Assessing the Order of Dependence for Partially Exchangeable Binary Data

Abstract

The problem we consider is how to assess the order of serial dependence within partially exchangeable binary sequences. We obtain exact conditional tests comparing any two orders by finding the conditional distribution of data given certain transition counts. These tests are facilitated with a new Monte Carlo scheme. Asymptotic tests are also discussed. In particular, we show that the likelihood ratio tests have an asymptotic chi-squared distribution, thus generalizing the results of Billingsley for the particular case of Markov chains. We apply these methods to several datasets, and perform a simulation to study their properties.