A New Lifetime Distribution For Series System: Model, Properties and Application

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1 Joural of Moder Applied Statistical Methods Volume 7 Issue Article 3 08 A New Lifetime Distributio For Series System: Model, Properties ad Applicatio Adil Rashid Uiversity of Kashmir, Sriagar, Idia, adilstat@gmail.com Zahooor Ahmad Uiversity of Kashmir, Sriagar, Idia, zahoorstat@gmail.com T R. Ja Uiversity of Kashmir, Sriagar, Idia, drtrja@gmail.com Follow this ad additioal works at: Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical Theory Commos Recommeded Citatio Rashid, A., Ahmad, Z., & Ja, T. R. (08). A New Lifetime Distributio for Series System: Model, Properties ad Applicatio. Joural of Moder Applied Statistical Methods, 7(), ep535. doi: 0.37/jmasm/ This Regular Article is brought to you for free ad ope access by the Ope Access Jourals at DigitalCommos@WayeState. It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods by a authorized editor of DigitalCommos@WayeState.

2 Joural of Moder Applied Statistical Methods May 08, Vol. 7, No., ep535 doi: 0.37/jmasm/ Copyright 08 JMASM, Ic. ISSN A New Lifetime Distributio for Series System: Model, Properties ad Applicatio Adil Rashid Uiversity of Kashmir Sriagar, Idia Zahoor Ahmad Uiversity of Kashmir Sriagar, Idia T. R. Ja Uiversity of Kashmir Sriagar, Idia A ew lifetime distributio for modelig system lifetime i series settig is proposed that embodies most of the compoud lifetime distributio. The reliability aalysis of paret ad of sub-models has also bee discussed. Various mathematical properties that iclude momet geeratig fuctio, momets, ad order statistics have bee obtaied. The ewlyproposed distributio has a fleible desity fuctio; more importatly its hazard rate fuctio ca take up differet shapes such as bathtub, upside dow bathtub, icreasig, ad decreasig shapes. The ukow parameters of the proposed geeralized family have bee estimated through MLE techique. The stregth ad usefuless of the proposed family was tested o a real life data set ad it is quite clear from the statistical aalysis that proposed family offers a better fit. Keywords: Geeralized Lidley distributio, power series distributio, compoudig, hazard fuctio, survival fuctio Itroductio The modelig of lifetime data has received prime attetio from researchers for the last decade. May cotiuous probability models such as epoetial, gamma, ad Weibull were frequetly used i statistical literature to aalyze the lifetime data, but these probability models caot be used efficietly to model lifetime data that is bathtub shaped ad have uimodal failure rates. To overcome this problem, researchers have focused their attetio o compoudig mechaisms which are a iovative way to costruct suitable, fleible, ad alterative models to fit the lifetime data of differet types. doi: 0.37/jmasm/ Accepted: Jue 5, 07; Published: Jue 7, 08. Correspodece: Adil Rashid, adilstat@gmail.com

3 RASHID ET AL Cosider a system with N compoets, where N is a discrete radom variable with domai N,,. The lifetime of the i th compoet is a cotiuous radom variable, say Xi, that may follow ay oe of the lifetime distributios such as epoetial, gamma, Weibull, Lidley, etc. The suitable discrete distributios for N may be geometric, zero trucated Poisso, or power series distributios i geeral. The lifetime of such a system i series ad parallel combiatio is defied ad deoted by a o-egative radom variable Y mix i or i Y ma X i i N N, respectively. With this i mid, Adamidis ad Loukas (998) costructed a two parameter lifetime distributio by compoudig the epoetial distributio with the geometric distributio, called the Epoetial Geometric (EG) distributio. Kus (007) obtaied a compoud of the epoetial distributio with that of Poisso ad it was amed the Epoetial Poisso (EP) distributio. Tahmasbi ad Rezaei (008) obtaied the Epoetial Logarithmic (EL) distributio by usig the same compoudig mechaism. The eteded EG distributio was cosidered by Adamidis, Dimitrakopoulou, ad Loukas (005). Power series distributios cotai several classical discrete distributios as special cases, therefore Chahkadi ad Gajali (009) itroduced a compoud class of Epoetial Power Series (EPS) distributios which cotai several compoud distributios as special cases. It is kow that the Weibull distributio cotais the epoetial distributio as a special case; i view of this, Morais ad Baretto-Souza (0) replaced the epoetial distributio with a Weibull distributio i the compoudig mechaism of the EPS distributio ad obtaied a compoud class of Weibull Power Series (WPS) distributios, which cotais EPS distributio as a special case. Zakerzadeh ad Mahmoudi (0) obtaied a two parameter lifetime distributio by compoudig a Lidley distributio with a geometric distributio. Recetly, a oe parameter Lidley distributio has bee used frequetly to model lifetime data because it has bee observed i several research papers that this distributio performs ecelletly well whe it comes to fit the lifetime data. Adil ad Ja (06) itroduced a ew family of lifetime distributios which is obtaied by compoudig Lidley distributio with power series distributio. This ew family of cotiuous lifetime distributios was called the Lidley Power Series (LPS) distributio. The proposed LPS family of compoud distributios cotais several lifetime distributios as its special cases that are very fleible ad able to accommodate differet types of data sets sice the probability desity fuctio ad hazard rate ca take o differet forms such as icreasig, decreasig, ad upside dow bathtub 3

4 A NEW LIFETIME DISTRIBUTION FOR SERIES SYSTEM shapes which have bee show through graphs for some selected values of parameters, ad the potetiality of LPS family has bee tested statistically by usig it to model some real life data set. There are may cotiuous distributios i statistics that ca be used to model lifetime data; amog them, the most popular are gamma, log ormal, ad Weibull distributios. The Weibull distributio has bee used etesively by researchers to model lifetime data because of its closed form for survival fuctio. However, all these lifetime model suffer from a major drawback, i.e., oe of the them ehibit bathtub shapes for their hazard rate fuctios ad hece caot be used efficietly to model real life data that has bathtub shape for hazard rate fuctio. Nadarajah, Bakouch, ad Tahmasbi (0) proposed a ew lifetime distributio called the geeralized Lidley distributio that removes all of these metioed drawbacks. It was show that the geeralized Lidley distributio has a attractive feature of allowig for mootoically decreasig, mootoically icreasig, ad bathtub shaped hazard rate fuctios, while ot allowig for costat hazard rate fuctios. These importat features of the geeralized Lidley distributio attracted the attetio of researchers Adil ad Ja (06), who replaced the Lidley distributio with a geeralized Lidley distributio i the compoudig mechaism of the LPS distributio. Costructio of the Family Let X,, X be a idepedet ad idetically distributed (iid) radom variables followig the geeralized Lidley distributio due to Nadarajah et al. (0), whose desity fuctio is give by g ( ; ) e e ( ), 0,, 0 () Here, the ide N is itself a discrete radom variable followig a zero trucated power series distributio with probability fuctio give by where a depeds oly o, a P N,,, C ( ) 4

5 RASHID ET AL Table. Useful quatities of some power series distributios Distributio a C(θ) C'(θ) C''(θ) C - (θ) θ Poisso! - e θ e θ e θ log(θ ) θ (0, ) Logarithmic - -log( θ) ( θ) - ( θ) - e -θ θ (0, ) Geometric θ( θ) - ( θ) - ( θ) -3 θ(θ ) - θ (0, ) m Biomial ( ) (θ ) m m(θ ) m ( -) m m - ( θ ) m (θ ) /m θ (0, ) C a ad θ is such that C(θ) is fiite. Table shows useful quatities of some zero trucated power series distributios such as Poisso, logarithmic, geometric, ad biomial (with m beig the umber of replicas). Let mi N X X. The coditioal cumulative distributio fuctio of i i X() N is give by ad G G G X N X N e a p ( X (), N ) e, 0, C The family of geeralized Lidley power series distributios is defied by the margial cumulative distributio fuctio of X(): 5

6 A NEW LIFETIME DISTRIBUTION FOR SERIES SYSTEM 6 F e C C e C a () Desity, Survival, ad Hazard Rate Fuctio The probability desity fuctio of a family of the Lidley power series (GLPS) distributio ca be obtaied by differetiatig () both sides with respect to. e f e C C e (3) If α i (3), it reduces to a Lidley power series distributio. Alteratively, the probability desity fuctio of the GLPS distributio ca also be obtaied usig the joit probability fuctio of X() ad N:, g, e e C e X N a Therefore, desity fuctio of a family of the GLPS distributio is defied by the margial desity of X():

7 RASHID ET AL 7 e f e C C e which is same as (3). The survival fuctio of the GLPS distributio is give by C e S C ad the hazard fuctio is e e C e h C e Propositio. The geeralized Lidley distributio is the limitig case of the proposed distributio whe θ 0. Proof. From the cumulative distributio fuctio of the GLPS distributio, C e lim F lim C e lim a a

8 A NEW LIFETIME DISTRIBUTION FOR SERIES SYSTEM 8 as C a (4) ad, usig L Hospital s rule, it follows that 0 0 e lim F lim e a a which is the distributio fuctio of the geeralized Lidley distributio. Propositio. The desities of proposed distributio ca be epressed as a ifiite liear combiatio of desities of st order statistics of the geeralized Lidley distributio f P g, N where g, e e e Proof. It is kow that C a

9 RASHID ET AL Therefore, f ( ) e e ( N ) ( ) P g, a e C (5) where g(, ), defied as i the statemet of the propositio, is the pdf of X() mi(x, X,, X). Therefore the desities of the proposed distributio ca be epressed as a ifiite liear combiatio of the st order statistics of the geeralized Lidley distributio. Momet Geeratig Fuctio The momet geeratig fuctio (mgf) of the proposed distributio ca be obtaied from (5): where M X ( t) ( N ) ( t) M P M X t is the momet geeratig fuctio of st order statistics of the geeralized Lidley distributio: 0 X M X ( t) e ( ) e e t e d 9

10 A NEW LIFETIME DISTRIBUTION FOR SERIES SYSTEM j t M X ( t) e ( ) e j 0 j 0 Hece As e j j t ( ) e ( ) e d j 0 j 0 j j k tk ( ) e ( ) j 0 k 0 j k 0 k ki j k j 0 k 0 i 0 j k i 0 ( ) k j k j k i j 0 k 0 i 0 j j k k t ki e d ki ( ) j k d k i k i k i k t k t k i k ki a j k MX ( t) C( ) j 0 k 0 i 0 j k i k ( X ) ( ) j k ( k i k i )( k t ) ( k i ) k i ( k t) j k j k E j k i m k i ( k ) ( ) k i k i j 0 k 0 i 0 m k i ( k ) m k i m k i k d 0

11 RASHID ET AL the m th momet of proposed distributio about origi is m m ( ) E X P N g d 0 P ( N ) E X k a j k j k ( ) j 0 k 0 i 0 C( ) j k i m k i ( k ) ( ) k i k i m k i ( k ) m k i m k i (6) Order Statistics ad Their Momets Let X, X,, X be a radom sample from the GLPS distributio, ad let X: < X: < < X : deote the correspodig order statistics. The pdf of the i th order statistic Xi: is give by C e!f fi : ( i)!( i )! C( ) C e C( ) i i (7) From d k i k i f F F k i d

12 A NEW LIFETIME DISTRIBUTION FOR SERIES SYSTEM the associated CDF is F i : i i! k i! i! k i k 0 k C e C( ) k i (8) Alteratively equatio (8) ca be writte as i : F i i! k i! i! k i k 0 k C e C( ) k i The epressio for the r th momet of i th order statistic Xi: with CDF (8) ca be obtaied, usig a well-kow result due to Barakat ad Abdelkadir (004), as C( ) X r d k i r k r E( i : ) C e k k i i k 0 where r,, 3, ad i,,,. k

13 RASHID ET AL Parameter Estimatio Let X,, XN be a radom sample with observed values,, from a GLPS distributio, ad let Θ (α, λ, θ) T be the ukow parameter vector. The loglikelihood fuctio is give by l l (, Θ) log log log log log i i log C( ) i ( ) log e i i i i log C e i The correspodig score fuctios are C e l C ( ) e C( ) i C e C e l e i C e log e log e i i i 3

14 A NEW LIFETIME DISTRIBUTION FOR SERIES SYSTEM l ( ) e i i ( ) i e e ( ) i ( ( )( )) C e e C e The maimum likelihood estimate of Θ, say ˆΘ, is obtaied by solvig the oliear system of equatios l l l Θ,, 0 T The solutio of this o-liear system of equatio ca be foud umerically by usig R. Some Special Sub-Models Geeralized Lidley Poisso Distributio The Poisso distributio is a special case of the power series distributio for C(θ) e θ ; therefore the cdf of a compoud of the geeralized Lidley Poisso (GLP) distributio is obtaied by usig the same argumets i (): e e F, 0 e The associated pdf, hazard, ad survival fuctios are give by, respectively, 4

15 Desity Desity RASHID ET AL e e ( ) f e e ( ) e h S e e e e e ( ) e e e e λ0., ɑ,θ4 λ0.6, ɑ,θ.4 λ0., ɑ.,θ5 λ.0 θ. θ.5 θ Figure. The fleibility of desity of the GLP ad LP distributios for some selected values of parameters 5

16 Hazard Hazard A NEW LIFETIME DISTRIBUTION FOR SERIES SYSTEM λ0.3, ɑ,θ4 λ0.7, ɑ4,θ4 λ.5, ɑ6,θ λ0.5 θ0.95 θ0.9 θ Figure. The fleibility of the hazard rate fuctios of the GLP ad LP distributios for some selected values of parameters for, α, λ > 0 ad 0 < θ <, respectively. The GLP distributio is very fleible i terms of its desity ad hazard rate fuctio, which is also corroborated by Figure ad Figure. The epressio for the m th momet of a radom variable followig the geeralized Lidley Poisso distributio becomes, by takig a! - ad C(θ) e θ i (6), k m j k j k E ( X ) ( e ) ( ) j 0 k 0 i 0! j k i for α, λ > 0 ad 0 < θ <. m k i ( k ) ( ) k i k i m k i ( k ) m k i m k i 6

17 Desity Desity RASHID ET AL λ0.7, ɑ,θ4 λ0.7, ɑ,θ.4 λ0.7, ɑ9,θ5 λ0. θ0.8 θ0.5 θ Figure 3. The fleibility of the desity fuctio of the GLL ad LL distributios for some selected values of parameters Geeralized Lidley Logarithmic Distributio The logarithmic distributio is a special case of the PSD whe C(θ) -log( θ), ad a compoud of the geeralized Lidley logarithmic (GLL) distributio follows from () from the above C(θ): log e F log ( ) The associated pdf, hazard rate, ad survival fuctios are 7

18 A NEW LIFETIME DISTRIBUTION FOR SERIES SYSTEM f h S ( ) e log ( ) e e e e ( ) e ( ) log e log e log ( ) The desity ad hazard rate fuctios of the GLL distributio take differet shapes for differet values of parameters, which is evidet from Figures 3 ad 4. The epressio for the m th momet of a radom variable followig a geeralized Lidley logarithmic distributio becomes, by takig a - ad C(θ) -log( θ) i (6), m j k E( X ) log( ) for α, λ > 0 ad 0 < θ <. k k i j k k i j 0 k 0 i 0 j k i m k i ( k ) ( ) m k i ( k ) m k i m k i 8

19 Hazard Hazard RASHID ET AL λ3, ɑ70,θ4 λ0.5, ɑ5,θ7 λ, ɑ6,θ59 λ0.5 θ0.95 θ0.9 θ Figure 4. The fleibility of the hazard rate fuctio of the GLL ad LL distributios for some selected values of parameters Geeralized Lidley Geometric Distributio The geometric distributio is a particular case of the PSD whe C(θ) θ( θ) -, ad a compoud of the geeralized Lidley geometric distributio followed from (): e F ( ), 0 e The associated pdf, hazard rate, ad survival fuctios are 9

20 A NEW LIFETIME DISTRIBUTION FOR SERIES SYSTEM 0 Figure 5. The fleibility of the desity of the GLG ad LG distributios for some selected values of parameters e e f e e e h e e e S e (9) Desity λ0., ɑ,θ4 λ0.6, ɑ,θ.4 λ0., ɑ.,θ Desity λ0. θ0.8 θ0.5 θ0.3

21 RASHID ET AL Hazard Hazard λ0.3, ɑ,θ.4 λ0.03, ɑ0.9,θ0.6 λ.0, ɑ.6,θ0.0 λ0.3 θ0. θ0. θ Figure 6. The fleibility of the hazard rate fuctio of the GLG ad LG distributios for some selected values of parameters for, α, λ > 0 ad 0 < θ <, respectively. Figure 5 ad Figure 6 show that the desity ad hazard rate fuctios of the GLG distributio take differet shapes for differet values of parameters. The epressio for the m th momet of a radom variable followig a geeralized Lidley geometric (GLG) distributio becomes, by takig a ad C(θ) θ( θ) - i (6), k m j k E( X ) ( ) j 0 k 0 i 0 ( ) j k i for α, λ > 0 ad 0 < θ <. k i j k m k i ( k ) ( ) k i m k i ( k ) m k i m k i

22 Desity A NEW LIFETIME DISTRIBUTION FOR SERIES SYSTEM Histogram of GLG GLP GLL LG LP LL Figure 7. Fittig of GLP, GLG, GLL, LG, LP, ad LL distributios to the cacer data Geeralized Lidley Biomial Distributio The biomial distributio is a particular case of the PSD for C(θ) (θ ) m, with m a positive iteger, ad a compoud of geeralized Lidley biomial (GLB) distributio follows from (): e F, 0 m ( ) The associated pdf, hazard rate, ad survival fuctios are m

23 RASHID ET AL e ( ) m ( ) ( ) m f e h m ( ) e e S e m e e m m m ( ) e m respectively, for, λ, α > 0 ad 0 < θ <. The epressio for the r th momet of a radom variable followig geeralized Lidley biomial distributio becomes, by a m ad C(θ) (θ ) m i (6), takig k r m j k j k E( X ) ( ) j 0 k 0 i 0 j k i m k i ( k ) ( ) k i k i m k i ( k ) m k i m k i for m, α, λ > 0, ad 0 < θ <. The ew family of LPS distributios due to Adil ad Ja (06) is obtaied by takig α i (). Sub-models GLP, GLG, GLL, ad GLB are ew compoud distributios. The graphical represetatio of the desity fuctio ad hazard rate fuctio of all the sub-models of the proposed family have also bee preseted. 3

24 A NEW LIFETIME DISTRIBUTION FOR SERIES SYSTEM Table. Parameter estimates, log likelihood, ad Akaike Iformatio Criterio (AIC) Model Maimum Likelihood Estimates Log-likelihood AIC GLG λˆ 0.06, θˆ 0.9, α ˆ GLP λˆ 0.08, θˆ 3.65, α ˆ GLL λˆ 0., θˆ 0.96, α ˆ LG λ ˆ 0.07, θ ˆ LP λ ˆ 0., θ ˆ LL λ ˆ 0., θ ˆ Applicatio The applicability of the proposed model will be eplored further. A compariso amog its sub-classes will also be udertake usig a real life data set based o the remissio times (i moths) of a radom sample of 8 bladder cacer patiets as reported i Lee ad Wag (003). The parameter estimates, log likelihood, ad Akaike Iformatio Criterio (AIC) are show i Table. Coclusio A ew lifetime distributio was proposed for series settig that ot oly geeralizes the Lidley power series family of distributio, but has a fleible desity fuctio; more importatly, its hazard fuctio ca take up differet shapes such as bathtub, upside dow bathtub, icreasig, ad decreasig shapes. The potetial of the proposed family has bee show by fittig it to the real life data set, ad it is clear from the statistical aalysis that the proposed family offers a better fit. Ackowledgemets We are very thakful to the Editors commets ad the publishig committee for efficietly readig the paper that truly improved the stadard of paper. Refereces Adamidis, K., Dimitrakopoulou, T., & Loukas, S. (005). O a geeralizatio of the epoetial-geometric distributio. Statistics & Probability Letters, 73(3), doi: 0.06/j.spl

25 RASHID ET AL Adamidis, K., & Loukas, S. (998). A lifetime distributio with decreasig failure rate. Statistics & Probability Letters, 39(), doi: 0.06/s067-75(98)000- Adil, R., & Ja, T. R. (06). A ew family of lifetime distributios with real life applicatios. Iteratioal Joural of Mathematics ad Statistics, 7(), Barakat, H. M., & Abdelkader, Y. H. (004). Computig the momets of order statistics from oidetical radom variables. Statistical Methods ad Applicatios, 3(), 5-6. doi: 0.007/s Chahkadi, M., & Gajali, M. (009). O some lifetime distributios with decreasig failure rate. Computatioal Statistics & Data Aalysis, 53(), doi: 0.06/j.csda Kus, C. (007). A ew lifetime distributio. Computatioal Statistics & Data Aalysis, 5(9), doi: 0.06/j.csda Lee, E. T., & Wag, J. (003). Statistical methods for survival data aalysis (3rd ed.). NewYork, NY: Wiley. doi: 0.00/ Morais, A. L., & Barreto-Souza, W. (0). A compoud class of Weibull ad power series distributios. Computatioal Statistics & Data Aalysis, 55(3), doi: 0.06/j.csda Nadarajah, S., Bakouch, S. H., & Tahmasbi, R. (0). A geeralized Lidley distributio. Sakhya B, 73(), doi: 0.007/s Tahmasbi, R., & Rezaei, S. (008). A two-parameter lifetime distributio with decreasig failure rate. Computatioal Statistics & Data Aalysis, 5(8), doi: 0.06/j.csda Zakerzadeh, H., & Mahmoudi, E. (0). A ew two parameter lifetime distributio: Model ad properties. Upublished mauscript, Departmet of Statistics, Yazd Uiversity, Yazd, Ira. Retrieved from 5