Variable selection of linear programming discriminant estimator

In this paper, the variable selection property will be studied for the linear programming discriminant (LPD) estimator, denoted by β^n$\hat{\beta }_n$ with n being the sample size. The LPD estimator is used in high-dimensional linear discriminant analysis under the assumption that the Bayes direction β:=Σ-1μ∈Rp$\beta :=\Sigma ^{-1}\mu \in \mathbb {R}^p$ is sparse which has support T. More exactly, we will study the property P({ Sign (β^n)= Sign (β)})→1$\mathbb {P}(\lbrace \textrm {Sign}(\hat{\beta }_n)=\textrm {Sign}(\beta)\rbrace)\rightarrow 1$ as n → ∞, which means sign consistency. A sufficient condition will be proposed under which the sign consistency property holds as log (p) ⩽ cn for small enough c > 0. The result is also non asymptotic. Our result gives optimal bounds on n and min a ∈ T|βa| and an optimal bound on |β^-β|∞$|\hat{\beta }-\beta |_{\infty }$.