Minimum average deviance estimation for sufficient dimension reduction

Sufficient dimension reduction methods aim to reduce the dimensionality of predictors while preserving regression information relevant to the response. In this article, we develop Minimum Average Deviance Estimation (MADE) methodology for sufficient dimension reduction. The purpose of MADE is to generalize Minimum Average Variance Estimation (MAVE) beyond its assumption of additive errors to settings where the outcome follows an exponential family distribution. As in MAVE, a local likelihood approach is used to learn the form of the regression function from the data and the main parameter of interest is a dimension reduction subspace. To estimate this parameter within its natural space, we propose an iterative algorithm where one step utilizes optimization on the Stiefel manifold. MAVE is seen to be a special case of MADE in the case of Gaussian outcomes with a common variance. Several procedures are considered to estimate the reduced dimension and to predict the outcome for an arbitrary covariate value. Initial simulations and data analysis examples yield encouraging results and invite further exploration of the methodology.