Bayesian estimation and prediction for Weibull model ...

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Bayesian estimation and prediction for Weibull model with progressive censoring a

Syuan-Rong Huang & Shuo-Jye Wu

b

a

Graduate Institute of Management Sciences, Tamkang University, Tamsui, New Taipei City, Taiwan, 25137, ROC b

Department of Statistics, Tamkang University, Tamsui, New Taipei City, Taiwan, 25137, ROC Version of record first published: 27 Jul 2011.

To cite this article: Syuan-Rong Huang & Shuo-Jye Wu (2012): Bayesian estimation and prediction for Weibull model with progressive censoring, Journal of Statistical Computation and Simulation, 82:11, 1607-1620 To link to this article: http://dx.doi.org/10.1080/00949655.2011.588602

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Journal of Statistical Computation and Simulation Vol. 82, No. 11, November 2012, 1607–1620

Bayesian estimation and prediction for Weibull model with progressive censoring

Downloaded by [Tamkang University] at 23:43 08 November 2012

Syuan-Rong Huanga * and Shuo-Jye Wub a Graduate Institute of Management Sciences, Tamkang University, Tamsui, New Taipei City, Taiwan 25137, ROC; b Department of Statistics, Tamkang University, Tamsui, New Taipei City, Taiwan 25137, ROC

(Received 10 March 2010; final version received 12 May 2011) This article presents the statistical inferences on Weibull parameters with the data that are progressively type II censored. The maximum likelihood estimators are derived. For incorporation of previous information with current data, the Bayesian approach is considered. We obtain the Bayes estimators under squared error loss with a bivariate prior distribution, and derive the credible intervals for the parameters of Weibull distribution. Also, the Bayes prediction intervals for future observations are obtained in the one- and two-sample cases. The method is shown to be practical, although a computer program is required for its implementation.A numerical example is presented for illustration and some simulation study are performed. Keywords: Bayes estimator; credible interval; Lindley’s approximation; maximum likelihood estimation; progressive type II censoring

1.

Introduction

The Weibull distribution is useful in modelling the lifetime of a product, such as ball bearings, automobile components and electrical insulation, and is also used widely in biological and medical application. The probability density function (pdf) and cumulative distribution function (cdf) of the Weibull distribution, respectively, are given by f (x|ν, β) = and

ν β

 ν−1   ν  x x , exp − β β

  ν  x F(x|ν, β) = 1 − exp − , β

x>0

x > 0,

(1)

(2)

where ν > 0 is the shape parameter and β > 0 is the scale parameter. For ν < 1(> 1), the hazard function of the Weibull distribution is decreasing (increasing), and it would reduce to a constant as *Corresponding author. Email: [email protected]

ISSN 0094-9655 print/ISSN 1563-5163 online © 2012 Taylor & Francis http://dx.doi.org/10.1080/00949655.2011.588602 http://www.tandfonline.com

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S.-R. Huang and S.-J. Wu

ν = 1. Let θ = 1/β ν . We can rewrite the pdf and cdf in Equations (1) and (2) to be, respectively, f (x|θ, ν) = θ νx ν−1 exp(−θ x ν ),

x>0

(3)

and

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F(x|θ , ν) = 1 − exp(−θ x ν ),

x > 0.

(4)

In this study, we consider a censoring scheme called progressive type II censoring. This scheme is a generalization of type II censoring. The progressive type II censoring allows for units to be removed form the life test at the point other than the final termination point, but this allowance is not permitted in a traditional type II censoring. Under a progressive type II censoring scheme, n units are placed on test at time zero, and m failures are going to be observed. When the first failure time x1 is observed, r1 of the surviving units are randomly selected and removed. At the second observed failure time x2 , r2 of the surviving units are randomly selected and removed. This experiment terminates at the time when the mth failure xm is observed and the remaining rm = n − r1 − r2 − · · · − rm−1 − m surviving units are all removed. The m ordered observed failure times x = (x1 , x2 , . . . , xm ) are called progressively type II censored sample of size m from a sample of size n with censoring scheme r = (r1 , r2 , . . . , rm ). Extensive publications can be found in the literature which discuss the statistical inference for progressively censored data under various lifetime distributions. Some of them are [1–18].A recent account on progressive censoring can be found in the book by Balakrishnan and Aggarwala [19] or in the review article by Balakrishnan [20]. In this article, our main purpose is to study the Bayes estimation for the parameters of Weibull distribution under the assumption of a bivariate prior. We also discuss the Bayes prediction problems for future observations. The rest of this article is organized as follows. In Section 2, the maximum likelihood estimators (MLE) of the parameters are introduced, and the Bayes estimators under squared error loss are also derived. In Section 3, the Bayes prediction problems are discussed. In Section 4, the proposed methods are applied to a numerical example for illustration. In Section 5, some simulation studies are carried out to investigate the performance of the proposed estimation and prediction methods. Some conclusions are made in Section 6.

2.

Parameter estimations

In this section, we are going to derive the MLEs of the parameters from a progressively type II censored Weibull sample. We will also obtain the Bayes estimators under squared error loss and the credible intervals for the parameters of Weibull distribution. 2.1.

Maximum likelihood estimation

Suppose that the lifetimes of the units being tested have a Weibull distribution. The pdf is given in Equation (3). Let x = (x1 , . . . , xm ) be a progressively type II censored sample from a Weibull distribution, with censoring scheme r = (r1 , . . . , rm ). Based on the observed data (x, r), the likelihood function is l(θ , ν|x) = k

m 

f (xi )[1 − F(xi )]ri

i=1

= k(θ ν)

 m

m  i=1

ν−1 xi

exp −θ

m  i=1

(ri +

1)xiν

,

(5)

Journal of Statistical Computation and Simulation

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where k = n(n − 1 − r1 )(n − 2 − r1 − r2 ) · · · (n − m − r1 − · · · − rm−1 ). The log-likelihood function may then be written as

m m   L(θ, ν|x) = ln l(θ , ν|x) = ln k + m ln(θ ν) + (ν − 1) ln xi − θ (ri + 1)xiν . (6) i=1

i=1

Taking derivatives with respect to θ and ν of (6) and equating them to zero, we obtain the likelihood equations for θ and ν to be ∂L(θ, ν|x) m  = − (ri + 1)xiν = 0, ∂θ θ i=1

(7)

 ∂L(θ, ν|x) m  = + ln xi − θ (ri + 1)xiν ln xi = 0. ∂ν ν i=1 i=1

(8)

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m

and

m

m

Equation (7) yields the MLE of θ to be θˆ =



+ 1)xiνˆ m

m i=1 (ri

−1 .

(9)

Substituting Equation (9) into Equation (8), the MLE of ν can be obtained by solving the nonlinear equation  m m  (ri + 1)xiνˆ ln xi m m ln xi − = 0. (10) + i=1 m νˆ νˆ i=1 (ri + 1)xi i=1 Because Equation (10) cannot be explicitly solved, a numerical method such as Newton–Raphson iteration must be used to obtain the MLE νˆ . The Newton–Raphson algorithm needs the second derivatives of the log-likelihood. Sometimes the computations of the derivatives are complicated. Balakrishnan and Kateri [21] proposed a graphical method as an alternative which does not require the second differentiations. Kernane and Raizah [22] used the fixed point approach to solve the MLE. They also proved that the graphical method reduce in fact to the fixed point solution. The fixed point iteration is easy to implement and does not require the derivation of a given function. Its convergence speed is also faster than the Newton–Raphson iteration. The asymptotic normal distribution for the MLEs can be derived in the usual way. From the log-likelihood function in (6), we have ∂ 2 L(θ , ν|x) m = − 2, ∂θ 2 θ m  ∂ 2 L(θ , ν|x) (ri + 1)xiν ln xi =− ∂θ ∂ν i=1 and

 ∂ 2 L(θ , ν|x) m = − − θ (ri + 1)xiν ln xi . ∂ν 2 ν2 i=1

(11) (12)

m

(13)

The Fisher’s information matrix I(θ , ν) is then obtained by taking expectations of minus equaˆ νˆ ) is approximately bivariate tions (11)–(13). Under some mild regularity conditions, the MLEs (θ, normal with mean (θ , ν) and covariance matrix I −1 (θ , ν).

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S.-R. Huang and S.-J. Wu

In practice, a simpler and equally valid procedure is to use the approximation ˆ ˆ )), ˆ νˆ ) ∼ N((θ , ν), I −1 (θ, 0 (θ , ν where I 0 (θˆ , νˆ ) is the observed information matrix ⎡ 2 ∂ L(θ , ν|x) ⎢− ∂θ 2 I 0 (θˆ , νˆ ) = ⎢ ⎣ ∂ 2 L(θ , ν|x) − ∂θ ∂ν

⎤ ∂ 2 L(θ , ν|x) ∂θ∂ν ⎥ ⎥ . 2 ∂ L(θ , ν|x) ⎦ − ∂ν 2 (θˆ ,ˆν )

−

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2.2. Bayes estimators In this subsection, we investigate the Bayes estimators for parameters θ and ν. We need to assume the prior distribution of the unknown parameters for the Bayesian inference. As mentioned by Arnold and Press [23], from a Bayesian viewpoint, there is clearly no way in which one can say that one prior is better than any other. We will make some discussions about the selection of prior distributions in Section 2.4. Here, following the suggestion of [24–26], we assume that θ and ν have a joint prior of the form π(θ , ν) =

d c ba θ c+a−1 ν a−1 exp {−θ(d + bν)} , (c)(a)

(14)

where a, b, c and d are all positive real numbers. It follows, from Equations (5) and (14), that the joint posterior distribution of θ and ν is given by

 m ν−1 c+a+m−1 a+m−1  ( m θ ν i=1 xi ) ν π(θ, ν|x) = , (15) exp −θ d + bν + (ri + 1)xi (c + a + m)(a, c, x) i=1 where

 (a, c, x) = 0

∞

 ν a+m−1 ( m xi )ν−1 m i=1 dν. [d + bν + i=1 (ri + 1)xiν ]c+a+m

By integrating out θ and ν, respectively, the marginal posterior distributions of ν and θ are  ν−1 a+m−1 ( m ν i=1 xi ) m π(ν|x) = (16) (a, c, x)[d + bν + i=1 (ri + 1)xiν ]c+a+m and π(θ |x) =

θ c+a+m−1

 ∞ m  ν ν−1 a+m−1 ν exp{−θ [d + bν + m i=1 (ri + 1)xi ]} dν 0 ( i=1 xi ) . (c + a + m)(a, c, x)

(17)

Under squared error loss, the Bayes estimators of ν and θ are the expectations of Equations (16) and (17), respectively. That is, νˆ B = E(ν|x) =

(a + 1, c − 1, x) (a, c, x)

and

(18)

(a, c + 1, x) . (19) (a, c, x) Because the Bayes estimators involve with the complicated integral function (a, c, x), we consider using the Lindley’s approximation to calculate the approximate Bayes estimators. θˆB = E(θ |x) = (c + a + m)

Journal of Statistical Computation and Simulation

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Lindley [27] proposed an approximation procedure to evaluate the ratio of two integrals such that ∞∞ 0 u(θ, ν)L(θ , ν|x)π(θ , ν) dθ dν ∞ E(u(θ , ν)|x) = 0  ∞ . 0 0 L(θ , ν|x)π(θ , ν) dθ dν This approximation procedure has been used by several authors to obtain the approximate Bayes estimators for various distributions, such as [28–31]. In a two-parameter case, using notation u(λ) = (λ1 , λ2 ), the posterior mean can be approximated by E(u(λ)|x) ≈ u(λˆ ) + 21 (A + l30 B12 + l03 B21 + l21 C12 + l12 C21 ) + p1 A12 + p2 A21 ,

(20)

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where A=

2 2  

∂ i+j L

lij =

wij τij ,

pi =

∂p , ∂λi

wi =

∂u(λ) , ∂λi

Aij = wi τii + wj τji ,

,

j

∂λi1 ∂λ2

i=1 j=1

wij =

i, j = 0, 1, 2, 3,

∂ 2 u(λ) , ∂λi ∂λj

i + j = 3,

p = ln π(λ1 , λ2 ),

Bij = (wi τii + wj τij )τii ,

and

Cij = 3wi τii τij + wj (τii τjj + 2τij2 ).

Here τij is the (i, j)th entry of the inverse of the observed information matrix and L is the loglikelihood function of the observed data. All the quantities of unknown (λ1 , λ2 ) in Equation (20) are evaluated at the MLEs (λˆ 1 , λˆ 2 ). Let u(λ) = (θ, ν). One has p1 =

c+a−1 − d, θˆ

l30 =

2m , θˆ 3

l12 = −

l03

m 

a−1 − θˆ b, νˆ m  2m = 3 − θˆ (ri + 1)xiνˆ (ln xi )3 , νˆ i=1 p2 =

(ri + 1)xiνˆ (ln xi )2 ,

l21 = 0,

i=1

τ11 =

I22 , 2 I11 I22 − I12

τ22 =

I11 , 2 I11 I22 − I12

and

τ12 = τ21 =

−I12 , 2 I11 I22 − I12

where Iij is the (i, j)th entry of the observed information matrix. When u(λ) = θ, we have w1 = 1, 2 , B21 = τ21 τ22 , w2 = 0 and wij = 0, i, j = 1, 2. Therefore, A = 0, A12 = τ11 , A21 = τ12 , B12 = τ11 2 C12 = 3τ11 τ12 and C21 = τ22 τ11 + 2τ21 . When u(λ) = ν, we have w1 = 0, w2 = 1 and wij = 0, 2 2 i, j = 1, 2. Hence, A = 0, A12 = τ21 , A21 = τ22 , B12 = τ12 τ11 , B21 = τ22 , C12 = τ11 τ22 + 2τ12 and C21 = 3τ22 τ21 . Based on Lindley’s approximation, the approximate Bayes estimators of θ and ν are, respectively,

  m  1 2m 2 2m θˆL = θˆ + τ + − θˆ (ri + 1)xiνˆ (ln xi )3 τ21 τ22 2 θˆ 3 11 νˆ 3 i=1   m  2 − (ri + 1)xiνˆ (ln xi )2 τ22 τ11 + 2τ21  +

i=1

   c+a−1 a−1 − d τ11 + − θˆ b τ12 νˆ θˆ

(21)

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and

S.-R. Huang and S.-J. Wu



 m  1 2m 2m νˆ 3 2 ˆ νˆ L = νˆ + τ12 τ11 + −θ (ri + 1)xi (ln xi ) τ22 2 θˆ 3 νˆ 3 i=1   m  νˆ 2 − (ri + 1)xi (ln xi ) (3τ22 τ21 ) 

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+

2.3.

i=1

   a−1 c+a−1 − d τ21 + − θˆ b τ22 . νˆ θˆ

(22)

Credible intervals

A 100(1 − γ )% Bayesian credible interval for a function of the parameters g(ν, θ) is any interval (Lg , Ug ) satisfying  Ug π(g(ν, θ )|x) dg(ν, θ) = 1 − γ . Lg

The two-sided interval (Lg , Ug ) can be chosen in different ways. Generally, the limits are chosen so that the tail areas are each γ /2. From Equation (16) and after some algebraic computation, the 100(1 − γ )% credible interval (Lν , Uν ) for ν is given by the solutions of the two equations: I (a, c, x, Lν ) γ I (a, c, x, Uν ) γ = and =1− , (a, c, x) 2 (a, c, x) 2  w m  m where I (a, c, x, w) = 0 ( i=1 xi )ν−1 ν a+m−1 /[d + bν + i=1 (ri + 1)xiν ]c+a+m dν. Similarly, the 100(1 − γ )% credible interval (Lθ , Uθ ) for θ satisfies the following equations:  ∞ m 1 ( i=1 xi )ν−1 ν a+m−1 I (ξ1 , c + a + m) γ  dν = ν c+a+m (a, c, x) 0 [d + bν + m (r + 1)x ] 2 i i=1 i and



∞

m

ν γ I (ξ2 , c + a + m) m dν = 1 − , ν c+a+m [d + bν + i=1 (ri + 1)xi ] 2 0  ξ c+a+m−1 −z  where I (ξ , c + a + m) = (1/(c + a + m)) 0 z e dz, ξ1 = Lθ [d + bν + m i=1 (ri + m 1)xiν ], and ξ2 = Uθ [d + bν + i=1 (ri + 1)xiν ]. 1 (a, c, x)

(

i=1 xi )

ν−1 a+m−1

2.4. Discussion of selection of prior distributions The prior distribution summarizes a priori subjective beliefs about the parameters and plays a very important role in Bayesian statistics. Arnold and Press [23] indicated that there is clearly no way in which one can say that one prior is better than any other. They also mentioned that it is more frequently the case that we elect to restrict attention to a given flexible family of prior distributions and we choose one from that family which seems to best match our personal beliefs. For the Weibull distribution discussed in this paper, both parameters ν and θ are unknown. One needs to assume that prior information on ν and θ can be adequately represented by a joint distribution π(θ, ν) for a Bayesian analysis. It is known that when both Weibull parameters are unknown, the continuous conjugate priors do not exist (see, e.g. [32]). Thus, in the literature, efforts have been devoted in finding a suitable prior for (θ , ν). An important consideration is the

Journal of Statistical Computation and Simulation

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dependence between θ and ν. For example, Kundu [14] assumed that θ and ν are independent, but Ahmadi and Doostparast [24] proposed a bivariate prior density in which θ and ν are dependent. For the case of independent priors, one usually selects the marginal univariate distributions of parameters and then assumes that they are independent. In [14], it is assumed that the prior distribution of θ is Gamma distribution with a pdf

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π(θ ) =

d c c−1 −dθ θ e , (c)

θ > 0,

(23)

where c and d are the hyper-parameters. No specific form of the prior distribution of ν is assumed; however, it is assumed that the probability density function of ν is log-concave and it is independent of θ. The approximation of Lindley was used to calculate the Bayes estimates and the Gibbs sampling procedure was used to construct the credible intervals. As mentioned in [33], all full conditional distributions of the posterior are needed to be ‘simple’ for operating Gibbs sampling efficiently. To use the Gibbs sampling, Kundu [14] assumed that the prior of ν is log-concave and required that the conditional density of ν, given the data, is log-concave for further development. Note that univariate priors cannot model statistical dependency between parameters ν and θ. However, such a dependency can be described by a bivariate prior distribution. Suppose that the prior knowledge about the parameters ν and θ is adequately represented by the conditional prior distribution, suggested by Jaheen and Al Harbi [34], with some modification, which is given by π(θ , ν) = π(θ)π(ν|θ ), where π(θ ) is a Gamma density function defined in Equation (23), and π(ν|θ ) is a Gamma density function π(ν|θ ) =

(θ b)a a−1 −(θ b)ν ν e , (a)

ν > 0.

Thus, we can obtain a bivariate prior distribution for ν and θ which was given in Equation (14) and was also studied by Ahmadi and Doostparast [24]. Under this bivariate prior distribution, we can find that the marginal prior distribution of ν is π(ν) =

(c + a) c a ν a−1 , d b (d + bν)c+a (c)(a)

ν > 0.

The second derivative of log-prior is d2 (c + a)b2 ν 2 − (a − 1)(d + bν)2 log π(ν) = . 2 dν ν 2 (d + bν)2 It can be seen that the prior distribution of ν is not a log-concave function for all possible values of hyper-parameters a, b, c and d. In addition, it is also hard to find the values of a, b, c and d such that π(ν) is log-concave. In addition, from the posterior distribution of ν in Equation (16), we can find that the second derivative of log π(ν|x) is d2 a+m−1 log π(ν|x) = − dν 2 ν2  m ν xi )2 xiν ][d + bν + m [ i=1 (ri + 1)(ln  i=1 (ri + 1)xi ] m 2 ν 2 −[b + i=1 (ri + 1)(ln xi ) xi ]  − . ν 2 (c + a + m)−1 [d + bν + m i=1 (ri + 1)xi ] It is easy to see that, as b → 0, we have (d2 /dν 2 ) log π(ν|x) < 0. That is, π(ν|x) is log-concave when b is small. However, as b → ∞, one has (d2 /dν 2 ) log π(ν|x) > 0. Thus, π(ν|x) is not log-concave when b is large. From the above discussion, we see that the requirement of logconcave of prior and posterior cannot be fulfilled in our study. Therefore, the approach proposed by Kundu [14] cannot be applied completely.

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3.

S.-R. Huang and S.-J. Wu

Bayes prediction interval

Predicting the future observations on the basis of the known information is an important topic in statistics. Bayesian approach is useful in predicting the future observations by using the predictive distribution. In the following two subsections, we will investigate the one- and two-sample prediction intervals in the Bayesian framework.

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3.1. One-sample prediction In a progressively type II censoring scheme, the life test is conducted till the mth failure occurs and the remaining rm = n − m − r1 − · · · − rm−1 surviving units are removed. In practice, one might be interested to know the lifetimes of the rm removed surviving units. Let Ys = Xm+s , s = 1, 2, . . . , rm , represent the failure times of the remaining units. From [35], the conditional pdf of Ys given X = x can be obtained as f (ys |x, θ, ν) = κ[R(xm ) − R(ys )]s−1 [R(ys )]rm −s [R(xm )]−rm f (ys )  s−1   s−1 =κ (−1)j exp{−θ(rm − s + j + 1)(ysν − xmν )}νθ ysν−1 , j j=0

(24)

where R(·) = 1 − F(·|θ , ν) is the  survival function of the Weibull distribution and κ is a normalizing constant satisfying f( ys |x, θ , ν) dys = 1. By forming the product of Equations (24) and (15), and integrating out over the set {(ν, θ); 0 < ν < ∞, 0 < θ < ∞}, the one-sample predictive distribution is obtained as  ∞ ∞ f (ys |x) = f (ys |x, θ , ν)π(θ, ν|x) dθ dν 0

0

 s−1  κ(c + a + m)  s − 1 = (−1)j j (a, c, x) j=0   ∞ ν−1 a+m [( m ν i=1 xi )ys ] m × dν. ν [d + bν + (r + 1)x + (r − s + j + 1)(ysν − xmν )]c+a+m+1 m 0 i i=1 i The corresponding one-sample predictive survival function is ∞ f (ys |x) dys P(Ys > w|x) = w∞ xm f (ys |x) dys s−1 s−1 (−1)j Q1 (w) (a, c, x) j=0 j = , s−1 s−1 j (r (−1) m − s + j + 1) j=0 j where

 Q1 (w) = 0

∞

 ( m xi )ν−1 ν a+m−1 (rm − s + j + 1)−1 m i=1 dν. [d + bν + i=1 (ri + 1)xiν + (rm − s + j + 1)(wν − xmν )]c+a+m

Therefore, we can obtain the two-sided 100(1 − γ )% one-sample prediction interval (L1 , U1 ) for Ys by solving the following two equations: P(Ys > U1 |x) =

γ 2

and P(Ys > L1 |x) = 1 −

γ . 2

Journal of Statistical Computation and Simulation

1615

3.2. Two-sample prediction In many business and engineering applications, the experimenters usually wish to predict the future observations in a population, based on existing data. That is, the experimenters are interested in predicting the sth failure time in a future sample of size N from the same lifetime distribution. A two-sample scheme is used in which the informative sample is a progressively type II censored sample and T1 < · · · < TN are the order statistics of a future sample. Let Ti , i = 1, 2, . . . , N be the order statistics in a sample of size N with lifetimes distributed as Equation (3). The pdf of the sth order statistic is given by N! [1 − exp(−θtsν )]s−1 [exp(−θ tsν )]N−s νθtsν−1 exp(−θ tsν ) (s − 1)!(N − s)!  s−1   N!νθtsν−1 s−1 = (−1)j exp{−θ(N − s + j + 1)tsν }. j (s − 1)!(N − s)! j=0

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f (ts |θ , ν) =

(25)

By forming the product of Equations (25) and (15), and integrating out over the set {(ν, θ); 0 < ν < ∞, 0 < θ < ∞}, the two-sample predictive distribution is obtained as  ∞ ∞ f (ts |ν, θ )π(θ, ν|x) dθ dν f (ts |x) = 0

0

 s−1   N!(c + a + m) s−1 = (−1)j j (s − 1)!(N − s)!(a, c, x) j=0   ∞ ν−1 a+m [( m ν i=1 xi )ts ]  × dν. m ν [d + bν + (r + 1)x + (N − s + j + 1)tsν ]c+a+m+1 0 i i=1 i The corresponding two-sample predictive survival function is  ∞ f (ts |x) dts P(Ts > w|x) = w

 s−1   s−1 N! (−1)j Q2 (w), = j (s − 1)!(N − s)!(a, c, x) j=0 where  Q2 (w) = 0

∞

 ( m xi )ν−1 ν a+m−1 i=1 dν. m (N − s + j + 1)[d + bν + i=1 (ri + 1)xiν + (N − s + j + 1)wν ]c+a+m

Thus, we can obtain the two-sided 100(1 − γ )% two-sample prediction interval (L2 , U2 ) for Ts by solving the following two equations: P(Ts > U2 |x) =

4.

γ 2

and P(Ts > L2 |x) = 1 −

γ . 2

Illustrative example

To illustrate the use of the methods proposed in this article, the following example is discussed. Nelson [36, p. 228, Table 6.1] reported the data on the time to breakdown of an insulating fluid in

1616

S.-R. Huang and S.-J. Wu Table 1.

Progressively type II censored sample generated from the times to breakdown data Nelson (1982).

i

1

2

3

4

5

6

7

8

xi ri

0.1900 0

0.7800 0

0.9599 3

1.3100 0

2.7799 3

4.8496 0

6.5000 0

7.3500 5

Table 2.

One-sample and two-sample 95% prediction intervals.

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One-sample

Two-sample

s

L1

U1

length

L2

U2

length

1 2 3 4 5

7.3972 7.8039 8.5748 9.7558 11.7698

21.1081 37.6112 65.1679 119.2170 273.9140

13.7110 29.8073 56.5931 109.4612 262.1442

0.0198 0.3505 1.2459 2.8250 5.5134

9.1956 19.9033 39.5629 81.7508 213.3753

9.1758 19.5528 38.3170 78.9257 207.8618

an accelerated test conducted at different test voltages. In analyzing this data set, [36] considered a Weibull distribution. For the purposes of illustrating the methods discussed in this article, a progressively type II censored sample of size m = 8 was randomly selected from the n = 19 observations recorded at 34 kV in the table given in [36], as given byViveros and Balakrishnan [37]. The observations and censoring scheme are reported in Table 1. From Equations (9) and (10), the MLEs of ν and θ are νˆ = 0.9743 and θˆ = 0.1148, respectively. We assume that the prior distribution of (ν, θ ) is a bivariate distribution described in (14). We choose the hyperparameters to be a = 0.1, b = 0.5, c = 1.5, and d = 15. From Equations (18) and (19), the Bayes estimates of ν and θ are νˆ B = 0.9736 and θˆB = 0.1167, respectively. Based on Lindley’s approximation, from Equations (21) and (22), the Bayes estimates of ν and θ are νˆ L = 0.9849 and θˆL = 0.1121, respectively. The 95% credible intervals for ν and θ are (0.5155, 1.5277) and (0.0370, 0.2458), respectively. Table 2 shows the 95% one-sample Bayes prediction intervals for the last five removed surviving units. Consider a future sample of size N = 5 from the same distribution. The 95% two-sample Bayes prediction intervals for the sth, 1 ≤ s ≤ 5, failure times are also reported in Table 2. It can be seen that the length of prediction interval becomes wider when s increases.

5.

Simulation study

In this section, we present some simulation results to evaluate the performance of proposed methods for different censoring schemes and parameter combinations. For simplicity of notation, we will denote these censoring schemes, for instance, by (20,5,4*0,15) which represents the censoring scheme r = (0, 0, 0, 0, 15) with n = 20 and m = 5. In a Weibull distribution, without loss of generality, one might take β = 1 because β is the scale parameter. However, β = 1 leads to θ = 1 for any value of ν through the relation θ = 1/β ν . For comparative purpose, therefore, we take β = 4 and ν = (0.5, 1.0, 1.5) such that all the dispositions of parameters are (ν, θ) = (0.5, 0.5), (1.0, 0.25) and (1.5, 0.125). To make the comparison meaningful, the hyperparameters (a, b, c, d) are determined by setting the marginal prior means to be equal to the true values of parameters. That is, c/d = θ and ad/(c − 1)b = ν. For various dispositions of parameters, the hyperparameters are (1, 6, 3, 6) for (ν, θ ) = (0.5, 0.5), (1, 8, 3, 24) for (ν, θ) = (1.0, 0.25), and (1, 6, 3, 12) for (1.5, 0.125), respectively.

MLE

Bayes

Lindley

θ

Censoring scheme

ν

θ

ν

θ

ν

θ

0.5

0.5

1 (20,15,14*0,5) 2 (20,15,5,14*0) 3 (30,15,14*0,15) 4 (30,15,15,14*0) 5 (30,15,1*15) 6 (30,20,19*0,10) 7 (30,20,10,19*0)

0.5630 (0.0239) 0.5453 (0.0160) 0.5643 (0.0261) 0.5400 (0.0136) 0.5540 (0.0197) 0.5449 (0.0160) 0.5315 (0.0101)

0.5138 (0.0307) 0.5121 (0.0325) 0.5327 (0.0297) 0.5201 (0.0316) 0.5282 (0.0280) 0.5123 (0.0192) 0.5100 (0.0231)

0.5421 (0.0174) 0.5314 (0.0120) 0.5355 (0.0176) 0.5297 (0.0108) 0.5330 (0.0147) 0.5310 (0.0129) 0.5236 (0.0082)

0.4953 (0.0138) 0.4955 (0.0142) 0.4982 (0.0111) 0.4972 (0.0138) 0.4988 (0.0118) 0.4978 (0.0106) 0.4976 (0.0123)

0.5072 (0.0109) 0.4956 (0.0075) 0.5187 (0.0138) 0.5006 (0.0073) 0.5144 (0.0113) 0.5118 (0.0099) 0.4996 (0.0060)

0.5612 (0.0181) 0.5751 (0.0205) 0.5435 (0.0132) 0.5700 (0.0197) 0.5517 (0.0155) 0.5434 (0.0136) 0.5575 (0.0168)

1.0

0.25

1 (20,15,14*0,5) 2 (20,15,5,14*0) 3 (30,15,14*0,15) 4 (30,15,15,14*0) 5 (30,15,1*15) 6 (30,20,19*0,10) 7 (30,20,10,19*0)

1.1242 (0.0950) 1.0939 (0.0650) 1.1338 (0.1103) 1.0805 (0.0530) 1.1082 (0.0766) 1.0919 (0.0649) 1.0600 (0.0384)

0.2447 (0.0106) 0.2497 (0.0122) 0.2501 (0.0084) 0.2532 (0.0112) 0.2508 (0.0087) 0.2465 (0.0070) 0.2513 (0.0088)

1.0797 (0.0538) 1.0616 (0.0365) 1.0893 (0.0744) 1.0580 (0.0334) 1.0717 (0.0531) 1.0662 (0.0449) 1.0439 (0.0251)

0.2443 (0.0041) 0.2456 (0.0042) 0.2456 (0.0038) 0.2459 (0.0040) 0.2457 (0.0038) 0.2458 (0.0035) 0.2477 (0.0038)

0.9425 (0.0248) 0.9411 (0.0191) 0.9880 (0.0382) 0.9599 (0.0184) 0.9859 (0.0306) 0.9805 (0.0256) 0.9653 (0.0156)

0.3024 (0.0067) 0.3098 (0.0076) 0.2859 (0.0050) 0.3033 (0.0067) 0.2885 (0.0054) 0.2856 (0.0049) 0.2963 (0.0061)

1.5

0.125

1 (20,15,14*0,5) 2 (20,15,5,14*0) 3 (30,15,14*0,15) 4 (30,15,15,14*0) 5 (30,15,1*15) 6 (30,20,19*0,10) 7 (30,20,10,19*0)

1.6854 (0.2122) 1.6397 (0.1455) 1.6977 (0.2358) 1.6159 (0.1179) 1.6604 (0.1669) 1.6336 (0.1397) 1.5967 (0.0874)

0.1207 (0.0043) 0.1251 (0.0048) 0.1206 (0.0031) 0.1278 (0.0044) 0.1230 (0.0033) 0.1219 (0.0029) 0.1249 (0.0033)

1.6038 (0.0821) 1.5843 (0.0571) 1.6200 (0.1149) 1.5781 (0.0542) 1.5983 (0.0839) 1.5878 (0.0691) 1.5692 (0.0428)

0.1216 (0.0011) 0.1222 (0.0011) 0.1219 (0.0011) 0.1229 (0.0011) 0.1224 (0.0011) 0.1227 (0.0010) 0.1229 (0.0010)

1.2764 (0.0841) 1.3216 (0.0537) 1.3449 (0.0636) 1.3658 (0.0397) 1.3847 (0.0442) 1.3824 (0.0379) 1.3996 (0.0265)

0.1683 (0.0028) 0.1687 (0.0028) 0.1583 (0.0020) 0.1646 (0.0024) 0.1576 (0.0021) 0.1568 (0.0020) 0.1597 (0.0022)

ν

Journal of Statistical Computation and Simulation

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Table 3. Average estimates and estimated risks (in parentheses) of MLEs, Bayes estimates and Lindley’s approximations.

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To generate the progressive type II censored data of Weibull distribution, we use the following algorithm proposed by Balakrishnan and Aggarwala [19]. The steps are as follows: 1. Generate m independent exponential random variables Z1 , Z2 , . . . , Zm . 2. Given the censoring scheme r = (r1 , r2 , . . . , rm ), one can set Wi =

Z1 Z2 Zi + + ··· + , n n − r1 − 1 n − r1 − r2 − · · · − ri−1 − i + 1

i = 1, . . . , m.

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3. Finally, using the given values of parameters (ν, θ), one can obtain the progressive type II censored data Xi = F −1 [1 − exp(−Wi )], for i = 1, . . . , m, where F −1 (·) is the inverse function of cdf defined in Equation (4). 5.1.

Point estimation

A Monte Carlo simulation is performed to compute the average estimates and estimated risks of the MLEs, Bayes estimators and Lindley’s approximations. All the results were computed over 10,000 simulated samples and listed in Table 3. It shows that the average estimates of MLEs and Bayes estimators are all close to the true parameters. The Bayes estimates are better than MLEs and Lindley’s approximations for all the considered cases. The Lindley’s approximations also performs well except for small value of parameter θ . Comparing the censoring schemes 1 with 2, 3 with 4, and 6 with 7, respectively, the three estimators of parameter ν (θ) have smaller (larger) estimated risks in schemes 2, 4 and 7 than those in schemes 1, 3 and 7. 5.2.

Prediction interval

In this subsection, we will investigate the performance of prediction intervals in terms of coverage probabilities and average prediction lengths. We report the results based on 10,000 simulated samples from a Weibull distribution for each censoring scheme. First, we consider the one-sample prediction problem. We compute the prediction intervals for the failure times of the first three removed units at the end of life test, that is, the prediction intervals of Y1 , Y2 and Y3 in Section 3.1. Table 4 shows that the coverage probabilities are all close to the desired level of 0.95. The prediction interval is wider the further Ys is from Xm . We now consider the two-sample prediction problem. Suppose that a future sample of size N = 3 from the same distribution is obtained. We wish to compute the prediction intervals for T1 , T2 and T3 in Section 3.2. The coverage probabilities presented in Table 5 shows that they are all close to the nominal level of 0.95. Comparing schemes 1 with 2, 3 with 4 and 6 with 7, we find Table 4.

Coverage probabilities and average lengths (in parentheses) of one-sample prediction intervals. ν

Censoring scheme

Y1

Y2

Y3

0.5

0.5

1 (20,15,14*0,5) 3 (30,15,14*0,15) 6 (30,20,19*0,10)

0.9514 (16.2660) 0.9526 (2.3241) 0.9505 (5.0894)

0.9530 (40.1617) 0.9542 (4.4627) 0.9488 (9.7245)

0.9552 (92.8442) 0.9579 (7.2730) 0.9516 (15.9422)

1.0

0.25

1 (20,15,14*0,5) 3 (30,15,14*0,15) 6 (30,20,19*0,10)

0.9488 (4.0376) 0.9510 (1.2929) 0.9532 (1.8235)

0.9494 (7.6099) 0.9506 (2.1632) 0.9536 (3.0443)

0.9492 (12.7285) 0.9520 (3.0704) 0.9533 (4.3405)

1.5

0.125

1 (20,15,14*0,5) 3 (30,15,14*0,15) 6 (30,20,19*0,10)

0.9488 (2.1426) 0.9477 (0.9013) 0.9532 (1.0937)

0.9492 (3.7089) 0.9492 (1.4402) 0.9496 (1.7470)

0.9493 (5.6244) 0.9483 (1.9529) 0.9486 (2.3810)

θ

Journal of Statistical Computation and Simulation Table 5.

Coverage probabilities and average lengths (in parentheses) of two-sample prediction intervals. θ

censoring scheme

T1

T2

T3

0.5

0.5

1 (20,15,14*0,5) 2 (20,15,5,14*0) 3 (30,15,14*0,15) 4 (30,15,15,14*0) 5 (30,15,1*15) 6 (30,20,19*0,10) 7 (30,20,10,19*0)

0.9494 (8.2013) 0.9498 (8.0863) 0.9556 (9.3826) 0.9514 (8.0389) 0.9507 (8.5198) 0.9533 (7.8508) 0.9491 (7.5836)

0.9478 (40.7744) 0.9486 (34.4148) 0.9516 (60.6229) 0.9484 (34.2119) 0.9542 (44.2207) 0.9540 (38.0456) 0.9455 (30.6368)

0.9506 (319.0400) 0.9510 (201.0973) 0.9523 (669.2523) 0.9515 (196.5204) 0.9510 (340.1805) 0.9504 (267.5441) 0.9516 (159.1722)

1.0

0.25

1 (20,15,14*0,5) 2 (20,15,5,14*0) 3 (30,15,14*0,15) 4 (30,15,15,14*0) 5 (30,15,1*15) 6 (30,20,19*0,10) 7 (30,20,10,19*0)

0.9443 (5.6053) 0.9494 (5.5471) 0.9475 (5.9256) 0.9519 (5.5387) 0.9464 (5.6961) 0.9468 (5.4594) 0.9523 (5.4051)

0.9495 (11.9097) 0.9448 (11.0245) 0.9509 (14.1531) 0.9527 (11.0054) 0.9461 (12.3753) 0.9464 (11.3930) 0.9484 (10.4979)

0.9478 (30.0567) 0.9437 (24.9633) 0.9467 (41.6631) 0.9487 (24.7528) 0.9473 (31.6298) 0.9469 (27.6527) 0.9497 (22.7443)

1.5

0.125

1 (20,15,14*0,5) 2 (20,15,5,14*0) 3 (30,15,14*0,15) 4 (30,15,15,14*0) 5 (30,15,1*15) 6 (30,20,19*0,10) 7 (30,20,10,19*0)

0.9474 (4.8553) 0.9478 (4.8025) 0.9472 (5.0159) 0.9515 (4.8013) 0.9456 (4.8801) 0.9472 (4.7570) 0.9432 (4.7127)

0.9456 (7.4889) 0.9490 (7.1217) 0.9471 (8.3067) 0.9464 (7.1339) 0.9431 (7.6386) 0.9429 (7.2405) 0.9472 (6.8630)

0.9474 (13.0514) 0.9443 (11.6939) 0.9483 (15.8395) 0.9461 (11.7127) 0.9426 (13.4865) 0.9423 (12.3580) 0.9451 (10.9586)

ν

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1619

that the schemes 2, 4 and 7 have shorter average prediction lengths than those of schemes 1, 3 and 6, respectively.

6.

Conclusion

In this study, we consider the point estimations and prediction intervals of the Weibull distribution based on progressive type II censoring data. MLEs and Bayes estimators are applied to estimate the parameters of lifetime distribution. Under squared error loss function, the Bayes estimators are derived. Because of the complicated numerical integration in calculating posterior mean, we provide the Lindley’s approximation procedure to calculate the approximate Bayes estimators. Another main subject of this paper is to obtain the Bayes prediction intervals for the unobserved failures in the one-sample and two-sample cases. A numerical example is used to illustrate the proposed methods, and we also assess the estimators and prediction intervals by performing a Monte Carlos simulation. The results indicate that our proposed methods work well. References [1] H.-K. Yuen and S.-K. Tse, Parameters estimation for Weibull distributed lifetimes under progressive censoring with random removals, J. Statist. Comput. Simul. 55 (1996), pp. 57–71. [2] M.A.M.A. Mousa, Inference and prediction for Pareto progressively censored data, J. Statist. Comput. Simul. 71 (2001), pp. 163–181. [3] M.A.M. Ali Mousa and Z.F. Jaheen, Bayesian prediction for progressively censored data from the Burr model, Statist. Pap. 43 (2002), pp. 587–593. [4] Z.F. Jaheen, Prediction of progressive censored data from the gompertz model, Commun. Statist. Simul. Comput. 32 (2003), pp. 663–676. [5] A.J. Fernández, On estimating exponential parameters with general type II progressive censoring, J. Statist. Plann. Inference 121 (2004), pp. 135–147. [6] M.A.M. Ali Mousa and S.A. AL-Sagheer, Bayesian prediction for progressively type-II censored data from the Rayleigh model, Commun. Statist. Theory Meth. 34 (2005), pp. 2353–2361.

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