  A major family of sufficient dimension reduction (SDR) methods, called
inverse regression, commonly require the distribution of the predictor $X$ to
have a linear $E(X|\beta^\mathsf{T}X)$ and a degenerate
$\mathrm{var}(X|\beta^\mathsf{T}X)$ for the desired reduced predictor
$\beta^\mathsf{T}X$. In this paper, we adjust the first and second-order
inverse regression methods by modeling $E(X|\beta^\mathsf{T}X)$ and
$\mathrm{var}(X|\beta^\mathsf{T}X)$ 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
$\sqrt{n}$-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.
