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In this paper, we address the problem of testing independence between two high-dimensional random vectors. Our approach involves a series of max-sum tests based on three well-known classes of rank-based correlations. These correlation classes encompass several popular rank measures, including Spearman's $\rho$, Kendall's $\tau$, Hoeffding's D, Blum-Kiefer-Rosenblatt's R and Bergsma-Dassios-Yanagimoto's $\tau^*$.The key advantages of our proposed tests are threefold: (1) they do not rely on specific assumptions about the distribution of random vectors, which flexibility makes them available across various scenarios; (2) they can proficiently manage non-linear dependencies between random vectors, a critical aspect in high-dimensional contexts; (3) they have robust performance, regardless of whether the alternative hypothesis is sparse or dense.Notably, our proposed tests demonstrate significant advantages in various scenarios, which is suggested by extensive numerical results and an empirical application in RNA microarray analysis.

Testing cross-sectional independence in panel data models is of fundamental importance in econometric analysis with high-dimensional panels. Recently, econometricians began to turn their attention to the problem in the presence of serial dependence. The existing procedure for testing cross-sectional independence with serial correlation is based on the sum of the sample cross-sectional correlations, which generally performs well when the alternative has dense cross-sectional correlations, but suffers from low power against sparse alternatives. To deal with sparse alternatives, we propose a test based on the maximum of the squared sample cross-sectional correlations. Furthermore, we propose a combined test to combine the p-values of the max based and sum based tests, which performs well under both dense and sparse alternatives. The combined test relies on the asymptotic independence of the max based and sum based test statistics, which we show rigorously. We show that the proposed max based and combined tests have attractive theoretical properties and demonstrate the superior performance via extensive simulation results. We apply the two new tests to analyze the weekly returns on the securities in the S\&P 500 index under the Fama-French three-factor model, and confirm the usefulness of the proposed combined test in detecting cross-sectional independence.

Testing for multi-dimensional white noise is an important subject in statistical inference. Such test in the high-dimensional case becomes an open problem waiting to be solved, especially when the dimension of a time series is comparable to or even greater than the sample size. To detect an arbitrary form of departure from high-dimensional white noise, a few tests have been developed. Some of these tests are based on max-type statistics, while others are based on sum-type ones. Despite the progress, an urgent issue awaits to be resolved: none of these tests is robust to the sparsity of the serial correlation structure. Motivated by this, we propose a Fisher's combination test by combining the max-type and the sum-type statistics, based on the established asymptotically independence between them. This combination test can achieve robustness to the sparsity of the serial correlation structure,and combine the advantages of the two types of tests. We demonstrate the advantages of the proposed test over some existing tests through extensive numerical results and an empirical analysis., Comment: 84 pages

For a set of dependent random variables, without stationary or the strong mixing assumptions, we derive the asymptotic independence between their sums and maxima. Then we apply this result to high-dimensional testing problems, where we combine the sum-type and max-type tests and propose a novel test procedure for the one-sample mean test, the two-sample mean test and the regression coefficient test in high-dimensional setting. Based on the asymptotic independence between sums and maxima, the asymptotic distributions of test statistics are established. Simulation studies show that our proposed tests have good performance regardless of data being sparse or not. Examples on real data are also presented to demonstrate the advantages of our proposed methods.

The stochastic block model is one of the most studied network models for community detection. It is well-known that most algorithms proposed for fitting the stochastic block model likelihood function cannot scale to large-scale networks. One prominent work that overcomes this computational challenge is Amini et al.(2013), which proposed a fast pseudo-likelihood approach for fitting stochastic block models to large sparse networks. However, this approach does not have convergence guarantee, and is not well suited for small- or medium- scale networks. In this article, we propose a novel likelihood based approach that decouples row and column labels in the likelihood function, which enables a fast alternating maximization; the new method is computationally efficient, performs well for both small and large scale networks, and has provable convergence guarantee. We show that our method provides strongly consistent estimates of the communities in a stochastic block model. As demonstrated in simulation studies, the proposed method outperforms the pseudo-likelihood approach in terms of both estimation accuracy and computation efficiency, especially for large sparse networks. We further consider extensions of our proposed method to handle networks with degree heterogeneity and bipartite properties.

We consider a testing problem for cross-sectional dependence for high-dimensional panel data, where the number of cross-sectional units is potentially much larger than the number of observations. The cross-sectional dependence is described through a linear regression model. We study three tests named the sum test, the max test and the max-sum test, where the latter two are new. The sum test is initially proposed by Breusch and Pagan (1980). We design the max and sum tests for sparse and non-sparse residuals in the linear regressions, respectively.And the max-sum test is devised to compromise both situations on the residuals. Indeed, our simulation shows that the max-sum test outperforms the previous two tests. This makes the max-sum test very useful in practice where sparsity or not for a set of data is usually vague. Towards the theoretical analysis of the three tests, we have settled two conjectures regarding the sum of squares of sample correlation coefficients asked by Pesaran (2004 and 2008). In addition, we establish the asymptotic theory for maxima of sample correlations coefficients appeared in the linear regression model for panel data, which is also the first successful attempt to our knowledge. To study the max-sum test, we create a novel method to show asymptotic independence between maxima and sums of dependent random variables. We expect the method itself is useful for other problems of this nature. Finally, an extensive simulation study as well as a case study are carried out. They demonstrate advantages of our proposed methods in terms of both empirical powers and robustness for residuals regardless of sparsity or not., Comment: 106 pages, 1 figure