Let A be a p-variate real Wishart matrix on n degrees of freedom with identity covariance. The distribution of the largest eigenvalue in A has important applications in multivariate statistics. Consider the asymptotics when p grows in proportion to n, it is known from Johnstone (2001) that after centering and scaling, these distributions approach the orthogonal Tracy-Widom law for real-valued data, which can be numerically evaluated and tabulated in software.
Under the same assumption, we show that more carefully chosen centering and scaling constants improve the accuracy of the distributional approximation by the Tracy-Widom limit to second order: O(min(n,p)^{-2/3}). Together with the numerical simulation, it implies that the Tracy-Widom law is an attractive approximation to the distributions of these largest eigenvalues, which is important for using the asymptotic result in practice. We also provide a parallel accuracy result for the smallest eigenvalue of A when n > p.
In this paper, we study a high-dimensional random matrix model from nonparametric statistics called the Kendall rank correlation matrix, which is a natural multivariate extension of the Kendall rank correlation coefficient. We establish the Tracy-Widom law for its largest eigenvalue. It is the first Tracy-Widom law for a nonparametric random matrix model, and also the first Tracy-Widom law for a high-dimensional U-statistic.
Let $\bY =\bR+\bX$ be an $M\times N$ matrix, where $\bR$ is a rectangular diagonal matrix and $\bX$ consists of $i.i.d.$ entries. This is a signal-plus-noise type model. Its signal matrix could be full rank, which is rarely studied in literature compared with the low rank cases. This paper is to study the extreme eigenvalues of $\bY\bY^*$. We show that under the high dimensional setting ($M/N\rightarrow c\in(0,1]$) and some regularity conditions on $\bR$ the rescaled extreme eigenvalue converges in distribution to Tracy-Widom distribution ($TW_1$).
Let $\mathbb{A}_p=\frac{\mathbb{Y}\mathbb{Y}^*}{m}$ and $\mathbb{B}_p=\frac{\mathbb{X}\mathbb{X}^*}{n}$ be two independent random matrices where $\mathbb{X}=(X_{ij})_{p \times n}$ and $\mathbb{Y}=(Y_{ij})_{p \times m}$ respectively consist of real (or complex) independent random variables with $\mathbb{E}X_{ij}=\mathbb{E}Y_{ij}=0$, $\mathbb{E}|X_{ij}|^2=\mathbb{E}|Y_{ij}|^2=1$. Denote by $λ_{1}$ the largest root of the determinantal equation $\det(λ\mathbb{A}_p-\mathbb{B}_p)=0$. We establish the Tracy-Widom type universality for $λ_{1}$ under some moment conditions on $X_{ij}$ and $Y_{ij}$ whe…
Let $X$ be an $M\times N$ random matrix consisting of independent $M$-variate elliptically distributed column vectors $\mathbf{x}_{1},\dots,\mathbf{x}_{N}$ with general population covariance matrix $Σ$. In the literature, the quantity $XX^{*}$ is referred to as the sample covariance matrix after scaling, where $X^{*}$ is the transpose of $X$. In this article, we prove that the limiting behavior of the scaled largest eigenvalue of $XX^{*}$ is universal for a wide class of elliptical distributions, namely, the scaled largest eigenvalue converges weakly to the same limit regardless of the distrib…
We establish a quantitative version of the Tracy--Widom law for the largest eigenvalue of high dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix $X^*X$ converge to its Tracy--Widom limit at a rate nearly $N^{-1/3}$, where $X$ is an $M \times N$ random matrix whose entries are independent real or complex random variables, assuming that both $M$ and $N$ tend to infinity at a constant rate. This result improves the previous estimate $N^{-2/9}$ obtained by Wang [73]. Our proof relies on a Green function comp…
We study the sample covariance matrix for real-valued data with general population covariance, as well as MANOVA-type covariance estimators in variance components models under null hypotheses of global sphericity. In the limit as matrix dimensions increase proportionally, the asymptotic spectra of such estimators may have multiple disjoint intervals of support, possibly intersecting the negative half line. We show that the distribution of the extremal eigenvalue at each regular edge of the support has a GOE Tracy-Widom limit. Our proof extends a comparison argument of Ji Oon Lee and Kevin Schn…
Detection of the number of signals corrupted by high-dimensional noise is a fundamental problem in signal processing and statistics. This paper focuses on a general setting where the high-dimensional noise has an unknown complicated heterogeneous variance structure. We propose a sequential test which utilizes the edge singular values (i.e., the largest few singular values) of the data matrix. It also naturally leads to a consistent sequential testing estimate of the number of signals. We describe the asymptotic distribution of the test statistic in terms of the Tracy-Widom distribution. The te…
We derive efficient recursive formulas giving the exact distribution of the largest eigenvalue for finite dimensional real Wishart matrices and for the Gaussian Orthogonal Ensemble (GOE). In comparing the exact distribution with the limiting distribution of large random matrices, we also found that the Tracy-Widom law can be approximated by a properly scaled and shifted Gamma distribution, with great accuracy for the values of common interest in statistical applications.
The distributions of the largest and the smallest eigenvalues of a $p$-variate sample covariance matrix $S$ are of great importance in statistics. Focusing on the null case where $nS$ follows the standard Wishart distribution $W_p(I,n)$, we study the accuracy of their scaling limits under the setting: $n/p\rightarrow γ\in(0,\infty)$ as $n\rightarrow \infty$. The limits here are the orthogonal Tracy--Widom law and its reflection about the origin. With carefully chosen rescaling constants, the approximation to the rescaled largest eigenvalue distribution by the limit attains accuracy of order ${…
We study the rate of convergence for the largest eigenvalue distributions in the Gaussian unitary and orthogonal ensembles to their Tracy-Widom limits. We show that one can achieve an $O(N^{-2/3})$ rate with particular choices of the centering and scaling constants. The arguments here also shed light on more complicated cases of Laguerre and Jacobi ensembles, in both unitary and orthogonal versions. Numerical work shows that the suggested constants yield reasonable approximations, even for surprisingly small values of N.
The greatest root statistic arises as the test statistic in several multivariate analysis settings. Suppose there is a global null hypothesis that consists of different independent sub-null hypotheses, and suppose the greatest root statistic is used as the test statistic for each sub-null hypothesis. Such problems may arise when conducting a batch MANOVA or several batches of pairwise testing for equality of covariance matrices. Using the union-intersection testing approach and by letting the problem dimension tend to infinity faster than the number of batches, we show that the global null can…
We consider sample covariance matrices of the form $\mathcal{Q}=(Σ^{1/2}X)(Σ^{1/2} X)^*$, where the sample $X$ is an $M\times N$ random matrix whose entries are real independent random variables with variance $1/N$ and where $Σ$ is an $M\times M$ positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of $\mathcal{Q}$ when both $M$ and $N$ tend to infinity with $N/M\to d\in(0,\infty)$. For a large class of populations $Σ$ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of $\mathcal{Q}$ is giv…
Let $A$ and $B$ be independent, central Wishart matrices in $p$ variables with common covariance and having $m$ and $n$ degrees of freedom, respectively. The distribution of the largest eigenvalue of $(A+B)^{-1}B$ has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that $m$ and $n$ grow in proportion to $p$. We show that after centering and scaling, the distribution is approximated to second-order, $O(p^{-2/3})$, by the Tracy--Widom law. The results are obtained for both complex and then real-valued data by using methods of random matrix theory …
Let the sample correlation matrix be $W=YY^T$, where $Y=(y_{ij})_{p,n}$ with $y_{ij}=x_{ij}/\sqrt{\sum_{j=1}^nx_{ij}^2}$. We assume $\{x_{ij}: 1\leq i\leq p, 1\leq j\leq n\}$ to be a collection of independent symmetric distributed random variables with sub-exponential tails. Moreover, for any $i$, we assume $x_{ij}, 1\leq j\leq n$ to be identically distributed. We assume $0<p<n$ and $p/n\rightarrow y$ with some $y\in(0,1)$ as $p,n\rightarrow\infty$. In this paper, we provide the Tracy-Widom law ($TW_1$) for both the largest and smallest eigenvalues of $W$. If $x_{ij}$ are i.i.d. standard norma…
Consider two random vectors $\mathbf C_1^{1/2}\mathbf x \in \mathbb R^p$ and $\mathbf C_2^{1/2}\mathbf y\in \mathbb R^q$, where the entries of $\mathbf x$ and $\mathbf y$ are i.i.d. random variables with mean zero and variance one, and $\mathbf C_1$ and $\mathbf C_2$ are $p \times p$ and $q\times q$ deterministic population covariance matrices. With $n$ independent samples of $(\mathbf C_1^{1/2}\mathbf x,\mathbf C_2^{1/2}\mathbf y)$, we study the sample correlation between these two vectors using canonical correlation analysis. We denote by $S_{xx}$ and $S_{yy}$ the sample covariance matrices …
We consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity. More precisely, let $X$ be an $n\times p$ matrix, and let its rows be i.i.d. complex normal vectors with mean 0 and covariance $Σ_p$. We show that for a large class of covariance matrices $Σ_p$, the largest eigenvalue of $X^*X$ is asymptotically distributed (after recentering and rescaling) as the Tracy--Widom distribution that appears in the study of the Gaussian unitary ensemble. We give…
Let X be a n*p matrix and l_1 the largest eigenvalue of the covariance matrix X^{*}*X. The "null case" where X_{i,j} are independent Normal(0,1) is of particular interest for principal component analysis. For this model, when n, p tend to infinity and n/p tends to gamma in (0,\infty), it was shown in Johnstone (2001) that l_1, properly centered and scaled, converges to the Tracy-Widom law. We show that with the same centering and scaling, the result is true even when p/n or n/p tends to infinity. The derivation uses ideas and techniques quite similar to the ones presented in Johnstone (2001). …
We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely toward that edge and fluctuates according to the Tracy-Widom law at the scale $N^{2/3}$. Moreover, given several generic positive edges, we establish that the associated extremal ei…
The correlated Wishart model provides a standard tool for the analysis of correlations in a rich variety of systems. Although much is known for complex correlation matrices, the empirically much more important real case still poses substantial challenges. We put forward a new approach, which maps arbitrary statistical quantities, depending on invariants only, to invariant Hermitian matrix models. For completeness we also include the quaternion case and deal with all three cases in a unified way. As an important application, we study the statistics of the largest eigenvalue and its limiting dis…
Let $\widetilde{X}_{M\times N}$ be a rectangular data matrix with independent real-valued entries $[\widetilde{x}_{ij}]$ satisfying $\mathbb {E}\widetilde{x}_{ij}=0$ and $\mathbb {E}\widetilde{x}^2_{ij}=\frac{1}{M}$, $N,M\to\infty$. These entries have a subexponential decay at the tails. We will be working in the regime $N/M=d_N,\lim_{N\to\infty}d_N\neq0,1,\infty$. In this paper we prove the edge universality of correlation matrices ${X}^{\dagger}X$, where the rectangular matrix $X$ (called the standardized matrix) is obtained by normalizing each column of the data matrix $\widetilde{X}$ by it…
We derive Painlevé--type expressions for the distribution of the $m^{th}$ largest eigenvalue in the Gaussian Orthogonal and Symplectic Ensembles in the edge scaling limit. This work generalizes to general $m$ the $m=1$ results of Tracy and Widom [23]. The results of Johnstone and Soshnikov (see [15], [19]) imply the immediate relevance of our formulas for the $m^{th}$ largest eigenvalue of the appropriate Wishart distribution.
In a recent study of large non-null sample covariance matrices, a new sequence of functions generalizing the GUE Tracy-Widom distribution of random matrix theory was obtained. This paper derives Painlevé formulas of these functions and use them to prove that they are indeed distribution functions. Applications of these new distribution functions to last passage percolation, queues in tandem and totally asymmetric simple exclusion process are also discussed. As a part of the proof, a representation of orthogonal polynomials on the unit circle in terms of an operator on a discrete set is present…
