Tensor Stein-rules in a generalized tensor regression model

In this paper, we consider an estimation problem in a generalized tensor regression model with multi-mode covariates. We generalize the main results in recent literature in four ways. First, we weaken assumptions underlying the main results of the previous works. In particular, the dependence structure of the error and covariates are as weak as an L2− mixingale array, and the error term does not need to be uncorrelated with regressors. Second, we consider a more general constraint than the one considered in previous works. Third, we establish the asymptotic properties of the tensor estimators. Specifically, we derive the joint asymptotic distribution of the unrestricted tensor estimator (UE) and restricted tensor estimator (RE). Fourth, we propose a class of shrinkage-type estimators in the context of tensor regression, and under a general loss function, we derive sufficient conditions for which the shrinkage estimators dominate the UE. In addition to these interesting contributions, we derive a kind of functional central limit theorem for mixingale vector-valued and we establish some identities which are useful in studying the risk dominance of shrinkage-type tensor estimators. Finally, to illustrate the application of the proposed methods, we corroborate the results by some simulation studies of binary, Normal and Poisson data and we analyze a multi-relational network and neuro-imaging datasets.