### 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) |
---|---|

Pages (from-to) | 109-125 |

Number of pages | 17 |

Journal | Metrika |

Volume | 48 |

Issue number | 2 |

State | Published - 1998 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Metrika*,

*48*(2), 109-125.

**Asymptotic properties of maximum likelihood estimates for a bivariate exponential distribution and mixed censored data.** / Chen, Di; Lu, Jye Chyi; Hughes-Oliver, Jacqueline M.; Li, Chin-Shang.

Research output: Contribution to journal › Article

*Metrika*, vol. 48, no. 2, pp. 109-125.

}

TY - JOUR

T1 - Asymptotic properties of maximum likelihood estimates for a bivariate exponential distribution and mixed censored data

AU - Chen, Di

AU - Lu, Jye Chyi

AU - Hughes-Oliver, Jacqueline M.

AU - Li, Chin-Shang

PY - 1998

Y1 - 1998

N2 - 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.

AB - 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.

KW - Asymptotic theory

KW - Concomitants of order statistics

KW - Life-testing

KW - Marshall-Olkin distribution

KW - Maximum likelihood estimation

UR - http://www.scopus.com/inward/record.url?scp=0041180617&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041180617&partnerID=8YFLogxK

M3 - Article

VL - 48

SP - 109

EP - 125

JO - Metrika

JF - Metrika

SN - 0026-1335

IS - 2

ER -