Abstract: In this article, we propose a flexible model-free approach to the regression analysis of a tensor response and a vector predictor. Without specifying the specific form of the regression mean function, we consider two closely related statistical problems: (i) estimation of the dimension reduction subspace that captures all the variations in the regression mean function, and (ii) hypothesis testing of whether the conditional expectation of a linear dimension reduction of the response given the predictor is invariant to the changes in the predictor. We propose a new nonparametric metric called tensor martingale difference divergence, and study its statistical properties. Built on this new metric, we develop computationally efficient estimation and asymptotically valid testing procedures. We demonstrate the efficacy of our method through both simulations and two real data applications for macroeconomics and e-commerce.

Key words and phrases: Martingale difference divergence, nonlinearity, sufficient dimension reduction, tensor decomposition, tensor regression, wild bootstrap.