A comparison of the bias in four versions of the product-limit estimator

Abstract

SUMMARY The bias of the product-limit estimator depends entirely on the method used to ‘complete’ the estimator beyond the last observation when it is censored. The bias has been investigated for the upper (Gill, 1980) and lower (Efron, 1967) bounds on these methods. Rather than using a completion method, investigators in practice commonly truncate the estimator using data-driven rules. The bias under two common truncation rules are considered: (i) truncating at the last observation if it is censored, and (ii) truncating at the ( n - l)th order statistic. A number of inequalities between these biases are derived, and properties of the bias functions are proved. This investigation shows that the method used for estimation near the last observation plays a key role in the resulting bias, a fact which has implications for comparing or pooling of such estimators.