arXiv subjects

Zeng Li

Publications and source records attributed to Zeng Li.

27 records · Page 2Linked to original sources

arXiv2021

Asymptotic normality for eigenvalue statistics of a general sample covariance matrix when $p/n \to \infty$ and applications

The asymptotic normality for a large family of eigenvalue statistics of a general sample covariance matrix is derived under the ultra-high dimensional setting, that is, when the dimension to sample size ratio $p/n \to \infty$. Based on this CLT result, we first adapt the covariance matrix test problem to the new ultra-high dimensional context. Then as a second application, we develop a new test for the separable covariance structure of a matrix-valued white noise. Simulation experiments are conducted for the investigation of finite-sample properties of the general asymptotic normality of eigenvalue statistics, as well as the second test for separable covariance structure of matrix-valued white noise.

stat.ME↗

arXiv2020

Provable More Data Hurt in High Dimensional Least Squares Estimator

This paper investigates the finite-sample prediction risk of the high-dimensional least squares estimator. We derive the central limit theorem for the prediction risk when both the sample size and the number of features tend to infinity. Furthermore, the finite-sample distribution and the confidence interval of the prediction risk are provided. Our theoretical results demonstrate the sample-wise nonmonotonicity of the prediction risk and confirm "more data hurt" phenomenon.

stat.ML↗

arXiv2019

Central Limit Theorem for Linear Spectral Statistics of Large Dimensional Kendall's Rank Correlation Matrices and its Applications

This paper is concerned with the limiting spectral behaviors of large dimensional Kendall's rank correlation matrices generated by samples with independent and continuous components. We do not require the components to be identically distributed, and do not need any moment conditions, which is very different from the assumptions imposed in the literature of random matrix theory. The statistical setting in this paper covers a wide range of highly skewed and heavy-tailed distributions. We establish the central limit theorem (CLT) for the linear spectral statistics of the Kendall's rank correlation matrices under the Marchenko-Pastur asymptotic regime, in which the dimension diverges to infinity proportionally with the sample size. We further propose three nonparametric procedures for high dimensional independent test and their limiting null distributions are derived by implementing this CLT. Our numerical comparisons demonstrate the robustness and superiority of our proposed test statistics under various mixed and heavy-tailed cases.

math.ST↗

arXiv2019

Asymptotic joint distribution of extreme eigenvalues and trace of large sample covariance matrix in a generalized spiked population model

This paper studies the joint limiting behavior of extreme eigenvalues and trace of large sample covariance matrix in a generalized spiked population model, where the asymptotic regime is such that the dimension and sample size grow proportionally. The form of the joint limiting distribution is applied to conduct Johnson-Graybill-type tests, a family of approaches testing for signals in a statistical model. For this, higher order correction is further made, helping alleviate the impact of finite-sample bias. The proof rests on determining the joint asymptotic behavior of two classes of spectral processes, corresponding to the extreme and linear spectral statistics respectively.

math.ST↗

arXiv2018

On testing for high-dimensional white noise

Testing for white noise is a classical yet important problem in statistics, especially for diagnostic checks in time series modeling and linear regression. For high-dimensional time series in the sense that the dimension $p$ is large in relation to the sample size $T$, the popular omnibus tests including the multivariate Hosking and Li-McLeod tests are extremely conservative, leading to substantial power loss. To develop more relevant tests for high-dimensional cases, we propose a portmanteau-type test statistic which is the sum of squared singular values of the first $q$ lagged sample autocovariance matrices. It, therefore, encapsulates all the serial correlations (upto the time lag $q$) within and across all component series. Using the tools from random matrix theory and assuming both $p$ and $T$ diverge to infinity, we derive the asymptotic normality of the test statistic under both the null and a specific VMA(1) alternative hypothesis. As the actual implementation of the test requires the knowledge of three characteristic constants of the population cross-sectional covariance matrix and the value of the fourth moment of the standardized innovations, non trivial estimations are proposed for these parameters and their integration leads to a practically usable test. Extensive simulation confirms the excellent finite-sample performance of the new test with accurate size and satisfactory power for a large range of finite $(p,T)$ combinations, therefore ensuring wide applicability in practice. In particular, the new tests are consistently superior to the traditional Hosking and Li-McLeod tests.

math.ST↗

arXiv2018

Joint CLT for eigenvalue statistics from several dependent large dimensional sample covariance matrices with application

Let $\mathbf{X}_n=(x_{ij})$ be a $k \times n$ data matrix with complex-valued, independent and standardized entries satisfying a Lindeberg-type moment condition. We consider simultaneously $R$ sample covariance matrices $\mathbf{B}_{nr}=\frac1n \mathbf{Q}_r \mathbf{X}_n \mathbf{X}_n^*\mathbf{Q}_r^\top,~1\le r\le R$, where the $\mathbf{Q}_{r}$'s are nonrandom real matrices with common dimensions $p\times k~(k\geq p)$. Assuming that both the dimension $p$ and the sample size $n$ grow to infinity, the limiting distributions of the eigenvalues of the matrices $\{\mathbf{B}_{nr}\}$ are identified, and as the main result of the paper, we establish a joint central limit theorem for linear spectral statistics of the $R$ matrices $\{\mathbf{B}_{nr}\}$. 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, though 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 ($R=1$).

math.ST↗

arXiv2016

Testing the Sphericity of a covariance matrix when the dimension is much larger than the sample size

This paper focuses on the prominent sphericity test when the dimension $p$ is much lager than sample size $n$. The classical likelihood ratio test(LRT) is no longer applicable when $p\gg n$. Therefore a Quasi-LRT is proposed and asymptotic distribution of the test statistic under the null when $p/n\rightarrow\infty, n\rightarrow\infty$ is well established in this paper. Meanwhile, John's test has been found to possess the powerful {\it dimension-proof} property, which keeps exactly the same limiting distribution under the null with any $(n,p)$-asymptotic, i.e. $p/n\rightarrow[0,\infty]$, $n\rightarrow\infty$. All asymptotic results are derived for general population with finite fourth order moment. Numerical experiments are implemented for comparison.

stat.ME↗

arXiv2015

Identifying the number of factors from singular values of a large sample auto-covariance matrix

Identifying the number of factors in a high-dimensional factor model has attracted much attention in recent years and a general solution to the problem is still lacking. A promising ratio estimator based on the singular values of the lagged autocovariance matrix has been recently proposed in the literature and is shown to have a good performance under some specific assumption on the strength of the factors. Inspired by this ratio estimator and as a first main contribution, this paper proposes a complete theory of such sample singular values for both the factor part and the noise part under the large-dimensional scheme where the dimension and the sample size proportionally grow to infinity. In particular, we provide the exact description of the phase transition phenomenon that determines whether a factor is strong enough to be detected with the observed sample singular values. Based on these findings and as a second main contribution of the paper, we propose a new estimator of the number of factors which is strongly consistent for the detection of all significant factors (which are the only theoretically detectable ones). In particular, factors are assumed to have the minimum strength above the phase transition boundary which is of the order of a constant; they are thus not required to grow to infinity together with the dimension (as assumed in most of the existing papers on high-dimensional factor models). Empirical Monte-Carlo study as well as the analysis of stock returns data attest a very good performance of the proposed estimator. In all the tested cases, the new estimator largely outperforms the existing estimator using the same ratios of singular values.

stat.ME↗

arXiv2014

On singular value distribution of large dimensional auto-covariance matrices

Let $(\varepsilon_j)_{j\geq 0}$ be a sequence of independent $p-$dimensional random vectors and $τ\geq1$ a given integer. From a sample $\varepsilon_1,\cdots,\varepsilon_{T+τ-1},\varepsilon_{T+τ}$ of the sequence, the so-called lag $-τ$ auto-covariance matrix is $C_τ=T^{-1}\sum_{j=1}^T\varepsilon_{τ+j}\varepsilon_{j}^t$. When the dimension $p$ is large compared to the sample size $T$, this paper establishes the limit of the singular value distribution of $C_τ$ assuming that $p$ and $T$ grow to infinity proportionally and the sequence satisfies a Lindeberg condition on fourth order moments. Compared to existing asymptotic results on sample covariance matrices developed in random matrix theory, the case of an auto-covariance matrix is much more involved due to the fact that the summands are dependent and the matrix $C_τ$ is not symmetric. Several new techniques are introduced for the derivation of the main theorem.

stat.ME↗