Some properties of bivariate empirical hazard processes under random censoring

In Campbell (1982, IMS Lecture Notes--Monograph Series Vol. 2, pp. 243-256, IMS, Hayward, CA) and Campbell and Földes (1982, Proceedings, Internat. Colloq. Nonparametric Statist. Inform., 1980, North-Holland, New York) some asymptotic properties of bivariate empirical hazard processes under random censoring are given. Taking the representation of the empirical hazard process for bivariate randomly censored samples in Campbell, op. cit., as a starting point and restricting attention to strong properties, we obtain a speed of strong convergence for the weighted bivariate empirical hazard processes as well as a speed of strong uniform convergence for bivariate hazard rate estimators. Our approach is based on a local fluctuation inequality for the bivariate hazard process and differs from the martingale methods quite often used in the univariate case.