A Cramér-von Mises statistic for randomly censored data

SUMMARY The asymptotic distributions of Cramer-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 Crame6r-von Mises statistic. An important problem in the analysis of survival data is whether the observed survival pattern for a cohort of individuals can be closely explained by a particular mathematical model with readily interpretable characteristics. Because the data arising from survival studies are often censored, because of deaths from competing causes or failure to observe individuals until time of death, statistical procedures formally relying upon the empirical distribution function can instead be defined in terms of the standard life table, or productlimit, estimate of the survival distribution. Our aim in this paper is to examine a Oram6r-von Mises type statistic for tests of goodness of fit, based on the product-limit empirical distribution function, when the data are subject to random censorship. In ? 2 we derive its asymptotic distribution under particular models of censorship; we investigate also the asymptotic properties of the components of the Cramer-von Mises statistic, which have been shown by Durbin & Knott (1972) to be of particular value in testing for specific aspects of the data. In ? 3 we study briefly the asymptotic power of this Cramer-von Mises statistic and its components against normal mean shift and variance shift alternatives, exponential scale shifts, and Weibull alternatives to exponentiality. In ? 4 we examine survival data with competing risks. We have derived the asymptotic distribution of a Cramer-von Mises type statistic only for a particular distributional form of random censorship; nevertheless, we believe that this pattern of censorship often does occur, and we are confident that our methods can be extended to other random censorship patterns.