Transfer learning is an emerging paradigm for leveraging multiple sources to improve the statistical inference on a single target. In this paper, we propose a novel approach named residual importance weighted transfer learning (RIW-TL) for high-dimensional linear models built on penalized likelihood. Compared to existing methods such as Trans-Lasso that selects sources in an all-in-all-out manner, RIW-TL includes samples via importance weighting and thus may permit more effective sample use. To determine the weights, remarkably RIW-TL only requires the knowledge of one-dimensional densities dependent on residuals, thus overcoming the curse of dimensionality of having to estimate high-dimensional densities in naive importance weighting. We show that the oracle RIW-TL provides a faster rate than its competitors and develop a cross-fitting procedure to estimate this oracle. We discuss variants of RIW-TL by adopting different choices for residual weighting. The theoretical properties of RIW-TL and its variants are established and compared with those of LASSO and Trans-Lasso. Extensive simulation and a real data analysis confirm its advantages.

Datasets containing both categorical and continuous variables are frequently encountered in many areas, and with the rapid development of modern measurement technologies, the dimensions of these variables can be very high. Despite the recent progress made in modelling high-dimensional data for continuous variables, there is a scarcity of methods that can deal with a mixed set of variables. To fill this gap, this paper develops a novel approach for classifying high-dimensional observations with mixed variables. Our framework builds on a location model, in which the distributions of the continuous variables conditional on categorical ones are assumed Gaussian. We overcome the challenge of having to split data into exponentially many cells, or combinations of the categorical variables, by kernel smoothing, and provide new perspectives for its bandwidth choice to ensure an analogue of Bochner's Lemma, which is different to the usual bias-variance tradeoff. We show that the two sets of parameters in our model can be separately estimated and provide penalized likelihood for their estimation. Results on the estimation accuracy and the misclassification rates are established, and the competitive performance of the proposed classifier is illustrated by extensive simulation and real data studies.

We introduce a unified and computationally efficient framework for regression on network data, addressing limitations of existing models that require specialized estimation procedures or impose restrictive decay assumptions. Our Network Propagation Regression (NPR) models outcomes as functions of covariates propagated through network connections, capturing both direct and indirect effects. NPR is estimable via ordinary least squares for continuous outcomes and standard routines for binary, categorical, and time-to-event data, all within a single interpretable framework. We establish consistency and asymptotic normality under weak conditions and develop valid hypothesis tests for the order of network influence. Simulation studies demonstrate that NPR consistently outperforms established approaches, such as the linear-in-means model and regression with network cohesion, especially under model misspecification. An application to social media sentiment analysis highlights the practical utility and robustness of NPR in real-world settings.

The conditional randomization test (CRT) provides a principled approach to conditional independence (CI) testing, guaranteeing exact type-I error control when the true conditional distribution is known. In practice, however, this distribution must be estimated, and estimation errors can inflate type-I errors, while high dimensionality and limited sample sizes can reduce power. Although external and unlabeled data offer the potential to improve CI testing, naive integration that ignores distributional heterogeneity can compromise type-I error control and fail to enhance power. We propose CRT*, a novel framework that robustly integrates external and unlabeled datasets to enhance CI testing in heterogeneous scenarios. CRT* employs smooth residual-bootstrap (SRB) with transfer learning for conditional distribution estimation, combined with adaptive data fusion via an optimal convex combination of test statistics. We theoretically establish that the SRB-based estimator converges to the true conditional distribution in expected total variation distance. Furthermore, even in high-dimensional regimes, CRT* maintains valid type-I error control and achieves strictly higher power than standard CRT without external data. Simulations and RNA-seq breast cancer data analyses demonstrate that CRT* substantially improves power while maintaining type-I error control in heterogeneous settings.

We propose a flexible dual functional factor model for modelling high-dimensional functional time series. In this model, a high-dimensional fully functional factor parametrisation is imposed on the observed functional processes, whereas a low-dimensional version (via series approximation) is assumed for the latent functional factors. We extend the classic principal component analysis technique for the estimation of a low-rank structure to the estimation of a large covariance matrix of random functions that satisfies a notion of (approximate) functional "low-rank plus sparse" structure; and generalise the matrix shrinkage method to functional shrinkage in order to estimate the sparse structure of functional idiosyncratic components. Under appropriate regularity conditions, we derive the large sample theory of the developed estimators, including the consistency of the estimated factors and functional factor loadings and the convergence rates of the estimated matrices of covariance functions measured by various (functional) matrix norms. Consistent selection of the number of factors and a data-driven rule to choose the shrinkage parameter are discussed. Simulation and empirical studies are provided to demonstrate the finite-sample performance of the developed model and estimation methodology.

This paper introduces Graphon Attachment Network Models (GAN-M), a novel framework for modeling evolving networks with rich structural dependencies, grounded in graphon theory. GAN-M provides a flexible and interpretable foundation for studying network formation by leveraging graphon functions to define attachment probabilities, thereby combining the strengths of graphons with a temporal perspective. A key contribution of this work is a methodology for learning structural changes in these networks over time. Our approach uses graph counts—frequencies of substructures such as triangles and stars—to capture shifts in network topology. We propose a new statistic designed to learn changes in the resulting piecewise polynomial signals and develop an efficient method for change detection, supported by theoretical guarantees. Numerical experiments demonstrate the effectiveness of our approach across various network settings, highlighting its potential for dynamic network analysis.

Asymmetric relational data are becoming increasingly prevalent in diverse fields, underscoring the need for developing directed network models to address the complex challenges posed by the unique structure of such data. Unlike undirected models, directed models can capture reciprocity, the tendency of nodes to form mutual links. This work addresses a fundamental question: what is the effective sample size for modelling reciprocity? We examine this question by analysing the Bernoulli model with reciprocity, allowing for varying sparsity levels between non-reciprocal and reciprocal effects. We then extend this framework to a model that incorporates node-specific heterogeneity and link-specific reciprocity using covariates. Our findings reveal the intriguing interplay between non-reciprocal and reciprocal effects in sparse networks. We propose a straightforward inference procedure based on maximum likelihood estimation that operates without prior knowledge of sparsity levels, whether covariates are included or not.

Reciprocity–the tendency of individuals to form mutual ties–is a fundamental structural feature of many directed networks. Despite its ubiquity, reciprocity remains insufficiently integrated into statistical network models, particularly in relation to covariate information. In this paper, we introduce the R^2-Model, a novel and flexible framework that explicitly models reciprocity while incorporating covariate effects. Built upon a generalized p_1 model, our framework accommodates both network sparsity and node heterogeneity, offering the most comprehensive parametrization of reciprocity to date–capturing not only its baseline level but also how it systematically varies with observed covariates. To address the challenges posed by high dimensionality and nuisance parameters, we develop a conditional likelihood estimator that isolates and consistently estimates the reciprocity effects. We establish its theoretical guarantees, including consistency, asymptotic normality, and minimax optimality under broad sparsity regimes. Extensive simulations and real-world applications demonstrate the R^2-Model's flexibility, interpretability, and strong finite-sample performance, highlighting its practical utility for uncovering covariate-driven patterns of reciprocity in directed networks.

Graphons offer a powerful framework for modeling large-scale networks, yet estimation remains challenging. We propose a novel approach that leverages a low-rank additive representation, yielding both a low-rank connection probability matrix and a low-rank graphon--two goals rarely achieved jointly. Our method resolves identification issues and enables an efficient sequential algorithm based on subgraph counts and interpolation. We establish consistency and demonstrate strong empirical performance in terms of computational efficiency and estimation accuracy through simulations and data analysis.

Clustered and longitudinal data are pervasive in scientific studies, from prenatal health programs to clinical trials and public health surveillance. Such data often involve non-Gaussian responses–including binary, categorical, and count outcomes–that exhibit complex correlation structures driven by multilevel clustering, covariates, over-dispersion, or zero inflation. Conventional approaches such as mixed-effects models and generalized estimating equations (GEEs) can capture some of these dependencies, but they are often too rigid or impose restrictive assumptions that limit interpretability and predictive performance. We investigate generalized correlation regression (GCR), a unified framework that models correlations directly as functions of interpretable covariates while simultaneously estimating marginal means. By applying a generalized z-transformation, GCR guarantees valid correlation matrices, accommodates unbalanced cluster sizes, and flexibly incorporates covariates such as time, space, or group membership into the dependence structure. Through applications to modern prenatal care, a longitudinal toenail infection trial, and clustered health count data, we show that GCR not only achieves superior predictive performance over standard methods, but also reveals family-, community-, and individual-level drivers of dependence that are obscured under conventional modeling. These results demonstrate the broad applied value of GCR for analyzing binary, count, and categorical data in clustered and longitudinal settings.

Large Language Models (LLMs) demonstrate strong performance on mathematical problems when prompted with Chain-of-Thought (CoT), yet it remains unclear whether this success stems from search, rote procedures, or rule-consistent reasoning. To address this, we propose modeling CoT as a certain rule-based stochastic process over directed acyclic graphs (DAGs), where nodes represent intermediate derivation states and edges encode rule applications. Within this framework, we introduce logical closeness, a metric that quantifies how well a model's CoT trajectory (i.e., the LLM's final output) adheres to the DAG structure, providing evaluation beyond classical PASS@k metrics. Building on this, we introduce the DAG-MATH CoT format and construct a benchmark that guides LLMs to generate CoT trajectories in this format, thereby enabling the evaluation of their reasoning ability under our framework. Across standard mathematical reasoning datasets, our analysis uncovers statistically significant differences in reasoning fidelity among representative LLM families-even when PASS@k is comparable-highlighting gaps between final-answer accuracy and rule-consistent derivation. Our framework provides a balance between free-form CoT and formal proofs systems, offering actionable diagnostics for LLMs reasoning evaluation. Our benchmark and code are available at https://github.com/YuanheZ/DAG-MATH.

In randomized controlled trials without interference, regression adjustment is widely used to enhance the efficiency of treatment effect estimation. This paper extends this efficiency principle to settings with network interference, where a unit's response may depend on the treatments assigned to its neighbors in a network. We make three key contributions: (1) we establish a central limit theorem for a linear regression-adjusted estimator and prove its optimality in achieving the smallest asymptotic variance within a class of linear adjustments; (2) we develop a novel, consistent estimator for the asymptotic variance of this linear estimator; and (3) we propose a nonparametric estimator that integrates kernel smoothing and trimming techniques, demonstrating its asymptotic normality and its optimality in minimizing asymptotic variance within a broader class of nonlinear adjustments. Extensive simulations validate the superior performance of our estimators, and a real-world data application illustrates their practical utility. Our findings underscore the power of regression-based methods and reveal the potential of kernel-and-trimming-based approaches for further enhancing efficiency under network interference.

The social characteristics of players in a social network are closely associated with their network positions and relational importance. Identifying those influential players in a network is of great importance as it helps to understand how ties are formed, how information is propagated, and, in turn, can guide the dissemination of new information. Motivated by a Sina Weibo social network analysis of the 2021 Henan Floods, where response variables for each Sina Weibo user are available, we propose a new notion of supervised centrality that emphasizes the task-specific nature of a player's centrality. To estimate the supervised centrality and identify important players, we develop a novel sparse network influence regression by introducing individual heterogeneity for each user. To overcome the computational difficulties in fitting the model for large social networks, we further develop a forward-addition algorithm and show that it can consistently identify a superset of the influential Sina Weibo users. We apply our method to analyze three responses in the Henan Floods data: the number of comments, reposts, and likes, and obtain meaningful results. A further simulation study corroborates the developed method.

Dynamic network data analysis requires joint modelling individual snapshots and time dynamics. This paper proposes a new two-way heterogeneity model towards this goal. The new model equips each node of the network with two heterogeneity parameters, one to characterize the propensity of forming ties with other nodes and the other to differentiate the tendency of retaining existing ties over time. Though the negative log-likelihood function is non-convex, it is locally convex in a neighbourhood of the true value of the parameter vector. By using a novel method of moments estimator as the initial value, the consistent local maximum likelihood estimator (MLE) can be obtained by a gradient descent algorithm. To establish the upper bound for the estimation error of the MLE, we derive a new uniform deviation bound, which is of independent interest. The usefulness of the model and the associated theory are further supported by extensive simulation and the analysis of some real network data sets.

Correlated data are ubiquitous in today's data-driven society. While regression models for analyzing means and variances of responses of interest are relatively well-developed, the development of these models for analyzing the correlations is largely confined to longitudinal data, a special form of sequentially correlated data. This paper proposes a new method for the analysis of correlations to fully exploit the use of covariates for general correlated data. In a renewed analysis of the Classroom data, a highly unbalanced multilevel clustered data with within-class and within-school correlations, our method reveals informative insights on these structures not previously known. In another analysis of the malaria immune response data in Benin, a longitudinal study with time-dependent covariates where the exact times of the observations are not available, our approach again provides promising new results. At the heart of our approach is a new generalized z-transformation that converts correlation matrices constrained to be positive definite to vectors with unrestricted support, and is order-invariant. These two properties enable us to develop regression analysis incorporating covariates for the modelling of correlations via the use of maximum likelihood.

The bootstrap is a widely used procedure for statistical inference because of its simplicity and attractive statistical properties. However, the vanilla version of bootstrap is no longer feasible computationally for many modern massive datasets due to the need to repeatedly resample the entire data. Therefore, several improvements to the bootstrap method have been made in recent years, which assess the quality of estimators by subsampling the full dataset before resampling the subsamples. Naturally, the performance of these modern subsampling methods is influenced by tuning parameters such as the size of subsamples, the number of subsamples, and the number of resamples per subsample. In this paper, we develop a novel hyperparameter selection methodology for selecting these tuning parameters. Formulated as an optimization problem to find the optimal value of some measure of accuracy of an estimator subject to computational cost, our framework provides closed-form solutions for the optimal hyperparameter values for subsampled bootstrap, subsampled double bootstrap and bag of little bootstraps, at no or little extra time cost. Using the mean square errors as a proxy of the accuracy measure, we apply our methodology to study, compare and improve the performance of these modern versions of bootstrap developed for massive data through simulation study. The results are promising.

Network data are ubiquitous in modern science and humanity. This paper concerns a new generative model, suitable for sparse networks commonly observed in practice, to capture degree heterogeneity and homophily, two stylized features of a typical network. The former is achieved by differentially assigning parameters to individual nodes, while the latter is materialized by incorporating covariates. Similar models in the literature for heterogeneity often include as many nodal parameters as the number of nodes, leading to over-parametrization. For parameter estimation, we use the penalized likelihood method with an (cid:96) 1 penalty on the nodal parameters, immediately connecting our estimation procedure to the LASSO literature. We highlight the differences of our approach to the LASSO method for logistic regression, emphasizing the feasibility of our model to conduct inference for sparse networks, study the finite-sample error bounds on the excess risk and the (cid:96) 1 -error of the resulting estimator, and develop a central limit theorem for the parameter associated with the covariates. Simulation and data analysis corroborate the developed theory. As a by-product of our main theory, we study what we call the Erdős-Rényi model with covariates and develop the associated statistical inference for sparse networks, which can be of independent interest.

Directed networks are conveniently represented as graphs in which ordered edges encode interactions between vertices. Despite their wide availability, there is a shortage of statistical models amenable for inference, specially when contextual information and degree heterogeneity are present. This paper presents an annotated graph model with parameters explicitly accounting for these features. To overcome the curse of dimensionality due to modelling degree heterogeneity, we introduce a sparsity assumption and propose a penalized likelihood approach with $\ell_1$-regularization for parameter estimation. We study the estimation and selection consistency of this approach under a sparse network assumption, and show that inference on the covariate parameter is straightforward, thus bypassing the need for the kind of debiasing commonly employed in $\ell_1$-penalized likelihood estimation. Simulation and data analysis corroborate our theoretical findings.

Contemporary time series data often feature objects connected by a social network that naturally induces temporal dependence involving connected neighbours. The network vector autoregressive model is useful for describing the influence of linked neighbours, while recent generalizations aim to separate influence and homophily. Existing approaches, however, require either correct specification of a time series model or accurate estimation of a network model or both, and rely exclusively on least-squares for parameter estimation. This paper proposes a new autoregressive model incorporating a flexible form for latent variables used to depict homophily. We develop a first-order differencing method for the estimation of influence requiring only the influence part of the model to be correctly specified. When the part including homophily is correctly specified admitting a semiparametric form, we leverage and generalize the recent notion of neighbour smoothing for parameter estimation, bypassing the need to specify the generative mechanism of the network. We develop new theory to show that all the estimated parameters are consistent and asymptotically normal. The efficacy of our approach is confirmed via extensive simulations and an analysis of a social media dataset.

Steel manufacturing is a long and complicated process including refining, casting, and rolling; hundreds of process parameters can potentially influence the mechanical properties of final products. This complexity results in significant challenges in correlating input parameters with final mechanical properties. Machine learning models, neural networks and XGBoost, have been used in the prediction of mechanical properties, however, interpretability remains an issue, especially in the case of neural networks. In this study, a statistical method - iGATE is utilised to reduce dimension of inputs in predicting mechanical properties of hot-rolled steel plates. It is found that iGATE can successfully extract the key features and reduce the dimension of inputs while maintaining a high prediction accuracy. With relative errors lower than 5%, XGboost with full inputs has the best prediction performance. With reduced input dimensions, interference of irrelevant features diminishes, and the ranking of important key features is more reliable. The iGATE methodology offers industry opportunities to identify the key input parameters in terms of materials chemistry and process variables to optimise mechanical properties of rolled plates.