## The Annals of Statistics

### Asymptotic theory for the first projective direction

Michael G. Akritas

#### Abstract

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

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

Dates
Revised: January 2016
First available in Project Euclid: 12 September 2016

https://projecteuclid.org/euclid.aos/1473685272

Digital Object Identifier
doi:10.1214/16-AOS1438

Mathematical Reviews number (MathSciNet)
MR3546447

Zentralblatt MATH identifier
1349.62123

#### Citation

Akritas, Michael G. Asymptotic theory for the first projective direction. Ann. Statist. 44 (2016), no. 5, 2161--2189. doi:10.1214/16-AOS1438. https://projecteuclid.org/euclid.aos/1473685272

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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).