### Abstract

We study two estimators of the mean function of a counting process based on "panel count data." The setting for "panel count data" is one in which n independent subjects, each with a counting process with common mean function, are observed at several possibly different times during a study. Following a model proposed by Schick and Yu, we allow the number of observation times, and the observation times themselves, to be random variables. Our goal is to estimate the mean function of the counting process. We show that the estimator of the mean function proposed by Sun and Kalbfleisch can be viewed as a pseudo-maximum likelihood estimator when a non-homogeneous Poisson process model is assumed for the counting process. We establish consistency of both the nonparametric pseudo maximum likelihood estimator of Sun and Kalbfleisch and the full maximum likelihood estimator, even if the underlying counting process is not a Poisson process. We also derive the asymptotic distribution of both estimators at a fixed time t, and compare the resulting theoretical relative efficiency with finite sample relative efficiency by way of a limited Monte-Carlo study.

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

Pages (from-to) | 779-814 |

Number of pages | 36 |

Journal | Annals of Statistics |

Volume | 28 |

Issue number | 3 |

State | Published - Dec 1 2000 |

Externally published | Yes |

### Fingerprint

### Keywords

- Algorithm
- Asymptotic distributions
- Consistency
- Convex minorant
- Counting process
- Current status data
- Empirical processes
- Interval censoring
- Iterative
- Maximum likelihood
- Monte-carlo
- Pseudo likelihood
- Relative efficiency

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

*Annals of Statistics*,

*28*(3), 779-814.

**Two estimators of the mean of a counting process with panel count data.** / Wellner, Jon A.; Zhang, Ying.

Research output: Contribution to journal › Article

*Annals of Statistics*, vol. 28, no. 3, pp. 779-814.

}

TY - JOUR

T1 - Two estimators of the mean of a counting process with panel count data

AU - Wellner, Jon A.

AU - Zhang, Ying

PY - 2000/12/1

Y1 - 2000/12/1

N2 - We study two estimators of the mean function of a counting process based on "panel count data." The setting for "panel count data" is one in which n independent subjects, each with a counting process with common mean function, are observed at several possibly different times during a study. Following a model proposed by Schick and Yu, we allow the number of observation times, and the observation times themselves, to be random variables. Our goal is to estimate the mean function of the counting process. We show that the estimator of the mean function proposed by Sun and Kalbfleisch can be viewed as a pseudo-maximum likelihood estimator when a non-homogeneous Poisson process model is assumed for the counting process. We establish consistency of both the nonparametric pseudo maximum likelihood estimator of Sun and Kalbfleisch and the full maximum likelihood estimator, even if the underlying counting process is not a Poisson process. We also derive the asymptotic distribution of both estimators at a fixed time t, and compare the resulting theoretical relative efficiency with finite sample relative efficiency by way of a limited Monte-Carlo study.

AB - We study two estimators of the mean function of a counting process based on "panel count data." The setting for "panel count data" is one in which n independent subjects, each with a counting process with common mean function, are observed at several possibly different times during a study. Following a model proposed by Schick and Yu, we allow the number of observation times, and the observation times themselves, to be random variables. Our goal is to estimate the mean function of the counting process. We show that the estimator of the mean function proposed by Sun and Kalbfleisch can be viewed as a pseudo-maximum likelihood estimator when a non-homogeneous Poisson process model is assumed for the counting process. We establish consistency of both the nonparametric pseudo maximum likelihood estimator of Sun and Kalbfleisch and the full maximum likelihood estimator, even if the underlying counting process is not a Poisson process. We also derive the asymptotic distribution of both estimators at a fixed time t, and compare the resulting theoretical relative efficiency with finite sample relative efficiency by way of a limited Monte-Carlo study.

KW - Algorithm

KW - Asymptotic distributions

KW - Consistency

KW - Convex minorant

KW - Counting process

KW - Current status data

KW - Empirical processes

KW - Interval censoring

KW - Iterative

KW - Maximum likelihood

KW - Monte-carlo

KW - Pseudo likelihood

KW - Relative efficiency

UR - http://www.scopus.com/inward/record.url?scp=0034359461&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034359461&partnerID=8YFLogxK

M3 - Article

VL - 28

SP - 779

EP - 814

JO - Annals of Statistics

JF - Annals of Statistics

SN - 0090-5364

IS - 3

ER -