  In a cross-sectional study, adolescent and young adult females were asked to
recall the time of menarche, if experienced. Some respondents recalled the date
exactly, some recalled only the month or the year of the event, and some were
unable to recall anything. We consider estimation of the menarcheal age
distribution from this interval censored data. A~complicated interplay between
age-at-event and calendar time, together with the evident fact of memory fading
with time, makes the censoring informative. We propose a model where the
probabilities of various types of recall would depend on the time since
menarche. For parametric estimation we model these probabilities using
multinomial regression function. Establishing consistency and asymptotic
normality of the parametric MLE requires a bit of tweaking of the standard
asymptotic theory, as the data format varies from case to case. We also provide
a non-parametric MLE, propose a computationally simpler approximation, and
establish the consistency of both these estimators under mild conditions. We
study the small sample performance of the parametric and non-parametric
estimators through Monte Carlo simulations. Moreover, we provide a graphical
check of the assumption of the multinomial model for the recall probabilities,
which appears to hold for the menarcheal data set. Our analysis shows that the
use of the partially recalled part of the data indeed leads to smaller
confidence intervals of the survival function.
