On the sphericity test with large-dimensional observations

In this paper, we propose corrections to the likelihood ratio test and John's test for sphericity in large-dimensions. New formulas for the limiting parameters in the CLT for linear spectral statistics of sample covariance matrices with general fourth moments are first established. Using these formulas, we derive the asymptotic distribution of the two proposed test statistics under the null. These asymptotics are valid for general population, i.e. not necessarily Gaussian, provided a finite fourth-moment. Extensive Monte-Carlo experiments are conducted to assess the quality of these tests with a comparison to several existing methods from the literature. Moreover, we also obtain their asymptotic power functions under the alternative of a spiked population model as a specific alternative.Comment: 37 pages, 3 figure

On singular values distribution of a large auto-covariance matrix in the ultra-dimensional regime

Let $(\varepsilon_{t})_{t>0}$ be a sequence of independent real random vectors of $p$-dimension and let $X_T=\sum_{t=s+1}^{s+T}\varepsilon_t\varepsilon^T_{t-s}/T$ be the lag-$s$ ($s$ is a fixed positive integer) auto-covariance matrix of $\varepsilon_t$. This paper investigates the limiting behavior of the singular values of $X_T$ under the so-called {\em ultra-dimensional regime} where $p\to\infty$ and $T\to\infty$ in a related way such that $p/T\to 0$. First, we show that the singular value distribution of $X_T$ after a suitable normalization converges to a nonrandom limit $G$ (quarter law) under the forth-moment condition. Second, we establish the convergence of its largest singular value to the right edge of $G$. Both results are derived using the moment method.Comment: 32 pages, 2 figure

On Two Simple and Effective Procedures for High Dimensional Classification of General Populations

In this paper, we generalize two criteria, the determinant-based and trace-based criteria proposed by Saranadasa (1993), to general populations for high dimensional classification. These two criteria compare some distances between a new observation and several different known groups. The determinant-based criterion performs well for correlated variables by integrating the covariance structure and is competitive to many other existing rules. The criterion however requires the measurement dimension be smaller than the sample size. The trace-based criterion in contrast, is an independence rule and effective in the "large dimension-small sample size" scenario. An appealing property of these two criteria is that their implementation is straightforward and there is no need for preliminary variable selection or use of turning parameters. Their asymptotic misclassification probabilities are derived using the theory of large dimensional random matrices. Their competitive performances are illustrated by intensive Monte Carlo experiments and a real data analysis.Comment: 5 figures; 22 pages. To appear in "Statistical Papers

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

Gaussian fluctuations for linear spectral statistics of large random covariance matrices

Consider a $N\times n$ matrix $\Sigma_n=\frac{1}{\sqrt{n}}R_n^{1/2}X_n$, where $R_n$ is a nonnegative definite Hermitian matrix and $X_n$ is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear statistics of the eigenvalues $$\operatorname {Trace}f \bigl(\Sigma_n\Sigma_n^*\bigr)=\sum_{i=1}^Nf(\lambda_i),\qquad (\lambda_i)\ eigenvalues\ of\ \Sigma_n\Sigma_n^*,$$ are shown to be Gaussian, in the regime where both dimensions of matrix $\Sigma_n$ go to infinity at the same pace and in the case where $f$ is of class $C^3$, that is, has three continuous derivatives. The main improvements with respect to Bai and Silverstein's CLT [Ann. Probab. 32 (2004) 553-605] are twofold: First, we consider general entries with finite fourth moment, but whose fourth cumulant is nonnull, that is, whose fourth moment may differ from the moment of a (real or complex) Gaussian random variable. As a consequence, extra terms proportional to $\vert \mathcal{V}\vert ^2=\bigl|\mathbb{E}\bigl(X_{11}^n\bigr) ^2\bigr|^2$ and $\kappa=\mathbb{E}\bigl \vert X_{11}^n\bigr \vert ^4-\vert {\mathcal{V}}\vert ^2-2$ appear in the limiting variance and in the limiting bias, which not only depend on the spectrum of matrix $R_n$ but also on its eigenvectors. Second, we relax the analyticity assumption over $f$ by representing the linear statistics with the help of Helffer-Sj\"{o}strand's formula. The CLT is expressed in terms of vanishing L\'{e}vy-Prohorov distance between the linear statistics' distribution and a Gaussian probability distribution, the mean and the variance of which depend upon $N$ and $n$ and may not converge.Comment: Published at http://dx.doi.org/10.1214/15-AAP1135 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

High-dimensional limits of eigenvalue distributions for general Wishart process

In this article, we obtain an equation for the high-dimensional limit measure of eigenvalues of generalized Wishart processes, and the results is extended to random particle systems that generalize SDEs of eigenvalues. We also introduce a new set of conditions on the coefficient matrices for the existence and uniqueness of a strong solution for the SDEs of eigenvalues. The equation of the limit measure is further discussed assuming self-similarity on the eigenvalues.Comment: 28 page