Haoran Li (李浩然)

About

I am an assistant professor in the Department of Mathematics and Statistics at Auburn University. Before joining Auburn, I was a term assistant professor in the Department of Statistics at Columbia University (2019–2022). I received my Ph.D. in Statistics from the University of California, Davis in 2019, advised by Debashis Paul and Alexander Aue.

Research interests

My research focuses on high-dimensional statistical inference, random matrix theory and its applications, time series analysis, change-point detection, and climate data analysis.

Professional experience

Education

Publications

1. Ridge-regularized largest root test for high-dimensional general linear hypotheses
   H. Li
   The Annals of Statistics (2026).
   Abstract

   A fundamental problem in multivariate analysis is testing general linear hypotheses for regression coefficients in a multivariate linear model. This framework encompasses a wide range of well-studied tasks, including MANOVA, joint significance testing of predictors, and detection of trends or seasonal effects. Among classical approaches, Roy's largest root test is particularly effective for detecting concentrated signals, relying on the largest eigenvalue of an F matrix constructed from residual covariance matrices. However, in high-dimensional settings, these matrices often become ill-conditioned or singular, rendering the test infeasible. To address this, we propose a ridge-regularized Roy's test that stabilizes the covariance estimation via a ridge term. We establish the asymptotic Tracy-Widom distribution of the largest eigenvalue of the regularized F-matrix under a high-dimensional regime, where both the dimension and hypotheses are comparable to the sample size, assuming only finite-moment conditions. A computationally efficient procedure is developed to estimate the associated centering and scaling parameters. We further analyze the power of the test under a class of low-rank alternatives and examine the influence of the regularization parameter. The method demonstrates strong performance in simulations and is applied to data from the Human Connectome Project to assess associations between volumetric brain measurements and behavioral variables.

   Abstract source (arXiv version)
2. An adaptable generalization of Hotelling’s T² test in high dimension
   H. Li, A. Aue, D. Paul, J. Peng, and P. Wang
   The Annals of Statistics, 48(3), 1815–1847 (2020).
   Abstract

   We propose a two-sample test for detecting the difference between mean vectors in a high-dimensional regime based on a ridge-regularized Hotelling's T². To choose the regularization parameter, a method is derived that aims at maximizing power within a class of local alternatives. We also propose a composite test that combines the optimal tests corresponding to a specific collection of local alternatives. Weak convergence of the stochastic process corresponding to the ridge-regularized Hotelling's T² is established and used to derive the cut-off values of the proposed test. Large sample properties are verified for a class of sub-Gaussian distributions. Through an extensive simulation study, the composite test is shown to compare favorably against a host of existing two-sample test procedure in a wide range of settings. The performance of the proposed test procedure is illustrated through an application to a breast cancer data set where the goal is to detect the pathways with different DNA copy number alterations across breast cancer subtypes.

   Abstract source (arXiv version)

   Best Student Paper Award, International Chinese Statistical Association (ICSA), 2017.

3. Testing general linear hypotheses under a high-dimensional multivariate regression model with spiked noise covariance
   H. Li, A. Aue, D. Paul, and J. Peng
   Journal of the American Statistical Association, 119(548), 2799–2810 (2024).
   Abstract

   We consider the problem of testing linear hypotheses under a multivariate regression model with a high-dimensional response and spiked noise covariance. The proposed family of tests consists of test statistics based on a weighted sum of projections of the data onto the estimated latent factor directions, with the weights acting as the regularization parameters. We establish asymptotic normality of the test statistics under the null hypothesis. We also establish the power characteristics of the tests and propose a data-driven choice of the regularization parameters under a family of local alternatives. The performance of the proposed tests is evaluated through a simulation study. Finally, the proposed tests are applied to the Human Connectome Project data to test for the presence of associations between volumetric measurements of human brain and behavioral variables. Supplementary materials for this article are available online including a standardized description of the materials available for reproducing the work.

   Paper source
4. High-dimensional general linear hypothesis tests via non-linear spectral shrinkage
   H. Li, A. Aue, and D. Paul
   Bernoulli, 26(4), 2541–2571 (2020).
   Abstract

   We are interested in testing general linear hypotheses in a high-dimensional multivariate linear regression model. The framework includes many well-studied problems such as two-sample tests for equality of population means, MANOVA and others as special cases. A family of rotation-invariant tests is proposed that involves a flexible spectral shrinkage scheme applied to the sample error covariance matrix. The asymptotic normality of the test statistic under the null hypothesis is derived in the setting where dimensionality is comparable to sample sizes, assuming the existence of certain moments for the observations. The asymptotic power of the proposed test is studied under various local alternatives. The power characteristics are then utilized to propose a data-driven selection of the spectral shrinkage function. As an illustration of the general theory, we construct a family of tests involving ridge-type regularization and suggest possible extensions to more complex regularizers. A simulation study is carried out to examine the numerical performance of the proposed tests.

   Abstract source (arXiv version)
5. Regularized fingerprinting with linearly optimal weight matrix in detection and attribution of climate change
   Y. Li and H. Li
   Journal of Climate, 39(19), 5625–5641 (2026).
   Abstract

   Climate change detection and attribution play a central role in establishing the causal influence of human activities on global warming. The dominant framework, optimal fingerprinting, is a linear errors-in-variables model in which each covariate is subject to measurement error with covariance proportional to that of the regression error. The reliability of such analyses depends critically on accurate inference of the regression coefficients. The optimal weight matrix for estimating these coefficients is the precision matrix of the regression error, which is typically unknown and must be estimated from climate model simulations. However, existing regularized optimal fingerprinting approaches often yield underestimated uncertainties and overly narrow confidence intervals that fail to attain nominal coverage, thereby compromising the reliability of analysis. In this paper, we first propose consistent variance estimators for the regression coefficients within the class of linear shrinkage weight matrices, addressing undercoverage in conventional methods. Building on this, we derive a linearly optimal weight matrix that directly minimizes the asymptotic variances of the estimated scaling factors. Numerical studies confirm improved empirical coverage and shorter interval lengths. When applied to annual mean temperature data, the proposed method produces narrower, more reliable intervals and provides new insights into detection and attribution across different regions.

   Abstract source (arXiv version)
6. Analysis of the limiting spectral distribution of large random matrices of the Marčenko–Pastur type
   H. Li
   Random Matrices: Theory and Applications, 14(03), 2550007 (2025).
   Abstract

   Consider the random matrix Wₙ = Bₙ + n⁻¹Xₙ^*AₙXₙ, where Aₙ and Bₙ are Hermitian matrices of dimensions p × p and n × n, respectively, and Xₙ is a p × n random matrix with independent and identically distributed entries of mean 0 and variance 1. Assume that p and n grow to infinity proportionally, and that the spectral measures of Aₙ and Bₙ converge as p, n → ∞ towards two probability measures 𝒜 and ℬ. Building on the groundbreaking work of Marčenko and Pastur (1967), which demonstrated that the empirical spectral distribution of Wₙ converges towards a probability measure F characterized by its Stieltjes transform, this paper investigates the properties of F when ℬ is a general measure. We show that F has an analytic density at the region near where the Stieltjes transform of ℬ is bounded. The density closely resembles C√|x − x₀| near certain edge points x₀ of its support for a wide class of 𝒜 and ℬ. We provide a complete characterization of the support of F. Moreover, we show that F can exhibit discontinuities at points where ℬ is discontinuous.

   Abstract source (arXiv version)
7. A machine learning framework with Shapley’s additive explanations to assess solder joint reliability for electronic packaging
   Q. Qasaimeh, H. Li, S. Hamasha, and J. Liu
   Journal of Electronic Materials, 54(9), 7586–7608 (2025).
   Abstract

   Assessing the reliability of solder joints is a significant challenge in the electronics manufacturing industry, as numerous factors affect their integrity and performance. Traditionally, accelerated life tests (ALTs) are used for evaluating solder joint reliability, and survival analysis models such as Weibull and the Cox proportional hazards model (Cox-PHM) are widely used to develop life prediction models based on ALT data. The rise of machine learning (ML) models, including random survival forest, extreme gradient boosting (XGB), and survival support vector machines (SSVMs), offers promising data-driven alternatives, especially given their potential for higher predictive accuracy. However, their interpretability remains a concern for the electronics manufacturing community. In this study, we conducted systematic research to integrate multiple ML algorithms and Shapley’s additive explanation (SHAP) techniques to model solder joint reliability in thermal cycling tests from various impacting factors and to extract knowledge from the ML models for interpretability. The ML approaches demonstrate superior predictive performance compared to traditional survival analysis models. For instance, XGB achieves the highest c-index of 0.88 on the testing dataset, indicating strong discriminative power. Similarly, the KSSVM model yields the lowest test MAPE of 15.26%, reflecting excellent accuracy in predicting cycles to failure. The GB model also performs well, with a c-index of 0.88 and test MAPE of 15.31%, highlighting the reliability of boosting-based approaches. While traditional models like Cox-PHM and Weibull yield c-indices around 0.87 and 0.85, respectively, they fall short in prediction error, with MAPEs exceeding 20%. These findings confirm the advantages of advanced ML models in capturing complex patterns in reliability data. Furthermore, SHAP analysis enhances model transparency by revealing how critical features—such as component type, solder material, and aging duration—interact to drive failure predictions, offering insight beyond what conventional models can provide.

   Paper source
8. Interpretable machine learning models can outperform statistical models in solder joint reliability
   Q. Qasaimeh, H. Li, S. Hamasha, J. Evans, and J. Liu
   23rd IEEE ITherm Conference (2024).
   Abstract

   Weibull regression and the Cox proportional hazard model (Cox-PHM) are commonly employed for analyzing the solder joint reliability due to their simplicity and meaningful results. Recently, various machine learning (ML) models have been applied and shown similar or better performance than conventional approaches, yet they do not incur widespread adoption in reliability modeling because of their limited transparency and interpretability. In this research, we compare the predictive capabilities of Weibull regression and Cox-PHM with ML techniques (specifically, Random Survival Forests [RSF] and Gradient Boosting [GB]) using experimental data from an accelerated thermal cycling reliability test and calculating the concordance index (c-index) as an evaluation metric to assess the effectiveness of a model in predicting the sequence of failure times in the tests. The findings with our dataset showed that ML-based models have a similar performance with the traditional Weibull regression (c-index = 0.808), and GB can even outperform with a high c-index value of 0.848. The superior performance of ML models can be attributed to their capacity to capture nonlinearities and intricate relationships within the data. Additionally, we utilized feature importance scores to understand which factors/variables significantly influence the model's prediction. Such interpretable ML methods can provide clear insights into how models make predictions, thereby enhancing trust and the adoption of innovative ML techniques in electronics manufacturing.

   Paper source
9. Detecting structural changes in high-dimensional multivariate regression models
   H. Li
   Under review · arXiv:2609.28462.
   Abstract

   We study structural-change testing in multivariate linear regression when the response dimension is proportional to the sample size and the number of predictors is fixed. Despite its relevance to applications across a broad range of fields, this problem remains underexplored. The alternatives of interest allow multiple unknown changes in a prescribed linear contrast of the predictor effects. We construct a least-squares-based Wald statistic standardized by the residual covariance estimator and scan it over candidate change-point segmentations. We propose flexible segment scans for both single and multiple change points, together with a discretized multiscale scan that reduces the computational cost. When the response dimension and sample size diverge proportionally, we establish weak convergence of the normalized statistic process to a centered Gaussian process, yielding implementable critical values for all three scans. We further characterize their asymptotic power under local alternatives, explicitly describing how the signal, design, and high-dimensional aspect ratio determine the limiting power. The finite-sample performance is examined through simulation studies. We apply the proposed methods to filtered and standardized returns from U.S. equity portfolios to investigate structural changes in their exposures to the Fama–French factors.

   Abstract source (arXiv version)
10. Fingerprint analysis for climate change detection and attribution under a latent factor model
    H. Li and Y. Li
    Under review · arXiv:2609.13488.
    Abstract

    Detection and attribution (D&A) analyses provide a statistical framework for quantifying the contribution of external forcings to observed climate change. Optimal fingerprinting, the primary approach for D&A, is formulated as an errors-in-variables regression with a high-dimensional covariance structure. Reliable inference is challenging because covariance matrices must be estimated from a limited number of control simulations, and climate models may exhibit variability patterns that differ from those of the observed climate system. We develop a spiked optimal fingerprinting framework that exploits the spiked structure of dominant climate variability modes while providing stable covariance estimation in high-dimensional settings. The proposed method develops bias-corrected spiked covariance estimation for constructing adaptive weight matrices and incorporates variability inflation adjustments to account for model–observation differences in internal variability. We further develop a valid uncertainty quantification procedure for the scaling factor estimators and a residual consistency test for assessing model adequacy. Numerical studies demonstrate improved estimation accuracy and uncertainty quantification compared with existing practical approaches. Applied to annual mean near-surface temperature data, the proposed framework produces shorter and better-calibrated confidence intervals and leads to different detection and attribution conclusions in several regions, providing new insights into the interpretation of climate attribution results.

    Abstract source (arXiv version)
11. Adaptable fingerprinting with nonlinear shrinkage for climate change detection and attribution under variance heterogeneity
    H. Li and Y. Li
    Under review · arXiv:2608.12496.
    Abstract

    Detection and attribution of climate change relies on fingerprinting–a linear errors-in-variables regression framework in which both predictors and responses exhibit internal variability governed by a proportional covariance structure, subject to a variability inflation factor. Accurate estimation of the scaling factors (regression coefficients) depends on inferring the precision matrix of the regression errors from limited climate model control runs. In high-dimensional settings, existing approaches often overlook the variance inflation of the predictors and suffer from imprecise precision matrix estimates, yielding biased estimators, underestimated uncertainties, and confidence intervals with poor coverage. We propose a nonlinear, rotation-invariant shrinkage framework for estimating the precision matrix that restores the asymptotic optimality of the total least squares estimator in high-dimensional regimes. Our procedure jointly estimates the scaling factors and the variability inflation factor, thereby correcting estimation bias, and incorporates consistent variance estimators to enable valid uncertainty quantification. We also develop a residual consistency test to assess model adequacy. Numerical studies demonstrate precise estimation, improved confidence interval coverage, and higher efficiency. Applied to annual mean near-surface air temperature data from 1951–2020, our method produces narrower and more reliable confidence intervals, yielding refined attribution results.

    Abstract source (arXiv version)
12. Adaptable regularized CCA tests for independence of high-dimensional random vectors
    H. Li
    Under review · arXiv:2607.10500.
    Abstract

    We propose an adaptable testing procedure for independence between two high-dimensional random vectors. The method incorporates ridge regularization and principal component-based dimension reduction into the canonical correlation analysis (CCA) framework, thereby stabilizing classical test statistics in high-dimensional settings. Depending on the reduced dimension, we develop both a regularized likelihood ratio test and a regularized largest-root test to accommodate different testing scenarios. We establish the asymptotic behavior of the proposed procedures under both the null hypothesis and representative alternatives, and further develop a data-driven method for selecting the regularization parameter. Extensive simulation studies demonstrate favorable finite-sample performance across a broad range of settings.

    Abstract source (arXiv version)
13. Adaptable high-dimensional change-point detection via ridge regularization
    H. Li and H. Xu
    Under review · arXiv:2605.24838.
    Abstract

    We study the problem of detecting multiple change points in the mean vectors of an independent sequence of high-dimensional observations. Targeting at detecting dense alternatives in the regime where the dimension is comparable to the sample size, we propose a family of ridge-regularized CUSUM statistics built upon the adaptable ridge-regularized Hotelling's T² test of Li et al. (Ann. Statist. 48 (2020) 1815–1847). Ridge regularization provides stable covariance normalization while allowing the tests to adapt to the underlying population covariance structure. To accommodate multiple change points without prior knowledge of their number or locations, we scan over a multi-scale collection of adjacent segments. We further enhance adaptability by aggregating information across a collection of ridge regularization parameters through the maximum of the corresponding test statistics. Under mild conditions, we establish the limiting distributions of the proposed statistics under the null hypothesis and a class of local alternatives. Extensive simulation studies demonstrate their finite-sample performance.

    Abstract source (arXiv version)

Ongoing research

1. ENSO-aligned variability mismatches between observations and simulations alter climate fingerprint detection
   H. Li, Y. Li, and C. Li
   Work in progress.

   Examines how mismatches in ENSO-aligned variability between observations and simulations affect climate fingerprint detection.

2. Functional connectome classification in the tangent space of Bures–Wasserstein geometry
   M. Zirpoli, J. Zheng, and H. Li
   Work in progress.

   Studies functional connectome classification in the tangent space of Bures–Wasserstein geometry.

3. Local laws and edge fluctuations for deformed quadratic matrices
   L. Chen and H. Li
   Work in progress.

   Studies local laws and spectral edge fluctuations for deformed quadratic matrices.

Teaching

Selected courses taught at Auburn University and Columbia University.

Auburn University

• STAT 7600/7610: Statistical Theory and Methods I/II (Ph.D. program core courses)
• STAT 7850: Theory of Statistical Inference (Graduate)
• STAT 7860: Applied Time Series Analysis (Graduate)
• STAT 7780: Survival Analysis (Graduate)
• STAT 3600: Probability and Statistics I (Undergraduate)

Columbia University

• GR5263: Time Series & Statistical Inferences (M.A. in Financial Mathematics)
• GR5221: Time Series Analysis (M.A. in Statistics)
• UN1001: Elementary Statistics (Undergraduate)

Software

ARHT: An R package implementing the adaptable regularized Hotelling’s T² test for high-dimensional mean testing, proposed by Li et al. (2020a). GitHub

ReguGLH: An R package implementing the general linear hypothesis tests proposed by Li et al. (2020b). It supports linear shrinkage (ridge), polynomial shrinkage, and general shrinkage, and provides both individual and composite testing procedures. GitHub

RidgeLRGLH: An R package implementing the ridge-regularized largest root test proposed by Li (2026). It tests a large number of simultaneous linear hypotheses in high-dimensional multivariate regression. GitHub

spikedGLHT: An R package implementing the calibrated likelihood ratio tests for high-dimensional general linear hypotheses under a spiked covariance model, proposed by Li et al. (2024). GitHub