The Annals of Statistics

Projected spline estimation of the nonparametric function in high-dimensional partially linear models for massive data

Heng Lian, Kaifeng Zhao, and Shaogao Lv

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Abstract

In this paper, we consider the local asymptotics of the nonparametric function in a partially linear model, within the framework of the divide-and-conquer estimation. Unlike the fixed-dimensional setting in which the parametric part does not affect the nonparametric part, the high-dimensional setting makes the issue more complicated. In particular, when a sparsity-inducing penalty such as lasso is used to make the estimation of the linear part feasible, the bias introduced will propagate to the nonparametric part. We propose a novel approach for estimation of the nonparametric function and establish the local asymptotics of the estimator. The result is useful for massive data with possibly different linear coefficients in each subpopulation but common nonparametric function. Some numerical illustrations are also presented.

Article information

Source
Ann. Statist., Volume 47, Number 5 (2019), 2922-2949.

Dates
Received: April 2018
Revised: July 2018
First available in Project Euclid: 3 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1564797868

Digital Object Identifier
doi:10.1214/18-AOS1769

Mathematical Reviews number (MathSciNet)
MR3988777

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

Keywords
Asymptotic normality B-splines local asymptotics profiled estimation

Citation

Lian, Heng; Zhao, Kaifeng; Lv, Shaogao. Projected spline estimation of the nonparametric function in high-dimensional partially linear models for massive data. Ann. Statist. 47 (2019), no. 5, 2922--2949. doi:10.1214/18-AOS1769. https://projecteuclid.org/euclid.aos/1564797868


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