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Universality for the largest eigenvalue of sample covariance matrices with general population
, 2013
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Spectral statistics of large dimensional Spearman’s rank correlation matrix and its application. arXiv preprint arXiv:1312.5119
, 2013
"... Abstract. Let Q = (Q1,..., Qn) be a random vector drawn from the uniform distribution on the set of all n! permutations of {1, 2,..., n}. Let Z = (Z1,..., Zn), where Zj is the mean zero variance one random variable obtained by centralizing and normalizing Qj, j = 1,..., n. Assume that Xi, i = 1,... ..."
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Abstract. Let Q = (Q1,..., Qn) be a random vector drawn from the uniform distribution on the set of all n! permutations of {1, 2,..., n}. Let Z = (Z1,..., Zn), where Zj is the mean zero variance one random variable obtained by centralizing and normalizing Qj, j = 1,..., n. Assume that Xi, i = 1,..., p are i.i.d. copies of 1√ p Z and X = Xp,n is the p × n random matrix with Xi as its ith row. Then Sn = XX ∗ is called the p × n Spearman’s rank correlation matrix which can be regarded as a high dimensional extension of the classical nonparametric statistic Spearman’s rank correlation coefficient between two independent random variables. In this paper we will establish a CLT for the linear spectral statistics of this nonparametric random matrix model in the scenario of high dimension supposing that p = p(n) and p/n → c ∈ (0,∞) as n→∞. We propose a novel evaluation scheme to estimate the core quantity in Anderson and Zeitouni’s cumulant method in [1] to bypass the so called joint cumulant summability. In addition, we raise a twostep comparison approach to obtain the explicit formulae for the mean and covariance functions in the CLT. Relying on this CLT we then construct a distributionfree statistic to test complete independence for components of random vectors. Owing to the nonparametric property, we can use this test on generally distributed random variables including the heavytailed ones. 1.
DistributionFree Tests of Independence with Applications to Testing More Structures
, 2014
"... We consider the problem of testing mutual independence of all entries in a ddimensional random vector X = (X1,..., Xd) T based on n independent observations. For this, we consider two families of distributionfree test statistics that converge weakly to an extreme value type I distribution. We furt ..."
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We consider the problem of testing mutual independence of all entries in a ddimensional random vector X = (X1,..., Xd) T based on n independent observations. For this, we consider two families of distributionfree test statistics that converge weakly to an extreme value type I distribution. We further study the powers of the corresponding tests against certain alternatives. In particular, we show that the powers tend to one when the maximum magnitude of the pairwise Pearson’s correlation coefficients is larger than C log d/n for some absolute constant C. This result is rate optimal. As important examples, we show that the tests based on Kendall’s tau and Spearman’s rho are rate optimal tests of independence. For further generalization, we consider accelerating the rate of convergence via approximating the exact distributions of the test statistics. We also study the tests of two more structural hypotheses: mdependence and data homogeneity. For these, we propose two rankbased tests and show their optimality.