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1. Empirical likelihood test for a large-dimensional mean vector

   https://doi.org/10.1093/biomet/asaa005  

   Cui, Xia; Li, Runze; Yang, Guangren; Zhou, Wang (March 2020, Biometrika)

   null (Ed.)

   Summary This paper is concerned with empirical likelihood inference on the population mean when the dimension $$p$$ and the sample size $$n$$ satisfy $$p/n\rightarrow c\in [1,\infty)$$. As shown in Tsao (2004), the empirical likelihood method fails with high probability when $p/n>1/2$ because the convex hull of the $$n$$ observations in $$\mathbb{R}^p$$ becomes too small to cover the true mean value. Moreover, when $p> n$, the sample covariance matrix becomes singular, and this results in the breakdown of the first sandwich approximation for the log empirical likelihood ratio. To deal with these two challenges, we propose a new strategy of adding two artificial data points to the observed data. We establish the asymptotic normality of the proposed empirical likelihood ratio test. The proposed test statistic does not involve the inverse of the sample covariance matrix. Furthermore, its form is explicit, so the test can easily be carried out with low computational cost. Our numerical comparison shows that the proposed test outperforms some existing tests for high-dimensional mean vectors in terms of power. We also illustrate the proposed procedure with an empirical analysis of stock data. 

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2. A Statistically and Numerically Efficient Independence Test Based on Random Projections and Distance Covariance

   https://doi.org/10.3389/fams.2021.779841  

   Huang, Cheng; Huo, Xiaoming (January 2022, Frontiers in Applied Mathematics and Statistics)

   Testing for independence plays a fundamental role in many statistical techniques. Among the nonparametric approaches, the distance-based methods (such as the distance correlation-based hypotheses testing for independence) have many advantages, compared with many other alternatives. A known limitation of the distance-based method is that its computational complexity can be high. In general, when the sample size is n , the order of computational complexity of a distance-based method, which typically requires computing of all pairwise distances, can be O ( n 2 ). Recent advances have discovered that in the univariate cases, a fast method with O ( n  log  n ) computational complexity and O ( n ) memory requirement exists. In this paper, we introduce a test of independence method based on random projection and distance correlation, which achieves nearly the same power as the state-of-the-art distance-based approach, works in the multivariate cases, and enjoys the O ( nK  log  n ) computational complexity and O ( max{ n , K }) memory requirement, where K is the number of random projections. Note that saving is achieved when K < n / log  n . We name our method a Randomly Projected Distance Covariance (RPDC). The statistical theoretical analysis takes advantage of some techniques on the random projection which are rooted in contemporary machine learning. Numerical experiments demonstrate the efficiency of the proposed method, relative to numerous competitors. 

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3. Shifted BH methods for controlling false discovery rate in multiple testing of the means of correlated normals against two-sided alternatives

   https://doi.org/10.1016/j.jspi.2024.106238  

   Sarkar, Sanat K; Zhang, Shiyu (May 2025, Journal of Statistical Planning and Inference)

   For simultaneous testing of multivariate normal means with known correlation matrix against two-sided alternatives, this paper introduces new methods with proven finite-sample control of false discovery rate. The methods are obtained by shifting each p-value to the left and considering a Benjamini–Hochberg-type linear step-up procedure based on these shifted p-values. The amount of shift for each -value is appropriately determined from the correlation matrix to achieve the desired false discovery rate control. Simulation studies and real-data application show favorable performances of the proposed methods when compared with relevant competitors. 

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4. Graph Learning From Multivariate Dependent Time Series Via A Multi-Attribute Formulation

   https://doi.org/10.1109/ICASSP43922.2022.9746603  

   Tugnait, Jitendra K. (May 2022, ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP))

   We consider the problem of inferring the conditional independence graph (CIG) of a high-dimensional stationary multivariate Gaussian time series. In a time series graph, each component of the vector series is represented by distinct node, and associations between components are represented by edges between the corresponding nodes. We formulate the problem as one of multi-attribute graph estimation for random vectors where a vector is associated with each node of the graph. At each node, the associated random vector consists of a time series component and its delayed copies. We present an alternating direction method of multipliers (ADMM) solution to minimize a sparse-group lasso penalized negative pseudo log-likelihood objective function to estimate the precision matrix of the random vector associated with the entire multi-attribute graph. The time series CIG is then inferred from the estimated precision matrix. A theoretical analysis is provided. Numerical results illustrate the proposed approach which outperforms existing frequency-domain approaches in correctly detecting the graph edges. 

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5. RFtest: A Robust and Flexible Community-Level Test for Microbiome Data Powerfully Detects Phylogenetically Clustered Signals

   https://doi.org/10.3389/fgene.2021.749573  

   Zhang, Lujun; Wang, Yanshan; Chen, Jingwen; Chen, Jun (January 2022, Frontiers in Genetics)

   Random forest is considered as one of the most successful machine learning algorithms, which has been widely used to construct microbiome-based predictive models. However, its use as a statistical testing method has not been explored. In this study, we propose “Random Forest Test” (RFtest), a global (community-level) test based on random forest for high-dimensional and phylogenetically structured microbiome data. RFtest is a permutation test using the generalization error of random forest as the test statistic. Our simulations demonstrate that RFtest has controlled type I error rates, that its power is superior to competing methods for phylogenetically clustered signals, and that it is robust to outliers and adaptive to interaction effects and non-linear associations. Finally, we apply RFtest to two real microbiome datasets to ascertain whether microbial communities are associated or not with the outcome variables. 

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