Variance estimation for a ratio in the presence of imputed data

In this paper, we study the problem of variance estimation for a ratio of two totals when marginal random hot deck imputation has been used to fill in missing data. We consider two approaches to inference. In the first approach, the validity of an imputation model is required. In the second approach, the validity of an imputation model is not required but response probabilities need to be estimated, in which case the validity of a nonresponse model is required. We derive variance estimators under two distinct frameworks: the customary two-phase framework and the reverse framework.