On formulations of skew factor models: skew errors versus skew factors

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.