Statistica Sinica 21 (2011), 707-730
ASYMPTOTIC PROPERTIES
OF SUFFICIENT DIMENSION
REDUCTION WITH A DIVERGING NUMBER OF
PREDICTORS
Yichao Wu and Lexin Li
North Carolina State University
Abstract: We investigate asymptotic properties of a
family of sufficient dimension reduction estimators when the number
of predictors 
diverges to infinity with the sample size. We
adopt a general formulation of dimension reduction estimation
through least squares regression of a set of transformations of the
response. This formulation allows us to establish the consistency of
reduction projection estimation. We then introduce the SCAD max
penalty, along with a difference convex optimization algorithm, to
achieve variable selection. We show that the penalized estimator
selects all truly relevant predictors and excludes all irrelevant
ones with probability approaching one, meanwhile it maintains
consistent reduction basis estimation for relevant predictors. Our
work differs from most model-based selection methods in that it does
not require a traditional model, and it extends existing sufficient
dimension reduction and model-free variable selection approaches
from the fixed scenario to a diverging .
Key words and phrases: Central subspace, diverging
parameters; SCAD, sliced inverse regression.