In the context of sufficient dimension reduction (SDR), sliced inverse regression (SIR) is the first and perhaps one of the most popular tools to reduce the covariate dimension for high dimensional nonlinear regressions. Despite the fact that the performance of SIR is very insensitive to the number of slices when the covariate is low or moderate dimensional, our empirical studies indicate that, the performance of SIR relies heavily upon the number of slices when the covariate is high or ultrahigh dimensional. How to select the optimal number of slices for SIR is still a longstanding problem in the SDR literature, which is a crucial issue for SIR to be effective in high and ultrahigh dimensional regressions.
In this paper, we work with an improved version of SIR, the cumulative slicing estimation (CUME) method, which does not require selecting the optimal number of slices. We provide a general framework to analyze the phase transition phenomenon for the CUME method. We show that, without sparsity assumption, CUME is consistent if and only if $p/n\to 0$, where $p$ stands for the covariate dimension and $n$ stands for the sample size. If we make certain sparsity assumptions, then the thresholding estimate for the CUME method is consistent as long as $\log(p)/n\to0$. We demonstrate the superior performance of our proposals through extensive numerical experiments.