This paper presents a hierarchical Bayesian approach to the estimation of components’ reliability using a Weibull model for each component in series and parallel systems. A series system is a frame of components that works if and only if all its components are functional, that is, whenever one fails the system fails. As a dual frame, the parallel system fails if and only if all components are malfunctioning.

The literature dealing with the problem of estimating the reliability of series systems, or competing risks, is abundant. The work of Kaplan and Meier [1] is arguably the most celebrated work, where it was developed a nonparametric estimator using a frequentist approach. For a Bayesian counterpart, we draw the attention to Salinas-Torres et al. [2] and Polpo and Sinha [3] . Rodrigues et al. [4] performed a simulation study of three different methods to estimate the reliability of a series system. They compared the Kaplan-Meier estimator [1] , maximum likelihood estimator (MLE) and the Bayesian plug-in estimator (BPE) for Weibull reliability systems. Their results indicated that MLE and BPE are similar in quality and that both outperformed the Kaplan-Meier estimator. However, the construction of credible bounds was not addressed in their work.

For parallel systems, the literature is scarce. To the best of our knowledge, Polpo and Pereira [5] were the first to address the nonparametric reliability estimation in parallel systems and their components, using the Bayesian paradigm. Later on, Bhattacharya and Samaniego [6] proposed a frequentist nonparametric estimator for components’ reliability under a restrictive condition that all components are independent and identically distributed.

In a related work, Polpo et al. [7] presented the reliability estimation with Weibull models and non-informa- tive priors, also using the Bayesian paradigm. Their proposal had very demanding computational needs, and the described Markov Chain Monte Carlo (MCMC) algorithm suffered from convergence issues in many problem instances, making the estimation a difficult task. The authors realized later on that it is often impossible to evaluate the components’ reliability with that method because of such issues, and unfortunately one cannot predict when the algorithm will succeed. The present work aims at achieving a robust estimation procedure for the estimation problem. We suggest considering non-informative priors in a one-level hierarchical model as means to solve the estimation issues faced by the algorithm of Polpo et al. [7] . Moreover, an important goal of this work is to provide a simple way to build credible bounds for the reliability functions, giving a step-by-step algorithm that performs such analysis.

Similarly to Polpo et al. [7] and Rodrigues et al. [4] , we consider Weibull statistical models for the system’s reliability. Two independent gamma distributions are considered in the first hierarchical level. The gamma distributions are parametrized by their means and variances, instead of the more common parametrization with scale and shape. In the second level of the hierarchy, we choose two flat priors for the means, and fixed values for the variances of the gamma distributions in the first level. These hyper-parameters corresponding to variances in the first level can be seen as prior precision parameters. The means of the gamma distributions (first level of the hierarchy) can be viewed as the prior expectations of the parameters of the Weibull model. In this work, the posterior modes are taken as the Bayesian estimators of the gamma distributions.

The estimation of the reliability functions has three main steps: 1) we draw a sample from the posterior distribution of the Weibull parameters; 2) using the appropriate transformation, we build a sample from the reliability posterior distribution; and 3) locally, for each reliability time, we evaluate the posterior mean. The high posterior density (HPD) procedure was used to define the credible region for the reliability function. We emphasize that we are not using the plug-in estimator, but the posterior mean of the reliability function, which seems more suitable under the Bayesian paradigm.

This paper is organized as follows. Section 2 describes all functions that are involved in the estimation procedure. Section 3 provides the estimation procedure itself. Section 4 presents a simulation study that highlights the quality of the model and the proposed estimators, and final remarks and additional comments are given in Section 5. We note that an extended abstract of this work has appeared in the Brazilian Conference on Bayesian Statistics [8] .

We use the same notation as in Polpo et al. [7] . Consider a system of k components and let X_j, , be the sequence of failure times of all components. We assume that this sequence is composed of independent random variables with a (possibly) different Weibull distribution for each component. Recall that we only observe a random vector of two variables, namely, with for the series system and for the parallel system, with if, for. The quantity can be viewed as an indicator function of the component that caused the system failure.

Consider a sample of n independent and identically distributed systems (either all series or all parallel systems). The observations are represented by. The reliability function of the jth com-

ponent is given by,. Therefore, the reliability function is for the series system and for the parallel system.

We define random variables for the components’ reliability with Weibull distributions parametrized by, that is,

for, (shape) and (scale). Then, the likelihood function for the series system is given by

and for the parallel system,

where f is the density function of a random variable with Weibull distribution, , and is the indicator function of the set A.

The prior distributions were considered independent with, ,

, and and as known constants,. Then

where, and are the prior mean parameters,

and are the prior variance (precision) parameters, and,.

In this case, we have that the posterior distributions of series and parallel systems are, respectively,

and

where are the observed failure time of the system, are the indicators of which component failed, and the other quantities are as defined before.

For the estimation, we use the EM algorithm [9] to obtain the posterior mode as estimates of and, and the MCMC procedure to generate a sample from the posterior distribution of and. The algorithm steps for the estimation are briefly given as follows:

1) Choose the prior precision values and. Note that one can set to the same value all the precision values, that is,. We suggest the use of.

2) Choose the initial guess for the parameters to be estimated:, , and.

3) Using the initial guess, consider and as fixed values. Employing the MCMC tool, generate a sample (of size n_p) from the posterior distribution of and. We suggest the use of. It may also be necessary to use a “burn-in” and a “jump” to ensure convergence of the MCMC.

4) (Expectation step of the EM) Using the posterior sample of and obtained in Step 3, evaluate the mean of the likelihood function, obtaining a “mean” function of and.

5) (Maximization step of the EM) Find the values of and that maximize the function obtained in Step 4.

6) Update the initial guess of and with the values obtained in Step 5, and the values of and with their posterior mean obtained in Step 3.

7) Repeat Steps 3, 4, 5 and 6 until convergence of and is reached. We suggest the use of a tolerance value of three decimal places between the previous values of and and the current ones, in order to decide whether to stop iterating.

8) Once convergence of and is reached, use their values to generate a sample from the posterior distribution of and by applying the MCMC tool.

Using this algorithm, we obtain the posterior mode (the Bayesian estimate) of and, and a sample (of size) from the joint posterior distribution of and. If we estimate the reliability function of any component (let us arbitrarily choose component 1), then we can notice that the estimations of the other components’ reliability functions are very similar and could be omitted here. Consider that the sample from the post-

erior of the parameters of component 1 can be expressed as and. To ob-

tain the reliability estimates and credible regions, consider the functions, , where are Weibull distribution functions conditioned on and. Consequently, the posterior mean estimate of the component’s reliability can be expressed as

Hence, for each fixed t, is a sample from the posterior of the component 1’s distribution function and, to obtain the credible region, we can either use the quantiles of or evaluate the high posterior density credible interval. To estimate the mean reliability time, we have

where is the mean of a random variable with Weibull distribution, and is the gamma function. Note that similar procedures can be used to evaluate other quantities of interest; as an example, the posterior median could be evaluated.

We have introduced a Bayesian reliability statistical analysis using hierarchical models for the problem of estimating the reliability functions and credible bounds of series and parallel systems. The MCMC has shown good performance in terms of convergence, making the inference process simple and efficient. It shall be noted that this performance is not dependent on our choice of a “non-informative” scheme to define the prior hyper-para- meters. This is important because other researchers may want to fairly compare our method with other frequentist estimators. However, informative priors may very well produce additional improvements in the estimates. The Example 1 has shown good robustness in the sense that the model has performed well for all components in

both series and parallel systems. Another important aspect is that we can obtain credible bounds for the reliability function, task that is usually hard if one uses a plug-in estimator for the reliability function. The Example 2 provides an extensive simulation study with more than one hundred different scenarios. Overall, the model has performed very well for estimating the mean reliability time. Some open questions that should be addressed in future works are the development of hypothesis tests for the components, for instance, one can have interest in testing the hypothesis of equal means of all components (or a subset of components), and the extension of these ideas to more general coherent systems.