| With the advent of Big Data,high-dimensional data are becoming more common.However,many traditional statistical methods fail as the dimensionality of data increases.And data storage becomes an important issue.Therefore,the dimensionality reduction of highdimensional data is an issue of great interest to modern statisticians.The factor model is a widespread tool for dimensionality reduction and occupies an important position in multivariate statistical analysis.It can concentrate,summarize and extract useful information from high-dimensional data by introducing common factors.Matrix-valued data are increasingly common in practical applications,such as finance,economics,and biology.For this type of data,the traditional vector factor model will lead to a large amount of information loss.For this reason,the matrix factor model has been proposed,which can achieve dimensionality reduction of both rows and columns of the matrix and extract the information in a core factor matrix.Extensive research has been conducted on the matrix factor model.In this paper,we propose estimation methods for the matrix factor model from the perspective of the loss function.This paper focuses on the estimation of the matrix factor model,including the estimation of the factor loading matrics,factor score matrix and the number of factors.We obtain the estimators of the factor loading m atrics and factor matrix by m inimizing the squared loss,which is equivalent to the projected estimators(PE)of the matrix factor model proposed by[43].In other words,we provide the least-square interpretation of the PE of the matrix factor model,which parallels to the least-square interpretation of the PCA of the vector factor model.In addition,we obtain the convergence rate of the theoretical minimizers of the squared loss function under the condition that the idiosyncratic factors have sub-Gaussian tails.Considering the heavy-tailed distribution data,the estimators obtained using the squared loss function lack robustness.Therefore.we generalize the squared loss function to the Huber loss function and obtain a weighted projected estimators for estimating the parameters.Theoretically,we obtain the convergence rate of theoretical minimizers of the Huber loss function in the presence of idiosyncratic factors of(2+e)th moments.In addition,we propose an algorithm to robustly estimate the number of factors,inspired by the idea of the eigenvalue ratio.The simulation illustrates that Huber estimators perform more robustly than other state-of-the-art methods for heavy-tailed distribution data.In this paper,we study the parameter estimation of the matrix factor model,which is innovative in methodology,theory,and application.In terms of methodology,we give a least-square interpretation of the projected estimators of the matrix factor model and propose a new robust iterative algorithm for estimating the parameters.In addition,we propose a new method for estimating the number of factors.Theoretically,we obtain the convergence rate of both theoretical minimizers under certain conditions.In terms of application,we analyze the Fama-French portfolio dataset,estimate the number of factors and the loading matrics,and make predictions based on the proposed estimators.This application demonstrates the advantages of the proposed Huber estimators. |
