Ope Joual of Statistics 06 6 685-700 Published Olie August 06 i SciRes. http://www.scip.og/joual/ojs http://dx.doi.og/0.436/ojs.06.64058 A Class of Lidley ad Weibull Distibutios Said Hofa Alkai Depatmet of Quatitative Aalysis Kig Saud Uivesity Riyadh Saudi Aabia Received 0 July 06; accepted 6 August 06; published 9 August 06 Copyight 06 by autho ad Scietific Reseach Publishig Ic. This wok is licesed ude the Ceative Commos Attibutio Iteatioal Licese (CC BY. http://ceativecommos.og/liceses/by/4.0/ Abstact I this pape we itoduce a class of Lidley ad Weibull distibutios (LW that ae useful fo modelig lifetime data with a compehesive mathematical teatmet. The ew class of geeated distibutios icludes some well-kow distibutios such as expoetial gamma Weibull Lidley ivese gamma ivese Weibull ivese Lidley ad othes. We povide closed-fom expessios fo the desity cumulative distibutio suvival fuctio hazad ate fuctio momets momets geeatig fuctio quatile ad stochastic odeigs. Moeove we discuss maximum likelihood estimatio ad the algoithm fo computig the paametes estimates. Some sub models ae discussed as a illustatio with eal data sets to show the flexibility of this class. Keywods Class of Lidley ad Weibull Distibutios Lidley Distibutios Weibull Distibutios. Itoductio The suvival aalysis is impeative aspect fo statisticias egiees ad pesoel i othe scietific fields such as public health actuaial sciece biomedical studies demogaphy ad idustial eliability. Seveal lifetime distibutios have bee suggested i statistics liteatue fo modelig suvival data. Of these distibutios two types gabbed the attetio of the eseaches fo fittig lifetime data: Weibull distibutios ad Lidley distibutios. The choice betwee the two types is due to the atue of hazad ate. Extesive eseach Baghei et al. [] exists o Weibull ad its modificatios. O the othe had may types of Lidley distibutios ad modificatios have bee developed as alteatives to Weibull distibutios. Fo efeeces see Ghitay et al. [] ad Alkai [3]. The emaide of this pape is ogaized as follows: I Sectio we defie the class of Lidley ad Weibull (LW distibutios ad show that may existig distibutios belog to this class. The LW popeties such as suvival fuctio hazad ate fuctio momets momet geeatig fuctio quatile ad stochastic odeigs ae discussed i Sectio 3. I Sectio 4 some special cases of the LW class ae itoduced to show the flexibili- How to cite this pape: Alkai S.H. (06 A Class of Lidley ad Weibull Distibutios. Ope Joual of Statistics 6 685-700. http://dx.doi.og/0.436/ojs.06.64058
S. H. Alkai ty of this class i geeatig existig distibutios. Sectio 5 cotais the maximum likelihood estimates of the LW class ad the elevat asymptotic cofidece iteval. Two eal data sets ae itoduced i Sectio 6 to show the applicability of the LW class. I Sectio 7 we itoduce a coclusio to summaize the cotibutio of this pape.. The Class of Lidley ad Weibull Distibutios I this sectio we itoduce simple foms of cumulative distibutio fuctio (cdf ad pobability distibutio fuctio (pdf fo the LW class. Defiitio. Let H( x; η be a o-egative mootoically iceasig fuctio that depeds o a oegative paamete vecto η > 0 we defie the cdf fo ay adom vaiable of the LW class to be β H( x; η F ( x; βη = + H( x; η e ; η x > 0 β 0. + β The coespodig pdf becomes Ad fo Y H ( x η H( x; η f ( x; βη = ( + βh( x; η h( x; η e ; η x > 0 β 0. + β = ; the cdf ad pdf of LW become β H ( ( y; η FY y = F H y = + H y; η e ; η y > 0 β 0 + β H ( ( ( y fy y = f H y h y = + βh y h y e ; η y > 0 β 0. + β May Lidley types ad Weibull types of distibutios ae membes of the LW class depedig o the choice H x; η ad β. Some examples ae listed i Table. The pdf( ca be show as a mixtue of two distibutios as follows: of the fuctio ( ; βη = + ( f x pf x p f x whee H( x p = f( x = h( x e ad f( x = h( x H( x e + β H x. deped o the type of H( x 3. Geeal Popeties 3.. Suvival ad Hazad Fuctios Fo ay o-deceasig fuctio H( x the suvival fuctio (sf is give by β H( x; η s ( x = F ( x = + H( x η x > β ; e ; 0 + ad the associate hazad ate fuctio is give by ad Fo Y H ( x τ ( x ( x ( + βh( x + + f h x = = ; x > 0. s x β βh x = the suvival ad hazad ate fuctios ae give espectively by ( ( (3 (4. The shape ad the mode locatio of f ( x β H ( y; η sy y = + H y; η e ; y > 0 + β (7 (5 (6 686
S. H. Alkai Table. Some existig distibutios as examples of the LW class. Distibutio H( x β η Refeeces Expoetial x 0 - Johso et al. [4] Rayleigh ( x 0 x 0 - Rayleigh [5] Weibull ( x 0 x 0 Johso et al. [4] Modified Weibull ( x 0 x exp( λ x 0 [ λ ] Lai et al. [6] Weibull extesio ( x 0 λ exp( x λ 0 [ λ ] ie et al. [7] Gompetz ( x 0 exp ( x 0 Gompetz [8] Expoetial powe ( x 0 exp ( λx 0 [ λ ] Smith & Bai [9] b Che ( x 0 exp( x 0 b Che [0] x Pham ( x 0 ( a 0 [ a ] Pham [] Lidley ( x > 0 x - Lidley [] Ivese Lidley x - Shama et al. [3] Powe Lidley x Ghitay et al. [4] Geealized ivese Lidley x Shama et al. [5] Two paametes Lidley x β - Shake et al. [6] Exteded powe Lidley x β Alkai [3] Exteded ivese Lidley x β Alkai [7] τ Y ( y ( + β H ( y h ( y = ; y > 0. + β H ( y β e H ( y + β (8 3.. Momets ad Momet Geeatig Fuctio th The momets ad the momets geeatig fuctio (mgf fo a LW class ca be obtaied by diect itegatio as follows: u u β u u E( = x f ( x dx = H e du+ H ue d u + β + β 0 0 0 t tx M t = E e = e f x dx. 0 Usig the seies expasio tx tx e = the above expessio is educed to! = 0 t u u β u u M ( t = H e du+ H ue d u. = 0! + β 0 + β 0 As a special case if we let H( x η ; = x the 687
S. H. Alkai M (( ( ( β ( + β Γ + + + µ = ( t ad hece the mea ad the vaiace ae Fo ( η ( + ( + t + β + β = Γ = (! (0 β + β + β µ = Γ β ( + σ = ( + β ( + β + β Γ ( + β + β Γ. 4 β Y = H x; = x the The mea ad the vaiace the ae (( ( ( β ( + β Γ + µ = > = 0 ( + t + β β M Y ( t = Γ >. (4! β ( β β + µ = Γ > + β σ ( β ( ( β β = + + Γ 3.3. Quatile ad Stochastic Odeigs ( β +. (6 ( ( β + β Γ > Theoem. Let be a adom vaiable with pdf as i ( the quatile fuctio say Q( p is ( β + ( p = β β e Q p H W ( β+ whee β > 0 p ( 0 ad (. Poof: We have Q( p = F ( p p ( 0 which implies F( Q( p H( Q( p + β + βh Q( p e = ( + β( p W is the egative Lambet W fuctio. = p so by substitutio we get aisig both sides to β ad multiplyig by we have the egative Lambet equatio e ( β βh( Q( p β β β β ( p + β + βh Q p e = + β e is complete. Note that oe ca use the same poof above to obtai β β (9 ( ( (3 (5. Solvig this equatio fo Q( P the poof ( β + QY p = H W ( β+ e p. Stochastic odeig of positive cotiuous adom vaiables is a impotat tool fo judgig the compaative 688
S. H. Alkai behavio. A adom vaiable is said to be smalle tha a adom vaiable Y i the followig cotests: Stochastic ode ( st Y if F ( x FY ( x x; Y if h x h x x; Hazad ate ode ( h Y 3 Mea esidual life ode ( ml Y 4 Likelihood atio ode ( Y if m x m x x; Y if f x f x deceases i x. l Y The followig implicatios (Shaked & Shathikuma [8] ae well kow i that Y Y Y l h ml The followig theoem shows that all membes of the LW class ae odeed with espect to likelihood atio odeig. Theoem. Suppose LW ( β ad Y LW ( β the If H( x; η 0 β = β ad ( o if = ad β β the l Y ad hece h Y ml Y ad st Y. If H( x; η < 0 β = β ad ( o if = ad β β the l Y ad hece h Y ml Y ad st Y. Poof. We have ad Thus st Y. Y + β + β ( H( x = e ; H( x 0 β β + + f x H x f x H x f ( x ( x + β = + + + log log log log fy + β ( βh( x ( H( x log +. d f log x β h x β h x dx f x H x H x Y ( + βh( x ( + βh( x ( βh( x ( h( x = + β + β β β ( h( x = + Case If H( x η β = β ( = β β d f log ( x 0. dx fy ( x < This meas that l Y Case If H( x η β β ( β β d f log ( x 0. dx f ( x > This meas that l Y ad hece the Y 4. Special Cases 4.. Lidley Distibutio ; 0 ad o if ad ad hece Y Y ad Y. ; < 0 = ad o if = ad the Y Y ad Y. h ml st h ml st The oigial Lidley distibutio (L poposed by Lidley [] is a special case of LW class with H( x; ad β =. Usig ( the cdf of the Lidley distibutio is give by. η = x 689
S. H. Alkai The associated pdf usig ( is give by x FL ( x; = + x e ; x > 0. + x fl ( x; = ( + x e ; x > 0. + It ca be see that this distibutio is a mixtue of expoetial ( ad gamma distibutios. Accodig to foms (5 ad (6 the coespodig sf ad hf ae give espectively by ad x sl ( x; = + x e ; x > 0 + ( + x τl ( x; = ; x > 0. + + x A diect substitutio i (9 ad (0 with = β = gives us the distibutio: ( ( ( + Γ + + + µ = t + + M t = Γ. The mea ad the vaiace fom ( ad ( ae ( + ( + = (! ( + + µ = σ = + + + 3. Figue displays the plots of desity ad hazad ate fuctio of the Lidley distibutio. 4.. Powe Lidley Distibutio th momets ad mgf fo the Lidley Powe Lidley distibutio (PL itoduced by Ghitay et al. [4] is a special case of LW class with Figue. Plots of the pdf ad hf of the Lidley distibutio fo diffeet values of. 690
( ; H x η = x ad β =. Usig the cdf fom i ( the cdf of PL distibutio is give by x FPL ( x; = + x e ; x > 0. + The associated pdf usig ( is give by fpl ( x = ( + x x x > + x ; e ; 0. S. H. Alkai The PL distibutio is a mixtue distibutio of the Weibull distibutio (with shape paametes ad scale ad a geealized gamma distibutio (with shape paametes ad scale with mixig popotio p = ( +. The sf ad hf of the PL distibutio ae obtaied fom (5 ad (6 x spl ( x; = + x e ; x > 0 + ( + x x τ ( ; PL x = ; x > 0. + + x Figue shows the pdf ad hf of the PL distibutio of some selected choices of ad. th The ow momet ad the mgf of the PL distibutio usig (9 ad (0 ae give espectively by M (( ( ( ( + Γ + + + µ = ( t ( t + + = Γ. = (! ( + Theefoe the mea ad the vaiace of PL distibutio ae obtaied by diect substitutio i ( ad ( 4 Γ + + Γ + + + Γ + + µ = σ =. + + Figue. The pdf ad hf of the PL distibutio fo some selected choices of ad. 69
S. H. Alkai 4.3. Exteded Powe Lidley Distibutio Exteded powe Lidley distibutio (EPL itoduced by Alkai [3] is a special case of LW class with H x; η = x. Usig the cdf fom i ( the cdf of the EPL distibutio is give by β x FEPL ( x; β = + x e ; β x > 0. + β The associated pdf usig ( is give by fepl ( x ( x x x + β x ; β = + β e ; β > 0. We see that the EPL is a two-compoet mixtue of the Weibull distibutio (with shape ad scale ad a geealized gamma distibutio (with shape paametes ad scale with mixig popotio p = ( + β. The sf ad hf of the EPL distibutio ae obtaied as a diect substitutio i (5 ad (6 τ EPL β sepl x x x + β ( x x = + e ; > 0 ( + β x x = ; β x > 0. + β + β x Figue 3 shows the pdf ad hf of the EPL distibutio fo some choices of β ad. th The ow momet ad the mgf of the EPL distibutio usig (9 ad (0 ae give espectively by M ( t + β + β µ = Γ β ( + t + β + β = Γ. = (! β ( + Usig ( ad ( the mea ad the vaiace of the EPL distibutio ae give espectively by Figue 3. The pdf ad hf of the EPL distibutio fo some choices of β ad. 69
S. H. Alkai + β + β µ = Γ σ = ( β ( β β ( β β. + + + Γ 4 + + Γ β β ( + ( + 4.4. Ivese Lidley Distibutio Ivese Lidley (IL distibutio poposed by Shama et al. [3] is a special case of the LW class with H x; η = xy ; = H x; η ad β =. Usig the cdf fom i (3 the cdf of the IL distibutio is give by The associated pdf usig (4 is give by y FIL ( y; = + e ; y > 0. + y + y y fil ( y; = e ; y > 0. 3 + y We see that the IL is a two-compoet mixtue of the Weibull distibutio (with shape ad scale ad a gep = + β. The sf ad hf of the IL distibutio ae obtaied as a diect substitutio i (7 ad (8 ealized gamma distibutio (with shape paametes ad scale with mixig popotio τ IL y sil ( y = + e ; y > 0 + y ( y ( + y = ; y > 0. y y ( + y e Figue 4 shows the pdf ad hf of the IL distibutio fo some choices of. 4.5. The Geealized Ivese Lidley Distibutio The geealized ivese Lidley (GIL distibutio poposed by Shama et al. [5] is a special case of LW class H x; η = x Y = H x; η ad β =. Usig the cdf fom i (3 the cdf of the GIL is give by with y FGIL ( y; = + e ; y 0. > + y Figue 4. The pdf ad hf of the IL distibutio fo some selected choices of. 693
S. H. Alkai The associate pdf usig (4 is give by The associate hf usig (8 is give by fgil ( x ( x x x + x ; = + e ; > 0. ( + y τgil ( x; = ; x > 0. + y y ( + y e Figue 5 shows the pdf ad hf of the GIL distibutio of some selected choices of ad. th The ow momet of the geealized ivese Lidley distibutio usig (0 is give by ( + Γ( ( ( + µ = >. The mea ad the vaiace of the geealized ivese Lidley distibutio ae give espectively by ( + µ = Γ > + ( + ( ( + σ = Γ. > ( + Γ Γ 4.6. Exteded Ivese Lidley Distibutio The exteded ivese Lidley (EIL distibutio poposed by Alkai [7] is a special case of the LW class with H x; η = x. Usig the cdf fom i (3 the cdf of the EIL distibutio is give by β x F( x; β = + e ; β x > 0. + β x Figue 5. The pdf ad hf of the GIL distibutio fo some selected choices of ad. 694
S. H. Alkai The associated pdf usig (4 is give by β + x x f ( x; β = e ; β x > 0. + + β x We see that the EIL is a two-compoet mixtue of the ivese Weibull distibutio (with shape ad scale ad a geealized ivese gamma distibutio (with shape paametes ad scale with the mixig popotio p = ( + β. The hf of the EIL distibutio is give by τ EIL ( y ( β + y = ; β y > 0. + y y ( + β y e β Figue 6 shows the pdf ad hf of the EIL distibutio fo some choices of β ad. th The ow momet of the EIL distibutio usig (9 is give by ( + + β β µ = Γ > β Theefoe the mea ad the vaiace of the EIL distibutio ae give espectively by ( + β β µ = Γ > + β ( σ β β β ( ( β β =. + + Γ + Γ > ( β + 5. Estimatio ad Ifeece Let be a adom sample with obseved values x x fom the LW class with paametes ad Θ= βη be the p paamete vecto. The log likelihood fuctio is give by β η. Let Figue 6. The pdf ad hf of the EIL distibutio fo some choices of β ad. 695
S. H. Alkai the the scoe fuctio is give by l = log + log + H x + log h x H x + β i= i= i= ( β η T U Θ = l l l whee ( β ( i ( i ( i ( + β H( xi ; η i= H( xi ; ( ; ( ; β H xi η h xi η H( xi; η i H( x; = i= h( x; i= l = + β l = + β + β + β η l = + η + β η η η η η k i k i k k The maximum likelihood estimatio (MLE of Θ says ˆΘ is obtaied by solvig the oliea system U ( x; Θ = 0. This oliea system of equatios does ot have a closed fom. Fo iteval estimatio ad hypothesis tests o the model paametes we equie the obseved ifomatio matix I T I Iβ I η T Iβ Iββ Iβη Θ = Iη Iβη Iηη whee the elemets of I ( Θ ae the secod patial deivatives of U Θ. Ude stadad egula coditios fo lage sample appoximatio (Cox ad Hikley [9] that fulfilled fo the poposed model the distibutio of ˆΘ appoximately Np( Θ J( Θ with J( Θ = E I( Θ. Wheeve the paametes ae i the iteio ˆ N 0 J Θ of the paamete space but ot o the bouday the asymptotic distibutio of whee J lim I. ΘΘ is Θ = Θ is the uit ifomatio matix ad p is the umbe of paametes of the distibutio. The asymptotic multivaiate omal N ( ˆ p I Θ Θ distibutio of ˆΘ ca be used to appoximate co- fidece iteval fo the paametes ad fo the hazad ate ad suvival fuctios. A 00( γ asymptoticcofidece iteval fo paamete whee ii I is the ( stadad omal distibutio. 6. Applicatios Θ is give by i ii diagoal elemet of ( ˆ ˆ ii ˆ ii Θi Zγ I Θ i + Zγ I I Θ fo i = p ad p Z γ is the quatile γ of the I this sectio we itoduce two data sets as applicatios of the LW class. Fo the fist data set we fit L PL ad EPL models as well as the Two-paamete Lidley (TL ad the stadad Weibull (W. The fist data set was itoduced by Bade ad Piest [0] as the tesile stegth measuemets o 000 cabo fibe-impegated tows at fou diffeet gauge legths. The data is listed i Table. The MLEs of the paametes wee obtaied usig the expectatio-maximizatio (EM algoithm. The MLEs Kolmogoov-Smiov statistic (K-S with its espective p-value the maximized log likelihood fo the above distibutios ae listed i Table 3. The distibutios ae odeed i the table accodig to thei pefomace. The fitted desities ad the empiical distibutio vesus the fitted cumulative distibutios of all models fo this data ae show i Figue 7 ad Figue 8 espectively. 696
S. H. Alkai Table. Cabo fibe tesile stegth..3.34.479.55.700.803.86.865.944.958.966.997.006.0.07.055.063.098.40.79.4.40.53.70.7.74.30.30.359.38.38.46.434.435.478.490.5.54.535.554.566.570.586.69.633.64.648.684.697.76.770.773.800.809.88.8.848.880.954 3.0 3.067 3.084 3.090 3.096 3.8 3.33 3.433 3.585 3.585 Table 3. Paamete estimates K-S statistic p-value ad logl of cabo fibe tesile stegth. Distibutio ˆ ˆβ ˆ K-S p-value log L EPL 0.0584 98.9 3.733 0.049 0.9996 48.9 PL 0.0450-3.8678 0.044 0.9993 49.06 W 0.000-4.875 0.0 0.4685 50.65 TL 0.858 4504.4-0.364 0.000 05.7 L 0.6545 - - 0.40 0.000 9. Figue 7. Plot showig the fitted desities of the models listed i Table 3. Fo the secod data set we demostate the applicability of the IL GIL ad EIL as well as the ivese Weibull (IW ad the geealized ivese Weibull (GIW models. Table 4 epesets the flood levels fo the Susquehaa Rive at Haisbug Pesylvaia ove 0 fou-yea peiods fom 890 to 969. This data has bee used by seveal authos ad was iitially epoted by Dumoceaux & Atle []. The MLEs of the paametes the Kolmogoov-Smiov statistic (K-S with its espective p-value ad the maximized log likelihood (logl fo the above distibutios ae give i Table 5 accodig to thei pefomace. The fitted desities ad the empiical distibutio vesus the fitted cumulative distibutios of all models fo this data ae show i Figue 9 ad Figue 0 espectively. 7. Cocludig Remaks We defie a ew family of lifetime distibutios called the LW family of distibutios that geeates Lidley ad Weibull distibutios. The LW class cotais may lifetime subclasses ad distibutios. Vaious stadad mathematical popeties wee deived such as desity ad suvival hazad fuctios momets momet geeatig fuctio ad quatile fuctio ad wee itoduced i flexible ad useful foms. The maximum likelihood 697
S. H. Alkai Figue 8. Plot showig the fitted cdfs of the models listed i Table 3. Figue 9. Plot showig the fitted desities of the models listed i Table 5. Figue 0. Plot showig the fitted cdfs of the models listed i Table 5. 698
S. H. Alkai Table 4. Flood level data fo the Susquehaa Rive. 0.654 0.63 0.35 0.449 0.97 0.40 0.379 0.43 0.379 0.34 0.69 0.740 0.48 0.4 0.494 0.46 0.338 0.39 0.484 0.65 Table 5. Paamete estimates KS statistic P-Value ad logl of flood level data. Distibutio ˆ ˆβ ˆ K-S p-value log L EIL 0.05 4.0439.9573 0.395 0.83 6.475 GIL 0.0899-3.0763 0.445 0.7977 6.475 IW 0.03-4.873 0.545 0.763 6.096 GIW 0.030 4.37 0.807 0.560 0.750 6.097 IL 0.6345 - - 0.3556 0.07 0.5854 method was used fo paamete estimatio usig the EM algoithm. Fially some special models wee itoduced ad fitted to eal datasets to show the flexibility ad the beefits of the poposed class. Ackowledgemets The autho is highly gateful to the Deaship of Scietific Reseach at Kig Saud Uivesity epeseted by the Reseach Cete at the College of Busiess Admiistatio fo suppotig this eseach fiacially. Competig Iteests The autho declaes that thee wee o competig iteests. Refeeces [] Baghei S. Bahami E. ad Gajali M. (06 The Geealized Modified Weibull Powe Seies Distibutio: Theoy ad Applicatios. Computatioal Statistics ad Data Aalysis 94 36-60. http://dx.doi.og/0.06/j.csda.05.08.008 [] Ghitay M. Atieh B. ad Nadadajah S. (008 Lidley Distibutio ad Its Applicatios. Mathematics ad Computes i Simulatio 78 493-506. http://dx.doi.og/0.06/j.matcom.007.06.007 [3] Alkai S. (05 Exteded Powe Lidley Distibutio: A New Statistical Model fo No-Mootoe Suvival Data. Euopea Joual of Statistics ad Pobability 3 9-34. [4] Johso N. Kotz S. ad Balakisha N. (994 Cotiuous Uivaiate Distibutio Volume. Wiley New Yok. [5] Rayleigh J. (880 O the Result of a Lage Numbe of Vibatios of the Same Pitch ad of Abitay Phase. Philosophical Magazie 0 73-78. http://dx.doi.og/0.080/478644800866893 [6] Lai C. ie M. ad Muthy D. (003 A Modified Weibull Distibutio. IEEE Tasactios o Reliability 5 7-33. http://dx.doi.og/0.09/tr.00.805788 [7] ie M. Tag Y. ad Goh T. (00 A Modified Weibull Extesio with Bathtub-Shaped Failue Rate Fuctio. Reliability Egieeig ad System Safety 76 79-85. http://dx.doi.og/0.06/s095-830(0000-4 [8] Gompetz B. (85 O the Natue of the Fuctio Expessive of the Law of Huma Motality ad o a New Mode of Detemiig Life Cotigecies. Philosophical Tasactios of the Royal Society 5 53-585. http://dx.doi.og/0.098/stl.85.006 [9] Smith R. ad Bai L. (975 A Expoetial Powe Life-Testig Distibutio. Commuicatios i Statistics-Theoy ad Methods 4 469-48. http://dx.doi.og/0.080/0360975088763 [0] Che Z. (000 A New Two-Paamete Lifetime Distibutio with Bathtub Shape o Iceasig Failue Rate Fuctio. Statistics ad Pobability Lettes 49 55-6. http://dx.doi.og/0.06/s067-75(0000044-4 [] Pham H. (00 A Vtub-Shape Hazad Rate Fuctio with Applicatios to System Safety. Iteatioal Joual of 699
S. H. Alkai Reliability ad Applicatios 3-6. [] Lidley D. (958 Fiducial Distibutios ad Bays Theoem. Joual of the Royal Statistical Society 0 0-07. [3] Shama V. Sigh S. Sigh U. ad Agiwal V. (05 The Ivese Lidley Distibutio: A Stess-Stegth Reliability Model. Joual of Idustial ad Poductio Egieeig 3 6-73. http://dx.doi.og/0.080/6805.05.0590 [4] Ghitay M. Al-Mutaii D. Balakisha N. ad Al-Eezi I. (03 Powe Lidley Distibutio ad Associated Ifeece. Computatioal Statistics ad Data Aalysis 64 0-33. http://dx.doi.og/0.06/j.csda.03.0.06 [5] Shama V. Sigh S. Sigh U. ad Meovci F. (05 The Geealized Ivese Lidley Distibutio: A New Ivese Statistical Model fo the Study of Upside-Dow Bathtub Suvival Data. Commuicatios i Statistics-Theoy ad Methods Pepit. [6] Shake R. Shama S. ad Shake R. (03 A Two-Paamete Lidley Distibutio fo Modelig WAITING ad Suvival Time Seies Data. Applied Mathematics 4 363-368. http://dx.doi.og/0.436/am.03.4056 [7] Alkai S. (05 Exteded Ivese Lidley Distibutio: Popeties ad Applicatio. SpigePlus 4-7. http://dx.doi.og/0.86/s40064-05-489- [8] Shaked M. ad Shathikuma J. (994 Stochastic Odes ad Thei Applicatios. Academic Pess New Yok. [9] Cox D. ad Hikley D. (974 Theoetical Statistics. Chapma ad Hall Lodo. http://dx.doi.og/0.007/978--4899-887-0 [0] Bade M. ad Piest A. (98 Statistical Aspects of Fibe ad Budle Stegth i Hybid Composites. I: Hayashi T. Kawata S. ad Umekawa S. Eds. Pogess i Sciece ad Egieeig Composites ICCM-IV Tokyo 9-36. [] Dumoceaux R. ad Atle C. (973 Discimiatio betwee the Logomal ad Weibull Distibutio. Techometics 5 93-96. http://dx.doi.og/0.080/0040706.973.04894 Submit o ecommed ext mauscipt to SCIRP ad we will povide best sevice fo you: Acceptig pe-submissio iquiies though Email Facebook LikedI Twitte etc. A wide selectio of jouals (iclusive of 9 subjects moe tha 00 jouals Povidig 4-hou high-quality sevice Use-fiedly olie submissio system Fai ad swift pee-eview system Efficiet typesettig ad poofeadig pocedue Display of the esult of dowloads ad visits as well as the umbe of cited aticles Maximum dissemiatio of you eseach wok Submit you mauscipt at: http://papesubmissio.scip.og/ 700