Inferences about the distribution of time to HIV infection in infants are complicated because infection is a silent event and imperfect diagnostic tests are used to detect its occurrence, leading to false-positive and false-negative results. Nonparametric likelihood approaches are computationally hampered by a large number of parameters and a possibly nonconcave likelihood function. To overcome these difficulties, we develop one-sample and regression methods based on profile likelihood and Markov chain Monte Carlo techniques. The methods also provide a useful diagnostic for assessing the infection status of individual subjects, and are illustrated using results from a recent clinical trial for the prevention of mother-to-child HIV transmission.