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 ρ=limp/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[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.

In deconvolution in R^d, d≥1, with mixing density p(∈\mathscr{P}) and kernel h, the mixture density f_p(∈\mathscr{F}_p) is estimated with MDE f_{\hat{p}n}, having upper L1-error rate, an, in probability or in risk; \hat{p}_n∈\mathscr{P}. In one application, \mathscr{P} consists of L1-separable densities in R with differences changing sign at most J times and h(x−y) Totally Positive. When h is known and p is \tilde{q}-smooth, vanishing outside a compact in R^d, plug-in upper bounds are provided for the L2-error rate of \hat{p}_n and its [s]-th mixed partial derivative \hat{p}^{(s)}_n, via ∥f_{\hat{p}_n}−f_p∥_1, with rates (log a^{−1}_n)^{−N_1} and a^{N_2}_n, respectively, for h super-smooth and smooth; \tilde{q}∈R^+, [s] ≤ \tilde{q}, d≥1, N_1>0, N_2>0. For a_n ∼ (logn)^ζ⋅(n^{−δ}), the former rate is optimal for any δ>0 and the latter misses the optimal by the factor (logn)^ξ when δ=.5; ζ>0, ξ>0. N_1 and N_2 appear in optimal rates and lower error and risk bounds in the deconvolution literature.

For multiple index models, it has recently been shown that the sliced inverse regression (SIR) is consistent for estimating the sufficient dimension reduction (SDR) subspace if and only if the dimension p and sample size n satisfies that ρ=limp/n=0. Thus, when p is of the same or a higher order of n, additional assumptions such as sparsity have to be imposed in order to ensure consistency for SIR. By constructing artificial response variables made up from top eigenvectors of the estimated conditional covariance matrix, \widehat{var(𝔼[x|y])}, we introduce a simple Lasso regression method to obtain an estimate of the SDR subspace. The resulting algorithm, Lasso-SIR, is shown to be consistent and achieve the optimal convergence rate under certain sparsity conditions when p is of order o(n^2λ^2) where λ is the generalized signal noise ratio. We also demonstrate the superior performance of Lasso-SIR compared with existing approaches via extensive numerical studies and several real data examples.