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1. Finite Sample Change Point Inference and Identification for High-Dimensional Mean Vectors

   https://doi.org/10.1111/rssb.12406  

   Yu, Mengjia ; Chen, Xiaohui ( December 2020 , Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Cumulative sum (CUSUM) statistics are widely used in the change point inference and identification. For the problem of testing for existence of a change point in an independent sample generated from the mean-shift model, we introduce a Gaussian multiplier bootstrap to calibrate critical values of the CUSUM test statistics in high dimensions. The proposed bootstrap CUSUM test is fully data dependent and it has strong theoretical guarantees under arbitrary dependence structures and mild moment conditions. Specifically, we show that with a boundary removal parameter the bootstrap CUSUM test enjoys the uniform validity in size under the null and it achieves the minimax separation rate under the sparse alternatives when the dimension p can be larger than the sample size n.

   Once a change point is detected, we estimate the change point location by maximising the ℓ∞-norm of the generalised CUSUM statistics at two different weighting scales corresponding to covariance stationary and non-stationary CUSUM statistics. For both estimators, we derive their rates of convergence and show that dimension impacts the rates only through logarithmic factors, which implies that consistency of the CUSUM estimators is possible when p is much larger than n. In the presence of multiple change points, we propose a principled bootstrap-assisted binary segmentation (BABS) algorithm to dynamically adjust the change point detection rule and recursively estimate their locations. We derive its rate of convergence under suitable signal separation and strength conditions.

   The results derived in this paper are non-asymptotic and we provide extensive simulation studies to assess the finite sample performance. The empirical evidence shows an encouraging agreement with our theoretical results.

    

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2. Homogeneity Tests of Covariance for High-Dimensional Functional Data with Applications to Event Segmentation

   https://doi.org/10.1111/biom.13844  

   Zhong, Ping-Shou ( February 2023 , Biometrics)

   We consider inference problems for high-dimensional (HD) functional data with a dense number of T repeated measurements taken for a large number of p variables from a small number of n experimental units. The spatial and temporal dependence, high dimensionality, and dense number of repeated measurements pose theoretical and computational challenges. This paper has two aims; our first aim is to solve the theoretical and computational challenges in testing equivalence among covariance matrices from HD functional data. The second aim is to provide computationally efficient and tuning-free tools with guaranteed stochastic error control. The weak convergence of the stochastic process formed by the test statistics is established under the “large p, large T, and small n” setting. If the null is rejected, we further show that the locations of the change points can be estimated consistently. The estimator's rate of convergence is shown to depend on the data dimension, sample size, number of repeated measurements, and signal-to-noise ratio. We also show that our proposed computation algorithms can significantly reduce the computation time and are applicable to real-world data with a large number of HD-repeated measurements (e.g., functional magnetic resonance imaging (fMRI) data). Simulation results demonstrate both the finite sample performance and computational effectiveness of our proposed procedures. We observe that the empirical size of the test is well controlled at the nominal level, and the locations of multiple change points can be accurately identified. An application to fMRI data demonstrates that our proposed methods can identify event boundaries in the preface of the television series Sherlock. Code to implement the procedures is available in an R package named TechPhD.

    

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3. 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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4. High-dimensional analysis of variance in multivariate linear regression

   https://doi.org/10.1093/biomet/asad001  

   Lou, Zhipeng ; Zhang, Xianyang ; Wu, Wei Biao ( January 2023 , Biometrika)

   Summary In this paper, we develop a systematic theory for high-dimensional analysis of variance in multivariate linear regression, where the dimension and the number of coefficients can both grow with the sample size. We propose a new U-type statistic to test linear hypotheses and establish a high-dimensional Gaussian approximation result under fairly mild moment assumptions. Our general framework and theory can be used to deal with the classical one-way multivariate analysis of variance, and the nonparametric one-way multivariate analysis of variance in high dimensions. To implement the test procedure, we introduce a sample-splitting-based estimator of the second moment of the error covariance and discuss its properties. A simulation study shows that our proposed test outperforms some existing tests in various settings. 

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5. Homogeneity tests of covariance matrices with high-dimensional longitudinal data

   https://doi.org/10.1093/biomet/asz011  

   Zhong, Ping-Shou ; Li, Runze ; Santo, Shawn ( May 2019 , Biometrika)

   Summary This paper deals with the detection and identification of changepoints among covariances of high-dimensional longitudinal data, where the number of features is greater than both the sample size and the number of repeated measurements. The proposed methods are applicable under general temporal-spatial dependence. A new test statistic is introduced for changepoint detection, and its asymptotic distribution is established. If a changepoint is detected, an estimate of the location is provided. The rate of convergence of the estimator is shown to depend on the data dimension, sample size, and signal-to-noise ratio. Binary segmentation is used to estimate the locations of possibly multiple changepoints, and the corresponding estimator is shown to be consistent under mild conditions. Simulation studies provide the empirical size and power of the proposed test and the accuracy of the changepoint estimator. An application to a time-course microarray dataset identifies gene sets with significant gene interaction changes over time. 

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