Robust Tensor Factor Analysis

Abstract: We consider (robust) inference in the context of a factor model for tensor-valued sequences. We study the consistency of the estimated common factors and loadings space when using estimators based on minimising quadratic loss functions. Building on the observation that such loss functions are adequate only if sufficiently many moments exist, we extend our results to the case of heavy-tailed distributions by considering estimators based on minimising the Huber loss function, which uses an $L_{1}$ -norm weight on outliers. We show that such class of estimators is robust to the presence of heavy tails, even when only the second moment of the data exists. We also propose a modified version of the eigenvalue-ratio principle to estimate the dimensions of the core tensor and show the consistency of the resultant estimators without any condition on the relative rates of divergence of the sample size and dimensions. Extensive numerical studies are conducted to show the advantages of the proposed methods over the state-of-the-art ones especially under the heavy-tailed cases. An import/export dataset of a variety of commodities across multiple countries is analyzed to show the practical usefulness of the proposed robust estimation procedure. An R package ``RTFA" implementing the proposed methods is available on R CRAN.