Chapter 4. Location-Scale-Based Parametric Distributions. William Q. Meeker and Luis A. Escobar Iowa State University and Louisiana State University

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1 Chaper 4 Locaion-Scale-Based Parameric Disribuions William Q. Meeker and Luis A. Escobar Iowa Sae Universiy and Louisiana Sae Universiy Copyrigh W. Q. Meeker and L. A. Escobar. Based on he auhors ex Saisical Mehods for Reliabiliy Daa, John Wiley & Sons Inc January 13, h 41min 4-1

2 Chaper 4 Locaion-Scale-Based Parameric Disribuions Objecives Explain imporance of parameric models in he analysis of reliabiliy daa. Define imporan funcions of model parameer ha are of ineres in reliabiliy sudies. Inroduce he locaion-scale and log-locaion-scale families of disribuions. Describe he properies of he exponenial disribuion. Describe he Weibull and lognormal disribuions and he relaed underlying locaion-scale disribuions. 4-2

3 Moivaion for Parameric Models Complemens nonparameric echniques. Parameric models can be described concisely wih jus a few parameers, insead of having o repor an enire curve. I is possible o use a parameric model o exrapolae (in ime) o he lower or upper ail of a disribuion. Parameric models provide smooh esimaes of failure-ime disribuions. In pracice i is ofen useful o compare various parameric and nonparameric analyses of a daa se. 4-3

4 Funcions of he Parameers Cumulaive disribuion funcion (cdf) of T F(;θ) = Pr(T ), > 0. The p quanile of T is he smalles value p such ha F( p ;θ) p. Hazard funcion of T h(;θ) = f(;θ) 1 F(;θ), >

5 Funcions of he Parameers-Coninued The mean ime o failure, MTTF, of T (also known as expecaion of T) E(T) = f(;θ)d = [1 F(;θ)] d. 0 0 If 0 f(;θ)d =, we say ha he mean of T does no exis. The variance (or he second cenral momen) of T and he sandard deviaion Var(T) = SD(T) = 0 [ E(T)]2 f(;θ)d Var(T). Coefficien of variaion γ 2 γ 2 = SD(T) E(T). 4-5

6 Locaion-Scale Disribuions Y belongs o he locaion-scale family of disribuions if he cdf of Y can be expressed as ( ) y µ F(y;µ,) = Pr(Y y) = Φ, < y < where < µ < is a locaion parameer and > 0 is a scale parameer. Φ is he cdf of Y when µ = 0 and = 1 and Φ does no depend on any unknown parameers. Noe: The disribuion of Z = (Y µ)/ does no depend on any unknown parameers. 4-6

7 Imporance of Locaion-Scale Disribuions Imporance due o: Mos widely used saisical disribuions are eiher members of his class or closely relaed o his class of disribuions: exponenial, normal, Weibull, lognormal, loglogisic, logisic, and exreme value disribuions. Mehods of inference, saisical heory, and compuer sofware generaed for he general family can be applied o his large, imporan class of models. Theory for locaion-scale disribuions is relaively simple. 4-7

8 Examples of Exponenial Disribuions Cumulaive Disribuion Funcion Probabiliy Densiy Funcion F().5 f() Hazard Funcion h() θ γ

9 Exponenial Disribuion For T EXP(θ,γ), F(;θ,γ) = 1 exp γ θ f(;θ,γ) = 1 ( θ exp γ ) θ h(;θ,γ) = f(;θ,γ) 1 F(;θ,γ) = 1 θ, > γ, where θ > 0 is a scale parameer and γ is boh a locaion and a hreshold parameer. When γ = 0 one ges he well-known one-parameer exponenial disribuion. ( ) Quaniles: p = γ θlog(1 p). Momens: For ineger m > 0, E[(T γ) m ] = m!θ m. Then E(T) = γ +θ, Var(T) = θ

10 Moivaion for he Exponenial Disribuion Simples disribuion used in he analysis of reliabiliy daa. Has he imporan characerisic ha is hazard funcion is consan (does no depend on ime ). Popular disribuion for some kinds of elecronic componens (e.g., capaciors or robus, high-qualiy inegraed circuis). This disribuion would no be appropriae for a populaion of elecronic componens having failure-causing qualiy-defecs. Migh be useful o describe failure imes for componens ha exhibi physical wearou only afer expeced echnological life of he sysem in which he componen would be insalled. 4-10

11 Examples of Normal Disribuions Cumulaive Disribuion Funcion Probabiliy Densiy Funcion F().5 f() Hazard Funcion h() µ

12 For Y NOR(µ,) Normal (Gaussian) Disribuion ( ) y µ F(y;µ,) = Φ nor f(y;µ,) = 1 ( ) y µ φ nor, < y <. where φ nor (z) = (1/ 2π)exp( z 2 /2) and Φ nor (z) = z φ nor (w)dw are pdf and cdf for a sandardized normal (µ = 0, = 1). < µ < is a locaion parameer; > 0 is a scale parameer. Quaniles: y p = µ+φ 1 nor(p) where Φ 1 nor(p) is he p quanile for a sandardized normal. Momens: For ineger m > 0, E[(Y µ) m ] = 0 if m is odd, and E[(Y µ) m ] = (m)! m /[2 m/2 (m/2)!] if m is even. Thus E(Y) = µ, Var(Y) =

13 Examples of Lognormal Disribuions Cumulaive Disribuion Funcion Probabiliy Densiy Funcion F().5 f() Hazard Funcion h() µ

14 Lognormal Disribuion If T LOGNOR(µ,) hen log(t) NOR(µ,) wih [ log() µ F(;µ,) = Φ nor f(;µ,) = 1 [ log() µ φ nor ] ], > 0. φ nor and Φ nor are pdf and cdf for a sandardized normal. exp(µ) is a scale parameer; > 0 is a shape parameer. Quaniles: p = exp ( µ+φ 1 nor(p) ), where Φ 1 nor(p) is he p quanile for a sandardized normal. Momens: For ineger m > 0, E(T m ) = exp ( mµ+m 2 2 /2 ). E(T) = exp ( µ+ 2 /2 ), Var(T) = exp ( 2µ+ 2)[ exp( 2 ) 1 ]. 4-14

15 Moivaion for Lognormal Disribuion The lognormal disribuion is a common model for failure imes. I can be jusified for a random variable ha arises from he produc of a number of idenically disribued independen posiive random quaniies. I has been suggesed as an appropriae model for failure ime caused by a degradaion process wih combinaions of random raes ha combine muliplicaively. Widely used o describe ime o fracure from faigue crack growh in meals. Useful in modeling failure ime of a populaion elecronic componens wih a decreasing hazard funcion (due o a small proporion of defecs in he populaion). Useful for describing he failure-ime disribuion of cerain degradaion processes. 4-15

16 Examples of Smalles Exreme Value Disribuions Cumulaive Disribuion Funcion Probabiliy Densiy Funcion F() 1.5 f() Hazard Funcion 1.5 µ h()

17 Smalles Exreme Value Disribuion For Y SEV(µ,), ( y µ F(y;µ,) = Φ sev f(y;µ,) = 1 ( ) y µ φ sev h(y;µ,) = 1 ( ) y µ exp, < y <. Φ sev (z) = 1 exp[ exp(z)], φ sev (z) = exp[z exp(z)] are cdf and pdf for sandardized SEV (µ = 0, = 1). < µ < is a locaion parameer and > 0 is a scale parameer. Quaniles: y p = µ+φ 1 sev(p) = µ+log[ log(1 p)]. Mean and Variance: E(Y) = µ γ, Var(Y) = 2 π 2 /6, where γ.5772,π ) 4-17

18 Examples of Weibull Disribuions Cumulaive Disribuion Funcion Probabiliy Densiy Funcion F().5 f() Hazard Funcion h() β η

19 Weibull Disribuion Common Parameerizaion: F() = Pr(T ) = 1 exp f() = β η h() = β η ( ( ) β 1 exp η ) β 1, > 0 η ( ) β η ( ) β β > 0 is shape parameer; η > 0 is scale parameer. η Quaniles: p = η[ log(1 p)] 1/β. Momens: For ineger m > 0, E(T m ) = η m Γ(1+m/β). Then ( ) ( ( ) Γ 2 E(T) = ηγ 1+ 1, Var(T) = η [Γ β β β where Γ(κ) = 0 wκ 1 exp( w)dw is he gamma funcion. Noe: When β = 1 hen T EXP(η) )]

20 Alernaive Weibull Parameerizaion Noe: If T WEIB(µ,) hen Y = log(t) SEV(µ,). For T WEIB(µ,) hen ( ) β [ F(;µ,) = 1 exp log() µ = Φ sev η f(;µ,) = β ( ) β 1 ( ) β exp = 1 [ log() µ η η η φ sev where = 1/β, µ = log(η), and Quaniles: φ sev (z) = exp[z exp(z)] Φ sev (z) = 1 exp[ exp(z)]. p = η[ log(1 p)] 1/β = exp [ µ+φ 1 sev (p)] where Φ 1 sev(p) is he p quanile for a sandardized SEV (i.e., µ = 0, = 1). ] ] 4-20

21 Moivaion for he Weibull Disribuion The heory of exreme values shows ha he Weibull disribuion can be used o model he minimum of a large number of independen posiive random variables from a cerain class of disribuions. Failure of he weakes link in a chain wih many links wih failure mechanisms (e.g., creep or faigue) in each link acing approximaely independen. Failure of a sysem wih a large number of componens in series and wih approximaely independen failure mechanisms in each componen. The more common jusificaion for is use is empirical: he Weibull disribuion can be used o model failure-ime daa wih a decreasing or an increasing hazard funcion. 4-21

22 Examples of Larges Exreme Value Disribuions Cumulaive Disribuion Funcion Probabiliy Densiy Funcion F() 1.5 f() Hazard Funcion h() µ

23 Larges Exreme Value Disribuion When Y LEV(µ,), ( y µ F(y;µ,) = Φ lev f(y;µ,) = 1 ( y µ φ lev h(y;µ,) = ) ) exp ( y µ { exp [ exp ( y µ ) )] 1 }, < y <. where Φ lev (z) = exp[ exp( z)] and φ lev (z) = exp[ z exp( z)] are he cdf and pdf for a sandardized LEV (µ = 0, = 1) disribuion. < µ < is a locaion parameer and > 0 is a scale parameer. 4-23

24 Larges Exreme Value Disribuion - Coninued Quaniles: y p = µ log[ log(p)]. Mean and Variance: E(Y) = µ + γ, Var(Y) = 2 π 2 /6, where γ.5772,π Noes: The hazard is increasing bu is bounded in he sense ha lim y h(y;µ,) = 1/. If Y LEV(µ,) hen Y SEV( µ,). 4-24

25 Examples of Logisic Disribuions Cumulaive Disribuion Funcion Probabiliy Densiy Funcion F() f() Hazard Funcion µ h()

26 Logisic Disribuion For Y LOGIS(µ,), ( y µ F(y;µ,) = Φ logis f(y;µ,) = 1 ( ) y µ φ logis h(y;µ,) = 1 ( ) y µ Φ logis, < y <. < µ < is a locaion parameer; > 0 is a scale parameer. ) φ logis and Φ logis are pdf and cdf for a sandardized logisic disribuion defined by φ logis (z) = Φ logis (z) = exp(z) [1+exp(z)] 2 exp(z) 1+exp(z). 4-26

27 Logisic Disribuion-Coninued Quaniles: y p = µ + Φ 1 logis (p) = µ + ) log( p 1 p, where Φ 1 logis (p) = log[p/(1 p)] is he p quanile for a sandardized logisic disribuion. Momens: For ineger m > 0, E[(Y µ) m ] = 0 if m is odd, and E[(Y µ) m ] = 2 m (m!) [ 1 (1/2) m 1] i=1 (1/i) m if m is even. Thus E(Y) = µ, Var(Y) = 2 π

28 Examples of Loglogisic Disribuions Cumulaive Disribuion Funcion Probabiliy Densiy Funcion F().5 f() Hazard Funcion h() µ

29 Loglogisic Disribuion If Y LOGIS(µ,) hen T = exp(y) LOGLOGIS(µ,) wih [ log() µ F(;µ,) = Φ logis f(;µ,) = 1 [ log() µ φ logis h(;µ,) = 1 [ log() µ Φ logis ] ] ], > 0. exp(µ) is a scale parameer; > 0 is a shape parameer. Φ logis and φ logis are cdf and pdf for a LOGIS(0,1). 4-29

30 Loglogisic Disribuion-Coninued Quaniles: p = exp [ µ+φ 1 logis (p)] = exp(µ)[p/(1 p)]. Momens: For ineger m > 0, E(T m ) = exp(mµ)γ(1+m)γ(1 m). The m momen is no finie when m 1. For < 1, E(T) = exp(µ)γ(1+)γ(1 ), and for < 1/2, Var(T) = exp(2µ) [ Γ(1+2)Γ(1 2) Γ 2 (1+)Γ 2 (1 ) ]. 4-30

31 Oher Topics in Chaper 4 Pseudorandom number generaion. 4-31

32 Topics in Chaper 5 Parameric models wih hreshold parameers. Imporan disribuions used in reliabiliy ha can no be ranslaed ino locaion-scale disribuions: gamma, generalized gamma, ec. Finie (discree) mixure disribuions F(;θ) = ξ 1 F 1 (;θ 1 )+ +ξ k F k (;θ k ) where ξ i 0, and iξ i = 1 Compound (coninuous) mixure disribuions. If failure-imes of unis in a populaion are EXP(η) wih 1/η GAM(θ,κ), hen he uncondiional failure ime, T, of a uni seleced a random from he populaion has a Pareo disribuion of he form 1 F(;θ,κ) = 1 (1+θ) κ, >