Bayesian accelerated life test plans for series systems with Weibull component lifetimes.

*(English)*Zbl 1462.62609Summary: This article presents optimal Bayesian accelerated life test plans for series systems under Type-I censoring scheme. First, the component lifetimes are assumed to follow independent Weibull distributions. The scale parameters of Weibull lifetime distributions are related to the external stress variable through a general stress translation function. For a fixed number of design points, optimal Bayesian ALT plans are first obtained by solving constrained optimization problems under two different Bayesian design criteria. The global optimality of the resulting fixed-point optimal designs is then verified via the General Equivalence Theorem. This article also provides the optimized compromise ALT plans which are extremely useful in real-life applications. A detailed sensitivity analysis is then performed to find out the effect of various planning inputs on the resulting optimal Bayesian ALT plans. A simulation study is then conducted to visualize the resulting sampling variations from the optimal Bayesian ALT plans. Finally, this article considers a series system with dependent component lifetimes. Optimal ALT plans are obtained assuming a Gamma frailty model.

##### MSC:

62N03 | Testing in survival analysis and censored data |

62N05 | Reliability and life testing |

62F15 | Bayesian inference |

62K05 | Optimal statistical designs |

##### Keywords:

Bayesian \(C\)-optimality; Bayesian \(D\)-optimality; competing risks; gamma frailty model; general equivalence theorem; Weibull distribution##### Software:

SPLIDA
Full Text:
DOI

##### References:

[1] | Nelson, W. B., Accelerated Testing: Statistical Models, Test Plans and Data Analysis (2004), Wiley: Wiley New York |

[2] | Nelson, W. B., A bibliography of accelerated test plans, IEEE Trans. Reliab., 54, 194-197 (2005) |

[3] | Nelson, W. B., A bibliography of accelerated test plans part II - references, IEEE Trans. Reliab., 54, 370-373 (2005) |

[4] | Pascual, F., Accelerated life test planning with independent Weibull Competing risks with known shape parameter, IEEE Trans. Reliab., 56, 85-93 (2007) |

[5] | Pascual, F., Accelerated life test planning with independent Weibull competing risks, IEEE Trans. Reliab., 57, 435-444 (2008) |

[6] | Pascual, F., Accelerated life test planning with independent lognormal competing risks, J. Stat. Plann. Inference, 140, 1089-1100 (2010) · Zbl 1179.62147 |

[7] | Liu, X., Planning of accelerated life tests with dependent failure modes based on a gamma frailty model, Technometrics, 54, 398-409 (2012) |

[8] | Chernoff, H., Locally optimum designs for estimating parameters, Ann. Stat., 24, 586-602 (1953) · Zbl 0053.10504 |

[9] | Chaloner, K.; Verdinelli, I., Bayesian experimental design: a review, Stat. Sci., 10, 273-304 (1995) · Zbl 0955.62617 |

[10] | Chaloner, K.; Larntz, K., Bayesian design for accelerated life testing, J. Stat. Plan. Inference, 33, 245-259 (1992) · Zbl 0781.62149 |

[11] | Polson, N. G., A Bayesian perspective on the design of accelerated life tests, (Basu, A. P., Advances in Reliability (1993), Elsevier: Elsevier New York), 321-330 |

[12] | Erkanli, A.; Soyer, R., Simulation-based designs for accelerated life tests, J. Stat. Plan. Inference, 90, 335-348 (2000) · Zbl 1080.62541 |

[13] | Zhang, Y.; Meeker, W. Q., Bayesian methods for planning accelerated life tests, Technometrics, 48, 49-60 (2006) |

[14] | Xu, A.; Tang, Y., A Bayesian method for planning accelerated life testing, IEEE Trans. Reliab., 64, 1383-1392 (2015) |

[15] | Atkinson, A.; Donev, A.; Tobias, R., Optimum Experimental Designs, with SAS, Oxford Statistical Science Series (2007), Oxford University Press: Oxford University Press USA · Zbl 1183.62129 |

[16] | Roy, S.; Mukhopadhyay, C., Bayesian \(D\)-optimal ALT plans for series systems with competing exponential causes of failure, J. Appl. Stat., 43, 1477-1493 (2016) |

[17] | Meeker, W. Q.; Escobar, L. A., Statistical Methods for Reliability Data (1998), Wiley: Wiley New York · Zbl 0949.62086 |

[18] | Mukhopadhyay, C.; Roy, S., Bayesian accelerated life testing under competing log-location-scale family of causes of failure, Comput. Stat., 31, 89-119 (2016) · Zbl 1342.65052 |

[19] | David, H. A.; Moeschberger, M. L., The Theory of Competing Risks (1978), Griffin: Griffin London · Zbl 0434.62076 |

[20] | Fan, T.-H.; Hsu, T.-M., Constant stress accelerated life test on a multiple-component series system under Weibull lifetime distributions, Commun. Stat. Theory Methods, 43, 2370-2383 (2014) · Zbl 1462.62617 |

[21] | Chaloner, K.; Larntz, K., Optimal Bayesian design applied to logistic regression experiments, J. Stat. Plan. Inference, 21, 191-208 (1989) · Zbl 0666.62073 |

[22] | Fedorov, V. V., Theory of Optimal Experiments (1972), Academic Press: Academic Press New York |

[23] | Firth, D.; Hinde, J. P., On Bayesian \(D\)-optimum design criteria and the equivalence theorem in non-linear models, J. R. Stat. Soc. Ser. B, 59, 793-797 (1997) · Zbl 0886.62074 |

[24] | Varadhan, R., Numerical optimization in R: beyond optim, J. Stat. Softw., 60, 1-3 (2014) |

[25] | Clyde, M.; Chaloner, K., The equivalence of constrained and weighted designs in multiple objective design problems, J. Am. Stat. Assoc., 91, 1236-1244 (1996) · Zbl 0883.62079 |

[26] | Liu, X.; Tang, L.-C., Accelerated life test plans for repairable systems with multiple independent risks, IEEE Trans. Reliab., 59, 115-127 (2010) |

[27] | Roy, S.; Mukhopadhyay, C., Bayesian accelerated life testing under competing Weibull causes of failure, Commun. Stat. Theory Methods, 43, 2429-2451 (2014) · Zbl 1462.62630 |

[28] | Moeschberger, M. L., Life tests under dependent competing causes of failure, Technometrics, 16, 39-47 (1974) · Zbl 0277.62073 |

[29] | Tsiatis, A., A nonidentifiability aspect of the problem of competing risks, Proceedings of the National Academy of Sciences of the United States of America, 72, 20-22 (1975) · Zbl 0299.62066 |

[30] | Crowder, M., Classical Competing Risk (2001), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.