Graph { Theoretical Aspects of Bivariate Censored

Censored data, whether univariate or bivariate, and whether right{, left{ or interval cen-sored, can be represented by its intersection graph. Paying special attention to the bivariate case, we show how the maximal clique structure of such an intersection graph is related to the support of the nonparametric maximum likelihood estimate (NPMLE) of the cumulative distribution function (CDF) for such data. The CDF NPMLE places mass on the real representations of the maximal cliques and nowhere else. Graph algorithms can be used to deal with the identiication of maximal cliques. We distinguish between two types of non{uniqueness of the NPMLE. We label these rep-resentational non{uniqueness and mixture non{uniqueness. Representational non{uniqueness arises because the likelihood is not aaected by the distribution of the estimated mass within the real representation of a maximal clique. Mixture non{uniqueness arises when the masses themselves are not unique. We indicate how mixture non{uniqueness can arise only in the multivariate case. We link mixture non{uniqueness to the rank of the clique matrix and the structure of the clique intersection graph of the data. We then provide a brief overview of estimation techniques and nally we conclude by applying the methodology to three data sets.