Nonparametric estimation of composite functions
We study the problem of nonparametric estimation of a multivariate function g that can be represented as a composition of two unknown smooth functions f and G. We suppose that f and G belong to known smoothness classes of functions. We obtain the full description of minimax rates of estimation of g in terms of the smoothness parameters of f and G and propose rate optimal estimators for the sup-norm loss. For the construction of such estimators, we first prove an approximation result for composite functions that may have an independent interest, and then a result on adaptation to the local structure. Interestingly, the construction of rate optimal estimators for composite functions (with given, fixed, smoothness) needs adaptation, but not in the traditional sense: it is now adaptation to the local structure. We prove that composition models generate only two types of local structures: the local single-index model and the local model with roughness isolated to a single dimension (i.e., a model containing elements of both additive and single-index structure).