  Binomial data with unknown sizes often appear in biological and medical
sciences and are usually overdispersed. All previous methods used parametric
models and only considered overdispersion due to the variation of sizes. The
proposed semiparametric model considers overdispersion due to the variation of
sizes and that of probabilities. By doing this, it can include variations
caused by observations, missing covariates, and random measurement errors in
covariates. An Expectation Conditional Maximization algorithm is provided to
stabilize the loglikelihood optimization. Selecting the number of support
points of the mixing distributions and the bootstrap methods are also
discussed. Simulation is done to evaluate the performance of the proposed
model. Two real examples are used to illustrate the proposed model.
