Asymptotic theory for the first projective direction

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.