Abstract: Sufficient dimension-reduction (SDR) methods characterize the relationship between a response Y and the covariates x using a few linear combinations of the covariates. Extensive SDR techniques have been developed, among which, the inverse regression-based methods are perhaps the most appealing in practice, because they do not involve multi-dimensional smoothing and are easy to implement. However, these methods require two distributional assumptions on the covariates. In particular, the first-order methods, such as the sliced inverse regression, require the linear conditional mean (LCM) assumption, while the second-order methods, such as the sliced average variance estimation, also require the constant conditional variance (CCV) assumption. We check the validity of the LCM and CCV conditions using mean independence tests, which are facilitated by the martingale difference divergence. We propose a consistent bootstrap procedure to decide the critical values of the test. Monte Carlo simulations and an application to a horse mussels data set demonstrate the finite-sample performance of the proposed method.

Key words and phrases: Constant variance, dimension reduction, inverse regression, linear mean, mean independence.