Comparing large covariance matrices has important applications in modern genomics, where scientists are often interested in understanding whether relationships (e.g., dependencies or co-regulations) among a large number of genes vary between different biological states. We propose a computationally fast procedure for testing the equality of two large covariance matrices when the dimensions of the covariance matrices are much larger than the sample sizes. A distinguishing feature of the new procedure is that it imposes no structural assumptions on the unknown covariance matrices. Hence, the test is robust with respect to various complex dependence structures that frequently arise in genomics. We prove that the proposed procedure is asymptotically valid under weak moment conditions. As an interesting application, we derive a new gene clustering algorithm which shares the same nice property of avoiding restrictive structural assumptions for high-dimensional genomics data. Using an asthma gene expression dataset, we illustrate how the new test helps compare the covariance matrices of the genes across different gene sets/pathways between the disease group and the control group, and how the gene clustering algorithm provides new insights on the way gene clustering patterns differ between the two groups. The proposed methods have been implemented in an R-package HDtest and are available on CRAN.

We propose a new omnibus test for vector white noise using the maximum absolute auto- correlations and cross-correlations of the component series. Based on an approximation by the L∞-norm of a normal random vector, the critical value of the test can be evaluated by bootstrap- ping from a multivariate normal distribution. In contrast to the conventional white noise test, the new method is proved to be valid for testing departure from white noise that is not independent and identically distributed. We illustrate the accuracy and the power of the proposed test by simu- lation, which also shows that the new test outperforms several commonly used methods, including the Lagrange multiplier test and the multivariate Box–Pierce portmanteau tests, especially when the dimension of the time series is high in relation to the sample size. The numerical results also indicate that the performance of the new test can be further enhanced when it is applied to pre-transformed data obtained via the time series principal component analysis proposed by J. Chang, B. Guo and Q. Yao (arXiv:1410.2323). The proposed procedures have been implemented in an R package.

In this paper, joint asymptotics of powered maxima for a triangular array of bivariate Gaussian random vectors are considered. Under the Hüsler–Reiss condition, limiting distributions of powered maxima are derived. Furthermore, the second-order expansions of the joint distributions of powered maxima are established under the refined Hüsler–Reiss condition.

The official air-quality statistic reported that Beijing had a 9.9% decline in the annual concentration of PM2.5 in 2016. While this statistic offered some relief for the inhabitants of the capital, we present several analyses based on Beijing’s PM2.5 data of the past 4 years at 36 monitoring sites along with meteorological data of the past 7 years. The analyses reveal the air pollution situation in 2016 was not as rosy as the 9.9% decline would convey, and improvement if any was rather uncertain. The paper also provides an assessment on the city’s PM2.5 situation in the past 4 years.

Two-sample U -statistics are widely used in a broad range of applications, including those in the fields of biostatistics and econometrics. In this paper, we establish sharp Cramér-type moderate deviation theorems for Studentized two-sample U-statistics in a general framework, including the two-sample t-statistic and Studentized Mann–Whitney test statistic as prototypical exam- ples. In particular, a refined moderate deviation theorem with second-order accuracy is established for the two-sample t-statistic. These results extend the applicability of the existing statistical methodologies from the one-sample t-statistic to more general nonlinear statistics. Applications to two-sample large-scale multiple testing problems with false discovery rate control and the regularized bootstrap method are also discussed.

We consider an independence feature screening technique for identifying explanatory variables that locally contribute to the response variable in high- dimensional regression analysis. Without requiring a specific parametric form of the underlying data model, our approach accommodates a wide spectrum of nonparametric and semiparametric model families. To detect the local con- tributions of explanatory variables, our approach constructs empirical likeli- hood locally in conjunction with marginal nonparametric regressions. Since our approach actually requires no estimation, it is advantageous in scenarios such as the single-index models where even specification and identification of a marginal model is an issue. By automatically incorporating the level of variation of the nonparametric regression and directly assessing the strength of data evidence supporting local contribution from each explanatory vari- able, our approach provides a unique perspective for solving feature screen- ing problems. Theoretical analysis shows that our approach can handle data dimensionality growing exponentially with the sample size. With extensive theoretical illustrations and numerical examples, we show that the local inde- pendence screening approach performs promisingly.

The covariance matrices are essential quantities in econometric and statistical applications including port- folio allocation, asset pricing and factor analysis. Testing the entire covariance under high dimensionality endures large variability and causes a dilution of the signal-to-noise ratio and hence a reduction in the power. We consider a more powerful test procedure that focuses on testing along the super-diagonals of the high dimensional covariance matrix, which can infer more accurately on the structure of the co- variance. We show that the test is powerful in detecting sparse signals and parametric structures in the covariance. The properties of the test are demonstrated by theoretical analyses, simulation and empirical studies.

In this paper, higher-order expansions for distributions and densities of powered extremes of standard normal random sequences are established under an optimal choice of normalized constants. Our findings refine the related results in Hall (1980). Furthermore, it is shown that the rate of convergence of distributions/densities of normalized extremes depends in principle on the power index.

We consider a multivariate time series model which represents a high dimensional vector process as a sum of three terms: a linear regression of some observed regressors, a linear combination of some latent and serially correlated factors, and a vector white noise. We investigate the inference without imposing stationary conditions on the target multivariate time series, the regressors and the underlying factors. Furthermore we deal with the endogeneity that there exist correlations between the observed regressors and the unobserved factors. We also consider the model with nonlinear regression term which can be approximated by a linear regression function with a large number of regressors. The convergence rates for the estimators of regression coefficients, the number of factors, factor loading space and factors are established under the settings when the dimension of time series and the number of regressors may both tend to infinity together with the sample size. The proposed method is illustrated with both simulated and real data examples.

We show that, when the double bootstrap is used to improve performance of bootstrap meth- ods for bias correction, techniques based on using a single double-bootstrap sample for each single-bootstrap sample can produce third-order accuracy for much less computational expense than is required by conventional double-bootstrap methods. However, this improved level of per- formance is not available for the single double-bootstrap methods that have been suggested to construct confidence intervals or distribution estimators.