近似广义距离协方差的原分布
Approximating the null distribution of generalized distance covariance
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中文总结 AI 辅助
该研究针对广义距离协方差原分布的近似问题，提出基于双中心距离矩阵谱的渐近检验方法，结合自适应算法与收缩校正，降低计算成本且提升尾部精度，模拟中其经验一类错误收敛至名义水平。
中文摘要 AI 辅助
距离协方差的原分布通常通过置换法近似，当需要极小的p值时该方法计算成本过高；或通过将若干矩与参数族匹配近似，该方法在尾部精度不足。第三种选择是通过双中心距离矩阵的谱直接近似极限分布（卡方变量的加权和），该方法已用于基于核的检验，但缺乏严格论证。我们证明，对于可分度量空间上的一般负型距离类，经验谱能一致地近似极限原分布，因此可得到渐近有效的检验，该结果涵盖希尔伯特-施密特独立性准则作为特例。我们还提出一种自适应算法，通过部分特征分解确定p值的区间，将计算成本从O(n³)降至O(k n²)，并提出一种匹配前两阶矩的收缩校正方法。模拟实验表明，所提出的检验是唯一的非蒙特卡洛方法，其经验一类错误收敛到名义水平。
英文摘要
The null distribution of distance covariance is usually approximated by permutation, which is prohibitive when very small p-values are needed, or by matching a few moments to a parametric family, which is inaccurate in the tails. A third option is to approximate the limiting distribution, a weighted sum of chi-square variables, directly through the spectra of the doubly centred distance matrices. This is used for kernel-based tests but has lacked a rigorous justification. We prove that the empirical spectra give a uniformly consistent approximation of the limiting null distribution, and hence an asymptotically valid test, for a general class of distances of negative type on separable metric spaces. The result covers the Hilbert-Schmidt independence criterion as a special case. We also give an adaptive algorithm that brackets the p-value from a partial eigendecomposition, reducing the cost from $O(n^3)$ to $O(k n^2)$, and a shrinkage correction matching the first two moments. In simulations, the proposed tests are the only non-Monte-Carlo procedures whose empirical type I error converges to the nominal level.
