγ(i+1 (19 After substitutig (19 i (6, the r th raw momet of CBIIIPL distributio takes the followig form: E(X r = α2 (1+α 1 i=0 (i+2 2 r β (i+1γ(γ(i+1 (1 + α i Γ (1 r β {(3+αΓ(γ(i+1+ Γ(γ(i+2+r β (i+2γ(γ(i+2 }. Likewise, the momet geeratig fuctio ca be expressed as the followig: M x (t = tr r=0 α 2 (1+α 1 i=0 r! (i+2 r β (i+1γ(γ(i+1 2 (1 + α i Γ (1 r β {(3+αΓ(γ(i+1+ Γ(γ(i+2+r β (i+2γ(γ(i+2 }. The log-likelihood fuctio based o the observed sample of size from CBIIIPL distributio is give by: l = 2lα(1 + α l(1 + 3α + α 2 + lβγ (β + 1 l x i (γ + 1 l(1 + x β i + l (3 + α (1 + x β i γ 3 l(1 + α (1 + x β i γ. The first partial derivatives for the log-likelihood equatio with respect to α, γ ad β are give respectively as follows: = 2(2α+1 α α(α+1 + 1 1+3α+α 2 (3+2α 3+α (1+x i β γ 3 1+α (1+x β i γ. (20 β γ l(1+x i β = l(1 + x γ γ i β + (1+x i β γ l(1+xi β 3+α (1+x β i γ 3 (1+x i 1+α (1+x β i γ. (21 Ad = lx β β i + (γ + 1 x i β lx i 1+x β γ (1+x i β γ 1 xi β lx i i 3+α (1+x i β γ
154 Iteratioal Joural of Advaced Statistics ad Probability +3γ (1+x i β γ 1 x i β lx i 1+α (1+x i β γ. (22 To achieve estimatios via maximum likelihood method, it is ot easy to solve equatios l = 0, l l = 0 ad α γ β = 0, directly, so a iterative techique will be used. 5.3. Complemetary Weibull Poisso Lidley distributio The Weibull distributio is a importat for modelig ad lifetime data aalysis i biological, medical ad egieerig scieces. It ca therefore model a great variety of data ad life characteristics. It is used extesively i reliability applicatios to model failure times. The distributio fuctio of Weibull distributio with the shape parameter β ad scale parameter λ takes the followig form: H(x; λ, β = 1 e λ(xβ, x > 0 ad β, λ > 0. (23 The probability desity ad distributio fuctios of CWPL are determied from geeral expressio (2 ad (3 by direct substitutio of CDF (23 ad its correspodig desity fuctio as the followig: f(x; α, λ, β = α2 (1+α 2 βλ xβ 1 e λ(xβ (2+α+e λ(xβ Ad (α+e λ(xβ 3, x > 0, (24 F(x; α, λ, β = ( α 2 (1 e λ(xβ 1+3α+α 2 (α+e λ(xβ 2 (1 + α + (2 + α (α + e λ(xβ. (25 Clearly, from propositio (2; lim α F(x; α, λ, β 1 e λ(xβ = H(x; λ, β, that is; the Weibull distributio is the limitig case whe α. The probability desity fuctio of CWPL is displayed i Figure (5 for some selected values of parameter (a α = 0.5, 1.5, 2, λ = 1, 1.5, 2, β = 1.5 (b α = 0.5, 2, 3, λ = 1.5, 1, β = 1.5 Fig. 5: Plots of CWPL Desities Fuctio for Some Parameter Values Also, the reliability ad hazard rate fuctios of CWPL are as follows: R(x; α, λ, β = 1 Ad α2 (1 e λ(xβ 1+3α+α 2 (α+e λ(xβ 2 (1 + α + (2 + α (α + e λ(xβ,
Iteratioal Joural of Advaced Statistics ad Probability 155 υ(x; α, λ, β = α 2 (1+α 2 βλx β 1 e λ(xβ (2+α+e λ(xβ (α+e λ(xβ 3 α 1 2 (1 e λ(xβ (α+e λ(xβ 2 (1+α+(2+α(α+e λ(xβ. Figure (6 shows the shapes of hazard rate fuctios for CWPL for some selected value of α, β ad λ. Clearly, the plots of the hazard rate fuctio take decreasig form for the same value of β ad as the values of λ icrease (see Figure 6(a. The hazard rate fuctio takes icreasig form for large values of β ad λ (see Figure. 6(b. (a α = 1, 2, 3, λ = 0.5, 1, 1.5, β = 0.3 (b α = 0.1, 1, 10, λ = 3, β = 5, 6, 7 Fig. 6: Plots of the CWPL Hazard Rate Fuctios for Some Parameter Values The r th raw momets ad momet geeratig fuctio of CWPL distributio are obtaied, based o geeral expressios (6 ad (7, as follows: (i+2 2 E(X r = α2 (1+α 1 Ad, i=0 (1 + α i [ (3 + αi(i I(i + 1], M x (t = α2 (1+α 1 tr r=0 i=0 ( i+2 2 (1 + α i [ (3 + αi(i I(i + 1], Where, I(i = 0 βλx r+β 1 e λ(xβ (1 e λ(xβ i r! i dx = k=0 ( i k ( 1 k 1 1 r (k+1 (λ(k+1 β Γ (1 + r β. The log-likelihood fuctio based o observed radom sample of size for CWPL distributio is give by: l = 2lα(1 + α l(1 + 3α + α 2 + lλβ + (β 1 lx i λ (x i β + l (2 + α + e λ(x i β 3 l (α + e λ(x i β. The first partial derivatives for the log-likelihood equatio with respect to α, λ ad β are give respectively as follows; = 2(2α+1 α α(α+1 + [ 1 1+3α+α 2 (3+2α 2+α+e λ(x i β 3 ], (26 α+e λ(x i β 3e λ(x i β (xi β 2+α+e λ(x i β = (x λ λ i β [ e λ(x i β (x i β Ad α+e λ(x i β ], (27
156 Iteratioal Joural of Advaced Statistics ad Probability = + lx β β i λ (x i β l(x i λ (x i β l(x i e λ(x i + 3λ (x i β l(x i e λ(x i 2+α+e λ(x i β. (28 α+e λ(x i β The MLEs of the parameters α, λ ad β are obtaied umerically by solvig a system of oliear Equatios (26 to (28, after settig with zero as l = 0, l l = 0 ad = 0. As it seems, there is o closed solutio, so a extesive α λ β umerical solutio will be applied. 5.4. Complemetary iverse Weibull Poisso Lidley distributio The iverse Weibull distributio is aother life time probability distributio which ca be used i the reliability egieerig disciplie. The iverse Weibull distributio ca be used to model a variety of failure characteristics such as ifat mortality, useful life ad wear-out periods (see Kha et al. [10]. The cumulative distributio fuctio for iverse Weibull distributio with shape parameter β ad scale parameter λ takes the followig form: H(x; λ, β = e λ(x β, x > 0, β ad λ > 0. (29 The PDF of CIWPL is obtaied by direct substitutio of CDF (29 ad its PDF i PDF (2 ad CDF (3 as the followig: f(x; α, λ, β = α2 (1+α 2 βλx β 1 e λ(x β Ad F(x; α, λ, β = 3+α e λ(x β β (1+α e λ(x β 3, x > 0, α, β, γ > 0, (30 α2 (e λ(x β 1+3α+α 2 (1+α e λ(x β 2 (1 + α + (2 + α (1 + α e λ(x β. (31 Note that: as α, the iverse Weibull distributio is obtaied as the limitig distributio for CIWPL, i.e lim F(x; α, λ, β e λ(x β = H(x; λ, β which is the distributio fuctio of iverse Weibull distributio. α Figure (7 represets the PDF of CIWPL distributio for some selected values of parameter β (a α = 10, λ = 1.5, 2, 2.5, β = 1, 1.5, 2 (b α = 10, 20, 30, λ = 0.5, β = 0.1, 0.3, 0.5 Fig. 7: Plots of CIWPL Desities Fuctio for Some Parameter Values I additio, the reliability ad hazard rate fuctios of CIWPL are give, respectively, as follows: R(x; α, λ, β = 1 Ad υ(x; α, λ, β = α2 e λ(x β 1+3α+α 2 (1+α e λ(x β 2 (1 + α + (2 + α (`1 + α e λ(x β, α 2 (1+α 2 βλx β 1 e λ(x β 3+α e λ(x β (1+α e λ(x β 3 α 1 2 (e λ(x β 1+3α+α 2 (1+α e λ(x β 2 (1+α+(2+α(1+α e λ(x β.
Iteratioal Joural of Advaced Statistics ad Probability 157 Figure (8 represets the hazard rate fuctio for CIWPL for some selected values of α, β ad λ. It is quite clear from plots that for small values of β ad λ the hazard rate fuctio takes decreasig form (see Figure. 8(a. Also, for large values of β ad λ the hazard rate fuctio takes covex form (see Figure. 8(b. (a α = 1, 2, 3, λ = 0.1, 0.3, 0.5, β = 0.5 (b α = 10, 15, 20, λ = 1, β = 1.5, 2, 2.5 Fig. 8: Plots of the CIWPL Hazard Rate Fuctios for Some Parameter Values Stadard calculatios show that the r th momet populatio ad momet geeratig fuctio of CIWPL distributio are: E(X r = α2 (1+α 1 Ad, i=0 (i+2 2 (1 + α i Γ (1 r β λ r β {(3 + α(i + 1 r β 1 (i + 2 r β 1 }, M x (t = α2 (1+α 1 tr r=0 i=0 ( i+2 2 (1 + α i Γ (1 r r λ β {(3 + α(i + 1 r β 1 (i + 2 r β 1 }. β r! The log-likelihood fuctio based o observed radom sample of size is give by: l = 2lα(1 + α l(1 + 3α + α 2 + lλβ (β + 1 lx i λ (x i β + l (3 + α e λ(x i β 3 l (1 + α e λ(x i β. The first partial derivatives for the log-likelihood equatio with respect to α, λ ad β are give respectively as follows: = 2(2α+1 α α(α+1 + 1 1+3α+α 2 (3+2α 3+α e λ(x i β 1 3, (32 1+α e λ(x i β = (x λ λ i β + e λ(x i β (x i β 3 e λ(x i (33 Ad 3+α e λ(x i β β (xi β, 1+α e λ(x i β = lx β β i + λ (x i β l(x i λ(x i β l(x i e λ(x i + 3 λ (x i β l(x i e λ(x i (34 3+α e λ(x i β MLEs of the α, λ ad β ca be worked out by lettig the partial derivatives l are o exact solutios to o-liear equatios, so a iterative techique will be used. 6. Applicatio to real data β α, l λ β. 1+α e λ(x i β l ad to be zero. It is clear that there β I this sectio, CBXIIPL, CBIIIPL CWPL ad CIWPL models are fitted to real data. Table (1 cotais a real data set correspodig to remissio times (i moths of radom sample of 128 bladder cacer patiets which reported i Lee ad Wag [13].
158 Iteratioal Joural of Advaced Statistics ad Probability Table 1: The Remissio Times (i Moths of 128 Bladder Cacer 0.08 2.09 3.48 4.87 6.94 8.66 13.11 23.63 0.2 2.23 0.52 4.98 6.97 9.02 13.29 0.4 2.26 3.57 5.06 7.09 0.22 13.8 25.74 0.5 2.46 3.46 5.09 7.26 9.47 14.24 0.82 0.51 2.54 3.7 5.17 7.28 9.74 14.76 26.31 0.81 0.62 3.28 5.32 7.32 10.06 14.77 32.15 2.64 3.88 5.32 0.39 10.34 14.38 34.26 0.9 2.69 4.18 5.34 7.59 10.66 0.96 36.66 1.05 2.69 4.23 5.41 7.62 10.75 16.62 43.01 0.19 2.75 4.26 5.41 7.63 17.12 46.12 1.26 2.83 4.33 0.66 11.25 17.14 79.05 1.35 2.87 5.62 7.87 11.64 17.36 0.4 3.02 4.34 5.71 7.93 11.79 18.1 1.46 4.4 5.85 0.26 11.98 19.13 1.76 3.25 4.5 6.25 8.37 12.02 2.02 0.31 4.51 6.54 8.53 12.03 20.28 2.02 3.36 6.76 12.07 0.73 2.07 3.36 6.39 8.65 12.63 22.69 5.49 Method of the maximum likelihood is used to get estimates of the ukow parameters ad its stadard errors (SE for each model. To compare the four distributios, criteria like; Akaike iformatio criterio (AIC, Bayesia iformatio criterio (BIC, mius log-likelihood (-l ad Kolmogorov-Smirov (K-S goodess of fit test statistics will be cosidered for the above data set. Covergece coditio for the large samples is satisfied, so parameters values are replaced by their likelihood estimators to estimate K-S statistics. Table (2 shows parameter MLEs (with the correspodig stadard error i paretheses, values of log-likelihood, values of AIC, values of BIC ad P-values based o oe sample K-S statistics. Distributio CWPL CIWPL CBXIIPL CBIIIPL Table 2: Statistical Aalysis for Bladder Cacer Patiet's Data Estimates(SE -l AIC α = 8.691(22.116 λ = 0.162(0.058 402.192 810.383 β = 0.899(0.085 α = 0.079(0.044 λ = 0.198(0.109 410.769 827.538 β = 1.12(0.075 α = 0.106(0.073 γ = 2.084(0.909 404.442 814.884 β = 0.779(0.273 α = 0.197(0.101 γ = 2.89(0.362 406.338 818.676 β = 1.329(0.089 BIC 681.772 681.793 681.768 681.774 P-value 0.00 0.00 0.824 * 0.776 * The values i Table (2 idicate that the P-values for K-S statistics have the smallest values for the data set uder CBXIIPL ad CBIIIPL models with regard to the other models. The quatile-quatile or Q-Q plot is used to check the validity of the distributioal assumptio for the data. Figure (9 shows that the data seems CBXIIPL ad CBIIIPL distributios provide the best fit to this data
Iteratioal Joural of Advaced Statistics ad Probability 159 7. Coclusio Fig. 9: QQ Plots to Bladder Cacer Patiet's Data which Fitted by CBXIIPL, CBIIIPL, CWPL ad CIWPL Distributios I this article, a ew geeral class of lifetime distributios so called the complemetary Poisso Lidley class of distributios is proposed. This ew class exteds several sub-models. A mathematical treatmet of the ew class, icludig expressios for desity fuctio, momets ad momet geeratig fuctio are provided. Maximum likelihood iferece is implemeted for estimatig the model parameters. Some sub-model distributios are itroduced ad fitted to real data set to show the usefuless of the proposed class. Refereces [1] K. Adamidis ad S. Loukas, A lifetime distributio with decreasig failure rate, Statistics ad Probability Letters, 39, 35-42, 1998. http://dx.doi.org/10.1016/s0167-7152(9800012-1. [2] S. Alkari ad A. Oraby, A compoud class of Poisso ad lifetime distributios, Joural of Statistics Applicatios ad Probability, 1, 45-51, 2012. http://dx.doi.org/10.12785/jsap/010106. [3] A. Asgharzadeh, H. S. Bakouch, S. Nadarajah ad L. Esmaeili, A ew family of compoud lifetime distributios, Kyberetika, 50, 142-169, 2014. http://dx.doi.org/10.14736/kyb-2014-1-0142. [4] I. W. Burr, Cumulative frequecy fuctios, Aals of Mathematical Statistics, 13, 215-232, 1942. http://dx.doi.org/10.1214/aoms/1177731607. [5] I. W. Burr, Parameters for a geeral system of distributios to match a grid of α3 ad α4, Commuicatios i Statistics, 2, 1-21, 1973. http://dx.doi.org/10.1080/03610927308827052. [6] M. Chahkadi ad M. Gajali, o some lifetime distributios with decreasig failure rate, Computatioal Statistics ad Data Aalysis, 53, 4433-4440, 2009. http://dx.doi.org/10.1016/j.csda.2009.06.016. [7] G. M. Cordeiro ad R. B. Silva, The complemetary exteded Weibull power series class of distributios, Ciêcia e Natura, 36, 1-13, 2014. http://dx.doi.org/10.5902/2179460x13194. [8] D. J. Flores, P. Borges, G. Cacho ad F. Louzada, The complemetary expoetial power series distributio, Brazilia Joural of Probability ad Statistics, 27, 565-584, 2013. http://dx.doi.org/10.1214/11-bjps182. [9] T. C. Headrick, M. D. Pat ad Y. Sheg, O simulatig uivariate ad multivariate Burr type III ad type XII distributios, Applied Mathematical Scieces, 4(45, 2207 2240, 2010. [10] M. S. Kha, G. R. Pasha ad A. H. Pasha, Theoretical aalysis of iverse Weibull distributio, WSEAS Trasactios o Mathematics, 7, 30-38, 2008. [11] C. Kus, A ew lifetime distributio, Computatioal Statistics ad Data Aalysis, 51, 4497-4509, 2007. http://dx.doi.org/10.1016/j.csda.2006.07.017. [12] A. Leahu, B. G. Muteau ad S. Cataraciuc,S, Max-Erlag ad Mi-Erlag power series distributios as two ew families of lifetime distributio, Buletiul Academiei de Stiite a Republicii Moldova. Matematica, 2(75: 60-73, 2014. [13] E. T. Lee ad J. W. Wag, Statistical methods for survival data aalysis, 3rd editio, Wiley, New York, 2003. http://dx.doi.org/10.1002/0471458546. [14] S. R. Lidsay, G. R. Wood ad R. C. Woollos, Modellig the diameter distributio of forest stads usig the Burr distributio, Joural of Applied Statistics, 23(6, 609-619, 1996. http://dx.doi.org/10.1080/02664769623973. [15] F. Louzada, M. Roma ad V. G. Cacho, The complemetary expoetial geometric distributio: model, properties, ad a compariso with its couterpart, Computatioal Statistics ad Data Aalysis, 55, 2516 2524, 2011. http://dx.doi.org/10.1016/j.csda.2011.02.018.
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