arXiv subjects

Z. D. Bai

Publications and source records attributed to Z. D. Bai.

6 recordsLinked to original sources

arXiv2017

Order Determination of Large Dimensional Dynamic Factor Model

Consider the following dynamic factor model: $\mathbf{R}_t=\sum_{i=0}^q \mathbfΛ_i \mathbf{f}_{t-i}+\mathbf{e}_t,t=1,...,T$, where $\mathbfΛ_i$ is an $n\times k$ loading matrix of full rank, $\{\mathbf{f}_t\}$ are i.i.d. $k\times1$-factors, and $\mathbf{e}_t$ are independent $n\times1$ white noises. Now, assuming that $n/T\to c>0$, we want to estimate the orders $k$ and $q$ respectively. Define a random matrix $$\mathbfΦ_n(τ)=\frac{1}{2T}\sum_{j=1}^T (\mathbf{R}_j \mathbf{R}_{j+τ}^* + \mathbf{R}_{j+τ} \mathbf{R}_j^*),$$ where $τ\ge 0$ is an integer. When there are no factors, the matrix $Φ_{n}(τ)$ reduces to $$\mathbf{M}_n(τ) = \frac{1}{2T} \sum_{j=1}^T (\mathbf{e}_j \mathbf{e}_{j+τ}^* + \mathbf{e}_{j+τ} \mathbf{e}_j^*).$$ When $τ=0$, $\mathbf{M}_n(τ)$ reduces to the usual sample covariance matrix whose ESD tends to the well known MP law and $\mathbfΦ_n(0)$ reduces to the standard spike model. Hence the number $k(q+1)$ can be estimated by the number of spiked eigenvalues of $\mathbfΦ_n(0)$. To obtain separate estimates of $k$ and $q$ , we have employed the spectral analysis of $\mathbf{M}_n(τ)$ and established the spiked model analysis for $\mathbfΦ_n(τ)$.

math.ST↗

arXiv2015

Strong limit of the extreme eigenvalues of a symmetrized auto-cross covariance matrix

The auto-cross covariance matrix is defined as \[\mathbf{M}_n=\frac{1} {2T}\sum_{j=1}^T\bigl(\mathbf{e}_j\mathbf{e}_{j+τ}^*+\mathbf{e}_{j+ τ}\mathbf{e}_j^*\bigr),\] where $\mathbf{e}_j$'s are $n$-dimensional vectors of independent standard complex components with a common mean 0, variance $σ^2$, and uniformly bounded $2+η$th moments and $τ$ is the lag. Jin et al. [Ann. Appl. Probab. 24 (2014) 1199-1225] has proved that the LSD of $\mathbf{M}_n$ exists uniquely and nonrandomly, and independent of $τ$ for all $τ\ge 1$. And in addition they gave an analytic expression of the LSD. As a continuation of Jin et al. [Ann. Appl. Probab. 24 (2014) 1199-1225], this paper proved that under the condition of uniformly bounded fourth moments, in any closed interval outside the support of the LSD, with probability 1 there will be no eigenvalues of $\mathbf{M}_n$ for all large $n$. As a consequence of the main theorem, the limits of the largest and smallest eigenvalue of $\mathbf{M}_n$ are also obtained.

math.ST↗

arXiv2014

Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing

Sample covariance matrices are widely used in multivariate statistical analysis. The central limit theorems (CLT's) for linear spectral statistics of high-dimensional non-centered sample covariance matrices have received considerable attention in random matrix theory and have been applied to many high-dimensional statistical problems. However, known population mean vectors are assumed for non-centered sample covariance matrices, some of which even assume Gaussian-like moment conditions. In fact, there are still another two most frequently used sample covariance matrices: the MLE (by subtracting the sample mean vector from each sample vector) and the unbiased sample covariance matrix (by changing the denominator $n$ as $N=n-1$ in the MLE) without depending on unknown population mean vectors. In this paper, we not only establish new CLT's for non-centered sample covariance matrices without Gaussian-like moment conditions but also characterize the non-negligible differences among the CLT's for the three classes of high-dimensional sample covariance matrices by establishing a {\em substitution principle}: substitute the {\em adjusted} sample size $N=n-1$ for the actual sample size $n$ in the major centering term of the new CLT's so as to obtain the CLT of the unbiased sample covariance matrices. Moreover, it is found that the difference between the CLT's for the MLE and unbiased sample covariance matrix is non-negligible in the major centering term although the two sample covariance matrices only have differences $n$ and $n-1$ on the dominator. The new results are applied to two testing problems for high-dimensional data.

stat.ME↗

arXiv2011

Asymptotic properties of eigenmatrices of a large sample covariance matrix

Let $S_n=\frac{1}{n}X_nX_n^*$ where $X_n=\{X_{ij}\}$ is a $p\times n$ matrix with i.i.d. complex standardized entries having finite fourth moments. Let $Y_n(\mathbf {t}_1,\mathbf {t}_2,σ)=\sqrt{p}({\mathbf {x}}_n(\mathbf {t}_1)^*(S_n+σI)^{-1}{\mathbf {x}}_n(\mathbf {t}_2)-{\mathbf {x}}_n(\mathbf {t}_1)^*{\mathbf {x}}_n(\mathbf {t}_2)m_n(σ))$ in which $σ>0$ and $m_n(σ)=\int\frac{dF_{y_n}(x)}{x+σ}$ where $F_{y_n}(x)$ is the Marčenko--Pastur law with parameter $y_n=p/n$; which converges to a positive constant as $n\to\infty$, and ${\mathbf {x}}_n(\mathbf {t}_1)$ and ${\mathbf {x}}_n(\mathbf {t}_2)$ are unit vectors in ${\Bbb{C}}^p$, having indices $\mathbf {t}_1$ and $\mathbf {t}_2$, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence $Y_n(\mathbf {t}_1,\mathbf {t}_2,σ)$ converges weakly to a $(2m+1)$-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of $S_n$ is asymptotically close to that of a Haar-distributed unitary matrix.

math.PR↗

arXiv2007

On asymptotics of eigenvectors of large sample covariance matrix

Let \{$X_{ij}$\}, $i,j=...,$ be a double array of i.i.d. complex random variables with $EX_{11}=0,E|X_{11}|^2=1$ and $E|X_{11}|^4<\infty$, and let $A_n=\frac{1}{N}T_n^{{1}/{2}}X_nX_n^*T_n^{{1}/{2}}$, where $T_n^{{1}/{2}}$ is the square root of a nonnegative definite matrix $T_n$ and $X_n$ is the $n\times N$ matrix of the upper-left corner of the double array. The matrix $A_n$ can be considered as a sample covariance matrix of an i.i.d. sample from a population with mean zero and covariance matrix $T_n$, or as a multivariate $F$ matrix if $T_n$ is the inverse of another sample covariance matrix. To investigate the limiting behavior of the eigenvectors of $A_n$, a new form of empirical spectral distribution is defined with weights defined by eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights. Moreover, if \{$X_{ij}$\} and $T_n$ are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of $A_n$ are proved to have Gaussian limits, which suggests that the eigenvector matrix of $A_n$ is nearly Haar distributed when $T_n$ is a multiple of the identity matrix, an easy consequence for a Wishart matrix.

math.PR↗

arXiv2007

On the signal-to-interference ratio of CDMA systems in wireless communications

Let $\{s_{ij}:i,j=1,2,...\}$ consist of i.i.d. random variables in $\mathbb{C}$ with $\mathsf{E}s_{11}=0$, $\mathsf{E}|s_{11}|^2=1$. For each positive integer $N$, let $\mathbf{s}_k={\mathbf{s}}_k(N)=(s_{1k},s_{2k},...,s_{Nk})^T$, $1\leq k\leq K$, with $K=K(N)$ and $K/N\to c>0$ as $N\to\infty$. Assume for fixed positive integer $L$, for each $N$ and $k\leq K$, ${\boldsα}_k=(α_k(1),...,α_k(L))^T$ is random, independent of the $s_{ij}$, and the empirical distribution of $(α_1,...,α_K)$, with probability one converging weakly to a probability distribution $H$ on $\mathbb{C}^L$. Let ${\boldsβ}_k={\boldsβ}_k(N)=(α_k(1)\mathbf{s}_k^T,...,α_k(L)\m athbf{s}_k^T)^T$ and set $C=C(N)=(1/N)\sum_{k=2}^K{\bolds β}_k{\bolds β}_k^*$. Let $σ^2>0$ be arbitrary. Then define $SIR_1=(1/N){\boldsβ}^*_1(C+σ^2I)^{-1}{\boldsβ}_1$, which represents the best signal-to-interference ratio for user 1 with respect to the other $K-1$ users in a direct-sequence code-division multiple-access system in wireless communications. In this paper it is proven that, with probability 1, $SIR_1$ tends, as $N\to\infty$, to the limit $\sum_{\ell,\ell'=1}^L\barα_1(\ell) alpha_1(\ell')a_{\ell,\ell'},$ where $A=(a_{\ell,\ell'})$ is nonrandom, Hermitian positive definite, and is the unique matrix of such type satisfying $A=\bigl(c \mathsf{E}\frac{{\boldsα}{\bolds α}^*}{1+{\boldsα}^*A{\boldsα}}+σ^2I_L\bigr)^{-1}$, where ${\boldsα}\in \mathbb{C}^L$ has distribution $H$. The result generalizes those previously derived under more restricted assumptions.

math.PR↗