Flexible Density-on-Scalar Regression Models in Bayes Hilbert Spaces
We introduce functional additive regression models with probability density functions as response variables and scalar covariates. Existing functional additive models are not directly applicable, as the non-negativity and integration to one constraints of densities are not preserved under summation and scalar multiplication.
We thus formulate the regression model in a Bayes Hilbert space with respect to an arbitrary measure. This enables us to not only consider continuous densities, but also discrete and mixed densities. Estimation is based on a gradient boosting algorithm that allows for a variety of flexible effects.
We apply our framework to a motivating data set from the German Socio-Economic Panel Study (SOEP). We analyze densities of the woman’s share in a couple’s total labor income, including covariate effects for year, federal state and age of the youngest child. We show how to handle the challenge of mixed densities within our framework, as the income share is a continuous variable with discrete point masses at zero and one for single-income couples.
This is joint work with Eva-Maria Maier, Almond Stöcker and Bernd Fitzenberger