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Nonparametric estimation of composite functions
Annals of Statistics
We study the problem of nonparametric estimation of a multivariate function g:R^d→R that can be represented as a composition of two unknown smooth functions f:R→R and G:R^d→R. We suppose that f and G belong to known smoothness classes of functions, with smoothness γ and β, respectively. We obtain the full description of minimax rates of estimation of g in terms of γ and β, and propose rate-optimal estimators for the sup-norm loss. For the construction of such estimators, we first prove andoi:10.1214/08-aos611 fatcat:33k4tlsatzapdexfzgvm7kez44