2015/07/14 by Lin, Qian, Zhao, Zhigen, Liu, Jun S.
#62H25 (Secondary) #62J02 (Primary) #FOS: Mathematics #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.1507.03895
We provide here a framework to analyze the phase transition phenomenon of slice inverse regression (SIR), a supervised dimension reduction technique introduced by \citeLi:1991. Under mild conditions, the asymptotic ratio ρ= lim p/n is the phase transition parameter and the SIR estimator is consistent if and only if ρ= 0. When dimension p is greater than n, we propose a diagonal thresholding screening SIR (DT-SIR) algorithm. This method provides us with an estimate of the eigen-space of the covariance matrix of the conditional expectation var(E[\boldsymbolx|y]). The desired dimension reduction space is then obtained by multiplying the inverse of the covariance matrix on the eigen-space. Under certain sparsity assumptions on both the covariance matrix of predictors and the loadings of the directions, we prove the consistency of DT-SIR in estimating the dimension reduction space in high dimensional data analysis. Extensive numerical experiments demonstrate superior performances of the proposed method in comparison to its competitors.