THE ASYMPTOTIC DISTRIBUTION AND BERRY-ESSEEN BOUND OF A NEW TEST FOR INDEPENDENCE IN HIGH DIMENSION WITH AN APPLICATION TO STOCHASTIC OPTIMIZATION

By Weidong Liu, Zhengyan Lin, Qi-Man Shao. 2008

92 citations recorded by the source corpus.

Let $\mathbf{X}_1,...,\mathbf{X}_n$ be a random sample from a $p$-dimensional population distribution. Assume that $c_1n^{\alpha}\leq p\leq c_2n^{\alpha}$ for some positive constants $c_1,c_2$ and $\alpha$. In this paper we introduce a new statistic for testing independence of the $p$-variates of the population and prove that the limiting distribution is the extreme distribution of type I with a rate of convergence $O((\log n)^{5/2}/\sqrt{n})$. This is much faster than $O(1/\log n)$, a typical convergence rate for this type of extreme distribution. A simulation study and application to stochastic optimization are discussed.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c5bed0cb7dccbf76c344413aa207d117be7e1bf5

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Citations in the corpus

Gaussian Approximation for High Dimensional Time Series (2015, background): “See Li and Rosalsky (2006) and Liu et al. (2008) for some reﬁned results.”
Two-Sample Covariance Matrix Testing and Support Recovery in High-Dimensional and Sparse Settings (2013, unknown): “Further extensions and improvements are given in Zhou (2007), Liu, Lin, and Shao (2008), and Cai and Jiang (2011, 2012).”
Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, background): “In general, their result allowed wide regime cnα ≤ p ≤ Cnα but the moment condition also depends on α. Liu et al. (2008) also introduced some “intermediate” approximation to approximate the test statistic Ln with a much faster rate of convergence.”
Estimating Sparse Precision Matrix: Optimal Rates of Convergence and Adaptive Estimation (2012, background): “It follows from Liu, Lin and Shao (2008) and Cai and Jiang (2011 , which means that if δ 2, the true Ω lies outside of the feasible set with high probability and solving the corresponding minimization problem cannot lead to a good estimator of Ω. Remark 2.”
Large-Scale Correlation Screening (2011, methodology): “Tests of diagonal covariance structure are often based on the maximum sample correlation coefficient, which has recently been studied in the large p regime [12, 14, 15, 16, 21].”
Testing mutual independence in high dimensions with sums of squares of rank correlations (2015, background): “He assumed higher-order moment conditions that were weakened in subsequent work (Li et al., 2010, 2012, Liu et al., 2008, Zhou, 2007).”
Simultaneous Inference for High-Dimensional Linear Models (2016, background): “Instead, it is suggested to employ an “intermediate” approximation to improve the rate of convergence in the literature, e.g., Liu et al. (2008).”