ABSTRACT
Department of Statistics and Actuarial Science and Department of Biostatis-
tics, University of Iowa, Iowa City, Iowa, USA
Ying Zhang
Department of Biostatistics, University of Iowa, Iowa City, Iowa, USA
Lei Hua
Center for Biostatistics in AIDS Research, Harvard School of Public Health,
Boston, Massachusetts, USA
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.2 Consistent Information Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
9.3 Observed Information Matrix in Sieve MLE . . . . . . . . . . . . . . . . . . . . . 242
9.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
9.4.1 Cox Model for Interval-Censored Data . . . . . . . . . . . . . . . . . . 246
9.4.2 Poisson Proportional Mean Model for Panel Count
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
9.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
9.5.1 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
234 Interval-Censored Time-to-Event Data: Methods and Applications
9.5.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
9.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
In a regular parametric model, the maximum likelihood estimator (MLE) is
asymptotically normal with variance equal to the inverse of the Fisher infor-
mation, and the Fisher information can be estimated by the observed infor-
mation. This result provides large sample justification for the use of normal
approximation to the distribution of MLE. An important factor making this
approximation useful in statistical inference is that the observed information
can be readily computed and is consistent. In many situations, consistency of
the observed information follows directly from the law of large numbers and
consistency of MLE.
