In screening and surveillance studies, event times are interval censored. Besides, screening tests are imperfect so that the interval at which an event takes place may be uncertain. We describe an expectation–maximization algorithm to find the nonparametric maximum likelihood estimator of the cumulative incidence function of an event based on screening test data. Our algorithm has a closed‐form solution for the combined expectation and maximization step and is computationally undemanding. A simulation study indicated that the bias of the estimator tends to zero for large sample size, and its mean squared error is in general lower than the mean squared error of the estimator that assumes the screening test is perfect. We apply the algorithm to follow‐up data from women treated for cervical precancer. Copyright © 2017 John Wiley & Sons, Ltd.