Kernel probability estimation for binomial and multinomial data

Kernel-based smoothers have enjoyed considerable success in the estimation of both probability densities and event frequencies. Existing procedures can be modified to yield a similar kernel-based estimator of instantaneous probability over the course of a binomial or multinomial time series. The resulting nonparametric estimate can be described in terms of one bandwidth per outcome alternative, facilitating both the understanding and reporting of results relative to more sophisticated methods for binomial outcome estimation. Also described is a method for sample size estimation, which in turn can be used to obtain credible intervals for the resulting estimate given mild assumptions. One application of this analysis is to model response accuracy in tasks with heterogeneous trial types. An example is presented from a study of transitive inference, showing how kernel probability estimates provide a method for inferring response accuracy during the first trial following training. This estimation procedure is also effective in describing the multinomial responses typical in the study of choice and decision making. An example is presented showing how the procedure may be used to describe changing distributions of choices over time when eight response alternatives are simultaneously available.