### Abstract

When two-component parallel systems are tested, the data consist of Type-II censored data X_{(i)}, i = 1, . . . , n, from one component, and their concomitants Y_{[i]} randomly censored at X_{(r)}, the stopping time of the experiment. Marshall & Olkin's (1967) bivariate exponential distribution is used to illustrate statistical inference procedures developed for this data type. Although this data type is motivated practically, the likelihood is complicated, and maximum likelihood estimation is difficult, especially in the case where the parameter space is a non-open set. An iterative algorithm is proposed for finding maximum likelihood estimates. This article derives several properties of the maximum likelihood estimator (MLE) including existence, uniqueness, strong consistency and asymptotic distribution. It also develops an alternative estimation method with closed-form expressions based on marginal distributions, and derives its asymptotic properties. Compared with variances of the MLEs in the finite and large sample situations, the alternative estimator performs very well, especially when the correlation between X and Y is small.

Original language | English (US) |
---|---|

Pages (from-to) | 323-336 |

Number of pages | 14 |

Journal | Australian and New Zealand Journal of Statistics |

Volume | 42 |

Issue number | 3 |

State | Published - Sep 2000 |

Externally published | Yes |

### Fingerprint

### Keywords

- Asymptotics
- Bivariate exponential
- Censored data
- Reliability

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Australian and New Zealand Journal of Statistics*,

*42*(3), 323-336.

**Parameter estimation for bivariate shock models with singular distribution for censored data with concomitant order statistics.** / Chen, Di; Li, Chin-Shang; Lu, Jye Chyi; Park, Jinho.

Research output: Contribution to journal › Article

*Australian and New Zealand Journal of Statistics*, vol. 42, no. 3, pp. 323-336.

}

TY - JOUR

T1 - Parameter estimation for bivariate shock models with singular distribution for censored data with concomitant order statistics

AU - Chen, Di

AU - Li, Chin-Shang

AU - Lu, Jye Chyi

AU - Park, Jinho

PY - 2000/9

Y1 - 2000/9

N2 - When two-component parallel systems are tested, the data consist of Type-II censored data X(i), i = 1, . . . , n, from one component, and their concomitants Y[i] randomly censored at X(r), the stopping time of the experiment. Marshall & Olkin's (1967) bivariate exponential distribution is used to illustrate statistical inference procedures developed for this data type. Although this data type is motivated practically, the likelihood is complicated, and maximum likelihood estimation is difficult, especially in the case where the parameter space is a non-open set. An iterative algorithm is proposed for finding maximum likelihood estimates. This article derives several properties of the maximum likelihood estimator (MLE) including existence, uniqueness, strong consistency and asymptotic distribution. It also develops an alternative estimation method with closed-form expressions based on marginal distributions, and derives its asymptotic properties. Compared with variances of the MLEs in the finite and large sample situations, the alternative estimator performs very well, especially when the correlation between X and Y is small.

AB - When two-component parallel systems are tested, the data consist of Type-II censored data X(i), i = 1, . . . , n, from one component, and their concomitants Y[i] randomly censored at X(r), the stopping time of the experiment. Marshall & Olkin's (1967) bivariate exponential distribution is used to illustrate statistical inference procedures developed for this data type. Although this data type is motivated practically, the likelihood is complicated, and maximum likelihood estimation is difficult, especially in the case where the parameter space is a non-open set. An iterative algorithm is proposed for finding maximum likelihood estimates. This article derives several properties of the maximum likelihood estimator (MLE) including existence, uniqueness, strong consistency and asymptotic distribution. It also develops an alternative estimation method with closed-form expressions based on marginal distributions, and derives its asymptotic properties. Compared with variances of the MLEs in the finite and large sample situations, the alternative estimator performs very well, especially when the correlation between X and Y is small.

KW - Asymptotics

KW - Bivariate exponential

KW - Censored data

KW - Reliability

UR - http://www.scopus.com/inward/record.url?scp=0034371033&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034371033&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034371033

VL - 42

SP - 323

EP - 336

JO - Australian and New Zealand Journal of Statistics

JF - Australian and New Zealand Journal of Statistics

SN - 1369-1473

IS - 3

ER -