  Recently, Su and Cook proposed a dimension reduction technique called the
inner envelope which can be substantially more efficient than the original
envelope or existing dimension reduction techniques for multivariate
regression. However, their technique relied on a linear model with normally
distributed error, which may be violated in practice. In this work, we propose
a semiparametric variant of the inner envelope that does not rely on the linear
model nor the normality assumption. We show that our proposal leads to globally
and locally efficient estimators of the inner envelope spaces. We also present
a computationally tractable algorithm to estimate the inner envelope. Our
simulations and real data analysis show that our method is both robust and
efficient compared to existing dimension reduction methods in a diverse array
of settings.
