Parametric Partially Exchangeable Models for Multiple Binary Sequences

Fernando Quintana and Michael A. Newton .

Brazilian Journal of Probability and Statistics , 13 , 55-76, 1999.

Formerly Technical Report 969, Department of Statistics, University of Wisconsin, Madison.

Originally issued December 1996.

Abstract:

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 infinity 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 efficient estimates of the parameter also shown to hold when both the number of sequences and their lengths go to infinity. Conditions are given so that no growth rates are needed for individual sequence lengths.

Key words: exchangeability, hypoconvergence, multiple binary sequences, partial exchangeability, Polya sequences.