Change-point detection in a tensor regression model

In this paper, we consider an inference problem in a tensor regression model with one change-point. Specifically, we consider a general hypothesis testing problem on a tensor parameter and the studied testing problem includes as a special case the problem about the absence of a change-point. To this end, we derive the unrestricted estimator (UE) and the restricted estimator (RE) as well as the joint asymptotic normality of the UE and RE. Thanks to the established asymptotic normality, we derive a test for testing the hypothesized restriction. We also derive the asymptotic power of the proposed test and we prove that the established test is consistent. Beyond the complexity of the testing problem in the tensor model, we consider a very general case where the tensor error term and the regressors do not need to be independent and the dependence structure of the outer-product of the tensor error term and regressors is as weak as that of an \(\mathcal {L}^2-\) mixingale. Further, to study the performance of the proposed methods in small and moderate sample sizes, we present some simulation results that corroborate the theoretical results. Finally, to illustrate the application of the proposed methods, we test the non-existence of a change-point in some fMRI neuro-imaging data.