A Comparison of the Bias in Four Versions of the Product-Limit Estimator
研究了乘积限估计量在删失数据下因尾部处理方法不同而产生的偏差，比较了两种截断规则的偏差性质，对估计量的比较与合并有参考价值。
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.