Nonparametric Estimation of Survival Distributions with Censored Initiating Time, and Censored and Truncated Terminating Time: Application to Transfusion Data for Acquired Immune Deficiency Syndrome

The analysis of survival data with censored initiating time, and censored and truncated terminating time arises in some recent epidemiological studies. The transfusion‐related acquired immune deficiency syndrome (AIDS) data of the Centers for Disease Control (CDC) are a typical example. The initiating time in this case is the time of infection by the human immunodeficiency virus and is not observed for every patient either because of unrecorded transfusion times or multiple transfusions. The terminating time here is the onset of AIDS and is truncated, the result of being able to report within an observational period only a proportion of the infected cases which came down with AIDS in the time period. We consider nonparametric estimation of the survival as well as the initiating time distributions assuming that they are independent and non‐informatively censored. We propose a simple algorithm to obtain the maximum likelihood estimates for the discrete formulation of the problem and apply it to estimating the AIDS latency and infection distributions for four age groups of transfusion‐related AIDS from the CDC surveillance database.