Distributed Partially Linear Additive Models With a High Dimensional Linear Part

We study how the divide and conquer principle works in high-dimensional partially linear additive models when the dimension of the linear part is large compared to the sample size. We find that a two-stage approach works well in this setting. Using the lasso penalty, first a debiased profiled estimator for the linear part is averaged to obtain an estimator that has the optimal rate, which is further thresholded to recover sparsity after averaging. In the second stage, estimates of the nonparametric functions are obtained and averaged after plugging in the linear part estimate. Undermild assumptions, the nonparametric part achieved the oracle property in the sense that each, possibly of different smoothness, has the same asymptotic distribution as when the other component functions, as well as the linear coefficients, are known.