The asymptotic distributions of camér-von Mises type statistics based on the productlimit estimate of the distribution function of a certain class of randomly censored observations are derived; the asymptotic significance points of the statistics for various degrees of censoring are given. The statistics are also partitioned into orthogonal components in the manner of Durbin & Knott (1972). The asymptotic powers of the statistics and their components against normal mean and variance shifts, exponential scale shifts, and Weibull alternatives to exponentiality are compared. Data arising in a competing risk situation are examined, using the Cramér-von Mises statistic.