Distributed debiased estimation of high-dimensional partially linear models with jumps

In this paper, we focus on the estimations of both parameter vector and nonparametric component in a high-dimensional partially linear model with jumps within the framework of divide and conquer strategy. We find that a three-stage estimation procedure works well in this setting. Applying the lasso penalty and projected spline approximation, first a profiled estimator for the linear part and a projected spline estimator for the nonparametric part are obtained on each local machine. In the second stage, an efficient jump detection algorithm is developed to obtain the new knot sequence, and then based on this, the estimate of the nonparametric function is obtained and averaged after plugging in the linear part estimate on each local machine. The aggregated estimate of the nonparametric function is then computed by pooling these local estimates. In the third stage, a debiased lasso estimator is averaged to obtain a distributed debiased estimator of the linear part after plugging in the aggregated estimate of nonparametric function. Asymptotic properties of resultant estimators are established under some mild assumptions. Some simulations are conducted to illustrate the empirical performances of our proposed method.