  We discuss scalar-on-function regression models where all parameters of the
assumed response distribution can be modeled depending on covariates. We thus
combine signal regression models with generalized additive models for location,
scale and shape (GAMLSS). We compare two fundamentally different methods for
estimation, a gradient boosting and a penalized likelihood based approach, and
address practically important points like identifiability and model choice.
Estimation by a component-wise gradient boosting algorithm allows for high
dimensional data settings and variable selection. Estimation by a penalized
likelihood based approach has the advantage of directly provided statistical
inference. The motivating application is a time series of stock returns where
it is of interest to model both the expectation and the variance depending on
lagged response values and functional liquidity curves.
