A note on the Breslow survival estimator

摘要

This note investigates the structure of the bias and mean squared error (MSE) for a common estimator of the survivor function, namely the Breslow [Breslow, N.E., 1972, Discussion of Professor Cox's paper. Journal of the Royal Statistical Society, Series B, 34, 216-217.] survivor function estimator. Using Ito's change-of-variables formula, new formulas for the bias and MSE (hence variance) for this estimator are established. These formulas are subsequently used to investigate a conjecture of Fleming and Harrington [Fleming, T.R. and Harrington, D.P., 1984, Nonparametric estimation of the survival distribution in censored data. Communications in Statistics— Theory and Methods, 13, 2469-2486.]. In particular, we verify that the MSE of the Kaplan-Meier estimator [Kaplan, E.L. and Meier, P., 1958, Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53,457-481.] exceeds that of the Breslow estimator, whenever the true survival probability is bounded sufficiently far from zero.