This paper introduces a matrix quantile factor model for matrix-valued data with a low-rank structure. We estimate the row and column factor spaces via minimizing the empirical check loss function over all panels. We show the estimates converge at rate $1/\min\{\sqrt{p_1p_2}, \sqrt{p_2T},$ $\sqrt{p_1T}\}$ in average Frobenius norm, where $p_1$, $p_2$ and $T$ are the row dimensionality, column dimensionality and length of the matrix sequence. This rate is faster than that of the quantile estimates via ``flattening" the matrix model into a large vector model. Smoothed estimates are given and their central limit theorems are derived under some mild condition. We provide three consistent criteria to determine the pair of row and column factor numbers. Extensive simulation studies and an empirical study justify our theory.

翻译：本文提出了一种用于具有低秩结构的矩阵型数据的矩阵分位数因子模型。我们通过在所有面板上最小化经验检验损失函数来估计行和列因子空间。我们证明这些估计量在平均Frobenius范数下以$1/\min\{\sqrt{p_1p_2}, \sqrt{p_2T}, \sqrt{p_1T}\}$的速率收敛，其中$p_1$, $p_2$和$T$分别表示矩阵序列的行维度、列维度和长度。该速率优于通过将矩阵模型"展平"为大规模向量模型得到的分位数估计的收敛速率。我们给出了平滑估计量，并在温和条件下推导了其中心极限定理。我们提出了三种一致准则来确定行和列因子数的配对。广泛的模拟研究和实证分析验证了我们的理论。