On testing for high-dimensional white noise

Zeng, Li; Lam, Clifford; Yao, Jianfeng; Yao, Qiwei

doi:10.1214/18-aos1782

Cited by 29 publications

(39 citation statements)

References 30 publications

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“…t−k ) * }, k ≥ 0, (k is called the lag or the step) carry useful information about the model (3), specially through their spectral distributions. Some of the works that deal with limit spectral distributions, mostly for high-dimensional real-valued time series, and their use in statistical inference are, [2,3,4,5,26,41,24,23,6].…”

Section: Application To Statistical Hypothesis Testingmentioning

confidence: 99%

Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application

Bose

Hachem

2020

Journal of Multivariate Analysis

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Suppose X is an N × n complex matrix whose entries are centered, independent, and identically distributed random variables with variance 1/n and whose fourth moment is of order O(n −2 ). In the first part of the paper, we consider the non-Hermitian matrix XAX * −z, where A is a deterministic matrix whose smallest and largest singular values are bounded below and above respectively, and z = 0 is a complex number. Asymptotic probability bounds for the smallest singular value of this matrix are obtained in the large dimensional regime where N and n diverge to infinity at the same rate.In the second part of the paper, we consider the special case where A = J = [1 i−j=1 mod n ] is a circulant matrix. Using the result of the first part, it is shown that the limit spectral distribution of XJX * exists in the large dimensional regime, and we determine this limit explicitly. A statistical application of this result devoted towards testing the presence of correlations within a multivariate time series is considered. Assuming that X represents a C N -valued time series which is observed over a time window of length n, the matrix XJX * represents the one-step sample autocovariance matrix of this time series. Guided by the result on the limit spectral distribution of this matrix, a whiteness test against an MA correlation model for the time series is introduced. Numerical simulations show the excellent performance of this test.

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Section: Application To Statistical Hypothesis Testingmentioning

confidence: 99%

Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application

Bose

Hachem

2020

Journal of Multivariate Analysis

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“…Consider the Stieltjes transform

{\underset{m}{false}}_{τ} false(z false)

of the limiting spectral distribution of

{\overset{N}{false}}_{τ}

, by implementing the Silverstein equation , we can infer that

{\underset{m}{false}}_{τ} false(z false)

satisfies

\begin{matrix} z & = - \frac{1}{{\underset{m}{false}}_{τ} false(z false)} + \frac{1}{c} \int \frac{t}{1 + t {\underset{m}{false}}_{τ} false(z false)} d H_{τ} false(t false) \\ = \frac{1}{{\underset{m}{false}}_{τ} false(z false)} ((), - 1 + \frac{1}{c} - \frac{1}{c \sqrt{1 - {\underset{m}{false}}_{τ}^{2} false(z false)}}), \end{matrix}

where p / n → c >0 as n → ∞ , which coincides with the results in Bai & Wang () and Li et al ().…”

Section: Proofs Of the Main Theoremsmentioning

confidence: 99%

“…They become extremely conservative, losing size and power dramatically. In a very recent work, Li et al () looked into this high‐dimensional portmanteau test problem and proposed several new test statistics based on single‐lagged and multi‐lagged sample auto‐covariance matrices. More precisely, let's consider a p ‐dimensional time series modelled as a linear process

x_{t} = \sum_{l ⩾ 0} A_{l} z_{t - l},

where { z t } is a sequence of independent p ‐dimensional random vectors with independent components z t = ( z i t ) satisfying

normalE z_{i t} = 0, normalE {false| z_{i t} false|}^{2} = 1, normalE false| z_{i t} |^{4} < \infty

.…”

Section: Introductionmentioning

confidence: 99%

“…The lag‐τ sample auto‐covariance matrix is

{\overset{Σ}{false}}_{τ} = \frac{1}{n} true \sum_{t = 1}^{n} x_{t} {boldx}_{t - τ}^{*},

where x t = x n + t when t ≤0. Li et al () proposed a test statistic based on

{\overset{Σ}{false}}_{τ}

. For any given constant integer 1≤ τ ≤ q , their test statistic

{\overset{L}{false}}_{τ}

was designed to test the specific lag‐ τ autocorrelation of the sequence, that is,

{\tilde{L}}_{τ} = {falsefalse}_{\sum}^{j = 1} λ_{j, τ}^{2} = Tr ({\tilde{boldM}}_{τ}^{*} {\tilde{boldM}}_{τ}),

where { λ j , τ , j = 1,⋯, p } are the eigenvalues of

{\tilde{boldM}}_{τ} = \frac{1}{2} ({\hat{normalΣ}}_{τ} + {\hat{normalΣ}}_{τ}^{*}) = \frac{1}{2 n} {falsefalse}_{\sum}^{t = 1} ({boldx}_{t} {boldx}_{t - τ}^{*} + {boldx}_{t - τ} {boldx}_{t}^{*}),

which is the symmetrized lag‐τ sample auto‐covariance matrix .…”

Section: Introductionmentioning

confidence: 99%

“…Consequently, we need to resort to the joint CLT studied in this paper to characterize the correlations among

false{{\overset{L}{false}}_{τ}, 1 \leq τ \leq q false}

. It is worth noticing that Li et al () proposed another multi‐lagged test statistic U q by stacking a number of consecutive observation vectors. It will be shown in this paper that this test statistic U q is essentially much less powerful than

L_{q}

considered here due to the data loss caused by observation stacking.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Joint Central Limit Theorem for Eigenvalue Statistics from Several Dependent Large Dimensional Sample Covariance Matrices with Application

Zeng

Yao

2018

Scandinavian J Statistics

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Let X n D .x ij / be a k n data matrix with complex-valued, independent and standardized entries satisfying a Lindeberg-type moment condition. We consider simultaneously R sample covariance matrices B nr D 1 n Q r X n X n Q > r ; 1 Ä r Ä R, where the Q r 's are non-random real matrices with common dimensions p k .k p/. Assuming that both the dimension p and the sample size n grow to infinity, the limiting distributions of the eigenvalues of the matrices ¹B nr º are identified, and as the main result of the paper, we establish a joint central limit theorem (CLT) for linear spectral statistics of the R matrices ¹B nr º. Next, this new CLT is applied to the problem of testing a high-dimensional white noise in time series modelling. In experiments, the derived test has a controlled size and is significantly faster than the classical permutation test, although it does have lower power. This application highlights the necessity of such joint CLT in the presence of several dependent sample covariance matrices. In contrast, all the existing works on CLT for linear spectral statistics of large sample covariance matrices deal with a single sample covariance matrix (R D 1).

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Multi-task Framework for Vibration-Based Structural Damage Detection of Spatial Truss Structure Using Graph Learning

Dang,

Nguyen

2024

J. Vib. Eng. Technol.

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On testing for high-dimensional white noise

Cited by 29 publications

References 30 publications

Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application

Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application

Joint Central Limit Theorem for Eigenvalue Statistics from Several Dependent Large Dimensional Sample Covariance Matrices with Application

Multi-task Framework for Vibration-Based Structural Damage Detection of Spatial Truss Structure Using Graph Learning

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