## Abstract

Because of the exponential distribution assumption, many reliability databases recorded data in an aggregate way. Instead of individual failure times, each aggregate data point is a summation of a series of collective failures representing the cumulative operating time of one component position from system commencement to the last component replacement. The data format is different from traditional lifetime data and the statistical inference is challenging. We first model the individual component lifetime by a gamma distribution. Confidence intervals for the gamma shape parameter can be constructed using a scaled χ^{2} approximation to a modified ratio of the geometric mean to the arithmetic mean, while confidence intervals for the gamma rate and mean parameters, as well as quantiles, are obtained using the generalized pivotal quantity method. We then fit the data using the inverse Gaussian (IG) distribution, a useful lifetime model for failures caused by degradation. Procedures for point estimation and interval estimation of parameters are developed. We also propose an interval estimation method for the quantiles of an IG distribution based on the generalized pivotal quantity method. An illustrative example demonstrates the proposed inference methods. Supplementary materials for this article are available online.

Original language | English |
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Pages (from-to) | 115-125 |

Journal | Technometrics |

Volume | 59 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2 Jan 2017 |

Externally published | Yes |

## Keywords

- Aggregate data
- Confidence interval
- Gamma distribution
- Generalized pivotal quantity (GPQ)
- Inverse Gaussian distribution