  Sufficient dimension reduction reduces the dimensionality of data while
preserving relevant regression information. In this article, we develop Minimum
Average Deviance Estimation (MADE) methodology for sufficient dimension
reduction. It extends the Minimum Average Variance Estimation (MAVE) approach
of Xia et al. (2002) from continuous responses to exponential family
distributions to include Binomial and Poisson responses. Local likelihood
regression is used to learn the form of the regression function from the data.
The main parameter of interest is a dimension reduction subspace which projects
the covariates to a lower dimension while preserving their relationship with
the outcome. To estimate this parameter within its natural space, we consider
an iterative algorithm where one step utilizes a Stiefel manifold optimizer. We
empirically evaluate the performance of three prediction methods, two that are
intrinsic to local likelihood estimation and one that is based on the
Nadaraya-Watson estimator. Initial results show that, as expected, MADE can
outperform MAVE when there is a departure from the assumption of additive
errors.
