Minimax Optimal Estimation in High Dimensional Semiparametric Models

In this paper, we consider minimax optimal estimation of semiparametric models in high dimensional setting. Our particular focus is on partially linear additive models with high dimensional sparse vectors and smooth nonparametric functions. The minimax rate for Euclidean components is the typical sparse estimation rate, independent of nonparametric smoothness. However, the minimax lower bound for each nonparametric function is established as an interplay among dimensionality, sparsity and smoothness. Indeed, the minimax risk for smooth nonparametric estimation can be slowed down to the sparse estimation rate given sufficiently large smoothness or dimensionality. In the above setting, we construct a general class of penalized least square estimators which nearly achieve minimax lower bounds.