Estimating distributions in the presence of measurement error with replicate values

In many applications, observations from some distribution of interest are contaminated with errors. In this thesis we examine estimation of the underlying distribution in the presence of such errors. The nonparametric maximum likelihood estimator (NPMLE) of the mixing distribution of interest has been studied by various authors when there are no nuisance parameters in the model. This thesis examines the existence, finite support, and weak convergence of the NPMLE under more general conditions than those previously studied. Attention is then given to a particular model in which, on a unit, either replicates or a mean value are assumed normally distributed with expected value equal to the true value of interest and some variance. A variety of models are allowed for the variances, which may be fixed or random. A full maximum likelihood method and various pseudo methods are examined for estimating the distribution when nuisance parameters are in the model. Asymptotic properties are developed for some special cases using finite mixtures and some approaches outlined for handling more general cases. Two bootstrap methods are discussed for making inferences on cumulative probabilities and percentiles from the distribution of interest. The methods are demonstrated with an example in which the distribution of beta carotene intake is estimated