Recruitment of disease-free subjects into cohort studies and measurement of their time from exposure/infection to disease selectively excludes individuals (the unseen sample) who had earlier exposure and who have shorter times to disease. The unseen and observed samples may differ in other characteristics in addition to incubation period and exposure/infection time. For data with known truncation times, we develop non-parametric maximum likelihood estimates of the size, exposure/infection dates and distribution of incubation time in the unseen sample. We provide procedures to estimate and compensate for the biasing effects due to exclusion of the unseen sample in descriptive and survival analysis. We give consistency properties of these estimates and assess variability using bootstrap methods. One can use imputation to derive the above estimates from data with unknown truncation times that have been estimated parametrically. Application is made to an AIDS cohort study of over 5000 homosexual men. Important estimates obtained from this application are the annual seroconversion rates from 1978 to 1983, not otherwise obtainable in this study population.