On Identity Tests for High Dimensional Data Using RMT

In this work, we redefined two important statistics, the CLRT test (Bai et.al., Ann. Stat. 37 (2009) 3822-3840) and the LW test (Ledoit and Wolf, Ann. Stat. 30 (2002) 1081-1102) on identity tests for high dimensional data using random matrix theories. Compared with existing CLRT and LW tests, the new tests can accommodate data which has unknown means and non-Gaussian distributions. Simulations demonstrate that the new tests have good properties in terms of size and power. What is more, even for Gaussian data, our new tests perform favorably in comparison to existing tests. Finally, we find the CLRT is more sensitive to eigenvalues less than 1 while the LW test has more advantages in relation to detecting eigenvalues larger than 1.Comment: 16 pages, 2 figures, 3 tables, To be published in the Journal of Multivariate Analysi

Optimal feature selection for sparse linear discriminant analysis and its applications in gene expression data

This work studies the theoretical rules of feature selection in linear discriminant analysis (LDA), and a new feature selection method is proposed for sparse linear discriminant analysis. An $l_1$ minimization method is used to select the important features from which the LDA will be constructed. The asymptotic results of this proposed two-stage LDA (TLDA) are studied, demonstrating that TLDA is an optimal classification rule whose convergence rate is the best compared to existing methods. The experiments on simulated and real datasets are consistent with the theoretical results and show that TLDA performs favorably in comparison with current methods. Overall, TLDA uses a lower minimum number of features or genes than other approaches to achieve a better result with a reduced misclassification rate.Comment: 20 pages, 3 figures, 5 tables, accepted by Computational Statistics and Data Analysi

Measuring the subprime crisis contagion: Evidence of change point analysis of copula functions

In this paper, we first determine the existence of structural changes in the dependence between time series of equity index returns of two markets using the change point testing method. The method is based on Archimedean copula functions, which are able to comprehensively describe dependence characteristics of random variables. The degree of financial contagion between markets is subsequently estimated using the tail dependence coefficient of copula functions before and after the change point. We empirically test our method by investigating financial contagion during the subprime crisis between the US S&P 500 index and five Asian markets, namely China, Japan, Korea, Hong Kong and Taiwan. Our results show that a statistically significant change point exists in the dependence between the US market and all Asian stock markets except Taiwan. The upper tail dependence is larger after the time of change, implying the existence of contagion during the banking crisis between the US and the Asian economies. The degree of financial contagion is also estimated and found to be consistent with market events and media reports during that period

Strong convergence rate of estimators of change point and its application

Let {Xn,n[greater-or-equal, slanted]1} be an independent sequence with a mean shift. We consider the cumulative sum (CUSUM) estimator of a change point. It is shown that, when the rth moment of Xn is finite, for n[greater-or-equal, slanted]1 and r>1, strong convergence rate of the change point estimator is o(M(n)/n), for any M(n) satisfying that M(n)[short up arrow][infinity], which has improved the results in the literature. Furthermore, it is also shown that the preceding rate is still valid for some dependent or negative associate cases. We also propose an iterative algorithm to search for the location of a change point. A simulation study on a mean shift model with a stable distribution is provided, which demonstrates that the algorithm is efficient. In addition, a real data example is given for illustration.

A note on the convergence rate of the kernel density estimator of the mode

In this paper, the mode estimator based on the Parzen-Rosenblatt kernel estimator is considered [Parzen, E., 1962. On estimating probability density function and mode. The Annals of Mathematical Statistics 33, 1065-1076]. In light of Shi et al. [Shi, X., Wu, Y., Miao, B., 2009. Strong convergence rate of estimators of change point and its application. Computational Statistics & Data Analysis 53, 990-998], under mild conditions, we establish the relationship between the convergence rate of the mode estimator and the window width. In this way, we obtain a better convergence rate of the mode estimator.

Central limit theorem of random quadratics forms involving random matrices

Let and S=(s1,s2,...,sK) where random variables are i.i.d. with . The central limit theorem of the random quadratic forms is established, which arises from the application in wireless communications.Central limit theorem Large dimensional matrix Quadratic forms

On limit theorem for the eigenvalues of product of two random matrices

The existence of limiting spectral distribution (LSD) of the product of two random matrices is proved. One of the random matrices is a sample covariance matrix and the other is an arbitrary Hermitian matrix. Specially, the density function of LSD of SnWn is established, where Sn is a sample covariance matrix and Wn is Wigner matrix.Limiting spectral distribution Product of random matrices Large dimensional random matrices