arXiv subjects

Peijun Sang

Publications and source records attributed to Peijun Sang.

At least 19 recordsLinked to original sources

arXiv2026

Modeling positive and unlabeled data with a generalized additive density ratio model

This paper focuses on learning from positive and unlabeled (PU) data, where only some positives are labeled and the rest are mixed with negatives. Classical exponential tilting models guarantee identifiability by imposing a linear structure, but they can be severely misspecified when the true relationships are nonlinear. We propose a generalized additive density-ratio framework that retains identifiability while allowing nonlinear and feature-specific effects. The approach comes with a practical fitting algorithm and supporting theory that enable estimation and inference for the mixture proportion and other quantities of interest. In simulations and analyses of real datasets, the proposed method matches the standard exponential tilting method when the linear model is correct and delivers clear gains when it is not. Overall, the framework strikes a useful balance between flexibility and interpretability for PU data and provides principled tools for estimation, prediction, and uncertainty assessment.

stat.ME↗

arXiv2026

Functional Cox model for interval-censored data

Interval-censored data arise frequently in scientific studies, where the event of interest is known only to occur within a specific time interval. In such studies, functional covariates taking the form of continuous curves or spatial profiles are increasingly encountered, and it is of substantial scientific relevance to investigate how the trajectory of a functional covariate affects the event time. We formulate the effects of both scalar and functional covariates on the interval-censored event time through a functional Cox model. We consider penalized maximum likelihood estimation for this model and devise an EM algorithm to stably compute the parameter estimators. The resulting estimators for the regression parameters and linear functionals of the coefficient function are shown to be consistent and asymptotically normal, with limiting covariance matrices that attain the semiparametric efficiency bound and can be readily estimated through the profile likelihood method. Building upon these results, we construct a global test for the overall effect of the functional covariate. Finally, we assess the performance of the proposed methods through extensive simulation studies and present an application to data from the Alzheimer's Disease Neuroimaging Initiative.

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arXiv2026

Varying coefficient model for longitudinal data with informative observation times

Varying coefficient models are widely used to characterize dynamic associations between longitudinal outcomes and covariates. Existing work on varying coefficient models, however, all assumes that observation times are independent of the longitudinal outcomes, which is often violated in real-world studies with outcome-driven or otherwise informative visit schedules. Such informative observation times can lead to biased estimation and invalid inference using existing methods. In this article, we develop estimation and inference procedures for varying coefficient models that account for informative observation times. We model the observation time process as a general counting process under a proportional intensity model, with time-varying covariates summarizing the observed history. To address potential bias, we incorporate inverse intensity weighting into a sieve estimation framework, yielding closed-form coefficient function estimators via weighted least squares. We establish consistency, convergence rates, and asymptotic normality of the proposed estimators, and construct pointwise confidence intervals for the coefficient functions. Extensive simulation studies demonstrate that the proposed weighted method substantially outperforms the conventional unweighted method when observation times are informative. Finally, we provide an application of our method to the Alzheimer's Disease Neuroimaging Initiative study.

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arXiv2025

Learning Shared and Source-specific Subspaces across Multiple Data Sources for Functional Data

In the era of big data, integrating multi-source functional data to extract a subspace that captures the shared subspace across sources has attracted considerable attention. In practice, data collection procedures often follow source-specific protocols. Directly averaging sample covariance operators across sources implicitly assumes homogeneity, which may bias the recovery of both shared and source-specific variation patterns. To address this issue, we propose a projection-based data integration method that explicitly separates the shared and source-specific subspaces. The method first estimates source-specific projection operators via smoothing to accommodate the nonparametric nature of functional data. The shared subspace is then isolated by examining the eigenvalues of the averaged projection operator across all sources. If a source-specific subspace is of interest, we re-project the associated source-specific covariance estimator onto the subspace orthogonal to the estimated shared subspace, and estimate the source-specific subspace from the resulting projection. We further establish the asymptotic properties of both the shared and source-specific subspace estimators. Extensive simulation studies demonstrate the effectiveness of the proposed method across a wide range of settings. Finally, we illustrate its practical utility with an example of air pollutant data.

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arXiv2025

Order Determination for Functional Data

Dimension reduction is often necessary in functional data analysis, with functional principal component analysis being one of the most widely used techniques. A key challenge in applying these methods is determining the number of eigen-pairs to retain, a problem known as order determination. When a covariance function admits a finite representation, the challenge becomes estimating the rank of the associated covariance operator. While this problem is straightforward when the full trajectories of functional data are available, in practice, functional data are typically collected discretely and are subject to measurement error contamination. This contamination introduces a ridge to the empirical covariance function, which obscures the true rank of the covariance operator. We propose a novel procedure to identify the true rank of the covariance operator by leveraging the information of eigenvalues and eigenfunctions. By incorporating the nonparametric nature of functional data through smoothing techniques, the method is applicable to functional data collected at random, subject-specific points. Extensive simulation studies demonstrate the excellent performance of our approach across a wide range of settings, outperforming commonly used information-criterion-based methods and maintaining effectiveness even in high-noise scenarios. We further illustrate our method with two real-world data examples.

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arXiv2025

L1-Regularized Functional Support Vector Machine

In functional data analysis, binary classification with one functional covariate has been extensively studied. We aim to fill in the gap of considering multivariate functional covariates in classification. In particular, we propose an $L_1$-regularized functional support vector machine for binary classification. An accompanying algorithm is developed to fit the classifier. By imposing an $L_1$ penalty, the algorithm enables us to identify relevant functional covariates of the binary response. Numerical results from simulations and one real-world application demonstrate that the proposed classifier enjoys good performance in both prediction and feature selection.

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arXiv2025

Kernel-based Method for Detecting Structural Break in Distribution of Functional Data

We propose a novel method to detect and date structural breaks in the entire distribution of functional data. Theoretical guarantees are developed for our procedure under fewer assumptions than in the existing work. In particular, we establish the asymptotic null distribution of the test statistic, which enables us to test the null hypothesis at a certain significance level. Additionally, the limiting distribution of the estimated structural break date is developed under two situations of the break size: fixed and shrinking towards 0 at a specified rate. We further propose a unified bootstrap procedure to construct a confidence interval for the true structural break date for these two situations. These theoretical results are justified through comprehensive simulation studies in finite samples. We apply the proposed method to two real-world examples: Australian temperature data for detecting structural beaks and Canadian weather data for goodness of fit.

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arXiv2025

Efficient Estimation of Structural Models via Sieves

We propose a class of sieve-based efficient estimators for structural models (SEES), which approximate the solution using a linear combination of basis functions and impose equilibrium conditions as a penalty to determine the best-fitting coefficients. Our estimators avoid the need to repeatedly solve the model, apply to a broad class of models, and are consistent, asymptotically normal, and asymptotically efficient. Moreover, they solve unconstrained optimization problems with fewer unknowns and offer convenient standard error calculations. As an illustration, we apply our method to an entry game between Walmart and Kmart.

econ.EM↗

arXiv2025

Likelihood-based Nonparametric Receiver Operating Characteristic Curve Analysis in the Presence of Imperfect Reference Standard

In diagnostic studies, researchers frequently encounter imperfect reference standards with some misclassified labels. Treating these as gold standards can bias receiver operating characteristic (ROC) curve analysis. To address this issue, we propose a novel likelihood-based method under a nonparametric density ratio model. This approach enables the reliable estimation of the ROC curve, area under the curve (AUC), partial AUC, and Youden's index with favorable statistical properties. To implement the method, we develop an efficient expectation-maximization algorithm algorithm. Extensive simulations evaluate its finite-sample performance, showing smaller mean squared errors in estimating the ROC curve, partial AUC, and Youden's index compared to existing methods. We apply the proposed approach to a malaria study.

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arXiv2025

Two-sample inference for sparse functional data

We propose a novel test procedure for comparing mean functions across two groups within the reproducing kernel Hilbert space (RKHS) framework. Our proposed method is adept at handling sparsely and irregularly sampled functional data when observation times are random for each subject. Conventional approaches, which are built upon functional principal components analysis, usually assume a homogeneous covariance structure across groups. Nonetheless, justifying this assumption in real-world scenarios can be challenging. To eliminate the need for a homogeneous covariance structure, we first develop a linear approximation for the mean estimator under the RKHS framework; this approximation is a sum of i.i.d. random elements, which naturally leads to the desirable pointwise limiting distributions. Moreover, we establish weak convergence for the mean estimator, allowing us to construct a test statistic for the mean difference. Our method is easily implementable and outperforms some conventional tests in controlling type I errors across various settings. We demonstrate the finite sample performance of our approach through extensive simulations and two real-world applications.

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arXiv2025

Bootstrap Nonparametric Inference under Data Integration

We propose multiplier bootstrap procedures for nonparametric inference and uncertainty quantification of the target mean function, based on a novel framework of integrating target and source data. We begin with the relatively easier covariate shift scenario with equal target and source mean functions and propose estimation and inferential procedures through a straightforward combination of all target and source datasets. We next consider the more general and flexible distribution shift scenario with arbitrary target and source mean functions, and propose a two-step inferential procedure. First, we estimate the target-to-source differences based on separate portions of the target and source data. Second, the remaining source data are adjusted by these differences and combined with the remaining target data to perform the multiplier bootstrap procedure. Our method enables local and global inference on the target mean function without using asymptotic distributions. To justify our approach, we derive an optimal convergence rate for the nonparametric estimator and establish bootstrap consistency to estimate the asymptotic distribution of the nonparametric estimator. The proof of global bootstrap consistency involves a central limit theorem for quadratic forms with dependent variables under a conditional probability measure. Our method applies to arbitrary source and target datasets, provided that the data sizes meet a specific quantitative relationship. Simulation studies and real data analysis are provided to examine the performance of our approach.

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arXiv2024

Functional principal component analysis with informative observation times

Functional principal component analysis has been shown to be invaluable for revealing variation modes of longitudinal outcomes, which serves as important building blocks for forecasting and model building. Decades of research have advanced methods for functional principal component analysis often assuming independence between the observation times and longitudinal outcomes. Yet such assumptions are fragile in real-world settings where observation times may be driven by outcome-related reasons. Rather than ignoring the informative observation time process, we explicitly model the observational times by a counting process dependent on time-varying prognostic factors. Identification of the mean, covariance function, and functional principal components ensues via inverse intensity weighting. We propose using weighted penalized splines for estimation and establish consistency and convergence rates for the weighted estimators. Simulation studies demonstrate that the proposed estimators are substantially more accurate than the existing ones in the presence of a correlation between the observation time process and the longitudinal outcome process. We further examine the finite-sample performance of the proposed method using the Acute Infection and Early Disease Research Program study.

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arXiv2024

Variational Nonparametric Inference in Functional Stochastic Block Model

We propose a functional stochastic block model whose vertices involve functional data information. This new model extends the classic stochastic block model with vector-valued nodal information, and finds applications in real-world networks whose nodal information could be functional curves. Examples include international trade data in which a network vertex (country) is associated with the annual or quarterly GDP over certain time period, and MyFitnessPal data in which a network vertex (MyFitnessPal user) is associated with daily calorie information measured over certain time period. Two statistical tasks will be jointly executed. First, we will detect community structures of the network vertices assisted by the functional nodal information. Second, we propose computationally efficient variational test to examine the significance of the functional nodal information. We show that the community detection algorithms achieve weak and strong consistency, and the variational test is asymptotically chi-square with diverging degrees of freedom. As a byproduct, we propose pointwise confidence intervals for the slop function of the functional nodal information. Our methods are examined through both simulated and real datasets.

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arXiv2023

A Bayesian Collocation Integral Method for Parameter Estimation in Ordinary Differential Equations

Inferring the parameters of ordinary differential equations (ODEs) from noisy observations is an important problem in many scientific fields. Currently, most parameter estimation methods that bypass numerical integration tend to rely on basis functions or Gaussian processes to approximate the ODE solution and its derivatives. Due to the sensitivity of the ODE solution to its derivatives, these methods can be hindered by estimation error, especially when only sparse time-course observations are available. We present a Bayesian collocation framework that operates on the integrated form of the ODEs and also avoids the expensive use of numerical solvers. Our methodology has the capability to handle general nonlinear ODE systems. We demonstrate the accuracy of the proposed method through simulation studies, where the estimated parameters and recovered system trajectories are compared with other recent methods. A real data example is also provided.

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arXiv2023

Restricted Tweedie Stochastic Block Models

The stochastic block model (SBM) is a widely used framework for community detection in networks, where the network structure is typically represented by an adjacency matrix. However, conventional SBMs are not directly applicable to an adjacency matrix that consists of non-negative zero-inflated continuous edge weights. To model the international trading network, where edge weights represent trading values between countries, we propose an innovative SBM based on a restricted Tweedie distribution. Additionally, we incorporate nodal information, such as the geographical distance between countries, and account for its dynamic effect on edge weights. Notably, we show that given a sufficiently large number of nodes, estimating this covariate effect becomes independent of community labels of each node when computing the maximum likelihood estimator of parameters in our model. This result enables the development of an efficient two-step algorithm that separates the estimation of covariate effects from other parameters. We demonstrate the effectiveness of our proposed method through extensive simulation studies and an application to real-world international trading data.

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arXiv2023

Scalable inference in functional linear regression with streaming data

Traditional static functional data analysis is facing new challenges due to streaming data, where data constantly flow in. A major challenge is that storing such an ever-increasing amount of data in memory is nearly impossible. In addition, existing inferential tools in online learning are mainly developed for finite-dimensional problems, while inference methods for functional data are focused on the batch learning setting. In this paper, we tackle these issues by developing functional stochastic gradient descent algorithms and proposing an online bootstrap resampling procedure to systematically study the inference problem for functional linear regression. In particular, the proposed estimation and inference procedures use only one pass over the data; thus they are easy to implement and suitable to the situation where data arrive in a streaming manner. Furthermore, we establish the convergence rate as well as the asymptotic distribution of the proposed estimator. Meanwhile, the proposed perturbed estimator from the bootstrap procedure is shown to enjoy the same theoretical properties, which provide the theoretical justification for our online inference tool. As far as we know, this is the first inference result on the functional linear regression model with streaming data. Simulation studies are conducted to investigate the finite-sample performance of the proposed procedure. An application is illustrated with the Beijing multi-site air-quality data.

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arXiv2022

Order Statistics Approaches to Unobserved Heterogeneity in Auctions

We establish nonparametric identification of auction models with continuous and nonseparable unobserved heterogeneity using three consecutive order statistics of bids. We then propose sieve maximum likelihood estimators for the joint distribution of unobserved heterogeneity and the private value, as well as their conditional and marginal distributions. Lastly, we apply our methodology to a novel dataset from judicial auctions in China. Our estimates suggest substantial gains from accounting for unobserved heterogeneity when setting reserve prices. We propose a simple scheme that achieves nearly optimal revenue by using the appraisal value as the reserve price.

econ.EM↗

arXiv2022

Nonlinear function-on-function regression by RKHS

We propose a nonlinear function-on-function regression model where both the covariate and the response are random functions. The nonlinear regression is carried out in two steps: we first construct Hilbert spaces to accommodate the functional covariate and the functional response, and then build a second-layer Hilbert space for the covariate to capture nonlinearity. The second-layer space is assumed to be a reproducing kernel Hilbert space, which is generated by a positive definite kernel determined by the inner product of the first-layer Hilbert space for $X$--this structure is known as the nested Hilbert spaces. We develop estimation procedures to implement the proposed method, which allows the functional data to be observed at different time points for different subjects. Furthermore, we establish the convergence rate of our estimator as well as the weak convergence of the predicted response in the Hilbert space. Numerical studies including both simulations and a data application are conducted to investigate the performance of our estimator in finite sample.

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