Parametric Partially Exchangeable Models for Multiple Binary Sequences

###
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.