  We investigate the parameter estimation of regression models with fixed group
effects, when the group variable is missing while group related variables are
available. This problem involves clustering to infer the missing group variable
based on the group related variables, and regression to build a model on the
target variable given the group and eventually additional variables. Thus, this
problem can be formulated as the joint distribution modeling of the target and
of the group related variables. The usual parameter estimation strategy for
this joint model is a two-step approach starting by learning the group variable
(clustering step) and then plugging in its estimator for fitting the regression
model (regression step). However, this approach is suboptimal (providing in
particular biased regression estimates) since it does not make use of the
target variable for clustering. Thus, we claim for a simultaneous estimation
approach of both clustering and regression, in a semi-parametric framework.
Numerical experiments illustrate the benefits of our proposition by considering
wide ranges of distributions and regression models. The relevance of our new
method is illustrated on real data dealing with problems associated with high
blood pressure prevention.
