Bayesian Inference for S-Shaped Software Reliability Growth Models
LYNN KUO University of Connecticut, Storrs JAECHANG LEE Korea University, Seoul KIHEON CHOI Duksung Women's University, Seoul TAE YOUNG YANG University of Missouri, Columbia Key Words { Gamma intensity function; General order statistics; Gibbs sampling; Metropolis algorithm; Model selection; Nonhomogeneous Poisson process. Reader Aids { Purpose: Widen state of the art Special math needed for explanation: Stochastic processes, Statistical inference Special math needed to use results: Same Results useful to: Software reliability theoreticians Summary & Conclusions { Bayesian inference for a nonhomogeneous Poisson process with an S-shaped mean value function is studied. In particular, we consider the model proposed by Ohba, et al, and its generalization to a class of gamma distribution growth
1
curves. Two Gibbs sampling approaches are proposed to compute the Bayes estimates of the s-expected number of errors remaining and the current system reliability. One algorithm is a Metropolis within Gibbs algorithm. The other is a stochastic substitution algorithm with data augmentation. Model selection based on the posterior Bayes factor is studied. A numerical example with simulated data is given.
1. Introduction Many reliability growth models have been proposed to model the number of failures in software testing. These models include the Goel-Okumoto [12], Ohba-Yamada [20], [24] models. This paper presents a Bayesian methodology for the Ohba-Yamada model. The Ohba-Yamada model assumes that the number of errors M (t) discovered in software testing follows a nonhomogeneous Poisson process (NHPP) with the mean value function m(t) to be a multiple of a gamma distribution function with shape parameter equal to 2. From now on, we will call this process NHPP-gamma-2. The mean value function of the NHPP-gamma-2 is S-shaped to re ect that it is usually dicult to nd the faults in the software at the beginning of testing. After a learning period, the faults are found rapidly and then gradually slowly due to debugging. In addition to the NHPP-gamma-2, we also consider a more general class of NHPP with S-shaped mean value functions with an arbitrary shape parameter k (known) and an unknown scale parameter in the gamma distribution. This class denoted by the NHPP-gamma-k models, or k-Stage Erlangian growth curve models (Khoshgoftaar and Woodcock [15]), includes the Goel-Okumoto model where the shape parameter is 1, the Ohba-Yamada model where the shape parameter is 2. To compute Bayes estimates, the Gibbs sampler is proposed to evaluate the features of the posterior distribution. The Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm which samples variates according to a Markov chain with the stationary distribution as the desired posterior distribution. The transition measure of this Markov chain is usually a product of conditional densities. We propose two samplers: one is a Metropolis-within-Gibbs algorithm, where the scale parameter is generated by the Metropolis algorithm; the other is a stochastic substitution algorithm with data augmentation. To facilitate speci cation of 2
the conditional densities used in the latter algorithm, we employ a few tricks. The mean value function for the NHPP-gamma-k model can be obtained by a nite sum of Poison probabilities. We introduce a data augmentation technique with two latent variables: one is the number of faults remaining in the software; the other is for indicating which component of the Poisson probabilities in the mean value function would contribute in updating the scale parameter. In addition to Bayesian inference, we also explore the posterior Bayes factor criterion (Aitkin [1]) to select the best model among the generalized class of gamma intensity functions. The posterior Bayes factor compares the marginal likelihoods of the whole data set for two models with respect to their posterior distributions. We prefer the posterior Bayes factor to the Bayes factor because (1) it is less sensitive to prior variations; (2) it can be used with improper priors; (3) it is e ectively equivalent to a penalized likelihood criterion in many cases; and (4) it is easy to implement using the MCMC algorithm. Let us note the controversy about the posterior Bayes factor, from the discussion section in Aitkin [1], because it uses the data twice. Nevertheless, we explore its use in this paper because of its desirable features listed above. In order to make the usual posterior Bayes factor more comparable to the ratio of the log of the maximum likelihoods, we use the log of the marginal likelihoods instead of the marginal likelihoods in de ning the posterior Bayes factor. Kuo and Yang [16] consider a uni ed approach to the software reliability growth models. They model the failure times with two di erent classes. One is the general order statistics model (GOS); the other is the record value statistics model. In the GOS model as given in Raftery [21], we assume there is an unknown number of faults N at the beginning of software testing. We model the observed epochs of failures to be the rst n order statistics taken from N i.i.d. observations with density f supported in R . If f is an exponential density, then this is the Jelinski-Moranda [14] model. If f is a gamma density with shape parameter 2, then we call it the gamma{2 GOS model. If we further assume that N has a Poisson distribution with mean , then the uni ed theory yields that the number of failures detected in (0; t] for the gamma{2 GOS model follows a NHPP with mean value function [1 ? (1+ t)e? t]. This is the Ohba-Yamada model. We can consider other GOS models using di erent densities f , such as the Pareto, the Weibull, and the truncated extreme value densities. While Bayesian +
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inference for these models was treated in Kuo and Yang [16], the Ohba-Yamada model was not included because it requires a more complicated data augmentation technique. This paper focuses on Bayesian inference for the Ohba-Yamada model and the NHPP-gamma-k model for any xed k. Section 2 exhibits Gibbs samplers for evaluating the posterior distribution for the OhbaYamada model. Section 3 develops Gibbs samplers for the NHPP-gamma-k model. Section 4 discusses Bayes inference for the current reliability function and the s-expected number of remaining errors. Section 5 discusses a method for model selection. A numerical example is given in Section 6. Notation
M (t)
a Nonhomogeneous Poisson Process (NHPP) that models the cumulative number of errors detected until time t m(t) mean value function of the NHPP M (t) (t) dm(t)=dt, intensity function of the NHPP M (t) (t) s-expected number of errors remaining in the software at time t R(xjt) software reliability, i.e., the probability of no failures in (t; t + x] given that failures have been monitored until time t Dt the data set consists of observed ordered epochs of failures 0 < x < < xn < t Dxn the data set consists of observed ordered epochs of n failures 0 < x < < xn s-expected number of errors at the beginning of testing to be estimated error detection rate per error in the steady-state to be estimated ?( ; ) gamma distribution with shape parameter and scale parameter (mean = ) mult(p; N ) multinomial distribution with cell probability p and sample size N is distributed as / is proportional to 1
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Other standard notation is given in \Information for Readers & Authors" at the rear of each issue. 4
2. Gibbs Sampler for the NHPP-Gamma-2 Model We rst describe the likelihood function and the Gibbs sampler for the NHPP-gamma-2 model. Given the time truncated model, testing until time t, the ordered epochs of the observed n failures are denoted by x ; : : : ; xn. Therefore, the data set Dt consists of fn; x ; x ; : : :; xn; tg. Given the failure truncated model (observed until nth failure), the data set Dxn consists of fx ; x ; : : :; xng. The likelihood for the time truncated model is ! n Y (xi) e?m t : (2:1) LNHPP (; jDt) = 1
1
1
2
2
( )
i
=1
It is developed in many textbooks, for example, Cox and Lewis [7], Basawa and Prakasa Rao [3], and Crowder et al. [8]. The NHPP-gamma-2 model has the mean value function m(t) = 1 ? (1 + t)e? t : We often write m(t) for m(tj ; ) for short. A similar abbreviation is used for (tj ; ). For the prior distributions, we assume has a gamma distribution ?(c; d) and has a gamma distribution ?(a; b). Moreover, the two distributions are independent. The gamma priors are assumed for convenience. Nevertheless, they are quite versatile to re ect densities with increasing or decreasing failure rates. In fact, our Metropolis-withinGibbs algorithm is applicable for any prior for . The posterior density is ! n Y ? x xie i e? ?e? t t c? e?d a? e?b : (2:2) (; jDt) / 2
[1
(1+
)]
1
1
i
=1
For the failure truncated model, similar expressions to (2.1) and (2.2) can be applied with t replaced by xn. For Bayesian inference, we propose two Gibbs samplers. The Gibbs sampler is a Markov chain Monte Carlo (MCMC) technique. The transition distribution of this Markov chain is the product of several conditional densities. The stationary distribution of the chain is the posterior distribution we desire. We can also replicate this chain with independent starting points to obtain multiple samples from the posterior distribution. Please refer to Geman and Geman [11], Tanner and Wong [23], Gelfand and Smith [9], Tierney [22], Casella and George [4], Chen and Singpurwalla [5], and Kuo and Yang [17] for detailed discussions of Gibbs sampling. 5
We describe brie y the Gibbs sampler. Suppose we desire to estimate f (U ; U ; : : :; UpjDt), the posterior joint density of (U ; U ; : : :; Up); given the data Dt . The algorithm assumes that we can generate variates from the conditional densities f (Ui jfUj gj6 i ; Dt). The algorithm proceeds as follows. Start with initial values, U ; U ; : : : ; Up . Generate a value U from the conditional density f (U jU ; : : :; Up ; Dt): (2:3) 1
1
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=
(0)
(0)
1
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(0)
1
(1)
(0)
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(0)
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Similarly, generate a value U from the conditional density (1)
2
f (U jU ; U ; : : :; Up ; Dt ); 2
(1)
(0)
1
3
(0)
and continue until the value Up is generated from the conditional density (1)
f (UpjU ; : : :; Up? ; Dt): (1)
(1)
1
1
With this new realization of the values U = (U ; : : :; Up ) replacing the old values, we can continue to iterate until the I th iteration. Under very mild regularity conditions, this Markov chain converges to a stationary distribution for large I . The vector (U I ; : : :; UpI ) has a distribution that is approximately equal to f (U ; : : :; UpjDt). By starting with independent initial choices, we can replicate the above iterations S times. Let U i;s = (U i;s ; : : :; Upi;s ) denote the realization of U for the ith iteration and the sth replication. The posterior moments, functionals, and credible sets can be computed from the empirical measure assigning weight 1=S to each (U I;s ; : : : ; UpI;s ); s = 1; : : : ; S . Alternatively as suggested by Gelman and Rubin [10], one can use the the empirical measure assigning weight 2=(IS ) to each (U i;s ; : : :; Upi;s ); i = I + 1; : : : ; I and s = 1; : : : ; S , for suciently large S and even I . When the conditional densities are not easily identi ed, the Metropolis [19] algorithm or importance sampling methods can be employed. We describe the Metropolis algorithm brie y. Suppose we desire to sample a variate from the target density (2.3). Let us consider the generic density f (U jU ; : : : ; Up; Dt ) = R 1 ff((DDtjjUU ;;:::::;: ;UUp))((UU ;;UU ;;::::::;;UUp))dU ; (2:4) t p p ?1 where is the prior density of (U ; : : :; Up). Let f (U ) denote the conditional density in (2.4), suppressing the conditioning variables for brevity. Let us de ne a transition kernel q(U ; X ) which maps U to X . If U is a real (1)
(1)
(1)
1
( )
( )
1
1
(
)
(
)
(
1
(
)
(
)
1
(
)
(
)
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)
variable with range in (?1; 1), we can construct q such that X = U + 0Z with Z being the standard normal random variate and 0 re ecting the conditional variance of U in (2.4). Then the Metropolis algorithm proceeds as follows: step 1: start with any point U , and stage indicator j = 0; step 2: generate a point X according to the transition kernel q(U j ; X ); step 3: update U j to U j = X with probability p = minf1; f (X )=f (U j )g, stay at U j with probability 1 ? p; step 4: repeat steps 2 and 3 by increasing the stage indicator until the process reaches a stationary distribution. Chapter 9 of Hammersley and Handscomb [13] provides a discussion of why this algorithm works. Note that this algorithm is de ned by using the ratio of two values of (2.4). Therefore, all we need is to know the functional form of the likelihood and prior. This spares us the task of evaluating the normalizing constant. If U is a variable with range in R , we can use a transformation, such as U 0 = log U ; to map (0; 1) into (?1; 1), then use the above transition kernel and apply the Metropolis algorithm to the density of U 0. After one trajectory of the Metropolis algorithm is done (j is suciently large), then we transform U 0 back to the original scale by means of U = eU 0 . In the following we only specify the conditional densities used in the Gibbs sampler. Method 1. Metropolis-within-Gibbs algorithm: Let ; denote U ; U in the above discussion. Then sample given ; Dt by the Metropolis algorithm with the following target density, Pn f ( j; Dt) / a n? e? b i xi t e? t ; 1
2
1
(0)
1
( )
1
( )
( +1)
( )
1
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( )
1
+
1
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+2
1
( +
=1
)+ (1+
)
j ; Dt ?(n + c; d + 1 ? e? t(1 + t)). Method 2. Gibbs sampler with data augmentation: We introduce two latent variables that help us specify the conditional distributions used in the Gibbs sampler. Let F (tj ) = 1 ? e? t(1 + t). The expression exp (?F (tj )) in the posterior (2.2) usually prevents us from specifying a convenient form for the conditional density of given and Dt . Therefore, we introduce the latent variable N 0 = N ? n which has a Poisson distribution with parameter (1 ? F (tj )). Then, the posterior distribution 7
f (; jDt) can be obtained from the joint density f (; ; N 0jDt) by marginalization. The latter density is approximated from the Gibbs sample drawn from the following conditional densities iteratively: f (N 0 j; ; Dt); f (jN 0; ; Dt); and f ( jN 0; ; Dt). While the rst two densities can be speci ed straightforwardly, the third one is still dicult. Therefore, we introduce another latent variable denoted by z to facilitate the generation of z given ; N 0; Dt and the generation of given z; N 0; Dt . The latent variable z is introduced to convert an aggregated model into a product of its components, so the generation of given ; N 0; z; Dt is easy. We now list the conditional densities used in the transitional measure of the Markov chain. Note N 0; z; ; denote U ; : : :; U in the Gibbs sampler discussed earlier. 1
4
N 0j; ; z; Dt poim(N 0; (1 + t)e? t); zj; ; N 0; Dt binm(z; 1=(1 + t); N 0); j ; z; N 0; Dt ?(c + n + N 0; d + 1); and jz; ; N 0; Dt ?(a + 2n + N 0 ? z; b + tN 0 + Pni xi). =1
3. Gibbs Sampler for the Generalized Gamma Class The generalized gamma class is a class of NHPP with intensity functions k (tj; ) = ?(k) tk? e? t; where the shape parameter k in the gamma density is assumed known. Observe the mean value function for the NHPP-gamma-k model is 0 1 Zt kX ? ( t)j A m(t) = (u)du = @1 ? e? t j! : j 1
1
0
=0
This Poisson probability representation for the gamma distribution in m(t) will help us in de ning the data augmentation in the stochastic substitution step of the Gibbs sampler. We assume the same prior distributions as in the NHPP-gamma-2. Therefore, the posterior density for the NHPP-gamma-k model is ! n Y ? t Pk? j k k ? ? x i e? ?e j t =j c? e?d a? e?b : (3:1) xi e (; jDt ) / 1
[1
i
=1
8
1 ( =0
)
!]
1
1
We describe two Gibbs samplers that are similar to those for the NHPP-gamma-2 model. In the following we only list the conditional densities used in the Gibbs samplers. We assume k 3 because the Gibbs sampler for k = 1 is given in Kuo and Yang [16] and the Gibbs sampler for k = 2 is given in Section 2. Method 1. Metropolis-within-Gibbs algorithm: sample given ; Dt by the Metropolis algorithm with the following target density, kX ? n X f ( j; Dt) / a kn? expf? (b + xi) + e? t ( t)j =j !g; 1
+
1
j
i
=1
=0
j ; Dt ?(n + c; d + 1 ? e? t Pjk? ( t)j =j !). 1
=0
Method 2. Gibbs sampler with data augmentation: We will introduce two latent variables that are modi ed versions of the ones used for the NHPP-gamma-2. Let F (tj ) = 1 ? e? t Pjk? ( t)j =j !. We rst introduce the latent variable N 0 = N ? n which has a Poisson distribution with parameter (1 ? F (tj )). Although this is in the right direction, another data augmentation step is needed to facilitate the speci cation of the conditional density of . Let z = (z ; : : : ; zk? ), and p = (p ; : : :; pk? ), where 1
=0
0
1
0
1
j pj = Pk(? t) =jj! ; j ( t) =j ! for j = 0; : : : ; k ? 1. Then we will generate z from a multinomial distribution with parameters N 0 and cell probabilities (p ; : : :; pk? ). Note N 0 = z + + zk? . The latent variable z is introduced to convert an aggregated model into a product of its components, so the conditional distribution of given ; N 0; z; Dt is easily obtained. We now list the conditional densities used in the transitional measure of the Markov chain. N 0j; ; z; D poim(N 0; e? t Pk? ( t)j =j !); 1
=0
0
1
t
0
j
1
1
=0
zj; ; N 0; Dt mult((p ; : : :; pk? ); N 0); 0
1
j ; z; N 0; Dt ?(c + n + N 0; d + 1); and jz; ; N 0; Dt ?(a + kn + z + 2z + + (k ? 1)zk? ; b + tN 0 + Pni xi). 1
2
1
9
=1
4. Bayesian Inference In this section, we describe the Bayes estimators. In addition to making inference on the unknown parameters, we are interested in predicting the s-expected number of remaining errors and the current reliability function. For inference on the parameters, such as and , we can use the empirical measure of the sample generated by the Gibbs sampler (cf. Gelman and Rubin, [10]). Now we study prediction. We illustrate it by using the NHPP-gamma-2 model. Prediction for the generalized gamma models is done similarly with the appropriate mean value functions. The s-expected number of remaining errors is de ned to be (t) = m(1) ? m(t) = (1 + t)e? t for the NHPP-gamma-2 model. Therefore, the Bayes estimator for it with respect to the squared error loss is E ((t)jDt) = E (1 + t)e? tjDt : Let i;s ; i;s denote the variates for and drawn in the ith iteration and the sth replication of the Gibbs sampler. Then E ((t)jDt) can be estimated by means of the Monte Carlo integration I S X 2 X ^(t) = IS i;s (1 + i;s t)e? i;s t: (4:1) s i I (
(
=1
=
2
)
(
)
(
)
(
)
)
+1
Bayes inference for m(t) can be carried out similarly. For inference for the reliability function, it would be easier if we consider the failure truncated situation, i.e., testing until the nth failure. All the posterior distributions described in Sections 2 and 3 should be modi ed by replacing t with xn. Inference for the current reliability function evaluated at x distance away from xn can be obtained by
E (R(x)jDxn ) = E fE (P (Xn > xn + x)j ; ; Dxn )jDxn g = E fexp (?m(xn + x) + m(xn)) jDxn g ZZ n o = exp [(1 + (xn + x))e? xn x ? (1 + xn)e? xn ] (; jDxn )dd : (4.2) +1
(
+ )
Equation (4.2) can be estimated by the Monte Carlo integration similar to (4.1). The second equality of (4.2) is due to the Markovian property of the waiting times in NHPP (cf. Cinlar [6], p. 97). Bayesian credible regions for the unknown parameters and Bayesian con dence intervals for the functionals such as the reliability function and the s-expected number of remaining 10
errors can be computed using the quantiles of the unknown parameters and the functionals evaluated from the second half of the Gibbs sample.
5. Model Selection We have presented a class of NHPP-gamma-k. It is natural to ask what is the best k value for a particular data set. Khoshgoftaar and Woodcock [15] study the Akaike Information criterion [2] for selecting the best model. We explore the posterior Bayes factor criterion for model selection. Let us de ne the posterior log marginal likelihood to be the average of the log marginal likelihood with respect to the posterior distribution based on the whole data set Dt for the NHPP-gamma-k model: Z Ik = log (LNHPP ?k (; jDt)) k (; jDt)dd ; (5:1) where L (; jD ) is obtained from (2.1) with m(t) = 1 ? e? t Pk? ( t)j =j ! . Here NHPP ?k
t
j
1
=0
k is the posterior joint density of and for the NHPP-gamma-k model. The posterior log Bayes factor criterion says that we prefer the model with k to the model with k , if Ik < Ik . The integrals Ik and Ik can easily be approximated by applying the Monte Carlo integration technique; this is, for l = 1; 2, Ikl is estimated by averaging log (LNHPP ?kl (; jDt)) over the ; 's drawn in the second half of the iteration and all replications of the Gibbs sampler. 1
1
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2
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6. A Numerical Example Some numerical results for Bayesian inference and model selection are given here for a simulated data set. We simulate the data from the Ohba-Yamada (NHPP-gamma-2) model. Then we t the data with the NHPP-gamma-k models for k = 1; : : : ; 4. Model selection among the four models is also considered. To simulate the data of 100 failures from a NHPP with the mean function m(t) = ? t 1 ? e (1 + t) with = 100 and = 0:1, we use the IMSL RNNPP routine which applies the thinning method proposed by Lewis and Shedler [18]. The simulated data are plotted in Figure 1, xi versus i for i = 1; : : : ; 100. 11
We consider relatively di use proper priors so that the numerical results are more focused on the likelihood. The priors for and are ?(1; 0:001) and ?(1; 0:001), respectively. Given the above data, we consider the failure truncated situation (n = 100) in our analysis. We monitor the convergence of the Gibbs sampler using the Gelman and Rubin [10] method that uses the analysis of variance technique to determine whether further iterations are needed. We found 20,000 iterations to be large enough for the priors being considered. All the following numerical results are obtained with 20,000 iterations and 50 replications in the Gibbs sampler. Table 1 lists the posterior means for the parameters. The Bayes estimates f^; ^g for the NHPP-gamma-2 model (in the k = 2 column) are reasonably close to the true values that generate the data. In addition to tting the data set with the k = 2 model, we also t it with k = 1; 3; and 4. As expected, the Bayes estimates are comparable to the maximum likelihood estimates (computed by the MATLAB CONSTR routine) for each k because of the relatively di use prior. In model selection, we choose the best model among the four models. Table 2 lists the log of the maximum likelihoods and the posterior log marginal likelihoods. The table shows that, as we would expect, the NHPP-gamma-2 model is the best. The Bayes estimate for the current reliability function for the NHPP-gamma-2 model as a function of time, measured from the last observed failure time 58.51, is plotted in Figure 3. The Bayes estimate for the s-expected number of remaining errors (t) plotted as a function of t is given in Figure 4. To compare the Bayes estimate of the mean function m(t) to the known function 100 (1 ? exp(?0:1t)(1 + 0:1t)), we plot each of them as a function of t as in Figure 4. The gure shows that the Bayes estimate of the mean function is quite close to the true mean function.
References [1] M. Aitkin, \Posterior Bayes factors," J. Roy. Statist. Soc. Ser. B, 1991, pp 111-142. [2] H. Akaike, \A New Look at Statistical Model Identi cation," IEEE Trans. Automatic Control, AC-19, 1974, pp 716-723. 12
[3] I. V. Basawa, B. L. S. Prakasa Rao, Statistical Inference for Stochastic Processes, 1980; Academic Press. [4] G. Casella, E.I. George, \Explaining the Gibbs sampler," The American Statistician, 46, 1992, pp 167-174. [5] Y.P. Chen, N. D. Singpurwalla, \A Non-Gaussian Kalman lter model for tracking software reliability," Statistica Sinica, 4, 1994, pp 535-548. [6] E. Cinlar, Introduction To Stochastic Process, 1975; New Jersey, Prentice-Hall. [7] D.R. Cox, P. A. Lewis, Statistical Analysis of Series of Events, 1966; Methuen. [8] M.J. Crowder, A.C. Kimber, R.L. Smith, T.J. Sweeting, Statistical Analysis of Reliability, 1991; Chapman and Hall. [9] A.E. Gelfand, A.F.M. Smith, \Sampling-based approaches to calculating marginal densities," J. Amer. Statist. Assoc. 85, 1990, pp 398-409. [10] A.E. Gelman, D. Rubin , \Inference from iterative simulation using multiple sequences," Statistical Science, 7, 1992, pp 457-472. [11] S. Geman, D. Geman, \Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images," IEEE Trans. Pattern Analysis and Machine Intelligence, 6, 1984, pp 721-741. [12] A. L. Goel, K. Okumoto, \Timed dependent error detection rate model for software reliability and other performance measures," IEEE Trans. Reliability, 28, 1979, pp 206211. [13] J.M. Hammersley, D. C. Handscomb, Monte Carlo Methods, 1964; London, Chapman and Hall. [14] Z. Jelinski, and P. B. Moranda, \Software Reliability Research," in Statistical Computer Performance Evaluation, ed. W. Freiberger, 1972; Academic Press, pp 465-497. 13
[15] T. M. Khoshgoftaar, T. G. Woodcock, \Software reliability model selection, a case study," Proc. Int. Symp. on Software Reliability Eng., 1991, pp 183-191. [16] L. Kuo, T.Y. Yang, \Bayesian computation for nonhomogeneous Poisson processes in software reliability," To appear in J. Amer. Statist. Assoc. [17] L. Kuo, T.Y. Yang, \Bayesian computation of software reliability," J. Comput. Graph. Statist. 4, 1995, pp 65-82. [18] P. A. W. Lewis, G. S. Shedler, \Simulation of Nonhomogeneous Poisson Processes by Thinning," Naval Research Logistic Quarterly, 26, 1979, pp 403-413. [19] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, \Equation of state calculations by fast computing machines," J. Chem. Physics, 21, 1953, pp 10871092. [20] M. Ohba, S. Yamada, K. Takeda, S. Osaki, \S-shaped software reliability growth curve: how good is it?" 1982, COMPSAC'82, pp 38-44. [21] A. E. Raftery, \Inference and prediction for a general order statistic model with unknown population size," J. Amer. Statist. Assoc., 82, 1987, pp 1163-1168. [22] L. Tierney, \Markov chains for exploring posterior distributions," Ann. Statistist., 1994, pp 1701-1725. [23] M. Tanner, W. Wong, \The calculation of posterior distributions by data augmentation," J. Amer. Statist. Assoc., 82, 1987, pp 528-550. [24] S. Yamada, M. Ohba, S. Osaki, \S-shaped reliability growth modeling for software error detection," IEEE Trans. Reliability, 32, 1983, pp 475-478. [25] T. Y. Yang, L. Kuo, \ Bayesian Computation for the Superposition of Nonhomogeneous Poisson Processes," in revision for J. Amer. Stat. Association.
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AUTHORS Dr. Lynn Kuo; Dept. of Statistics; University of Connecticut; Storrs, Connecticut 062693120 USA. Internet:
[email protected] Lynn Kuo is an associate professor at the University of Connecticut. She received her B.A., M.A., and Ph.D. (1980) in Mathematics from the University of California at Los Angeles. She has been a system analyst at the Jet Propulsion Lab., a visiting scholar at Stanford University, and an adjunct professor at the Naval Postgraduate School. She has worked on Bayesian computation for software reliability, regression models, and survival analysis. Dr. Jaechang Lee; Dept. of Statistics; Korea University; Anam-dong, Sungbuk-ku, Seoul, Korea. Lee Jaechang was a president of Korean Statistical Association. He received a B.S. in Economics in 1967, an M.S. in Mathematics in 1969, and a Ph.D. in Statistics in 1972 from Ohio State University. He has done research in nonparametric statistics and reliability. Dr. Kiheon Choi; Dept of Statistics; Duksung Women's University; 419 Ssangmoon-dong, Tobong-ku, Seoul, Korea, 132-714. Kiheon Choi is chairman in the Statistics Department at Duksung Women's University, Seoul, Korea. He received a B.S. in Mathematics in 1978, an M.S. in Statistics in 1980 from Korea University, and a Ph.D. in Statistics in 1988 from the University of Michigan at Ann Arbor. He has done research in sequential analysis and Bayesian reliability. Dr. Tae Young Yang; Dept. of Statistics; University of Missouri; Columbia, Missouri 65211-5211 USA. Tae Young Yang is a visiting assistant professor at the University of Missouri. He received a B.S. in Mathematics in 1985 from Korea University, a M.S. in Statistics from the University of Vermont in 1987, and a Ph.D. in Statistics in 1994 from the University of Connecticut. He has done research in software reliability and Bayesian inference for stochastic point processes. He has been a visiting scholar at Stanford University. 15
Table 1: Comparison between the Bayes Estimates and the Maximum Likelihood Estimates k=1 Posterior Mean ^ = 110:65 ^ = 0:043 MLE ~ = 108:73 ~ = 0:043
k=2 ^ = 102:46 ^ = 0:107 ~ = 101:46 ~ = 0:106
k=3 ^ = 101:32 ^ = 0:166 ~ = 100:38 ~ = 0:164
k=4 ^ = 101:04 ^ = 0:222 ~ = 100:11 ~ = 0:221
Table 2: Model Selection k=1 k=2 k=3 k=4 Posterior Log Marginal Likelihood -24.51 -15.60 -20.44 -31.14 Log Maximum Likelihood -23.51 -14.63 -19.45 -30.16
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Figure 1: Simulated Data
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Figure 3: Bayes Estimate for the s-Expected Number of Remaining Errors
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Figure 4: Bayes Estimate of the Mean Function
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................................................. .. .. .. .. .. .. . .. ... .......... . . . . . . . ...... . . . . .... . . . .. ... . . ... . . .. ... .. .. . .. .. . .. solid line: Bayes estimate ... .. . dotted line: 100[1-exp(-0.1t)(1+0.1t)] .. . . .. . . .. . . .. . .. .... 0
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