Adaptive Estimation of High Dimensional Partially Linear Model
Year: 2017
Type: preprint
Abstract: 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 efficiently 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, which automatically adapts to the model via a tuning-insensitive bandwidth parameter. In many cases, the proposed procedure also proves to attain weaker scaling and noise requirements than the best existing ones. The proof rests on a general method for determining the estimation accuracy of a perturbed M-estimator and new U-statistics tools, which are of independent interest.
Source: arXiv (Cornell University)
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2619126893