Estimating component characteristics from system failure-time data.

*(English)*Zbl 1186.62122Summary: Suppose that failure times are available from a random sample of \(N\) systems of a given, fixed design with components which have i.i.d. life times distributed according to a common distribution \(F\). The inverse problem of estimating \(F\) from the data on observed system life times is considered. Using the known relationship between the system and component life time distributions via signature and domination theory, the nonparametric maximum likelihood estimator \(\widehat{\overline F}_N(t)\) of the component survival function \({\overline F}(t)\) is identified and shown to be numerically accessible in any application of interest. The asymptotic distribution of \(\widehat{\overline F}_N(t)\) is also identified, facilitating the construction of approximate confidence intervals for \({\overline F}(t)\) for \(N\) sufficiently large. Simulation results for samples of size \(N=50\) and \(N=100\) for a collection of five parametric life time models demonstrate the utility of the recommended estimator. Possible extensions beyond the i.i.d. framework are discussed in the concluding section.

##### MSC:

62N05 | Reliability and life testing |

62G05 | Nonparametric estimation |

62N02 | Estimation in survival analysis and censored data |

62E20 | Asymptotic distribution theory in statistics |

62G15 | Nonparametric tolerance and confidence regions |

65C60 | Computational problems in statistics (MSC2010) |

##### Keywords:

nonparametric maximum likelihood estimation; component life time distribution; inverse problems; reliability; system signatures; dominations; asymptotic distribution; approximate confidence intervals; identifiability
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\textit{D. Bhattacharya} and \textit{F. J. Samaniego}, Nav. Res. Logist. 57, No. 4, 380--389 (2010; Zbl 1186.62122)

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