  In the past few years, there have been a number of proposals for generalizing
the factor analysis (FA) model and its mixture version (known as mixtures of
factor analyzers (MFA)) using non-normal and asymmetric distributions. These
models adopt various types of skew densities for either the factors or the
errors. While the relationships between various choices of skew distributions
have been discussed in the literature, the differences between placing the
assumption of skewness on the factors or on the errors have not been closely
studied. This paper examines these formulations and discusses the connections
between these two types of formulations for skew factor models. In doing so, we
introduce a further formulation that unifies these two formulations; that is,
placing a skew distribution on both the factors and the errors.
