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1. Inference for change points in high-dimensional mean shift models

   Kaul, Abhishek; Michailidis, George (May 2025, Statistica sinica)

   We consider the problem of constructing confidence intervals for the locations of change points in a high-dimensional mean shift model. We develop a locally refitted least squares estimator and obtain component-wise and simultaneous rates of estimation of change points. The simultaneous rate is the sharpest available by at least a factor of log p, while the component-wise one is optimal. These results enable existence of limiting distributions for the locations of the change points. Subsequently, component-wise distributions are characterized under both vanishing and non-vanishing jump size regimes, while joint distributions of change point estimates are characterized under the latter regime, which also yields asymptotic independence of these estimates. We provide the relationship between these distributions, which allows construction of regime adaptive confidence intervals. All results are established under a high dimensional scaling, in the presence of diverging number of change points. They are illustrated on synthetic data and on sensor measurements from smartphones for activity recognition. 

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2. Gaussian differentially private robust mean estimation and inference

   https://doi.org/10.3150/23-BEJ1706  

   Yu, Myeonghun; Ren, Zhao; Zhou, Wen-Xin (November 2024, Bernoulli)

   In this paper, we propose differentially private algorithms for robust (multivariate) mean estimation and inference under heavy-tailed distributions, with a focus on Gaussian differential privacy. First, we provide a comprehensive analysis of the Huber mean estimator with increasing dimensions, including non-asymptotic deviation bound, Bahadur representation, and (uniform) Gaussian approximations. Secondly, we privatize the Huber mean estimator via noisy gradient descent, which is proven to achieve near-optimal statistical guarantees. The key is to characterize quantitatively the trade-off between statistical accuracy, degree of robustness and privacy level, governed by a carefully chosen robustification parameter. Finally, we construct private confidence intervals for the proposed estimator by incorporating a private and robust covariance estimator. Our findings are demonstrated by simulation studies. 

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3. Simultaneous Change Point Inference and Structure Recovery for High Dimensional Gaussian Graphical Models

   Liu, B.; Zhang, X.; Liu, Y. (October 2021, Journal of machine learning research)

   In this article, we investigate the problem of simultaneous change point inference and structure recovery in the context of high dimensional Gaussian graphical models with possible abrupt changes. In particular, motivated by neighborhood selection, we incorporate a threshold variable and an unknown threshold parameter into a joint sparse regression model which combines p l1-regularized node-wise regression problems together. The change point estimator and the corresponding estimated coefficients of precision matrices are obtained together. Based on that, a classifier is introduced to distinguish whether a change point exists. To recover the graphical structure correctly, a data-driven thresholding procedure is proposed. In theory, under some sparsity conditions and regularity assumptions, our method can correctly choose a homogeneous or heterogeneous model with high accuracy. Furthermore, in the latter case with a change point, we establish estimation consistency of the change point estimator, by allowing the number of nodes being much larger than the sample size. Moreover, it is shown that, in terms of structure recovery of Gaussian graphical models, the proposed thresholding procedure achieves model selection consistency and controls the number of false positives. The validity of our proposed method is justified via extensive numerical studies. Finally, we apply our proposed method to the S&P 500 dataset to show its empirical usefulness. 

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4. Nonparametric Composite Hypothesis Testing in an Asymptotic Regime

   https://doi.org/10.1109/JSTSP.2018.2865884  

   Li, Qunwei; Wang, Tiexing; Bucci, Donald J.; Liang, Yingbin; Chen, Biao; Varshney, Pramod K. (October 2018, IEEE Journal of Selected Topics in Signal Processing)

   We investigate the nonparametric, composite hypothesis testing problem for arbitrary unknown distributions in the asymptotic regime where both the sample size and the number of hypothesis grow exponentially large. Such asymptotic analysis is important in many practical problems, where the number of variations that can exist within a family of distributions can be countably infinite. We introduce the notion of discrimination capacity , which captures the largest exponential growth rate of the number of hypothesis relative to the sample size so that there exists a test with asymptotically vanishing probability of error. Our approach is based on various distributional distance metrics in order to incorporate the generative model of the data. We provide analyses of the error exponent using the maximum mean discrepancy and Kolmogorov–Smirnov distance and characterize the corresponding discrimination rates, i.e., lower bounds on the discrimination capacity, for these tests. Finally, an upper bound on the discrimination capacity based on Fano's inequality is developed. Numerical results are presented to validate the theoretical results. 

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5. Semiparametric Efficient Inference in Adaptive Experiments

   Cook, Thomas; Mishler, Alan; Ramdas, Aaditya (April 2024, Proceedings of Machine Learning Research)

   We consider the problem of efficient inference of the Average Treatment Effect in a sequential experiment where the policy governing the assignment of subjects to treatment or control can change over time. We first provide a central limit theorem for the Adaptive Augmented Inverse-Probability Weighted estimator, which is semiparametric efficient, under weaker assumptions than those previously made in the literature. This central limit theorem enables efficient inference at fixed sample sizes. We then consider a sequential inference setting, deriving both asymptotic and nonasymptotic confidence sequences that are considerably tighter than previous methods. These anytime-valid methods enable inference under data-dependent stopping times (sample sizes). Additionally, we use propensity score truncation techniques from the recent off-policy estimation literature to reduce the finite sample variance of our estimator without affecting the asymptotic variance. Empirical results demonstrate that our methods yield narrower confidence sequences than those previously developed in the literature while maintaining time-uniform error control. 

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