Most of the papers dealing with truncation have been confined to
one-sided (left or right) truncated data. However, in some applications,
doubly truncated data emerge. This is the case, for example, of
epidemiological registers corresponding to all detected cases within a
given calendar time period, but lacking information on non-diseased
(Stovring and Wang, 2005). Some exceptions in the methodological
literature are Martin and Betensky (2005), Bilker and Wang (1996), or
Betensky and Martin (2003). Efron and Petrosian (1999) formally
introduced the nonparametric maximum likelihood estimator of the
marginal distribution of a doubly truncated variable, insisting in the
fact that (unlike with one-sided truncation) there is no explicit form
for this key estimator. The NPMLE for doubly truncated data was
revisited in Shen (2008), who formally established its uniform
consistency and converge to a normal. Moreira and de Uña Alvarez (2009)
introduced a bootstrap approximation for the NPMLE. We revisit the
advances in estimation of a distribution function under double
truncation, including the situation in which the truncation times fall
on a straight line (case not covered by the theory in Shen (2008)). We
report Simulations and real data applications in which double truncation
naturally arises. |