Parametric Partially Exchangeable Models for Multiple Binary Sequences

In this paper we discuss models for multiple binary sequences using de Finetti's notions of exchangeability and partial exchangeability as a cornerstone for model building. Thus, we assume each sequence to be a Markov chain conditional on some random transition matrix. This construction is seen as an extension of models based on Polya urns. We consider two particular problems in which the distribution of random effects is indexed by a parameter . We first show that the likelihood hypoconverges to an "ideal" but unobserved likelihood of transition matrices, when sequence lengths go to 1 for a fixed number of experimental units. This permits us to study consistency of the likelihood maximizer to the ideal maximum likelihood estimator. Secondly, we introduce a "naive" estimation approach in which we estimate transition matrices and pretend these are actual draws from the mixing measure. Under smoothness conditions, the approach yields strongly consistent, asymptotically normal and efficie..