### Abstract

This article investigates asymptotic properties of the maximum likelihood estimators (MLE) of parameters in the bivariate exponential distribution (BVE) of Marshall and Olkin (1967) based on the following mixed censored data. In life-testing two-component parallel systems (A, B), a cost-saving procedure is to stop the testing experiment after observing the first r failure times of component A. The resulting data are (X_{(1)}, Y^{*}_{[1]}), (X_{(2)}, Y^{*}_{[2]}), . . . , (X_{(r)}, Y^{*}_{[r]}), (X^{*}_{(r+1)}, Y^{*}_{[r+1]}), . . . , (X^{*}_{(n)} Y^{*}_{[n]}), where X_{(1)} ≤ X_{(2)} ≤ ⋯ ≤ X_{(r)} are ordered lifetimes from component A, and Y_{[i]} is the concomitant order statistic corresponding to X_{(i)} from component B. The data X^{*}_{(r+1)}, . . . , X^{*}_{(n)}, and Y^{*}_{[i]}, i = 1, . . . , n are all censored at time X_{(r)} = x_{(r)}. Because of the complexity of the data type and the irregularity of the BVE distribution of (X, Y), there are no immediately applicable asymptotic results for the MLE. This article provides a rigorous treatment of the asymptotic behavior of the MLE, with a numerical example illustrating the mathematical derivations.

Original language | English (US) |
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Pages (from-to) | 109-125 |

Number of pages | 17 |

Journal | Metrika |

Volume | 48 |

Issue number | 2 |

State | Published - 1998 |

Externally published | Yes |

### Keywords

- Asymptotic theory
- Concomitants of order statistics
- Life-testing
- Marshall-Olkin distribution
- Maximum likelihood estimation

### ASJC Scopus subject areas

- Statistics and Probability

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## Cite this

*Metrika*,

*48*(2), 109-125.