The Annals of Statistics

Asymptotic theory for the first projective direction

Michael G. Akritas

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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.

Article information

Ann. Statist., Volume 44, Number 5 (2016), 2161-2189.

Received: September 2015
Revised: January 2016
First available in Project Euclid: 12 September 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62G20: Asymptotic properties
Secondary: 60G15: Gaussian processes 60G17: Sample path properties

Nonparametric regression dimension reduction index models asymptotic theory uniform convergence empirical processes Gaussian processes


Akritas, Michael G. Asymptotic theory for the first projective direction. Ann. Statist. 44 (2016), no. 5, 2161--2189. doi:10.1214/16-AOS1438.

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Supplemental materials

  • Supplement to “Asymptotic theory for the first projective direction”. The proofs of relations (5.7) and (5.10)–(5.15) are given in Akritas (2016).