  Finite mixture of Gaussian distributions provide a flexible semi-parametric
methodology for density estimation when the variables under investigation have
no boundaries. However, in practical applications variables may be partially
bounded (e.g. taking non-negative values) or completely bounded (e.g. taking
values in the unit interval). In this case the standard Gaussian finite mixture
model assigns non-zero densities to any possible values, even to those outside
the ranges where the variables are defined, hence resulting in severe bias. In
this paper we propose a transformation-based approach for Gaussian mixture
modelling in case of bounded variables. The basic idea is to carry out density
estimation not on the original data but on appropriately transformed data.
Then, the density for the original data can be obtained by a change of
variables. Both the transformation parameters and the parameters of the
Gaussian mixture are jointly estimated by the Expectation-Maximisation (EM)
algorithm. The methodology for partially and completely bounded data is
illustrated using both simulated data and real data applications.
