arXiv subjects

Hyemin Yeon

Publications and source records attributed to Hyemin Yeon.

10 recordsLinked to original sources

arXiv2026

Dimension-invariant uniform consistency of the empirical spatial distribution function and its associated spatial depth estimator

We provide a proof that the empirical spatial distribution estimator in $\mathbb R^d$ as well as the corresponding plug-in estimator of the spatial depth are uniformly $L^1$-consistent. The consistency rate only depends on the sample size $n$, not on the dimension $d$ or any tuning or regularization parameters. This is a rare property. The result of this note originates from a conversation with ChatGPT 5.4 Pro as part of some of our own earlier experiments on its mathematical reasoning capabilities.

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arXiv2026

Wild bootstrap for mean response inference in functional linear regression models

Functional regressors complicate inference in linear regression problems so that the bootstrap can play a useful role in quantifying uncertainty and calibrating intervals. The best bootstrap in practice, though, can depend on factors in the data as well as computational considerations and existing bootstraps can have limitations: residual bootstrap is computationally fast and simple but may fail when the errors are heterogeneous, while paired bootstrap applies more generally in functional linear regression at a cost of much higher computation. To bridge this gap, we develop a wild bootstrap method for functional linear regression, which is akin to a modified version of residual bootstrap but designed to have a wide scope of application like paired bootstrap, including to heteroscedastic errors. Its theoretical consistency is established and numerical studies suggest that wild bootstrap can provide accurate and computationally fast inference. Importantly, we also suggest a practical and effective approach of selecting truncation levels, specifically designed for mean response inference problems. The proposed bootstrap in functional linear regression is further illustrated through a weather data example, and an accompanying R package BTSinFLRM provides numerical implementations.

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arXiv2026

Conditional regularized halfspace depth for sparse functional data and its applications

Many functional datasets are observed sparsely and irregularly. Ordering such data is challenging because only limited information is available from each observation, while the underlying trajectories remain infinite-dimensional. This paper develops a novel depth notion for sparse functional data, called the conditional regularized halfspace depth (CRHD). CRHD is defined as the infimum of conditional halfspace probabilities of the underlying trajectory given the observed sparse measurements, thereby enabling depth evaluation directly at sparse observations without requiring trajectory reconstruction. We study several basic theoretical properties of CRHD that clarify its behavior as a depth measure. The proposed depth is applicable even to extremely sparsely observed functional data, overcoming key limitations of existing sparse functional depths that often rely on reconstructed curves. In addition, CRHD induces meaningful rankings for complex functional data. Its numerical performance is demonstrated through rank-based tests, and its practical utility is illustrated using an infant growth dataset.

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arXiv2026

Gaussian and bootstrap approximations for functional principal component regression

Asymptotic inference using functional principal component regression (FPCR) has long been considered difficult, largely because, upon any scalar scaling, the FPCR estimator fails to satisfy a central limit theorem, leading to the prevailing belief that it is unsuitable for direct statistical inference. In this paper, we revisit this traditional viewpoint by establishing a new result: upon suitable operator scaling, valid Gaussian and bootstrap approximations hold for the FPCR estimator. We apply this surprising finding to hypothesis testing for the significance of the slope function in functional regression models and demonstrate the strong numerical performance of the resulting tests. Our finding also provides a new framework for projection inference. While concise, our results yield powerful inferential tools for functional regression. We believe it paves the way for new lines of inferential methodology for more complex functional regression settings.

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arXiv2026

Inference for function-on-function regression: central limit theorem and residual bootstrap

We investigate asymptotic inference in a linear regression model where both response and regressors are functions, using an estimator based on functional principal components analysis. Although this approach is widely used in functional data analysis, there remains significant room for developing its asymptotic properties for function-on-function regression. Our study targets the mean response at a new regressor with two primary aims. First, we refine the existing central limit theorem by relaxing certain technical conditions, which include generalizing the scaling factor, resulting in incorporating a broader class of random functions beyond those having scores with independence or finite higher moments. Second, we introduce a residual bootstrap method that enhances the calibration of various confidence sets for quantities related to mean response, while its consistency is rigorously verified. Numerical studies compare the finite sample performance of both asymptotic and bootstrap approaches, demonstrating higher accuracy of the latter. To illustrate bootstrap inference for mean response, we apply it to the Canadian weather dataset.

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arXiv2026

Effective and flexible depth-based inference for functional parameters

For hypothesis testing of functional parameters, given a functional statistic $T_n$ and a functional depth $D$ with respect to the distribution $P_n$ of $T_n$, we propose the depth value $DT_n \equiv D(T_n;P_n)$ as a test statistic, which we refer to as a depth statistic. In practice, its sampling distribution is approximated by a resampling method such as bootstrap. While achieving accurate sizes, a test based on the proposed depth statistic produces stronger power, as it remains sensitive even to subtle variations arising from complex functional patterns in the alternatives. Moreover, it is broadly applicable to a broad range of inference problems for functional parameters, including two-sample tests, analysis of variance, regression, etc. We provide its theoretical guarantee under mild assumptions along with examples of bootstrap methods and functional depths that satisfy these conditions. Its effectiveness is thoroughly investigated through numerical studies under two popular frameworks: (i) two-sample functional mean tests and (ii) mean response inference for function-on-function regression. The proposed depth statistic is illustrated with two data examples: Canadian weather and German electricity prices datasets.

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arXiv2026

Projection depth for functional data: Practical issues, computation and applications

Statistical analysis of functional data is challenging due to their complex patterns, for which functional depth provides an effective means of reflecting their ordering structure. In this work, we investigate practical aspects of the recently proposed regularized projection depth (RPD), which induces a meaningful ordering of functional data while appropriately accommodating their complex shape features. Specifically, we examine the impact and choice of its tuning parameter, which regulates the degree of effective dimension reduction applied to the data, and propose a random projection-based approach for its efficient computation, supported by theoretical justification. Through comprehensive numerical studies, we explore a wide range of statistical applications of the RPD and demonstrate its particular usefulness in uncovering shape features in functional data analysis. This ability allows us to propose a novel RPD-based outlier detection method, and to demonstrate that the RPD outperforms competing depth-based approaches in tasks such as functional classification and two-sample hypothesis testing.

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arXiv2025

Projection depth for functional data: Theoretical properties

We introduce a novel projection depth for data lying in a general Hilbert space, called the regularized projection depth, with a focus on functional data. By regularizing projection directions, the proposed depth does not suffer from the degeneracy issue that may arise when the classical projection depth is naively defined on an infinite-dimensional space. Compared to existing functional depth notions, the regularized projection depth has several advantages: (i) it requires no moment assumptions on the underlying distribution, (ii) it satisfies many desirable depth properties including invariance, monotonicity, and vanishing at infinity, (iii) its sample version uniformly converges under mild conditions, and (iv) it generates a highly robust median. Furthermore, the proposed depth is statistically useful as it (v) does not produce ties in the induced ranks and (vi) effectively detects shape outlying functions. This paper focuses mainly on the theoretical properties of the regularized projection depth.

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arXiv2024

Regularized Halfspace Depth for Functional Data

Data depth is a powerful nonparametric tool originally proposed to rank multivariate data from center outward. In this context, one of the most archetypical depth notions is Tukey's halfspace depth. In the last few decades notions of depth have also been proposed for functional data. However, Tukey's depth cannot be extended to handle functional data because of its degeneracy. Here, we propose a new halfspace depth for functional data which avoids degeneracy by regularization. The halfspace projection directions are constrained to have a small reproducing kernel Hilbert space norm. Desirable theoretical properties of the proposed depth, such as isometry invariance, maximality at center, monotonicity relative to a deepest point, upper semi-continuity, and consistency are established. Moreover, the regularized halfspace depth can rank functional data with varying emphasis in shape or magnitude, depending on the regularization. A new outlier detection approach is also proposed, which is capable of detecting both shape and magnitude outliers. It is applicable to trajectories in $L^2$, a very general space of functions that include non-smooth trajectories. Based on extensive numerical studies, our methods are shown to perform well in terms of detecting outliers of different types. Three real data examples showcase the proposed depth notion.

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arXiv2024

Bootstrap inference in functional linear regression models with scalar response under heteroscedasticity

Inference for functional linear models in the presence of heteroscedastic errors has received insufficient attention given its practical importance; in fact, even a central limit theorem has not been studied in this case. At issue, conditional mean estimates have complicated sampling distributions due to the infinite dimensional regressors, where truncation bias and scaling issues are compounded by non-constant variance under heteroscedasticity. As a foundation for distributional inference, we establish a central limit theorem for the estimated conditional mean under general dependent errors, and subsequently we develop a paired bootstrap method to provide better approximations of sampling distributions. The proposed paired bootstrap does not follow the standard bootstrap algorithm for finite dimensional regressors, as this version fails outside of a narrow window for implementation with functional regressors. The reason owes to a bias with functional regressors in a naive bootstrap construction. Our bootstrap proposal incorporates debiasing and thereby attains much broader validity and flexibility with truncation parameters for inference under heteroscedasticity; even when the naive approach may be valid, the proposed bootstrap method performs better numerically. The bootstrap is applied to construct confidence intervals for centered projections and for conducting hypothesis tests for the multiple conditional means. Our theoretical results on bootstrap consistency are demonstrated through simulation studies and also illustrated with a real data example.

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