  In prevalent cohort studies where subjects are recruited at a cross-section,
the time to an event may be subject to length-biased sampling, with the
observed data being either the forward recurrence time, or the backward
recurrence time, or their sum. In the regression setting, it has been shown
that the accelerated failure time model for the underlying event time is
invariant under these observed data set-ups and can be fitted using standard
methodology for accelerated failure time model estimation, ignoring the
length-bias. However, the efficiency of these estimators is unclear, owing to
the fact that the observed covariate distribution, which is also length-biased,
may contain information about the regression parameter in the accelerated life
model. We demonstrate that if the true covariate distribution is completely
unspecified, then the naive estimator based on the conditional likelihood given
the covariates is fully efficient.
