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. |