Asymptotic theory for the first projective direction
The Annals of Statistics · 2016 · Volume 44 · Issue 5
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.
Keywords
60G15, 60G17, 62G08, 62G20, Asymptotic theory, Dimension reduction, Empirical processes, Gaussian processes, index models, Nonparametric regression, Uniform convergence
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