  In a regression analysis, suppose we suspect that there are several
heterogeneous groups in the population that a sample represents. Mixture
regression models have been applied to address such problems. By modeling the
conditional distribution of the response given the covariate as a mixture, the
sample can be clustered into groups and the individual regression models for
the groups can be estimated simultaneously. This approach treats the covariate
as deterministic so that the covariate carries no information as to which group
the subject is likely to belong to. Although this assumption may be reasonable
in experiments where the covariate is completely determined by the
experimenter, in observational data the covariate may behave differently across
the groups. Thus the model should also incorporate the heterogeneity of the
covariate, which allows us to estimate the membership of the subject from the
covariate.
  In this paper, we consider a mixture regression model where the joint
distribution of the response and the covariate is modeled as a mixture. Given a
new observation of the covariate, this approach allows us to compute the
posterior probabilities that the subject belongs to each group. Using these
posterior probabilities, the prediction of the response can adaptively use the
covariate. We introduce an inference procedure for this approach and show its
properties concerning estimation and prediction. The model is explored for the
functional covariate as well as the multivariate covariate. We present a
real-data example where our approach outperforms the traditional approach,
using the well-analyzed Berkeley growth study data.
