  Mixture models are well-known for their versatility, and the Bayesian
paradigm is a suitable platform for mixture analysis, particularly when the
number of components is unknown. Bhattacharya (2008) introduced a mixture model
based on the Dirichlet process, where an upper bound on the unknown number of
components is to be specified. Here we consider a Bayesian asymptotic framework
for objectively specifying the upper bound, which we assume to depend on the
sample size. In particular, we define a Bayesian analogue of the mean
integrated squared error (Bayesian MISE), and select that form of the upper
bound, and also that form of the precision parameter of the underlying
Dirichlet process, for which Bayesian MISE of a specific density estimator,
which is a suitable modification of the Polya-urn based prior predictive model,
converges at a desired rate. As a byproduct of our approach, we investigate
asymptotic choice of the precision parameter of the traditional Dirichlet
process mixture model; the density estimator we consider here is a modification
of the prior predictive distribution of Escobar & West (1995) associated with
the Polya urn model. Various asymptotic issues related to the two
aforementioned mixtures, including comparative performances, are also
investigated.
