A flexible distribution to handle response styles when modelling rating scale data

It is commonly known that the respondents to rating scale questions, when are not aware, can select their own response using only certain response categories regardless the item content. This behavior is described as response style. Thus, the observed response can be a real opinion or dictated by a response style behavior (uncertainty). Marginal models (HMMLU) for multivariate responses by Colombi et al., 2018, enables us to distinguish these two behaviors and allows to specify the distributions of uncertain responses.
We extend the class of HMMLU models with a new family of discrete distributions whose two parameters allow the uncertain distributions to be U-shaped, bell-shaped, unimodal, symmetric, skewed or uniform, for capturing different response styles.