### Abstract

Let Q = (Q_{1}, . . . , Q_{n}) be a random vector drawn from the uniform distribution on the set of all n! permutations of {1, 2, . . . , n}. Let Z = (Z_{1}, . . . , Z_{n}), where Zj is the mean zero variance one random variable obtained by centralizing and normalizing Qj , j = 1, . . . , n. Assume that X_{i} , i = 1, . . . ,p are i.i.d. copies of 1/√ p Z and X = X_{p,n} is the p × n random matrix with Xi as its ith row. Then S_{n} = 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 establish a CLT for the linear spectral statistics of this nonparametric random matrix model in the scenario of high dimension, namely, 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 [Ann. Statist. 36 (2008) 2553-2576] to bypass the so-called joint cumulant summability. In addition, we raise a two-step comparison approach to obtain the explicit formulae for the mean and covariance functions in the CLT. Relying on this CLT, we then construct a distribution-free 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 heavy-tailed ones.

Original language | English |
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Pages (from-to) | 2588-2623 |

Number of pages | 36 |

Journal | Annals of Statistics |

Volume | 43 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2015 Dec |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Annals of Statistics*,

*43*(6), 2588-2623. https://doi.org/10.1214/15-AOS1353