Fused Estimators of the Central Subspace in Sufficient Dimension Reduction

Abstract

When studying the regression of a univariate variable Y on a vector x of predictors, most existing sufficient dimension-reduction (SDR) methods require the construction of slices of Y to estimate moments of the conditional distribution of X given Y. But there is no widely accepted method for choosing the number of slices, while a poorly chosen slicing scheme may produce miserable results. We propose a novel and easily implemented fusing method that can mitigate the problem of choosing a slicing scheme and improve estimation efficiency at the same time. We develop two fused estimators—called FIRE and DIRE—based on an optimal inverse regression estimator. The asymptotic variance of FIRE is no larger than that of the original methods regardless of the choice of slicing scheme, while DIRE is less computational intense and more robust. Simulation studies show that the fused estimators perform effectively the same as or substantially better than the parent methods. Fused estimators based on other methods can be developed in parallel: fused sliced inverse regression (SIR), fused central solution space (CSS)-SIR, and fused likelihood-based method (LAD) are introduced briefly. Simulation studies of the fused CSS-SIR and fused LAD estimators show substantial gain over their parent methods. A real data example is also presented for illustration and comparison. Supplementary materials for this article are available online.

KEY WORDS:

• Cumulative mean estimation
• Inverse regression estimator
• Sliced average variance estimation
• Sliced inverse regression