论文标题
基质分位数模型
Matrix Quantile Factor Model
论文作者
论文摘要
本文介绍了具有低级别结构的基质值数据的矩阵分位数模型。我们通过用正交旋转约束最小化经验检查损失函数来估计行和色谱重因子空间。我们表明，估计值以$ $ $ $（\ min \ {p_1p_2，p_2t，p_1t \}）为单位的估计值在平均frobenius norm中收敛于{ - 1/2} $，其中$ p_1 $，$ p_2 $ and $ p_2 $和$ t $是行尺寸，列维度，列尺寸和矩阵序列的长度。该速率比通过``将矩阵模型''变成大型向量模型的量化估计速度快。为了得出中心极限定理，我们引入了一种新颖的增强的拉格朗日功能，这等于原始受约束的经验检查损失问题。围绕真实因素及其载荷的损失功能很容易导致分数函数的可行二阶扩展，并很容易地确定了载荷的平滑估计值。
This paper introduces a matrix quantile factor model for matrix-valued data with low-rank structure. We estimate the row and column factor spaces via minimizing the empirical check loss function with orthogonal rotation constraints. We show that the estimates converge at rate $(\min\{p_1p_2,p_2T,p_1T\})^{-1/2}$ in the average Frobenius norm, where $p_1$, $p_2$ and $T$ are the row dimensionality, column dimensionality and length of the matrix sequence, respectively. This rate is faster than that of the quantile estimates via ``flattening" the matrix model into a large vector model. To derive the central limit theorem, we introduce a novel augmented Lagrangian function, which is equivalent to the original constrained empirical check loss minimization problem. Via the equivalence, we prove that the Hessian matrix of the augmented Lagrangian function is locally positive definite, resulting in a locally convex penalized loss function around the true factors and their loadings. This easily leads to a feasible second-order expansion of the score function and readily established central limit theorems of the smoothed estimates of the loadings. 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.
