  We provide here a framework to analyze the phase transition phenomenon of
slice inverse regression (SIR), a supervised dimension reduction technique
introduced by \cite{Li:1991}. Under mild conditions, the asymptotic ratio
$\rho= \lim p/n$ is the phase transition parameter and the SIR estimator is
consistent if and only if $\rho= 0$. 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 $var(\mathbf{E}[\boldsymbol{x}|y])$. 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.
