Abstract: For right censored data, Efron (1981) has shown that his ``simple'' and ``obvious'' methods of bootstrapping are equivalent. We explain why this equivalence no longer holds for truncated data. Wang (1991) generalized Efron's ``obvious'' bootstrap method to data that are both left truncated and right censored, under the assumption that C>=T and C—T is independent of T, where T and C denote the (generic) censoring and truncation variables. We discuss how the ``obvious'' bootstrap method can be extended when this independence assumption is removed, and also develop an asymptotic theory of the ``simple'' bootstrap method for left truncated and right censored data, showing that the ``simple'' bootstrap approximations to the sampling distributions of various nonparametric statistics from these data are accurate to the order of Op(n-1).
Key words and phrases: Left truncation, right censoring, estimable functionals, Edgeworth expansions, bootstrap, asymptotic U-statistics.