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October 2016 Asymptotic theory for the first projective direction
Michael G. Akritas
Ann. Statist. 44(5): 2161-2189 (October 2016). DOI: 10.1214/16-AOS1438


For a response variable $Y$, and a $d$ dimensional vector of covariates $\mathbf{X}$, the first projective direction, $\mathbf{\vartheta}$, is defined as the direction that accounts for the most variability in $Y$. The asymptotic distribution of an estimator of a trimmed version of $\mathbf{\vartheta}$ has been characterized only under the assumption of the single index model (SIM). This paper proposes the use of a flexible trimming function in the objective function, which results in the consistent estimation of $\mathbf{\vartheta}$. It also derives the asymptotic normality of the proposed estimator, and characterizes the components of the asymptotic variance which vanish when the SIM holds.


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Michael G. Akritas. "Asymptotic theory for the first projective direction." Ann. Statist. 44 (5) 2161 - 2189, October 2016.


Received: 1 September 2015; Revised: 1 January 2016; Published: October 2016
First available in Project Euclid: 12 September 2016

zbMATH: 1349.62123
MathSciNet: MR3546447
Digital Object Identifier: 10.1214/16-AOS1438

Primary: 62G08 , 62G20
Secondary: 60G15 , 60G17

Keywords: Asymptotic theory , Dimension reduction , Empirical processes , Gaussian processes , index models , Nonparametric regression , Uniform convergence

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 5 • October 2016
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