Serially correlation binomial data with random cluster sizes occur frequently in environmental and health studies. Such data series have traditionally been analyzed using binomial state-space or hidden Markov models without appropriately accounting for the randomness in the cluster sizes. To characterize correlation and extra-variation arising from the random cluster sizes properly, we introduce a joint Poisson state-space modelling approach to analysis of binomial series with random cluster sizes. This approach enables us to model the marginal counts and binomial proportions simultaneously. An optimal estimation of our model has been developed using the orthodox best linear unbiased predictors. This estimation method is computationally efficient and robust since it depends only on the first- and second- moment assumptions of unobserved random effects. Our proposed approach is illustrated with analysis of birth delivery data.
Keywords: best linear unbiased predictor; logistic regression; over-dispersion; random effects; time series.