COMPARISON OF MODIFIED WEIBULL MODELS WHEN BAT HT …jaguar.fcav.unesp.br/RME/fasciculos/v25/v25_n2/A8_VicenteEdwin.pdf · COMPARISON OF MODIFIED WEIBULL MODELS WHEN BAT HT UB-SHAPED

COMPARISON OF MODIFIED WEIBULL MODELS WHEN

BAT HT UB-SHAPED FAILURE RAT ES

Vicente Garibay CANCHO3

E d w in M o ises M arco s OR T E GA 4

Glad ys D o ro tea Cacsire B AR R IGA5

ABSTRACT: In survival analysis applications, models with bathtub-shaped hazard

function are very important. The traditional W eibull model is, however, unable to

model lifetime data with a bathtub-shaped failure rate. Some ex tensions of the W eibull

distribution has hazard function in form of bathtub. In this paper, we considered some

of those distributions and developed a Bayesian methodolog y for analysis of lifetime

data with hazard in form bathtub. The Bayesian analysis is based on the M ark ov Chain

M onte Carlo (M CM C) methods. W e also, present some Bayesian criteria to choice

models considered. The methodolog y is illustrated with real as well as simulated data.

K E Y W O RD S: Bayesian inference; lifetime data; M CM C; bathtub curve.

1 Introduction

T h e W eibu ll d istribu tio n, is co m m o nly u sed fo r analyz ing o f lifetim e d ata. T h is

fam ily d istribu tio n acco m m o d ate co nstant, increasing and d ecreasing failu re rates.

Ho w ev er, th e W eibu ll d istribu tio n d o es no t p ro v id e a reaso nable p aram etric fi t fo r

so m e p ractical ap p licatio ns w h ere th e u nd erlying h azard rates m ay be bath tu b

sh ap es. In o rd er to ach iev e th ese beh av io rs fro m a sing le d istribu tio n and to

m o d el th e d ata sev eral m o d els w ere intro d u ced , to m o d el th is k ind o f d ata, as

3Departamento de Matemática Aplicada e Estat́ıstica, Instituto de Ciências Matemáticas e de

Computação – ICMC, U niv ersidade de S ão P aulo – U S P , Caix a P ostal 6 6 8 , CEP : 1 3 5 6 0 -9 7 0 , S ão

Carlos, S P , B rasil, E-mail: [email protected] de Ciências Ex atas, Escola S uperior de Ag ricultura L uiz de Q ueiroz – ES AL Q ,

U niv ersidade de S ão P aulo – U S P , Caix a P ostal 9 , CEP : 1 3 4 1 8 -9 0 0 , P iracicab a, S P , B rasil, E-mail:

ed w in @esalq.usp.br5Departamento de Estat́ıstica, CCET , U niv ersidade F ederal de S ão Carlos – U F S Car, Caix a P ostal

6 7 6 , CEP : 1 3 5 6 5 -9 0 5 , S ão Carlos, S P , B raz il, E-mail: glad yscacsire@yah oo .com.br

R ev . Mat. Estat., S ão P aulo, v .2 5 , n.2 , p.1 1 1 -1 3 6 , 2 0 0 7 1 1 1

the generalized gamma distributions proposed by Stacy (1962), the generalized Fdistributions proposed by P rentice (197 5 ), the IDB distribution proposed by Hjort(198 0). A good review of these models is presented in Rajarshi and Rajarshi (198 8 ).In recent years a new classes of models to be used for this kind of data is based inmodifications of the Weibull distribution. Mudholkar and Srivastava (1994 ) proposea new class of models generalizing the Weibull distribution given by the introductionof an additional shape parameter. This ex tended family provides models for a largenumber of survival or reliability problems, which includes unimodal, bathtub another classes of monotonous failure rate functions. Another modification of theWeibull distribution introduced by X ei and L ai (1995 ) is the additive Weibull model,which is additive in the sense that the failure rate function is ex pressed as the sumof two failure rate functions of Weibull type. Recently, X ei et al.(2002) proposed anew modification of the Weibull distribution allowing it to ex hibit bathtub shapedfailure rate functions. In general, the inferential part of these models is carried outusing the asymptotic distribution of the max imum likelihood estimators, which insituation when the sample is small, it might present diffi cult results to be justified.In this paper, we ex plored the techniq ue of MCMC to development a Bayesianinference for the Weibull ex tension models proposed by Mudholkar and Srivastava(1994 ), X ei and L ai (1995 ) and X ei et al.(2002). We also, present some Bayesiancriteria to choose between the models considered. In order to identify the type offailure rate of the lifetime data, many approaches have been proposed (see, Glaser,198 0). In this study, a graphical method based on the total time on test (TTT)transformed introduced by Barlow and Campo (197 5 ) will be used to illustrate thevariety hazard rate shapes. It has been shown that the hazard function of F (t)increases (decreases) if the scaled TTT-transform, φF (t) = H

−1F (t)/ H

−1F (1), where

H−1F (t) =∫ F−1(t)0

(1 − F (u))d u, 0 ≤ u ≤ 1, is concave (convex ). In addition, for adistribution with bathtub (unimodal) failure rate the TTT-transform is first convex(concave) and then concave (convex ).

In Section 2 presents some of the modified Weibull models with bathtub shapesfor the failure rate function. Section 3 discusses likelihood inference for the modifiedWeibull models. In Section 4 , the Bayesian approach based on MCMC methodologyis presented for fitting the modified Weibull models. Section 5 reviews model choiceby using the Bayes factor methodology. Finally, Section 6 illustrates the approachwith real and simulated data sets.

2 Some models with bathtub-shaped failure rate

Some models are introduced in the literature to analyze lifetimes T with failurerate function in the form of a bathtub. A special case is given by the family ofex ponentiated-Weibull (EW) distributions with parameters α, θ and σ . The failure

rate function h(t) = f(t)R(t) , where f(t) is the probability density function of T and

112 Rev. Mat. Estat., São Paulo, v.25, n.2, p.111-136, 2007

R(t) = P (T ≥ t) is the reliability function, is given by

h(t) =αθ

[

1 − exp{

−(

tσ

)α} ]θ−1exp

{

−(

tσ

)α} ( tσ

)α−1

σ(1 −[

1 − exp{

−(

tσ

)α} ]θ)

. (1)

This family of distributions includes the exponential distribution when α = θ = 1and the Weibull distribution when θ = 1.

The great fl exibility of this model to fit lifetime data, is given by the diff erentforms that the failure rate function (1) can take, that is, (i) If α ≥ 1 and αθ ≥ 1,then the failure rate function is monotonically increasing; (ii) If α ≤ 1 and αθ ≤ 1,then the failure rate function is monotonically decreasing; (iii) If α > 1 and αθ < 1,then the failure rate function is bathtub shaped; (iv) If α < 1 and αθ > 1, then wehave a unimodal failure rate function. In the figure 1, we have some case specialsfor hazard function (1).

0 20 40 60 80 100

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

t

h(t)

EW(0.5;0.5;100)EW(5;0.1;100)EW(0.6;12;0.25)EW(4;4;80)

Figure 1 - Some forms specials for hazard function for the exponetiated-Weibullfamily (E W (α, θ, σ)).

Xei, et al.(2002) introduce a new modified Weibull distribution that can alsobe seen as a generalization of the Weibull distribution. This model is capable ofmodeling bathtub-shaped and increasing failure rate lifetime data.

The probability density function for the modified Weibull distribution is given

Rev. Mat. Estat., São Paulo, v.25, n.2, p.111-136, 2007 113

by

f(t) = λβ

(

t

α

)β−1

exp

{

(

t

α

)β

+ λα[1 − e(tα

)β ]

}

, λ, α, β > 0, t ≥ 0. (2)

This family of distributions has the ordinary Weibull distribution as a special andasymptotic case and hence it can be considered as an extension of the Weibulldistribution.

For the new Weibull distribution, the reliability function is given by,

R(t) = exp{

λα[1 − e(tα

)β ]}

, (3)

When the scale parameter α becomes very large or approach infinity, we have thatR(t) ≈ exp

{

−λα1−βtβ}

, which is a standard two parameter Weibull distributionwith shape parameter β and scale parameter λα1−β .

The failure rate function has the following form:

h(t) = λβ

(

t

α

)β−1

exp

{

(

t

α

)β}

. (4)

The shape of the failure rate function depends only on the parameter β, that is, (i)If β ≥ 1 then the failure rate function is increasing; (ii) If β < 1, then we have abathtub shaped failure rate function. In the Figure 2 shows the plots of the failurerate function for some different parameter combinations.

It can be shown that the kth moment for the modified model is given by

µk = E(Tk) =

∫ 1

0

Q(u)kdu, (5)

where Q(u) = F−1(u) = α[log(1− lo g (1−u)λ α )]1/ β and F (.) is the distribution function

of the new model.When α approaches infinity (α −→ ∞ ), it can be shown that

µk = E(Tk) =

λkβ Γ (1 + k/β)

αk(β−1)

β

. (6)

Moreover, it can be shown that the mean time to failure (MTTF) is given by

µ =

∫

∞

0

exp{

λα[1 − e(tα

)β ]}

dt. (7)

The variance of the time to failure can be obtained by

V a r (T ) =

∫

∞

0

t exp{

λα[1 − e(tα

)β ]}

dt − µ2. (8)


0 50 100 150 200 250 300

510

1520

t

h(t)

MW(100,0.4,2)MW(100,0.6,2)MW(100,0.8,2)MW(100,1.0,2)MW(100,1.2,2)

Figure 2 - Some forms specials for hazard function for the modified Weibull family(MW (α, β, λ)).

The calculations in (7) and (8) includes an integral that does not have a closedform, requiring numerical integration. But when α becomes large, the MTTF andvariance are given, respectively, by

µ =λ

1β Γ(1 + 1/β)

α(β−1)

β

, and V ar(T ) =λ

2β Γ(1 + 2/β)

α2(β−1)

β

−λ

2β Γ2(1 + 1/β)

α2(β−1)

β

.

Another modification of the Weibull model is presented in Xei and Lai (1995),where the additive Weibull model is introduced. It is based on the simple idea ofcombining the failure rates of two Weibull distributions: one has a decreasing failurerate and another has an increasing failure rate. It has the failure rate function givenin the following form:

h(t) = αβ(αt)β−1 + γ δ (γ t)δ−1, t ≥ 0, α, β, γ , δ > 0. (9)

It follows that the failure rate function (9) is bathtub shaped if β > 1 and δ < 1. Inthe Figure 3 shows the plots of the failure rate function for some different parametercombinations. Other properties of this family of distributions can be found in Xeiand Lai (1995).


0 5 10 15

23

45

6

t

h(t)

AW(6,0.5,1,1)AW(0.1,5,5,0.9)AW(0.2,2,5,0.6)AW(0.2,5,5,0.6)

Figure 3 - Some forms specials for hazard function for the additive Weibull family(AW (α, β, γ, δ)).

3 Likelihood inference

We assume that the lifetimes are independently distributed, and alsoindependent from the censoring mechanism. Considering the type II censured case(Lawless, 2003), let t1 ≤ t2 ≤ · · · ≤ tr be the time failure times of the r failedcomponents from an experiment with n components. In this case the likelihoodfunction for θ, a parameter vector of dimension p is given by,

L(θ) ∝

r∏

i= 1

f(ti; θ)R(tr; θ)n−r (10)

where f(.) and R(.) are probability density function and realiability function,respectively of ti, i = 1, . . . , r.

In the case the modified Weibull model (2), θ = (α, β, λ), the likelihoodfunction for θ corresponding to the observed sample is given by,

L(θ) ∝ λrβr exp

{

(β − 1)

r∑

i= 1

log

(

tiα

)

+

r∑

i= 1

(

tiα

)β

+ nλα

−λα

r∑

i= 1

e(tiα

)β− (n − r)αλe(

trα

)β

}

(11)


with corresponding log-likelihood function given by

l(θ) ∝ r log(λ) + r log(β) + nλα + (β − 1)

r∑

i=1

log

(tiα

)

+

r∑

i=1

(tiα

)β− λα

r∑

i=1

e(tiα

)β− (n − r)λαe(

trα

)β (12)

The maximum likelihood estimator θ̂ of θ is obtained by maximizing thelikelihood (11) (or log-likelihood (12)), which results in solving the equations

∂ l(θ)

∂ α= −

r(β − 1)

α+ nλ −

1

α

r∑

i=1

(tiα

)β− λ

r∑

i=1

[e(

tiα

)β(

1 − (tiα

)β)]

−(n − r)λe(trα

)β(

1 − (trα

)β)

= 0 (13)

∂ l(θ)

∂ β=

r

β+

r∑

i=1

log

(tiα

)+

r∑

i=1

[(tiα

)βlog

(tiα

)]

−λα

r∑

i=1

[e(

tiα

)β(

tiα

)βlog

(tiα

)]

−(n − r)λαe(trα

)β(

trα

)βlog

(trα

)= 0

∂ l(θ)

∂ λ=

r

λ− α

r∑

i=1

e(tiα

)β− (n − r)αe(

trα

)β = 0.

These equations cannot be solved analytically so that statistical software such asOx or R can be used to solved them. In this paper, software Ox (MAXBFGSsubroutine) is used to compute the maximum likelihood estimator (MLE). A similarprocedure can be used to obtain the MLE of the parameters of the models (1) and(9).

Large sample inference for θ can be obtained from large sample properties ofthe maximum estimator θ̂, which leads to (Sen and Singer, 1993)

i1/2(θ̂)

(θ̂ − θ

)D

−→ N(0, I3), (14)

where i(θ̂) is the observed information matrix of θ evaluated at the maximum

likelihood estimator θ̂ and I3 denotes the identity matrix of dimension three.Following Arellano-Valle et al. (2005) we also propose selecting the model

presenting the best fit by inspection of information criteria such as Akaike’sInformation Criterion (AIC, −`(θ̂)/n + P/n), and the Hannan-Q uinn Criterion

(HQ , −`(θ̂)/n + log(log(n))P/n), where P is the number of free parameters in the


model. This approach can be used in practice to select the model that seems topresent the best fit between the models considered.

4 Bayesian inference

The Bayesian model is completed by specifying a prior distribution for θ. Thejoint posterior distribution is then obtained by applications of Bayes theorem,

π(θ|D) =L(θ)π(θ)∫L(θ)π(θ)dθ

, (15)

where D denotes the set observations, L(θ) is the likelihood function (10) and π(θ)is the prior density of θ. Inferences about sets of θ should be based on the jointposterior, while if interest on any particular parameter, say θ1, its marginal posteriordistribution is obtained by integration the others out of the joint posterior. Sincethe derivation of exact posterior densities is not feasible for the models describedin the previous section, we make use of the Markov Chain Monte Carlo techniquemethodology to obtain approximations for such densities.

4.1 The modified Weibull model

Considering the modified Weibull model (2), for a Bayesian analysis, we assumethe following prior density for α, β and λ

α ∼ Γ(α0, α1), with α0 and α1 know,

β ∼ Γ(β0, β1), with β0 and β1 know, (16)

λ ∼ Γ(λ0, λ1), with λ0 and λ1 know,

where Γ(a, b) denotes a gamma distributions with mean ab and varianceab2 . We

further assume independence among the parameters.

Thus combining the likelihood (11) and prior density (16), the joint posteriordensity of the α, β and λ is given by,

π(α, β, λ|D) ∝ λr+λ0−1βr+β0−1αα0−1 exp

{

(β − 1 )

r∑

i= 1

lo g

(

ti

α

)

+

r∑

i= 1

(

ti

α

)β

+nλα− λα

r∑

i= 1

e(tiα

)β− (n− r)αλe(

trα

)β− α1α− β1β − λ1λ

}

, (1 7 )

w h ere D d en o tes th e o b serv ed d a ta .

T o im plem en t th e M C M C m eth o d o lo g y , w e c o n sid er th e G ib b s w ith inM etro po lis sa m pler, w h ich req u ires th e d eriv a tio n o f th e c o m plete set o f c o n d itio n a l


posterior distributions. After some algebraic manipulations it follows that theconditional posterior densities are given by

π(α|β, λ, D) ∝ αα0−r(β−1)−1 exp

{

r∑

i=1

(

ti

α

)β

− α[λ1 + λ(A(α, β) − n)]

}

π(β|α, λ, D) ∝ βr+β0−1 exp

{

(β − 1)

r∑

i=1

log

(

ti

α

)

+

r∑

i=1

(

ti

α

)β

− αλA(α, β) − β1β

}

(18 )

λ|α, β, D ∼ Γ (r + λ0, λ1 + αA(α, β)),

where A(α, β) =r∑

i=1

e(tiα

)β +(n− r)e(trα

)β . S ince the conditional posteriors of α and

β in (18 ) do not present standard form, the use of the Metropolis-H asting sampleris required. The Gibbs sampler can be used for sampling the conditional for λ.

4.2 The exponentiated-Weibull model

F or the exponentiated-W eibull model (1), the lik elihood function (10 ) of θ =(α, θ, σ) is given by,

L(θ) ∝ αrθrσ−rα exp

{

−

r∑

i=1

(

ti

σ

)α}

r∏

i=1

tα−1i

(

1 − exp

{

−

(

ti

σ

)α} )θ−1

(19 )

×

[

1 −

(

1 − exp

{

−

(

ti

σ

)α} )θ]n−r

.

Assuming independence among the parameters, considerer the following priordensities for α, θ and σ :

α ∼ Γ (a1, b1), with a1 and b1 k now,

θ ∼ Γ (a2, b2), with a2 and b2 k now, (2 0 )

σ ∼ Γ (a3, b3), with a3 and b3 k now.

Thus combining the lik elihood (19 ) and prior density (2 0 ), the joint posterior of theα, θ and σ is given by,

π(α, θ, σ|D) ∝ θr+a2−1σa3−rα−1 exp

{

−

r∑

i=1

(

ti

σ

)α

− b1α − b2θ − b3σ

}

αr+a1−1r

∏

i=1

tα−1i Ai(α, σ)θ−1

[

1 − (Ar(α, σ))θ]n−r

, (2 1)


where α > 0, θ > 0, σ > 0, Ai(α, σ) =(

1 − exp{

−(

tiσ

)α} )

and D the set observeddata.

The conditional posterior densities for the Gibbs algorithm are given by

π(α|, θ, σ,D) ∝ αr+a1−1 exp

{

−

r∑

i=1

(

ti

σ

)α

− α(b1 −

r∑

i=1

log(ti))

}

A(α, θ, σ)

π(θ|α, σ, D) ∝ θr+a2−1 exp

{

θ

(

r∑

i=1

log(Ai(α, σ)) − b2

) }

×(1 − (Ar(α, σ))θ))n−r (22)

π(σ|α, θ, D) ∝ σa3−rα−1A(α, θ, σ) exp

{

−

r∑

i=1

(

ti

σ

)α

− b3σ

}

,

where A(α, θ, σ) =r∏

i=1

(Ai(α, σ))θ−1 [

1 − Ar(α, σ)θ]n−r

.

O bserve that we need to use the Metropolis-Hasting algorithm to generate thevariables, α, θ and σ from the conditional posterior density.

4.3 The additive Weibull model

For the additive Weibull model (9), θ = (α, β, γ, δ), the likelihood function(10), for θ is given by,

L(θ) ∝

r∏

i=1

[

αβ(αti)β−1 + γδ(γti)

δ−1]

exp

{

−αβ

(

r∑

i=1

tβi − (n − r)t

βr

) }

× exp

{

−γδ

(

r∑

i=1

tδi − (n − r)tδr

) }

. (23 )

Assuming independence among the parameters, considerer the following priordensities for α, θ and σ :

α ∼ Γ(c1, d1), with c1 and d1 know,

β ∼ Γ(c2, d2), with c2 and d2 know, (24 )

δ ∼ Γ(c3, d3), with c3 and d3 know,

γ ∼ Γ(c4, d4), with c3 and d3 know.

Combining (23 )-(24 ), the joint posterior density for α, β, δ and γ is given by,

π(α, β, γ, δ|D) ∝ αa1−1βa2−1δc3−1γc4−1 exp

{

−αβ(

r∑

i=1

tβi − (n − r)t

βr )

}

× exp

{

−γδ(

r∑

i=1

tδi − (n − r)tδr) − c1α − c2β − c3δ − c4γ

}

G(α, β, δ, γ), (25 )


where G(α, β, δ, γ) =r∏

i=1

[

αβ(αti)β−1 + γδ(γti)

δ−1]

.

The conditional posterior densities for the Gibbs algorithm are given by,

π(α|β, δ, γ, D) ∝ αa1−1 exp

{

−αβ(

r∑

i=1

tβi − (n − r)t

βr ) − c1α

}

G(α, β, δ, γ)

π(β|α, δ, γ, D) ∝ βa2−1 exp

{

−αβ(

r∑

i=1

tβi − (n − r)t

βr ) − c2β

}

G(α, β, δ, γ)

π(δ|α, β, γ, D) ∝ δc3−1 exp

{

−γδ(

r∑

i=1

tδi − (n − r)tδr) − c3δ

}

G(α, β, δ, γ)(26 )

π(γ|α, β, δ|D) ∝ γc4−1 exp

{

−γδ(

r∑

i=1

tδi − (n − r)tδr) − c4γ

}

G(α, β, δ, γ)

N otice that the conditional densities (26 ) require Metropolis-Hasting steps for theimplementations of MCMC methodology.

5 Model choice

Model determination is a fundamental issue in statistics. The literature onthe model assessment or checking and model selection presents many approaches,beginning with the B ayes factor approach. Several modifi cations of the B ayes factorare presented in the literature (see for example, Aitkin, 1981; B erger and P erichi,1996 ). Geisser and E ddy (1979) took a predictive approach based on cross validationmethods to obtain the pseudo-B ayes factor. This approach is described in somedetail next.

Consider a choice between two parametric models denoted by the joint densityf(t; θi, Mi) or the likelihood function L(θi; t, Mi), i = 1, 2. Suppose that wi is theprior probability of selecting the model Mi i = 1, 2, and f(t|Mi) is the predictivedistribution for the model Mi, that is,

f(t|Mi) =

∫

f(t|θi, Mi)π(θi|Mi)dθ (27)

where π(θi|Mi) is the prior distribution under model Mi. If t0 denotes the observeddata, then we choose the model yielding the larger wif(t0|Mi).

Often we set wi = 0.5, i = 1, 2, and compute the B ayes factor of the M1 withrespect to M2 as

B12 =f(t0|M1)

f(t0|Mi). (28)


The predictive distribution (27) can be approximated by its Monte Carlo estimateusing S generated samples from the prior density π(θi|Mi), that is,

f̂(t|Mi) =1

S

S∑

s=1

f(t|θ(s)i , Mi) (29)

where θ(s)i denotes the estimate for vector θi drawn at the s-th iteration. Some

modifications of the estimate (31) for the predictive density are proposed in theliterature (see, e.g., Newton and R aftery, 1994; Gelfand and D ey, 1994). For themodel selection we also could consider the conditional predictive ordinate (CPO)(see, e.g., Gelfand and D ey, 1994), given by

f(tr|t, Mi) =

∫

f(tr|θi, t(r), Mi)π(θi|t(r), Mi)dθi (30)

where t(r) = (t1, . . . , tr−1, tr+1, . . . , tn).U sing the generated Gibbs samples, equation (30) can be approximated by its

Monte Carlo estimate,

f̂(tr|t(r)Mi) =1

S

S∑

s=1

f(tr|t(r), θ(s)i , Mi) (31)

where θ(s)i denotes the estimate for vector θi drawn at the sth Gibbs sample. We

can use the obtained estimates cr(l) = f̂(tr|t(r)Mi) in model selection. In this way,we consider plots of cr(l) versus r (r = 1, . . . , n) for diff erent models; large values(on average) indicate the better model. An alternative to cr(l) plots is the is to

choose the model for which c(l) =n∏

r=1cr(l) (l index models) is largest. Geisser and

Eddy (1979) suggest that the product of the predictive densitiesn∏

r=1f(tr|t(r), Mi)

could be used in model selection as a relative indicator. If we have two models M1and M2, we have the ratio, which is called the pseudo Bayes factor,

P SBF =

n∏

r=1f(tr|t(r), M1)

n∏

r=1f(tr|t(r), M2)

. (32)

as an approximation to the Bayes factor. U sing samples generated by the MCMCprocess, pseudo Bayes factor (32) can be estimated by using equation (31).

Many other Bayesian criteria has been proposed in the literature, forinstance, the deviance information criterion (D IC) proposed by Spiegelhalter et al.(2002), and expected Akaike information criteria (EAIC) and expected Bayesianinformation criteria (EBIC) proposed in Brooks (2002). The criteria are based inthe posterior mean of the deviance: E (D(θ)) which is also a measure of fit that


can be approximated using samples generated by the MCMC process, consideringthe value of

Dbar =1

B

B∑

b=1

D(θ(b)),

where the index b represent the g-ith realization of a total of B realizations, and

D(θ) = −2lo g (f(t|θ)).

The criteria EAIC, EBIC and DIC can be estimated using MCMC output byconsidering

ÊAI C = Dbar + 2p,

ÊBI C = Dbar + plo g (N),

andD̂I C = Dbar + ρ̂D = 2Dbar − Dh at,

respectively, where p is the number of parameters in the model, N is the totalnumber of observations and ρD, namely the effective number of parameters. whichis defined as

E (D(θ)) − D(E(θ))

where D(E(θ)) is the deviance of posterior mean obtained when considering themean values of the generated posterior means of the model parameters, which isestimated by

Dh at = D

(1

B

B∑

b=1

θ(b)

).

Given the comparison of two alternative models, the model that fits better adata set is the model with smallest value of the DIC, EBIC and EAIC.

6 Some examples

6 .1 A n example w ith real data set

To illustrated the approach developed in the previous sections we consider thedata set presented in Aarset (1987). The data describe lifetimes for 50 industrialdevices put on life test at time zero. The TTT plot, that indicates a bathtub shapedfailure rate function, is presented in Figure 4.

Considering the data in Aarset, we fit the modified Weibull model to thedata set, using subroutine MaxBFGS in Ox (as the maximization approach), and


Table 1 - Lifetimes for the 50 devices

0.1 0.2 1 1 1 1 1 2 3 6 7 11 12 1818 18 18 18 21 32 36 40 45 46 47 50 55 6063 63 67 67 67 67 72 75 79 82 82 83 84 8484 85 85 85 85 85 86 86

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

r/n

TTT

Figure 4 - TTT plot for the 50 observations in Aarset (1987).

obtain the following maximum likelihood estimates (see Table 2): α̂ = 13.746645,

β̂ = 0.58770374 and λ̂ = 0.008759687. Since the estimate of β is smaller than one(with sd=0.06), we have strong indication that the data set presents bathtub shapedfailure rate.

These estimates which are presented in Table 2 are, however, different fromthe ones obtained by X ei et al. (2002) which are given by α̃ = 110.0909, β̃ = 0.8408and λ̃ = 0.0141.. For those values, the equations given in (13) are not satisfied, thatis, the left side of each equation is different from zero, as we show next:

∂ l(θ)

∂ α|(α̃,β̃,λ̃) = −0.0300265319,

∂ l(θ)

∂ β|(α̃,β̃,λ̃) = 0.11264210,


and

∂l(θ)

∂λ|(α̃,β̃,λ̃) = −11.294634.

Notice that specially the third equation tremendously deviates from zero. Hence,the authors (Xie et al., 2002) think they are using the correct estimates and indeedthey have highly incorrect estimates, so that further inference based on it like thelikelihood ratio statistics, for example, may present incorrect values which wouldput all further inference in tremendous jeopardy, leading to dangerously incorrectinferences.

Further analysis of this data set was performed in Mudholkar and Srivastava(1994) using the exponentiated Weibull model. Parameter estimation by themaximum likelihood approach under this model leads to the following estimates:α̂ = 4.69, θ̂ = 0.146 and σ̂ = 91.023. In Table 2, we present the maximumlikelihood estimates of the parameters of the aditive Weibull (9) and modifiedWeibull (2) models. The MAXBFGS subroutine in software Ox is used to computethe maximum likelihood estimates for both models.

Table 2 - Parameters estimates and AIC and HQ values

Model Parameter estimates AIC HQThe exponentiated Weibull α̂= 4.69 4.662275 4.6913

θ̂= 0.146σ̂= 91.023

The additive Weibull α̂= 0.011777 4.20193 4.23105

β̂= 82.343γ̂= 0.016217

δ̂= 0.7025The modified Weibull α̂= 13.746645 4.6929 4.7148

β̂= 0.58770374

λ̂= 0.00875968

Also in the Table 2, we presented the maximum likelihood estimates of AICand HQ for the three considered models. We observe that for the data set in Table1, AIC and HQ criteria indicate that the additive Weibull model is better than both,the exponentiated Weibull (EW) model and new modified Weibull (MW) model.However, the EW model and MW model seem to be equivalent, since the estimatesof the AIC and HQ criteria are similar, that is, the above criteria are not able todistinguish between them.


6.1.1 Bayesian inference

We consider first the modified Weibull model (2), considering the priordensities for α, β and λ given in (16), with α0 = 28, α1 = 2, β0 = 0.001,β1 = 0.001, λ0 = 0.001 and λ1 = 0.001, we generated two parallel independentruns of the Gibbs sampler chain with size 25000, for each parameter discardingthe first 5000 iterations, to eliminate the effect of the initial values and to avoidcorrelation problems, we considered a spacing of size 10, obtaining a sample of size2000 from each chain, we monitored the convergence of the Gibbs samples using theGelman and Rubin (1992) method that uses the analysis of variance technique iffurther iterations are needed. In the Table 3, we report posterior summaries for theparameters, and in Figure 5 we have the approximate marginal posterior densitiesconsidering the 4000 Gibbs samples. We also have in the table 3, the estimatedpotential scale reduction R̂ (see Gelman and Rubin, 1992) which is an index tocheck the convergence of the algorithm. Since R̂ < 1.1 for all parameters, it seemsthat the chains converge.

Table 3 - Posterior summaries the modified Weibull model

Parameters Mean S.D 95% Credible interval R̂α 13.77 2.51 (9.24; 19.07) 1.001β 0.582 0.0602 ( 0.4707 ; 0.7028) 1.001λ 0.00916 0.002608 ( 0.00488 ; 0.01501) 1.002

Considering now the exponentiated Weibull model, assuming the priordensities (20) with a1 = 0.001 b1 = 0.001, a2 = 0.001 b2 = 0.001, a3 = 0.001and b3 = 0.001, we have in the Table 4, the posterior summaries for the parametersof interest and in Figure 6, we have the approximate marginal posterior densities,based on two parallel chains of size 30000 after discarding the first 10000 iterations;to avoid correlation problems in the generated chains, the lag value was taken tobe 10, obtaining a sample of size 4000. We also have in the Table 4, the estimatedpotential scale reductions R̂ for all the parameters, which indicates that chainsconverge.

Table 4 - Posterior summaries for the exponentiated Weibull model

Parameters Mean S.D 95% Credible interval R̂α 2.293 0.3551 ( 1.478 ; 2.912) 1.002θ 0.3147 0.08137 ( 0.2034 ; 0.5159) 0.999σ 84.10 11.33 ( 63.96 ; 108.30) 1.003


α

Dens

ity

10 15 20

0.00

0.10

β

Dens

ity

0.4 0.5 0.6 0.7 0.8

02

46

λ

Dens

ity

0.005 0.015

050

100

150

Figure 5 - Approximate marginal posterior density for α, β and λ of modifiedWeibull model.

Finally, considering the additive Weibull model (9), assuming the priordensities (24) with c1 = 0.001, d1 = 0.001, c2 = 0.001, d2 = 0.001, c3 = 0.001,d3 = 0.001, c4 = 0.001, and d4 = 0.001, we have in the Table 5, the posteriorsummaries for the parameters of interest and in Figure 7 we have the approximatemarginal posterior densities, based on two parallel chains of size 30000 afterdiscarding the first 10000 iterations, to avoid correlation problems in the generatedchains, the lag value was taken to be 10, obtaining a sample of size 4000. We alsoobserve approximate convergence since, the estimated potential scale reductions R̂are close one for all parameters.

Table 5 - Posterior summaries the additive Weibull model

Parameters Mean S.D 95% Credible interval R̂α 0.01178 6.2685E-5 (0.01168 ; 0.01191) 1.000β 74.41 22.68 ( 36.02 ; 125.7) 0.9997γ 0.01614 0.004028 ( 0.009062 ; 0.0248) 1.002δ 0.6824 0.1051 ( 0.4928 ; 0.9013) 1.003


α

Dens

ity

0.5 1.5 2.5 3.5

0.00.4

0.8

θ

Dens

ity

0.2 0.4 0.6 0.8 1.0

01

23

4

σ

Dens

ity

40 60 80 120

0.00

0.02

Figure 6 - Approximate marginal posterior density for α, θ and σ of exponentiated-Weibull model.

Further, we have in Table 6, estimates for EAIC, EBIC and DIC based onthe simulated Gibbs samples. We observe that additive Weibull model is the bestmodel based on the tree criteria.

Table 6 - Comparison between EW, MW and AW

Model EAIC EBIC DICmodified Weibull 489.04 471.30 467.40additive Weibull 442.04 424.3 420.00exponentiated Weibull 499.55 475.92 471.71

From the generated Gibbs sampling, it can be show that the pseudo Bayesfactor of the modified Weibull (M1) with respect exponentiated weibull model (M2)is given by PSFB12 = 28.69802, indicating strong to the modified Weibull model.The pseudo Bayes factor of the modified Weibull (M1) with respect additive weibullmodel (M3) is given by PSFB13 = 1.277583×10

−10 and the the pseudo Bayes factorM3 with respect M1 is given PSFB31 = 4.451814× 10

−12, indicating strong to theadditive Weibull model.


α

Dens

ity

0.0116 0.0120 0.0124

020

0060

00

β

Dens

ity

0 50 100 150

0.000

0.010

α

Dens

ity

0.005 0.015 0.025 0.035

020

6010

0

δDe

nsity

0.4 0.6 0.8 1.0

01

23

4Figure 7 - Approximate marginal posterior density for α, β, γ and δ of additive

Weibull model.

In order to asses if the models considered are appropriated, in Figure 8 we plotthe empirical survival jointly with the fitted survival functions of modified Weibullmodel (M1), exponentiated-Weibull model (M2) and additive Weibull model (M3).The Figure 8 show that the M1 and M2 give similar fits for the survival curves.But, the model M3 is better fit to the of data.

The fitted hazard function of exponentiated-Weibull (EW), modified Weibull(MW) and additive Weibull (AW) distributions are shown in Figure 9. Notice thatall the estimates of hazard function have form of bathtub.

6.2 An example with censored data

In Table 7 we have a generated data set considering a exponentiated-Weibulldistribution (1) with α = 2, θ = 0.2 and σ = 100 and censored type II.

Considering the data in Table 7, we fits first the exponentiated Weibull model,assuming the prior densities (20) with a1 = 0.01 b1 = 0.01, a2 = 0.01 b2 = 0.001,a3 = 0.01 and b3 = 0.01, we have in the Table 8, the posterior summaries for theparameters of interest, based on two parallel chains of size 35000 after discardingthe first 5000 iterations; to avoid correlation problems in the generated chains, thelag value was taken to be 10, obtaining a sample of size 6000. We also have in theTable 8, the estimated potential scale reductions R̂ for all the parameters, which


0 20 40 60 80

0.00.2

0.40.6

0.81.0

t

Survi

val

Kaplan MeierEWAWMW

Figure 8 - Estimated of survival functions of M1, M2 and M3 and empirical survival.

Table 7 - Generated data with α = 2, θ = 0.2 and σ = 100.0.69 2.18 2.51 2.77 3.09 3.77 8.15 8.688.70 9.23 12.69 14.35 18.73 18.88 19.14 20.3124.27 25.03 26.78 34.14 37.24 48.17 55.75 57.1758.44 58.84 67.19 73.52 74.90 75.00 80.06 82.6087.46 92.79 94.34 101.59 101.95 102.71 104.98 115.18125.80 127.90 134.74 143.80 144.73 144.73+ 144.73+ 144.73+

144.73+ 144.73+

indicates that chains converge.

Considering now the modified Weibull model (2), considering the priordensities for α, β and λ given in (16), with α0 = 0.1, α1 = 0.1, β0 = 0.01,β1 = 0.01, λ0 = 0.1 and λ1 = 0.01, we generated two parallel independent runsof the Gibbs sampler chain with size 25000, for each parameter discarding the first5000 iterations, to eliminate the effect of the initial values and to avoid correlationproblems, we considered a spacing of size 10, obtaining a sample of size 2000from each chain, we monitored the convergence of the Gibbs samples using theGelman and Rubin (1992) method that uses the analysis of variance technique iffurther iterations are needed. In the Table 9, we report posterior summaries for the


0 20 40 60 80

0.00

0.05

0.10

0.15

t

h(t)

EWMWAW

Figure 9 - Estimated of hazard function of EW, MW, and AW models.

Table 8 - Posterior summaries for the exponentiated Weibull model

Parameters Mean S.D 95% Credible interval R̂α 1.451 0.341 ( 0.945 ; 2.266) 1.001θ 0.586 0.1391 ( 0.328 ; 0.866) 1.008σ 102.112 9.667 ( 85.031 ; 121.814) 1.002

parameters considering the 4000 Gibbs samples. We also have in the Table 3, theestimated potential scale reduction R̂ (see Gelman and Rubin, 1992) which is anindex to check the convergence of the algorithm. Since R̂ < 1.1 for all parameters,it seems that the chains converge.

Finally, considering the additive Weibull model (9), assuming the priordensities (24) with c1 = 0.001, d1 = 0.001, c2 = 0.001, d2 = 0.001, c3 = 0.001,d3 = 0.001, c4 = 0.001, and d4 = 0.001, we have in the Table 10, based on twoparallel chains of size 35000 after discarding the first 5000 iterations, to avoidcorrelation problems in the generated chains, the lag value was taken to be 10,obtaining a sample of size 6000. We also observe approximate convergence since,the estimated potential scale reductions R̂ are close one for all parameters.

We have in Table 11, estimates for EAIC, EBIC and DIC based on thesimulated Gibbs Samples. We observe that additive Weibull model is the best


Table 9 - Posterior summaries the modified Weibull modelParameters Mean S.D 95% Credible interval R̂α 75.2131 8.405 (59.4833; 92.2719) 1.007β 0.7883 0.1005 ( 0.6111 ; 0.9974) 1.000λ 0.0094 0.0014 (0.0069 ; 0.0123) 1.004

Table 10 - Posterior summaries the additive Weibull modelParameters Mean S.D 95% Credible interval R̂α 0.0053 0.0019 (0.0005; 0.0074) 1.000β 9.1169 3.2927 (4.0238 ; 16.2405) 1.007γ 0.0129 0.0028 ( 0.0077 ; 0.0187) 1.002δ 0.8574 0.1361 ( 0.5852 ;1.1312) 1.009

model based on the tree criteria.

Table 11 - Comparison between EW, MW and AWModel EAIC EBIC DICmodified Weibull 504.738 510.4741 501.416additive Weibull 482.4573 494.0174 478.6807exponentiated Weibull 480.9784 486.7145 478.9594

Further, it can be show that the pseudo Bayes factor of the modifiedWeibull (M1) with respect exponentiated weibull model (M2) is given byPSFB12 = 1.770107×10

−6, indicating strong to the exponentiated Weibull model.The pseudo Bayes factor of the modified Weibull (M1) with respect additive weibullmodel (M3) is given by PSFB13 = 1.065487 × 10

−6, indicating strong to theadditive Weibull model. The the pseudo Bayes factor M3 with respect M3 is givenPSFB13 = 1.661312, indicating the additive Weibull and exponentiated Weibullmodels are similarity.

In the Figure 10, shows the estimated of Monte Carlo of hazard function forexponentiated-Weibull (EW), modified Weibull (MW) and additive Weibull (AW)distributions including the true hazard rate function. Notice that the EW and AWgive similar fits for the survival curves.

Considering the exponentiated Weibull (EW), modified Weibull (MW) andadditive Weibull (AW) models we conducted a sensitivity analysis, considered avaried the value of hiperparameters. Table 12 shows that the proposed models arerobust for a wide range of noninformative priors.


0 50 100 150

0.00

0.02

0.04

0.06

0.08

0.10

t

h(t)

TRUEEWAWMW

Figure 10 - Estimated of hazard function of EW, MW, and AW models.

7 Concluding remarks

The paper discusses the use of Markov chain Monte Carlo methods as areasonable way to get Bayesian inference for analysis of lifetime data with bathtubshaped failure rate. Besides, in this paper is shown that the maximum likelihoodestimates obtained are different from the ones obtained in Xie et al. (2002), whichseem to be incorrect since they do not verify the likelihood equations. Therefore,further inference based on those estimates are in jeopardy. The Bayesian analysis ofa real data set seems to indicate that the modified Weibull distribution presents thebest fit than that of the exponentiated Weibull distribution. However, the additiveWeibull model seems to fit better the that set than the other two models.

*

A cknow ledgments

The authors thank the editorial board and a referee for comments andsuggestions on this work.


Table 12 - Sensitivity analysis with different hyperparameters on the prior of EW,MW and AW models for the simulated data.

Hiperparameters Parameters Mean S.D 95% IC

E W a1 = 0.01 , b1 = 0.001 α 1 .4 7 1 0.4 5 2 (0.9 6 7 ; 3 .1 03 )a2 = 0.01 , b2 = 0.001 θ 0.4 9 6 0.1 2 3 ( 0.2 8 1 ; 0.8 9 4 )a3 = 0.01 , b3 = 0.001 σ 1 01 .5 6 4 1 4 .1 3 1 (9 2 .3 8 7 ;1 2 4 .09 2 )

a1 = 0.001 , b1 = 0.0001 α 1 .5 1 2 0.6 7 5 (0.9 8 9 ; 3 .6 3 1 )a2 = 0.001 , b2 = 0.0001 θ 0.4 9 8 0.3 7 1 ( 0.3 1 8 ; 0.9 1 5 )a3 = 0.001 , b3 = 0.0001 σ 1 03 .5 6 4 1 4 .1 3 1 (9 0.3 8 7 ;1 3 3 .5 6 3 )

M W α0 = 0.01 , α1 = 0.001 α 7 4 .8 9 2 1 0.1 3 5 (5 8 .3 8 3 ; 9 4 .03 4 )β0 = 0.01 , β1 = 0.001 β 0.8 01 0.1 1 9 ( 0.5 4 9 ; 0.9 8 9 )λ0 = 0.01 , λ1 = 0.001 λ 0.008 9 0.001 8 (0.007 4 ; 0.02 01 )

α0 = 0.001 , α1 = 0.0001 α 7 5 .08 7 1 1 .02 3 (5 6 .9 7 5 ; 9 6 .08 7 )β0 = 0.001 , β1 = 0.0001 θ 0.7 8 5 0.2 1 9 ( 0.6 02 ; 0.9 9 9 )λ0 = 0.001 , λ1 = 0.0001 λ 0.009 1 0.002 5 (0.006 5 ;0.02 7 4 )

A W c1 = 0.01 , d1 = 0.001 α 0.006 1 0.002 8 (0.0004 ; 0.008 8 )c2 = 0.01 , d2 = 0.001 β 9 .6 9 1 3 3 .8 7 1 3 (4 .2 7 8 3 ; 1 7 .01 5 6 )c3 = 0.01 , d3 = 0.001 γ 0.01 3 1 0.005 5 ( 0.006 5 ; 0.01 9 7 )c4 = 0.01 , d4 = 0.001 δ 0.8 6 1 2 0.1 6 4 5 ( 0.5 9 1 3 ;2 .1 09 2 )

c1 = 0.001 , d1 = 0.0001 α 0.001 1 0.003 2 (0.0009 ; 0.001 01 )c2 = 0.001 , d2 = 0.0001 β 8 .9 8 1 4 3 .9 8 1 2 ( 5 .1 4 2 9 ; 1 9 .08 2 9 )c3 = 0.001 , d3 = 0.0001 γ 0.01 4 1 0.006 7 (0.004 4 ; 0.02 1 3 )c4 = 0.001 , d4 = 0.0001 δ 0.9 1 02 0.1 8 2 3 (0.3 8 7 4 ; 2 .8 2 1 9 )

CANCHO, V. G.; ORTEGA, E. M. M.; BARRIGA, G. D. C. Comparação demodelos W eib u lls modifi cados com tax a de falh a em forma de ” b an h eira” :u ma ab ordag em Bay esian a. Rev. Mat. Estat., S ão P au lo, v .2 5 , n .2 , p.1 1 1 -1 3 6 ,2 0 0 7 . Rev. Mat. Estat. (S ão P au lo), v . 2 5 , n .2 , p. 1 1 1 -1 3 6 , 2 0 0 7 .

R E S U M O : E m a p lic a ç õ e s d e a n a lise d e so b re v iv ê n c ia , m o d e lo s c o m fu n ç ã o d e risc o e m

fo rm a d e b a n h e ira sã o m u ito s im p o rta n te s. N ã o e n ta n to , o m o d e lo tra d ic io n a l W e ib u ll

n ã o m o d e la d a d o s c o m ta x a d e fa lh a e m fo rm a d e b a n h e ira . A lg u m a s e x te n sõ e s d a

d istrib u i̧c ã o W e ib u ll te m fu n ç ã o risc o e m fo rm a d e b a n h e ira . N e ste a rtig o , c o n sid e ra -se

a lg u m a s d e ssa s d istrib u i̧c õ e s e d e se n v o lv e -se u m a m e to d o lo g ia B a y e sia n a , p a ra a n á lise d e

d a d o s te m p o s d e v id a c o m ta x a d e fa lh a e m fo rm a d e b a n h e ira . A m e to d o lo g ia B a y e sia n a

é b a se a d a e m m é to d o s d e M o n te C a rlo v ia C a d e ia d e M a rk o v (M C M C ). T a m b é m ,

a p re se n ta -se a lg u n s c rité rio s B a y e sia n o s p a ra se le ç ã o d o s m o d e lo s c o n sid e ra d o s. A

m e to d o lo g ia é ilu stra d a c o m u m c o n ju n to d e d a d o s re a is in tro d u z id a s p o r A a rse t (1 9 8 7 )

e u m c o n ju n to d e d a d o s sim u la d o s.

P A L A V R A S -C H A V E : A lg o ritm o s M C M C ; in fe rê n c ia B a y e sia n a ; d a d o s d e te m p o s d e

v id a ; fu n ç ã o risc o e m fo rm a d e b a n h e ira .


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PRENTICE, R. L . Discrimination among some parametric models. B io m entrics,Washimgton, v.61, p.539 -54 4 , 19 75.

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SPIEGELHATER, D. J.; BEST, N. G.; CARLIN, B. P.; VAN DER LINDE, A.Bayesian measure of model complexity and fit (with discusssion), J. R. Stat. Soc.,London, B, v.64, p.583-639, 2002.

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Recebido em 08.05.2007.

Aprovado após revisão em 18.10.2007.


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