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.

翻译：本文考虑张量值序列因子模型中的（鲁棒）推断。研究基于最小化二次损失函数的估计量时，所估计公共因子与载荷空间的一致性。基于此类损失函数仅在足够多矩存在时适用的观察，我们将结果扩展至重尾分布情形，通过考虑基于最小化Huber损失函数的估计量——该损失函数对离群值采用$L_{1}$范数加权。我们证明，即使数据仅存在二阶矩，此类估计量对重尾分布仍具有鲁棒性。同时提出修正特征值比准则以估计核心张量维度，并在样本量与维度发散相对速率无任何约束条件下，证明所得估计量的一致性。大量数值实验表明，所提方法尤其在重尾情形下优于现有最优方法。通过分析跨国多种商品进出口数据集，展示所提鲁棒估计过程的实际应用价值。实现所提方法的R语言程序包"RTFA"已发布在R CRAN平台。