Electronic Journal of Statistics

Oracally efficient estimation for single-index link function with simultaneous confidence band

Lijie Gu and Lijian Yang

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Abstract

Over the last twenty-five years, various $\sqrt{n}$-consistent estimators have been devised for the coefficient vector in the popular semiparametric single-index model. In this paper, we prove under general assumptions that the kernel estimator of the link function by a univariate regression on the index variable is oracally efficient, namely, the estimator with the true single-index coefficient vector is asymptotically indistinguishable from that with any $\sqrt{n}$-consistent coefficient vector estimator. As a mathematical byproduct of the oracle efficiency, a simultaneous confidence band is constructed for the link function based on the oracally efficient kernel estimator. Simulation experiments corroborate the theoretical results. The proposed simultaneous confidence band is applied to analyze and test hypothesis about the Boston housing data.

Article information

Source
Electron. J. Statist., Volume 9, Number 1 (2015), 1540-1561.

Dates
Received: January 2015
First available in Project Euclid: 23 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1437658702

Digital Object Identifier
doi:10.1214/15-EJS1051

Mathematical Reviews number (MathSciNet)
MR3376116

Zentralblatt MATH identifier
1327.62254

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62G15: Tolerance and confidence regions

Keywords
Confidence band kernel link function oracle efficiency single-index

Citation

Gu, Lijie; Yang, Lijian. Oracally efficient estimation for single-index link function with simultaneous confidence band. Electron. J. Statist. 9 (2015), no. 1, 1540--1561. doi:10.1214/15-EJS1051. https://projecteuclid.org/euclid.ejs/1437658702


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