  In this paper, we study an additive model where the response variable is
Hilbert-space-valued and predictors are multivariate Euclidean, and both are
possibly imperfectly observed. Considering Hilbert-space-valued responses
allows to cover Euclidean, compositional, functional and density-valued
variables. By treating imperfect responses, we can cover functional variables
taking values in a Riemannian manifold and the case where only a random sample
from a density-valued response is available. This treatment can also be applied
in semiparametric regression. Dealing with imperfect predictors allows us to
cover various principal component and singular component scores obtained from
Hilbert-space-valued variables. For the estimation of the additive model having
such variables, we use the smooth backfitting method. We provide full
non-asymptotic and asymptotic properties of our regression estimator and
present its wide applications via several simulation studies and real data
applications.
