Maximum Likelihood Estimation of a Proportion from a Sample of Triplets

When estimating a proportion and only a sample of triplets is given, dependencies within the triplets are to be accounted for. Without assuming a distribution for the success count of the triplet, together with the proportion, as second and third parameter the correlations of 1st and 2nd order enter the model. We apply maximum likelihood estimation, and derive consistency by using that the triplet count is multinomially distributed, combined with the continuous mapping theorem. The asymptotic normality follows with the delta-method, resulting in closed-form expressions for the standard errors. As application we study caries prevalence of pre-school children from a sample to nursing schools. We compare the standard errors with those for assuming erroneously independence within the nursing schools. As to be suspected, the design `inflates' the standard error markedly.