Statistica Sinica 13(2003), 507-517

FISHER INFORMATION IN ORDERED RANDOMLY

CENSORED DATA WITH APPLICATIONS TO

CHARACTERIZATION PROBLEMS

Gang Zheng$^1$ and Joseph L. Gastwirth$^{2,3}$

$^1$National Heart, Lung, and Blood Institute,

$^2$National Cancer Institute and $^3$George Washington University

Abstract: The proportional hazards model of Koziol-Green is often considered in survival analysis. If the lifetime and censoring random variables are independent, the Koziol-Green model implies that the variable indicating whether the observation is censored or not does not contain Fisher information about the parameters of the underlying lifetime distribution. The Koziol-Green model, however, is not uniquely characterized by this result on the lack of Fisher information in the censoring indicator. Given the ordered randomly censored lifetimes with corresponding indicators, we obtain a necessary and sufficient condition, weaker than the Koziol-Green model, which ensures that a set of any number of censoring indicators does not contain Fisher information about the parameters of the lifetime distribution. Under this weaker condition, the results are applied to characterize the Weibull distribution within the class of scale parameter families of lifetime distributions and the factorization of the hazard function in terms of the Fisher information in randomly censored data.

Key words and phrases: Factorization, Fisher information, hazard function, Koziol-Green model, order statistics, time-dependent, Weibull family.