Many statistical applications involve regression with many predictors and problems arise when the number of predictors is large or they are very correlated.  Many methods have been proposed to reduce predictor dimensions.  A partial list is Principal Component Regression, Partial Least Square, Continuum Regression, LASSO, Factor Analysis, Latent variables and the inverse regression methods like SIR and SAVE.

The focus of this proposal is reduction of the dimension of the predictors from the Inverse Regression point of view.  Assuming that the inverse regression X|Y follows a normal distribution with covariance independent of Y, Cook (2007) was able to find the central subspace for the regression of Y|X.  This means the linear combinations of the predictors X that say everything in the regression of Y|X, in the sense that Y|X ~ Y|b^TX for b e R^dxp with d ≤ p the smallest possible number.  In this proposal we present the maximum likelihood estimators for such combinations as well as testing procedures for this setting.