大规模相关矩阵中块可交换结构的检测
Detection of Block-Exchangeable Structure in Large-Scale Correlation Matrices
AI总结：
本文针对高维相关矩阵提出一种基于 Kendall 秩相关的稳健算法，在无需预知簇数 K 的条件下检测块可交换结构，从而显著减少未知参数并提升有限样本估计效率。
摘要：
相关矩阵在多变量数据分析中无处不在。当变量数量 d 很大时，相关矩阵的样本估计通常噪声较大，并掩盖了潜在的依赖模式。我们考虑变量可以被分为 K 个具有可交换依赖性的簇的情形；这一假设在金融和计量经济学等应用中经常被提出。在这种部分可交换性条件下，相应的相关矩阵具有块结构，未知参数的数量从 d(d-1)/2 减少到至多 K(K+1)/2。我们提出一种基于 Kendall 秩相关的稳健算法来识别这些簇，无需事先知道 K，也不需要对边缘分布作除连续性之外的任何假设。当 K < d 时，相应的块结构估计器表现明显优于样本 Kendall 秩相关矩阵。即使在无结构的 K = d 情形下，新估计器在有限样本中也可以高效得多，尽管在渐近意义上没有增益。当数据分布为椭圆分布时，这些结果可推广到线性相关矩阵及其逆矩阵。该过程在金融股票收益率数据上进行了说明。
英文摘要：
Correlation matrices are omnipresent in multivariate data analysis. When the number d of variables is large, the sample estimates of correlation matrices are typically noisy and conceal underlying dependence patterns. We consider the case when the variables can be grouped into K clusters with exchangeable dependence; this assumption is often made in applications, e.g., in finance and econometrics. Under this partial exchangeability condition, the corresponding correlation matrix has a block structure and the number of unknown parameters is reduced from d(d-1)/2 to at most K(K+1)/2. We propose a robust algorithm based on Kendall's rank correlation to identify the clusters without assuming the knowledge of K a priori or anything about the margins except continuity. The corresponding block-structured estimator performs considerably better than the sample Kendall rank correlation matrix when K < d. The new estimator can also be much more efficient in finite samples even in the unstructured case K = d, although there is no gain asymptotically. When the distribution of the data is elliptical, the results extend to linear correlation matrices and their inverses. The procedure is illustrated on financial stock returns.
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