Adaptive Estimation of High Dimensional Partially Linear Model

Consider the partially linear model (PLM) with random design: $Y=X^T\beta^*+g(W)+u$, where $g(\cdot)$ is an unknown real-valued function, $X$ is $p$-dimensional, $W$ is one-dimensional, and $\beta^*$ is $s$-sparse. Our aim is to estimate $\beta^*$ based on $n$ i.i.d. observations of $(Y,X,W)$ with possibly $n<p$. The popular approaches and their theoretical properties so far have mainly been developed with an explicit knowledge of some function classes. In this paper, we propose an adaptive estimation procedure, with consistency and exact rates of convergence obtained in high dimensions under mild scaling requirements. Two surprising features are revealed: (i) the bandwidth parameter automatically adapts to the model and is actually tuning-insensitive; and (ii) the procedure could even maintain fast rate of convergence for Lipschitz class of unbounded support and $\alpha$-H\"older class of $\alpha\leq1/2$.