Knowledge of the timing of perinatal transmission of HIV would be valuable for the determination and evaluation of preventive treatments and would shed light on the mechanism of transmission. Estimation of the distribution of the time of perinatal transmission is difficult, however, because tests of infection status can only be undertaken after birth. DNA and RNA polymerase chain reaction (PCR) assays and HIV culture have been the most commonly used diagnostic tests for perinatal HIV infection. Such tests have high sensitivity and specificity, except when they are given shortly after infection. In this paper we use the time-dependent sensitivity of these diagnostic tests to make nonparametric and semiparametric inferences about the distribution of the time of perinatal HIV transmission as well as the cumulative probability of perinatal transmission. The methods are illustrated with data from a clinical trial conducted by the AIDS Clinical Trials group.