ABSTRACT
A common feature of many failure time data in epidemiological studies or reliability studies is that they are simultaneously censored and truncated. Additionally, survival or reliability data could be truncated. There are also various types of truncation. For example right-truncated data occur in registers. An acquired immune deficiency syndrome (AIDS) register only contains AIDS cases which have already been reported, which generates right-truncated samples of induction times. The chapter defines a marginal non-parametric likelihood for interval-censored and interval-truncated data first introduced in B. W. Turnbull and later studied by A. Alioum and D. Commenges and C. Huber and F. Vonta in connection also to parametric and semi-parametric models. It formulates a complete non-parametric likelihood for interval-censored and interval-truncated data. The chapter discusses consistency in Hellinger distance of the density of survival or reliability under regularity conditions. It considers as the total interval of observation time the interval.
