c Heldermann Verlag Economic Quality Control ISSN 0940-5151 Vol 18 2003, No. 2, 195 207 Bayesian Predictions for Exponentially Distributed Failure Times With One Change-Point Y. Abdel-Aty, J. Franz, M.A.W. Mahmoud Abstract: Suppose that a number of items is put on operation and that their life times are identically exponentially distributed up to a time a, when the operating conditions change resulting in a different failure intensity of the exponential distribution for those items which have survived the change-point a. Predictions are made for the lifetime of operating items on condition of observed failures of a part of the items before and after the change-point a. The predictions are based on an Bayes approach, which is briefly introduced. Numerical examples are given to illustrate the results. 1 Introduction The exponential distribution with a jump at the change-point a a > 0 is denoted Expα, β, a. The probability density function pdf and distribution function cdf for Expα, β, a are, { αe αx 0 x a fx = βe β αa βx 1 a<x { 1 e αx 0 x a F x = 1 e β αa βx 2 a<x where α>0, β > 0, a > 0. The exponential distribution with one jump change in failure intensity has been studied by many researches [See, for example, Mattews and Farewell 1982, Nguyen, Rogers and Walker 1984, Pham and Nguyen 1990] These papers refer to the detection of the change-point a; in contrast, in our investigations the jump point a is fixed and known. Suppose there are n items X = X 1,X 2,..., X n with identical lifetime distribution Expα, β, a put on operation. Assume that after observation of the first r 1 failure times x 1 x 2... x r1 with x r1 <aon the condition that there will be exactly k 1 >r 1 failures up to the change-point, a prediction is required on the failure times for some or all remaining k 1 r 1 items. Next assume that r 2 ordered failure times are observed with x r2 a and a prediction on the life time is required of some or all of the remaining n r 2 items.
196 Y. Abdel-Aty, J. Franz, M.A.W. Mahmoud In Lawless 1971 methods are given for constructing a prediction for the failure time of X r when the lifetime is Expα and the first s, s<r failure times of n items have been observed. In Lawless 1972 a related problem is discussed. A review on classical prediction methods may be found in Patel 1989, which covers the following lifetime distributions: discrete case: Poisson, Binomial and Negative Binomial continuous case: Normal, Lognormal, Exponential, Weibull, Gamma, Inverse Gaussian, Pareto, and IFR. Patel also presented nonparametric predictions, among other results. In Nagaraja 1995 besides predictions the following point predictors are discussed: Best linear unbiased predictor BLUP, best linear invariant predictor BLIP, maximum likelihood predictor MLP, and Bayes predictors. As Geisser 1993 has shown, the problem of prediction can be solved within the Bayesian framework. Several researchers have studied Bayesian prediction in the case of exponential distribution, among them are e.g. Dunsmore 1974, Geisser 1990, Lingappaiah [1978, 1979, 1986], AL-Hussaini and Jaheen 1999, AL-Hussaini [1999, 2001], Upadhyay and Pendey 1989. The two books by Aitcheson and Dunsmore 1975 and Geisser 1993 are concerned primarily with Bayes prediction. 2 Predictions For Failures Before the Change-Point Let X 1 <X 2 <... < X n denote the ordered lifetimes of the n items put on operation and consider the case that r 1 item have failed until x r1, while the remaining n r 1 items are still operating. The density function of X r+s on the condition of x =x 1,x 2,..., x r1 isgivenby: h Xr+s x x r1,α= s n r 1 s [F x F xr1 ] s 1 [1 F x] n r 1 s fx 3 [1 F x r1 ] n r 1 where 1 s n r Suppose the observed failure times are x 1 <x 2 <... < x r1 and we want to predict X r1 +s 1 on the condition that there are exactly k 1 failures before a and 1 s 1 k 1 r 1. For deriving a Bayes prediction, the likelihood function of α must be derived and a prior distribution of α must be specified. The likelihood function of α for given x =x 1,x 2,..., x r1 is as follows:
Bayesian Predictions for Exponentially Distributed Failure Times 197 Lα; x = = k 1! k 1 r 1! [1 F x r 1 ] k 1 r 1 r 1 i=1 fx i k 1! k 1 r 1! αr 1 e αu 1 4 where u 1 = r 1 x i +k 1 r 1 x r1. i=1 The prior distribution of α is selected to be a Gamma distribution with density function: π 1 α = αc 1 e αd d c c>0,d>0 5 Γc Then from 4 and 5 the posterior density of α is obtained: π 1α x = αc+r 1 1 e αd+u1 u 1 + d c+r 1 6 Γc + r 1 Inserting the distribution and density function of Expα into 3, the conditional density function of X r1 +s 1 on the condition of x 1,..., x r1 is obtained. After elementary manipulation one arrives at the following expression: s h x x,α=d Xr1 +s 1 1α c 1j e x xr 1 αk 1j 7 where D 1 = s k1 r 1 1 s 1 c 1j = s 1 1 j 1 j and k 1j = k 1 r 1 s 1 + j +1 2.1 Fixed Number of Failures Before the Change-Point The Bayesian joint probability density of X r1 +s 1,α is equal to the product of the posterior density 6 and the conditional density 7 of X r1 +s 1. The Bayesian prediction density function of X r1 +s 1 is obtained as density of the corresponding marginal distribution of X r1 +s 1 by integrating the joint density with respect to α yielding: p 1 x x =D 1 u 1 + d c+r 1 c 1j r 1 + c[x x r1 k 1j +u 1 + d] r 1+c+1 The so-called predictive reliability function of X r1 +s 1 is defined as follows P X r1 +s 1 t x = 1 t x r1 p 1 x xdx = D 1 u 1 + d c+r 1 8 c 1j k 1j 1 [t x r1 k 1j +u 1 + d] r 1+c 9
198 Y. Abdel-Aty, J. Franz, M.A.W. Mahmoud Any prediction for X r1 +s 1 with credibility level γ meets the following condition: P [Lx <X r1 +s 1 <Ux] γ 10 Assuming that symmetric error probabilities will not be far from the best solution, the lower and upper bound Lx andux of the prediction may be determined by solving the following two equations: P X r1 +s 1 Lx = 1+γ 2 11 P X r1 +s 1 Ux = 1 γ 2 Using 9 for solving 11 numerically, we obtain L and U. Numerical examples are given in Section 4. Next the so-called non informative prior is used for the parameter α. Note that this choice does not lead to a proper density function, because of the assumed unbounded parameter space. Thus we assume π 2 α 1 α Then, the posterior density of α is given by π 2α = αr 1 1 e αu 1 u r 1 1 Γr 1 and the bounds of the prediction are obtained from 9 and11 with c = d =0. 12 13 2.2 Random Number of Failures Before the Change-Point In reality the number of failures before the change-point is, of course, a random variable denoted by K 1. Each of the items fails with identical probability before the change-point, thus it is reasonable to assume that K 1 is binomially distributed: n P K 1 = k 1 =p k 1 = p k 1 q n k 1 k 1 =0, 1,..., n 14 where p + q =1. k 1 In Consul 1984 and Gupta and Gupta 1984 it is shown that the predictive density function of X r1 +s 1 for random K 1 is given by: 1 n g 1 x x = p k 1 p 1 x k 1 15 P K 1 s 1 + r 1 k 1 =s 1 +r 1 where p 1 x k 1 is the predictive density function of X r1 +s 1 when k 1 is fixed. From 8, 14 and 15 we obtain n g 1 x x = k 1 =s 1 +r 1 n k 1 p k 1 q n k 1 D 1 u 1 + d c+r 1 1 s 1+r i=0 n i c 1j r 1 +c [x x r1 k 1j +u 1 +d] r 1 +c+1 pi q n i 16
Bayesian Predictions for Exponentially Distributed Failure Times 199 The predictive reliability function of X r1 +s 1 is immediately obtained from 16: g1t x = P X r1 +s 1 t x n n k 1 p k 1 q n k1 c D 1j 1 [ t xr1 ] r1 k 1 =s 1 +r 1 k 1j u = 1 +d k +c 1j+1 17 1 s 1+r pi q n i i=0 n i Thus, for a random number of failures before the change-point the prediction bounds L and U are obtained by solving 11 based on 17. 3 Predictions For Failures After the Change-Point Next it is assumed that the failures x =x 1,...,x k1,x k1 +1...,x r2 18 are observed with x k1 <a x k1 +1 x r2. In this case a prediction for X r2 +s 2 after the change-point a is of interest. For the likelihood function for β we obtain: Lβ; x [1 F x r2 ] n r 2 r 2 i=k 1 +1 fx i = β r 2 k 1 e βu 2 19 where u 2 =n r 2 x r2 n k 1 a + r 2 x i i=k 1 +1 As prior distribution for the parameter β we again select a gamma prior distribution with density π 1 β = βc 1 e βd d c c>0,d>0 20 Γc yielding the following posterior density of β: π 1β = βc+r 2 k e βd+u2 u 2 + d c+r 1 k 1 21 Γc + r 2 k 1 The conditional density function of X r2 +s 2 on condition X r2 = x r2 is, h x β =D [F x F x r2 ] s2 1 [1 F x] n r 2 s 2 Xr2 +s 2 2 fx 22 [1 F x r2 ] n r 2 Inserting in?? the respective expressions of the Expβ, we obtain
200 Y. Abdel-Aty, J. Franz, M.A.W. Mahmoud h Xr2 +s 2 x β =D 2 β where n j = n r 2 s 2 + j +1 c 2j = s 2 1 j 1 j and D 2 = s n r2 2 s 2 c 2j e x r 2 +s 2 x r2 βn j 23 The Bayesian prediction density function of X r2 +s 2 is obtained by integrating the product of posterior density 21 and the conditional density 23 and integrating over β: p 2 x x = D 2 r 2 + c k 1 u 2 + d c+r 2 k 1 c 2j [x x r2 n j +u 2 + d] r 2+c k 1 +1 24 The predictive reliability function of X r2 +s 2 is immediately obtained from 24: P X r2 +s 2 t x = D 2 u 2 + d c+r 2 k 1 c 2j n j [t x r2 n j +u 2 + d] r 2+c k 1 25 The lower and upper L and U for a prediction with credibility level γ for X r2 +s 2 are obtained by solving 11 numerically using 25. 3.1 Random Number of Items Put on Operation We now consider the number of items put on operation being a random variable denoted N distributed according a truncated Poisson distribution with parameter δ: e δ δ n P n = n =1, 2,... 26 n! 1 e δ The predictive density function of X r2 +s 2 when N is random is given by: 1 g 3 x x = P np 2 x n; x 27 P N s 2 + r 2 n=s 2 +r 2 from 24, 26 and 27 the predictive reliability function is obtained as: P X r2 +s 2 t x = n=s 2 +r 2 e δ δ n n!1 e δ 1 s 2+r i=1 n j [ t xr2 u 2 +d n j+1 e δ δ i i!1 e δ D 2 c 2j ] r2 +c k 1 Solving 11 numerically by using 28 yields the desired prediction bounds L and U. 28
Bayesian Predictions for Exponentially Distributed Failure Times 201 4 Prediction Before the Change-Point Concerning a Future Operation Run In this section we will discuss the Bayesian prediction concerning a future batch of items Y 1 <Y 2 <... < Y m put on operation with identical distribution Expα, β,a ofthe failure times as the first batch X 1 <X 2 <... < X n assuming that the life times within and between the batches are independent. The density function of Y s for 1 s m is m h Y s y α, β =s [1 F y] m s [F y] s 1 fy 29 s where 1 s m. The prediction depends on the amount and type of observation available from the first sample. Suppose the observed failure times from the first sample are x 1 <x 2 <... < x r1 and we want to predict Y s1 with 1 s 1 k 2 where k 2 is the number of failures in the future batch before the change-point a. For the remainder it is assumed that the size of the first batch n and the number of failed items k 1 before the change point is fixed and known. 4.1 Fixed Number of Failures Before the Change-Point in the Future Batch The density function of Y s1 on condition of x =x 1,..., x r1 is obtained inserting the corresponding expressions of Expα into 29: h Y s1 y α =D 3 e αy k 2 s 1 1 e αy s αe αy 30 or h Y s1 y α =D 3 α c 1j e αk 2jy 31 where k 2j = k 2 s 1 + j +1andD 3 = s 1 k2 s 1. The Bayesian prediction density function of Y s1 is obtained as in the previous cases yielding: p 3 y x =D 3 u 1 + d c+r 1 c 1j r 1 + c [yk 2j +u 1 + d] r 1+c+1 32 From the density function 32 the predictive reliability function of Y s1 is obtained: D 3 c 1j P Y s1 t x = tk2j k 2j +1 r1 33 +c u 1 +d The lower and upper prediction bounds L and U with credibility level γ are obtained as numerical solution of 11 using the prediction reliability function 33.
202 Y. Abdel-Aty, J. Franz, M.A.W. Mahmoud 4.2 Random Number of Failures Before the Change-Point in the Future Batch Next assume more realistically that the number of failures before the change-point in the future batch is a random variable denoted K 2 and binomially distributed with parameters m and p, m P K 2 = k 2 =p k 2 = p k 2 q m k 2 for k 2 =0, 1,..., m 34 k 2 The predictive density function of Y s1 is n k g 4 y x = 2 =s 1 p k 2 p 3 y x,k 2 35 P K 2 s 1 where p 3 y x,k 2 is the predictive density function of Y s1 when the number of failures before the change-point is fixed and given by k 2. With 32 and 34 we obtain from 35 g 4 y x = m k 2 =s 1 m k 2 p k 2 q m k 2 s 1 1 1 [ s 1 1 i=0 j=1 m i D 3 u 1 +d c+r 1 c 1j r 1 +c [yk 2j +u 1 +d] r 1 +c+1 pi q m i ] With the density 36 the predictive reliability function of Y s1 is immediately obtained: m m k 2 p k 2 q m k 2 s 1 1 D 3 c 1j g4y k 2 =s 1 k 2j [ tk 2j u 1 +d +1]r 1 +c x =P Y t x = 37 1 s 1 1 pi q m i By means of 37 and solving 11 numerically the prediction bounds L and U are obtained. i=0 m i 36 5 Prediction After the Change-Point Concerning a Future Operation Run In this case we want to make a prediction for a failure after the change point Y s2, i.e., Y s2 >aand 1 s 2 m k 2. It is assumed that from the first batch only the observations x i >awith i = k 1 +1,..., n are available. The conditional density function corresponding to 1 yields for the likelihood function: Lx; a, β β r 2 k 1 e βu 2 38 where x =x k1 +1,...,x n andu 2 =n r 2 x r2 +n k 1 a + r 2 x i i=k 1 +1 In the following sections the prediction problem for different cases of k 2 and m are solved.
Bayesian Predictions for Exponentially Distributed Failure Times 203 5.1 Predictions For Fixed Batch-Size and Fixed Number of Failed Items After Change-Point The density function of Y s2 on condition x 1,..., x r2 is obtained by replacing F and f in 30 by the corresponding terms of Expβ. h Y s2 y β =D 4 e βy a m s 2 1 e βy a s βe βy a or h Ys 2 y β =D 4 β c 2j e βm jy a where m j = m k 2 s 2 + j +1andD 4 = s 2 m k2 s 2. The Bayesian prediction density function of Y s2 is obtained as marginal distribution with respect to the joint distribution of Y s2,β. p 4 y x =D 4 u 2 + d c+r 2 k 1 c 2j r 2 + c k 1 [y am j +u 2 + d] r 1+c k 1 41 +1 Thus, the predictive reliability function of Y s2 is as follows: P Y s2 t x = m j t amj u 2 +d +1 39 40 D 4 c 2j r1 +c k 1 42 The lower and upper prediction bounds L and U for credibility level γ for Y s2 are obtained by solving 11 numerically using 42. 5.2 Predictions For Random Batch-Size and Fixed Number of Failed Items After Change-Point Assume now that the size of the future batch is random and denoted M distributed according to a truncated Poisson distribution with parameter δ. e δ δ m P m = m =1, 2,... 43 m! 1 e δ Then the predictive density function of Y s2 is P mp 4 y m m=s g 6 y x = 2 44 P M s 2 Analogously as in the previous cases with 43, 44 and 45 the predictive reliability function of Y s2 is obtained. s 2 P Y s2 t x = m=s 2 e δ δ m m!1 e δ 1 m j t a u 2 +d m j+1 s 2 1 e δ δ i i=1 i!1 e δ D 4 c 2j r2 +c k 1 With 45 and solving 11 numerically the bounds L and U for given credibility level γ are obtained. 45
204 Y. Abdel-Aty, J. Franz, M.A.W. Mahmoud 6 Numerical Examples For the examples the change-point was set to be a =1.0and the size of the batch to be n = 20. For the Gamma-prior distribution of α we selected the parameter values c, d = 0.05, 0.1 and for the Gamma prior distribution of β the parameter values c, d =2, 10. The following values were generated: generated values for the distribution parameters α =0.5 andβ =0.2 lifetimes of the items in the batch x 1 =0.134910 x 2 =0.247596 x 3 =0.258166 x 4 =0.302317 x 5 =0.329222 x 6 =0.365428 x 7 =0.607163 x 8 =1.476456 x 9 =1.720737 x 10 =2.345809 x 11 =2.979479 x 12 =3.378253 x 13 =3.754769 x 14 =4.437662 x 15 =6.459277 x 16 =6.613639 x 17 =6.915906 x 18 =8.140099 x 19 =9.982992 x 20 =11.213757 Table 1: The generated lifetimes of the items put on operation. For the examples, the following values for the number of available observations were used: r 1 =5,r 2 = 17. Predictions for X 6, X 7 Based on x =x 1,...,x 5, predictions with credibility level γ =0.95 for X 6 and X 7 are calculated. First for fixed k 1 = 7 and subsequently for random number K 1 of failures before the change-point, where K 1 Bi20, 0.4. For the prior distribution of α the noninformative prior with c = d = 0 and the Gamma prior with c =0.05 and d =0.1 were used. The results are given in Table 2. c d L X6 U X6 L X7 U X7 k 1 =7 0 0.334122 1.38266.39134 2.98829.05.1.334235 1.42177.393958 3.08538 K 1 random 0 0.332097 1.374985.353712 1.9880 p =.4.05.1.332216 1.41462.35474 2.0498 Table 2: The prediction bounds L Xi x andu Xi x for i =6, 7. Predictions for X 18, X 20 Based on x =x 1,...,x 17, predictions with credibility level γ =0.95 for X 18 and X 20 are calculated. For given k 1 = 7 and fixed batch size n = 20 and subsequently for random number batch size N, wheren Po10, i.e., δ = 10. For the prior distribution of β the noninformative prior with c = d = 0 and the Gamma prior with c =2.0 and d =10.0 wereused.theresultsaregivenintable3.
Bayesian Predictions for Exponentially Distributed Failure Times 205 c d L X18 U X18 L X20 U X20 k 1 =7 0 0 6.95624 14.015208 8.4688 35.9221 2 10 6.9565 13.8426 8.4961 34.9858 N random 0 0 6.97074 23.876249 7.61137 31.50620 δ =10 2 10 6.9711 23.5310 7.6237 30.8096 Table 3: The prediction bounds L Xi x andu Xi x for i =18, 20. Predictions for Y 1, X 8 Based on x =x 1,...,x 5, predictions with credibility level γ =0.95 for Y 1 and Y 8 are calculated. First for fixed k 2 = 8 and subsequently for random number K 2 of failures before the change-point, where K 1 Bi20, 0.4, i.e, m = 20. For the prior distribution of α the noninformative prior with c = d = 0 and the Gamma prior with c =0.05 and d =0.1 wereused.theresultsaregivenintable4. c d L Y1 U Y1 L Y8 U Y8 k 2 =8 0 0.001225.26336.30636 3.9271.05.1.001275.273138.31959 4.0666 K 2 random 0 0.001246.311515.1819926 2.9413 p =.4.05.1.001298.323149.189825 3.0473 Table 4: The prediction bounds L Yi x andu Yi x for i =1, 8. Predictions for Y 9, Y 20 Based on x =x 1,...,x 17, predictions with credibility level γ =0.95 for Y 9 and Y 20 are calculated. For given k 2 = 8 and fixed batch size m = 20, then for random number K 2 with K 2 Bi0.4, 20 and subsequently for random number batch size M, where M Po10, i.e., δ = 10. For the prior distribution of β the noninformative prior with c = d = 0 and the Gamma prior with c =2.0 andd =10.0 wereused. Theresultsare given in Table 5. c d L Y9 U Y9 L Y20 U Y20 k 2 =8 0 0 1.00935 2.61829 6.4254 40.7746 2 10 1.0101 2.7316 6.5898 39.1305 K 2 random 0 0 1.00998 2.8686 4.39919 31.5401 p =.4 2 10 1.0100 2.8246 4.4972 30.4508 M random 0 0 1.012118 3.61522 4.09650 31.8731 δ =10 2 10 1.0122 3.5579 4.1803 30.7925 Table 5: The prediction bounds L Yi x andu Yi x for i =9, 20.
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Bayesian Predictions for Exponentially Distributed Failure Times 207 [17] Nguyen, H. T., Rogers, G. S. and Walker; E.A. 1984. Estimation in change point hazard rate model. Biometrica 71, 299-304. [18] Patel, J. K. 1989. Prediction intervals - A review.communic. Statist. Theory Meth. 18, 2393-2465. [19] Pham, D. T. and Nguyen, H. T. 1990. Strong consistency of the maximum likelihood estimator in the change point hazard rate model. Statistics 21, 203-216. [20] Upadhyay and Pendy 1989. Prediction limits for an exponential distributiom. A Bayes predictive distribution approach. IEEE Trans. Rel. 38, 599-602. Yahia Abdel-Aty Department of Mathematics, Faculty of Science Al-Azhar University, Nasr City Cairo 11884, Egypt, and Institut für Mathematische Stochastik Technische Universität Dresden D-01062 Dresden, Germany Jürgen Franz Institut für Mathematische Stochastik Technische Universität Dresden D-01062 Dresden, Germany M. A. W. Mahmoud Department of Mathematics, Faculty of Science Al-Azhar University, Nasr City Cairo 11884, Egypt