  In many complex applications, data heterogeneity and homogeneity exist
simultaneously. Ignoring either one will result in incorrect statistical
inference. In addition, coping with complex data that are non-Euclidean becomes
more common. To address these issues we consider a distributional data response
additive model in which the response is a distributional density function and
the individual effect curves are homogeneous within a group but heterogeneous
across groups, the covariates capturing the variation share common additive
bivariate functions. A transformation approach is first utilized to map density
functions into a linear space. We then apply the B-spline series approximating
method to estimate the unknown subject-specific and additive bivariate
functions, and identify the latent group structures by hierarchical
agglomerative clustering (HAC) algorithm. Our method is demonstrated to
identify the true latent group structures with probability approaching one. To
improve the efficiency, we further construct the backfitted local linear
estimators for grouped structures and additive bivariate functions in
post-grouping model. We establish the asymptotic properties of the resultant
estimators including the convergence rates, asymptotic distributions and the
post-grouping oracle efficiency. The performance of the proposed method is
illustrated by simulation studies and empirical analysis with some interesting
results.
