A major family of sufficient dimension reduction (SDR) methods, called
inverse regression, commonly require the distribution of the predictor X to
have a linear E(X∣βTX) and a degenerate
var(X∣βTX) for the desired reduced predictor
βTX. In this paper, we adjust the first and second-order
inverse regression methods by modeling E(X∣βTX) and
var(X∣βTX) under the mixture model assumption on X,
which allows these terms to convey more complex patterns and is most suitable
when X has a clustered sample distribution. The proposed SDR methods build a
natural path between inverse regression and the localized SDR methods, and in
particular inherit the advantages of both; that is, they are
n​-consistent, efficiently implementable, directly adjustable under the
high-dimensional settings, and fully recovering the desired reduced predictor.
These findings are illustrated by simulation studies and a real data example at
the end, which also suggest the effectiveness of the proposed methods for
nonclustered data