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This paper presents a hierarchical Bayesian approach to the estimation of components’ reliability (survival) using a Weibull model for each of them. The proposed method can be used to estimation with general survival censored data, because the estimation of a component’s reliability in a series (parallel) system is equivalent to the estimation of its survival function with right- (left-) censored data. Besides the Weibull parametric model for reliability data, independent gamma distributions are considered at the first hierarchical level for the Weibull parameters and independent uniform distributions over the real line as priors for the parameters of the gammas. In order to evaluate the model, an example and a simulation study are discussed.

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 [

For parallel systems, the literature is scarce. To the best of our knowledge, Polpo and Pereira [

In a related work, Polpo et al. [

Similarly to Polpo et al. [

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 [

We use the same notation as in Polpo et al. [_{j},

Consider a sample of n independent and identically distributed systems (either all series or all parallel systems). The observations are represented by

ponent is given by

We define random variables

for

and for the parallel system,

where f is the density function of a random variable with Weibull distribution,

The prior distributions were considered independent with

where

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

and

where

For the estimation, we use the EM algorithm [

1) Choose the prior precision values

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

3) Using the initial guess, consider _{p}) from the posterior distribution of

4) (Expectation step of the EM) Using the posterior sample of

5) (Maximization step of the EM) Find the values of

6) Update the initial guess of

7) Repeat Steps 3, 4, 5 and 6 until convergence of

8) Once convergence of

Using this algorithm, we obtain the posterior mode (the Bayesian estimate) of

erior of the parameters of component 1 can be expressed as

tain the reliability estimates and credible regions, consider the functions

Hence, for each fixed t,

where

Consider three random variables X_{1}, X_{2}, X_{3} such that X_{1} has Weibull distribution with mean 2 and variance 4, X_{2} has gamma distribution with mean 2 and variance 0.667, and X_{3} has log-normal distribution with mean 2.014 and variance 6.968. We have generated a sample (with size n = 100) of series systems with these three components and another sample (again with size n = 100) of parallel systems with the exactly same three components. The components were chosen in order to have similar means but different variances and, consequently, different distributions. We have used the same theoretical components in both simulations (series and parallel systems) to verify, in each situation, the differences that are due to the distinct system models with the available data. The simulated data have the following characteristics: 1) for the series systems, we have obtained 64%, 80%, and 56% of censured data for components 1, 2, and 3, respectively; and 2) for the parallel systems, we have observed 61%, 68% and 71%, respectively for the same three components. In this case, the main interest is in the estimation of the components’ reliability functions. Note that, with our simulated example, we have a huge amount of censored data, making it a challenging example.

As already said, the estimation procedures are performed using MCMC. We have discarded the first 10,000 samples (as burn-in) from the posterior to achieve the stationary measure and then have generated a sample from the posterior. To perform the estimation of the reliability functions and the credible region, we have used a sample of size 1000 from the posterior, which was obtained by discarding 10 samples (the jump between each final sample point). We have used

ment with series systems, we have obtained

these estimates, we have compared the “true” reliability of each component with the estimated reliability function.

It can be seen from the 95% credible bounds that the “true” reliability of each component was well estimated. We note however that the “true” reliability functions of the components are, for short (time) intervals, outside the 95% credible bounds. Considering that these are reasonably challenging examples, this situation is likely to happen in any estimation procedure (see

This example has a simulation study to show the quality of the proposed hierarchical Weibull model over many different conditions. We have considered 108 different scenarios that were built using three different sample sizes

To summarize and to compare the results of the simulation, we have evaluated the bias and the mean squared error (MSE) of the estimated mean reliability time, for each scenario. The results are presented in Tables 2-7. One can see from the results that all biases and MSEs are close to zero, indicating that, in all scenarios, the model has estimated well the true mean reliability time. Even in the most challenging scenarios, which are those with small sample size

. Posterior mean (sd) of some quantities involved in the estimation of the simulated examples

Series system estimates | |||
---|---|---|---|

β | η | ||

Component 1 | 1.26 (0.17) | 2.27 (0.41) | 2.13 (0.45) |

Component 2 | 3.98 (0.53) | 2.06 (0.12) | 1.87 (0.11) |

Component 3 | 1.40 (0.17) | 1.83 (0.22) | 1.68 (0.22) |

Parallel system estimates | |||

β | η | ||

Component 1 | 1.23 (0.15) | 2.60 (0.27) | 2.45 (0.22) |

Component 2 | 2.47 (0.32) | 2.25 (0.14) | 2.00 (0.13) |

Component 3 | 0.87 (0.11) | 1.98 (0.33) | 2.17 (0.29) |

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

Reliability of the components in the experiment with series systems: (a) Component 1; (b) Component 2; (c) Component 3

Reliability of the components in the experiment with parallel systems: (a) Component 1; (b) Component 2; (c) Component 3

. Bias and mean squared error (MSE) of the E(T) estimate for data generated from the Weibull distribution with right-censored data

Cens. | True E(T) | n = 30 | n = 100 | n = 1000 | |||
---|---|---|---|---|---|---|---|

Bias | MSE | Bias | MSE | Bias | MSE | ||

0% | 2 | −0.1300 | 0.2217 | −0.0426 | 0.0572 | −0.0057 | 0.0043 |

0% | 7 | 0.0293 | 0.1523 | 0.0026 | 0.0474 | −0.0095 | 0.0043 |

20% | 2 | −0.2813 | 0.4893 | −0.1242 | 0.1275 | −0.0200 | 0.0068 |

20% | 7 | −0.0522 | 0.2307 | −0.0233 | 0.0565 | −0.0148 | 0.0047 |

40% | 2 | −0.4293 | 0.7364 | −0.2394 | 0.2925 | −0.0239 | 0.0108 |

40% | 7 | −0.1773 | 0.3590 | −0.0661 | 0.0874 | −0.0162 | 0.0055 |

. Bias and mean squared error (MSE) of the E(T) estimate for data generated from the gamma distribution with right-censored data

Cens. | True E(T) | n = 30 | n = 100 | n = 1000 | |||
---|---|---|---|---|---|---|---|

Bias | MSE | Bias | MSE | Bias | MSE | ||

0% | 2 | −0.1407 | 0.2286 | −0.0191 | 0.0593 | −0.0003 | 0.0055 |

0% | 7 | 0.0006 | 0.1845 | 0.0081 | 0.0623 | 0.0098 | 0.0048 |

20% | 2 | −0.3829 | 0.6346 | −0.1132 | 0.1234 | −0.0560 | 0.0111 |

20% | 7 | 0.0569 | 0.1721 | 0.1121 | 0.0669 | 0.1049 | 0.0153 |

40% | 2 | −0.5068 | 0.9424 | −0.3084 | 0.3807 | −0.1174 | 0.0299 |

40% | 7 | 0.1389 | 0.2505 | 0.2372 | 0.1107 | 0.2336 | 0.0582 |

. Bias and mean squared error (MSE) of the E(T) estimate for data generated from the log-normal distribution with right-censored data

Cens. | True E(T) | n = 30 | n = 100 | n = 1000 | |||
---|---|---|---|---|---|---|---|

Bias | MSE | Bias | MSE | Bias | MSE | ||

0% | 2 | −0.1160 | 0.1757 | −0.0024 | 0.0396 | −0.0089 | 0.0050 |

0% | 7 | −0.0348 | 0.1551 | 0.0462 | 0.0429 | 0.0220 | 0.0052 |

20% | 2 | 0.0720 | 0.1410 | 0.2475 | 0.0779 | 0.2523 | 0.0655 |

20% | 7 | 0.0786 | 0.1492 | 0.1848 | 0.0637 | 0.1620 | 0.0296 |

40% | 2 | 0.2373 | 0.1795 | 0.4205 | 0.1981 | 0.4371 | 0.1928 |

40% | 7 | 0.2190 | 0.2094 | 0.3438 | 0.1532 | 0.3337 | 0.1147 |

. Bias and mean squared error (MSE) of the E(T) estimate for data generated from the Weibull distribution with left- censored data

Cens. | True E(T) | n = 30 | n = 100 | n = 1000 | |||
---|---|---|---|---|---|---|---|

Bias | MSE | Bias | MSE | Bias | MSE | ||

0% | 2 | −0.1300 | 0.2217 | −0.0426 | 0.0572 | −0.0057 | 0.0043 |

0% | 7 | 0.0293 | 0.1523 | 0.0026 | 0.0474 | −0.0095 | 0.0043 |

20% | 2 | 0.1030 | 0.1178 | 0.0034 | 0.0475 | −0.0060 | 0.0043 |

20% | 7 | 0.0387 | 0.1562 | −0.0001 | 0.0490 | −0.0094 | 0.0043 |

40% | 2 | −0.1813 | 0.2660 | −0.0471 | 0.0579 | −0.0065 | 0.0043 |

40% | 7 | 0.0301 | 0.1813 | 0.0040 | 0.0597 | −0.0099 | 0.0048 |

. Bias and mean squared error (MSE) of the E(T) estimate for data generated from the gamma distribution with left- censored data

Cens. | True E(T) | n = 30 | n = 100 | n = 1000 | |||
---|---|---|---|---|---|---|---|

Bias | MSE | Bias | MSE | Bias | MSE | ||

0% | 2 | −0.1407 | 0.2286 | −0.0192 | 0.0594 | −0.0003 | 0.0055 |

0% | 7 | 0.0006 | 0.1845 | 0.0081 | 0.0623 | 0.0098 | 0.0048 |

20% | 2 | 0.1029 | 0.1038 | 0.0448 | 0.0490 | 0.0008 | 0.0054 |

20% | 7 | 0.1108 | 0.2143 | 0.0950 | 0.0739 | 0.0932 | 0.0139 |

40% | 2 | −0.1934 | 0.2463 | −0.0245 | 0.0602 | −0.0029 | 0.0055 |

40% | 7 | 0.2185 | 0.2842 | 0.2105 | 0.1242 | 0.1850 | 0.0402 |

. Bias and mean squared error (MSE) of the E(T) estimate for data generated from the log-normal distribution with left-censored data

Cens. | True E(T) | n = 30 | n = 100 | n = 1000 | |||
---|---|---|---|---|---|---|---|

Bias | MSE | Bias | MSE | Bias | MSE | ||

0% | 2 | −0.1160 | 0.1757 | −0.0024 | 0.0396 | −0.0089 | 0.0050 |

0% | 7 | −0.0348 | 0.1551 | 0.0462 | 0.0429 | 0.0220 | 0.0052 |

20% | 2 | −0.0092 | 0.1253 | 0.0254 | 0.0405 | 0.0211 | 0.0052 |

20% | 7 | 0.1003 | 0.1701 | 0.1757 | 0.0784 | 0.1375 | 0.0240 |

40% | 2 | −0.1165 | 0.1937 | 0.0575 | 0.0441 | 0.0624 | 0.0086 |

40% | 7 | 0.2359 | 0.2727 | 0.3138 | 0.1548 | 0.2763 | 0.0825 |

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.