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Changliang Zou

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math.ST Methodology Statistics Theory Machine Learning

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preprint2026arXiv

Learning U-Statistics with Active Inference

$U$-statistics play a central role in statistical inference. In many modern applications, however, acquiring the labels required for $U$-statistics is costly. Motivated by recent advances in active inference, we develop an active inference framework for $U$-statistics that selectively queries informative labels to improve estimation efficiency under a fixed labeling budget, while preserving valid statistical inference. Our approach is built on the augmented inverse probability weighting $U$-statistic, which is designed to incorporate the sampling rule and machine learning predictions. We characterize the optimal sampling rule that minimizes its variance and design practical sampling strategies. We further extend the framework to $U$-statistic-based empirical risk minimization. Experiments on real datasets demonstrate substantial gains in estimation efficiency over baseline methods, while maintaining target coverage.

preprint2026arXiv

Model-Agnostic and Uncertainty-Aware Dimensionality Reduction in Supervised Learning

Dimension reduction is a fundamental tool for analyzing high-dimensional data in supervised learning. Traditional methods for estimating intrinsic order often prioritize model-specific structural assumptions over predictive utility. This paper introduces predictive order determination (POD), a model-agnostic framework that determines the minimal predictively sufficient dimension by directly evaluating out-of-sample predictiveness. POD quantifies uncertainty via error bounds for over- and underestimation and achieves consistency under mild conditions. By unifying dimension reduction with predictive performance, POD applies flexibly across diverse reduction tasks and supervised learners. Simulations and real-data analyses show that POD delivers accurate, uncertainty-aware order estimates, making it a versatile component for prediction-centric pipelines.

preprint2026arXiv

Stable Localized Conformal Prediction via Transduction

Existing evaluations of conformal prediction, such as prediction efficiency and test-conditional coverage, are defined in expectation over the calibration data. In practice, when only one calibration set of limited size is available, prediction sets often exhibit high variability in size, especially for methods with localization. We formalize this concern as set stability, defined as the variance of the conditional expectation of the set size given the calibration data. To improve stability without requiring additional target-task labels, we propose Stable Conformal Prediction (StCP), a transfer learning approach that utilizes labeled source-task data and unlabeled target data. Theoretically, we characterize the marginal coverage and stability of StCP; empirically, it delivers more stable prediction sets than standard conformal prediction methods, especially for those with localization, when calibration data are limited.

preprint2023arXiv

Statistical Inference for Ultrahigh Dimensional Location Parameter Based on Spatial Median

Motivated by the widely used geometric median-of-means estimator in machine learning, this paper studies statistical inference for ultrahigh dimensionality location parameter based on the sample spatial median under a general multivariate model, including simultaneous confidence intervals construction, global tests, and multiple testing with false discovery rate control. To achieve these goals, we derive a novel Bahadur representation of the sample spatial median with a maximum-norm bound on the remainder term, and establish Gaussian approximation for the sample spatial median over the class of hyperrectangles. In addition, a multiplier bootstrap algorithm is proposed to approximate the distribution of the sample spatial median. The approximations are valid when the dimension diverges at an exponentially rate of the sample size, which facilitates the application of the spatial median in the ultrahigh dimensional region. The proposed approaches are further illustrated by simulations and analysis of a genomic dataset from a microarray study.

preprint2022arXiv

Model-Free Statistical Inference on High-Dimensional Data

This paper aims to develop an effective model-free inference procedure for high-dimensional data. We first reformulate the hypothesis testing problem via sufficient dimension reduction framework. With the aid of new reformulation, we propose a new test statistic and show that its asymptotic distribution is $χ^2$ distribution whose degree of freedom does not depend on the unknown population distribution. We further conduct power analysis under local alternative hypotheses. In addition, we study how to control the false discovery rate of the proposed $χ^2$ tests, which are correlated, to identify important predictors under a model-free framework. To this end, we propose a multiple testing procedure and establish its theoretical guarantees. Monte Carlo simulation studies are conducted to assess the performance of the proposed tests and an empirical analysis of a real-world data set is used to illustrate the proposed methodology.

preprint2020arXiv

A New Procedure for Controlling False Discovery Rate in Large-Scale t-tests

This paper is concerned with false discovery rate (FDR) control in large-scale multiple testing problems. We first propose a new data-driven testing procedure for controlling the FDR in large-scale t-tests for one-sample mean problem. The proposed procedure achieves exact FDR control in finite sample settings when the populations are symmetric no matter the number of tests or sample sizes. Comparing with the existing bootstrap method for FDR control, the proposed procedure is computationally efficient. We show that the proposed method can control the FDR asymptotically for asymmetric populations even when the test statistics are not independent. We further show that the proposed procedure with a simple correction is as accurate as the bootstrap method to the second-order degree, and could be much more effective than the existing normal calibration. We extend the proposed procedure to two-sample mean problem. Empirical results show that the proposed procedures have better FDR control than existing ones when the proportion of true alternative hypotheses is not too low, while maintaining reasonably good detection ability.

preprint2014arXiv

Nonparametric maximum likelihood approach to multiple change-point problems

In multiple change-point problems, different data segments often follow different distributions, for which the changes may occur in the mean, scale or the entire distribution from one segment to another. Without the need to know the number of change-points in advance, we propose a nonparametric maximum likelihood approach to detecting multiple change-points. Our method does not impose any parametric assumption on the underlying distributions of the data sequence, which is thus suitable for detection of any changes in the distributions. The number of change-points is determined by the Bayesian information criterion and the locations of the change-points can be estimated via the dynamic programming algorithm and the use of the intrinsic order structure of the likelihood function. Under some mild conditions, we show that the new method provides consistent estimation with an optimal rate. We also suggest a prescreening procedure to exclude most of the irrelevant points prior to the implementation of the nonparametric likelihood method. Simulation studies show that the proposed method has satisfactory performance of identifying multiple change-points in terms of estimation accuracy and computation time.

preprint2012arXiv

Coordinate-independent sparse sufficient dimension reduction and variable selection

Sufficient dimension reduction (SDR) in regression, which reduces the dimension by replacing original predictors with a minimal set of their linear combinations without loss of information, is very helpful when the number of predictors is large. The standard SDR methods suffer because the estimated linear combinations usually consist of all original predictors, making it difficult to interpret. In this paper, we propose a unified method - coordinate-independent sparse estimation (CISE) - that can simultaneously achieve sparse sufficient dimension reduction and screen out irrelevant and redundant variables efficiently. CISE is subspace oriented in the sense that it incorporates a coordinate-independent penalty term with a broad series of model-based and model-free SDR approaches. This results in a Grassmann manifold optimization problem and a fast algorithm is suggested. Under mild conditions, based on manifold theories and techniques, it can be shown that CISE would perform asymptotically as well as if the true irrelevant predictors were known, which is referred to as the oracle property. Simulation studies and a real-data example demonstrate the effectiveness and efficiency of the proposed approach.

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