Projective independence tests in high dimensions: the curses and the cures
针对高维随机向量独立性检验中投影相关法计算复杂、零分布难解、功效下降的问题，提出改进权重函数和交叉验证特征筛选，将复杂度降至O{n²(p+q)}，并证明渐近零分布为标准正态。
Summary Testing independence between high-dimensional random vectors is fundamentally different from testing independence between univariate random variables. Taking the projection correlation as an example, it suffers from at least three problems. First, it has a high computational complexity of O{n3(p+q)}, where n, p and q are the sample size and dimensions of the random vectors; this limits its usefulness substantially when n is extremely large. Second, the asymptotic null distribution of the projection correlation test is rarely tractable; therefore, random permutations are often suggested as a means of approximating the asymptotic null distribution, which further increases the complexity of implementing independence tests. Third, the power performance of the projection correlation test deteriorates in high dimensions. To address these issues, the projection correlation is improved by using a modified weight function, which reduces the complexity to O{n2(p+q)}. We estimate the improved projection correlation with U-statistic theory. Importantly, its asymptotic null distribution is standard normal, thanks to the high dimesnionality of the random vectors. This expedites the implementation of independence tests substantially. To enhance the power performance in high dimensions, we propose incorporating a cross-validation procedure with feature screening into the projection correlation test. The implementation efficacy and power enhancement are confirmed through extensive numerical studies.