ASYMPTOTIC THEORY FOR THE GAMMA FRAILTY MODEL WITH DEPENDENT CENSORING

MICHAEL R. KOSOROK^{1}, JASON P. FINE^{1}, HONGYU JIANG^{2} AND RICK CHAPPELL^{1}

1Department of Statistics, University of Wisconsin, 600 Highland Avenue, Box 4675, Madison, WI 53792, U.S.A. 2Department of Biostatistics, Harvard School of Public Health, 655 Huntington Avenue, Boston, MA 02115, U.S.A.

(Received October 23, 2000; revised June 4, 2001)

Abstract. In many clinical studies, there are two dependent event times with one of the events being terminal, such as death, and the other being nonfatal, such as myocardial infarction or cancer relapse. Morbidity can be dependently censored by mortality, but not vice versa. Asymptotic theory is developed for simultaneous estimation of the marginal distribution functions in this semi-competing risks setting. We specify the joint distribution of the event times in the upper wedge, where the nonfatal event happens before the terminal event, with the popular gamma frailty model. The estimators are based on an adaptation of the self-consistency principle. To study their properties, we employ a modification of the functional delta-method applied to Z-estimators. This approach to weak convergence leads naturally to asymptotic validity of both the nonparametric and multiplier bootstraps, facilitating inference in spite of the complexity of the limiting distribution.

Key words and phrases: Bootstrap, dependent censoring, empirical processes, functional delta-method, gamma frailty model, U-statistics, weak convergence, Zestimators.