arXiv subjects

Stanislav Volgushev

Publications and source records attributed to Stanislav Volgushev.

At least 19 recordsLinked to original sources

arXiv2026

Limiting properties of monotone rearrangements of estimators when the truth is flat

Monotone rearrangements provide a simple way to enforce shape constraints of an estimator, but existing distributional theory does not cover flat regions, where the target induces no local ordering. We study rearranged estimators in two canonical flat settings. First, for a histogram estimator of the uniform density, we establish functional weak convergence of its non-decreasing rearrangement at the parametric rate on compact subsets of the interval $(0,1)$ after an additional deterministic centering. This result is strikingly different from what is known for strictly monotone densities. Second, we consider two rearranged estimators of rearranged copulas under independence, based on empirical-copula increments and on a checkerboard approximation. After appropriate centering and rescaling, both estimators converge weakly on $[0,1]^2$ to an integrated Gaussian process which was not known before. We further use these results to prove asymptotic normality for a broad class of rearranged copula-based dependence measures, which were recently discussed in Strothmann et al. (2024).

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arXiv2026

Adaptivity of the NPMLE to finitely discrete mixing distributions in Gaussian/Poisson mixtures

We study the nonparametric maximum likelihood estimator (NPMLE) for Gaussian and Poisson mixture models, assuming the support of the true mixing distribution lies in a fixed bounded set. In this setting, we establish exact parametric rates for both, marginal density estimation and the posterior mean when the true mixing distribution is finitely discrete. Moreover, we show that the NPMLE attains the optimal demixing rate previously known for overparameterized finite mixture models. Finally, we identify a new adaptivity phenomenon for inference: the likelihood ratio test statistic is asymptotically tight if and only if the true mixing distribution is finitely discrete.

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arXiv2026

Empirical tail dependence functions in high dimensions: uniform linearizations and inference

The analysis of extremal dependence in high dimensions is a key challenge in modern extreme-value statistics. Existing methodology primarily focuses on modeling and estimation of extremal dependence structures, often supported by concentration bounds for empirical tail quantities. However, comparatively little is known about general inferential procedures in high-dimensional extremes. In this paper, we develop foundational results that enable inference for rank-based empirical tail dependence coefficients, stable tail dependence functions, and functionals derived from them. We start by establishing finite-sample probability bounds that quantify the linearization error for such estimators uniformly over collections of coordinates. Moreover, we derive high-dimensional central limit theorems and establish the validity of multiplier bootstrap procedures for collections of empirical tail dependence statistics. Within an asymptotic framework, our results allow the dimension to grow exponentially with the effective sample size. We illustrate the usefulness of the results through two applications: uniform expansions and normal approximations for M-estimators of tail dependence parameters and inference for spatial isotropy based on collections of tail dependence functions.

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arXiv2026

Graph structure learning for stable processes

We introduce Ising-H\"usler-Reiss processes, a new class of multivariate L\'evy processes that allows for sparse modeling of the path-wise conditional independence structure between marginal stable processes with different stability indices. The underlying conditional independence graph is encoded as zeroes in a suitable precision matrix. An Ising-type parametrization of the weights for each orthant of the L\'evy measure allows for data-driven modeling of asymmetry of the jumps while retaining an arbitrary sparse graph. We develop consistent estimators for the graphical structure and asymmetry parameters, relying on a new uniform small-time approximation for L\'evy processes. The methodology is illustrated in simulations and a real data application to modeling dependence of stock returns.

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arXiv2025

On a surprising behavior of the likelihood ratio test in non-parametric mixture models

We study the likelihood ratio test in general mixture models where the base density is parametric, the null is a known fixed mixing distribution, and the alternative is a general mixing distribution supported on a bounded parameter space. For Gaussian location mixtures and Poisson mixtures, we show a surprising result: the non-parametric likelihood ratio test statistic converges to a tight limit if and only if the null distribution is a finite mixture, and diverges to infinity otherwise. We further demonstrate that the likelihood ratio test diverges for a fairly general class of distributions when the null mixing distribution is not finitely discrete.

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arXiv2025

Simultaneous hypothesis testing for comparing many functional means

Data with multiple functional recordings at each observational unit are increasingly common in various fields including medical imaging and environmental sciences. To conduct inference for such observations, we develop a paired two-sample test that allows to simultaneously compare the means of many functional observations while maintaining family-wise error rate control. We explicitly allow the number of functional recordings to increase, potentially much faster than the sample size. Our test is fully functional and does not rely on dimension reduction or functional PCA type approaches or the choice of tuning parameters. To provide a theoretical justification for the proposed procedure, we develop a number of new anti-concentration and Gaussian approximation results for maxima of $L^2$ statistics which might be of independent interest. The methodology is illustrated on the task-related cortical surface functional magnetic resonance imaging data from Human Connectome Project.

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arXiv2025

Universality of Benign Overfitting in Binary Linear Classification

The practical success of deep learning has led to the discovery of several surprising phenomena. One of these phenomena, that has spurred intense theoretical research, is ``benign overfitting'': deep neural networks seem to generalize well in the over-parametrized regime even though the networks show a perfect fit to noisy training data. It is now known that benign overfitting also occurs in various classical statistical models. For linear maximum margin classifiers, benign overfitting has been established theoretically in a class of mixture models with very strong assumptions on the covariate distribution. However, even in this simple setting, many questions remain open. For instance, most of the existing literature focuses on the noiseless case where all true class labels are observed without errors, whereas the more interesting noisy case remains poorly understood. We provide a comprehensive study of benign overfitting for linear maximum margin classifiers. We discover a phase transition in test error bounds for the noisy model which was previously unknown and provide some geometric intuition behind it. We further considerably relax the required covariate assumptions in both the noisy and noiseless cases. Our results demonstrate that benign overfitting of maximum margin classifiers holds in a much wider range of scenarios than was previously known and provide new insights into the underlying mechanisms.

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arXiv2024

Spectral Clustering with Variance Information for Group Structure Estimation in Panel Data

Consider a panel data setting where repeated observations on individuals are available. Often it is reasonable to assume that there exist groups of individuals that share similar effects of observed characteristics, but the grouping is typically unknown in advance. We first conduct a local analysis which reveals that the variances of the individual coefficient estimates contain useful information for the estimation of group structure. We then propose a method to estimate unobserved groupings for general panel data models that explicitly account for the variance information. Our proposed method remains computationally feasible with a large number of individuals and/or repeated measurements on each individual. The developed ideas can also be applied even when individual-level data are not available and only parameter estimates together with some quantification of estimation uncertainty are given to the researcher. A thorough simulation study demonstrates superior performance of our method than existing methods and we apply the method to two empirical applications.

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arXiv2023

Change-point Inference for High-dimensional Heteroscedastic Data

We propose a bootstrap-based test to detect a mean shift in a sequence of high-dimensional observations with unknown time-varying heteroscedasticity. The proposed test builds on the U-statistic based approach in Wang et al. (2022), targets a dense alternative, and adopts a wild bootstrap procedure to generate critical values. The bootstrap-based test is free of tuning parameters and is capable of accommodating unconditional time varying heteroscedasticity in the high-dimensional observations, as demonstrated in our theory and simulations. Theoretically, we justify the bootstrap consistency by using the recently proposed unconditional approach in Bucher and Kojadinovic (2019). Extensions to testing for multiple change-points and estimation using wild binary segmentation are also presented. Numerical simulations demonstrate the robustness of the proposed testing and estimation procedures with respect to different kinds of time-varying heteroscedasticity.

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arXiv2022

Structure learning for extremal tree models

Extremal graphical models are sparse statistical models for multivariate extreme events. The underlying graph encodes conditional independencies and enables a visual interpretation of the complex extremal dependence structure. For the important case of tree models, we develop a data-driven methodology for learning the graphical structure. We show that sample versions of the extremal correlation and a new summary statistic, which we call the extremal variogram, can be used as weights for a minimum spanning tree to consistently recover the true underlying tree. Remarkably, this implies that extremal tree models can be learned in a completely non-parametric fashion by using simple summary statistics and without the need to assume discrete distributions, existence of densities, or parametric models for bivariate distributions.

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arXiv2022

Fighting Noise with Noise: Causal Inference with Many Candidate Instruments

Instrumental variable methods provide useful tools for inferring causal effects in the presence of unmeasured confounding. To apply these methods with large-scale data sets, a major challenge is to find valid instruments from a possibly large candidate set. In practice, most of the candidate instruments are often not relevant for studying a particular exposure of interest. Moreover, not all relevant candidate instruments are valid as they may directly influence the outcome of interest. In this article, we propose a data-driven method for causal inference with many candidate instruments that addresses these two challenges simultaneously. A key component of our proposal involves using pseudo variables, known to be irrelevant, to remove variables from the original set that exhibit spurious correlations with the exposure. Synthetic data analyses show that the proposed method performs favourably compared to existing methods. We apply our method to a Mendelian randomization study estimating the effect of obesity on health-related quality of life.

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arXiv2022

Mirror Descent Strikes Again: Optimal Stochastic Convex Optimization under Infinite Noise Variance

We study stochastic convex optimization under infinite noise variance. Specifically, when the stochastic gradient is unbiased and has uniformly bounded $(1+κ)$-th moment, for some $κ\in (0,1]$, we quantify the convergence rate of the Stochastic Mirror Descent algorithm with a particular class of uniformly convex mirror maps, in terms of the number of iterations, dimensionality and related geometric parameters of the optimization problem. Interestingly this algorithm does not require any explicit gradient clipping or normalization, which have been extensively used in several recent empirical and theoretical works. We complement our convergence results with information-theoretic lower bounds showing that no other algorithm using only stochastic first-order oracles can achieve improved rates. Our results have several interesting consequences for devising online/streaming stochastic approximation algorithms for problems arising in robust statistics and machine learning.

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arXiv2021

The integrated copula spectrum

Frequency domain methods form a ubiquitous part of the statistical toolbox for time series analysis. In recent years, considerable interest has been given to the development of new spectral methodology and tools capturing dynamics in the entire joint distributions and thus avoiding the limitations of classical, $L^2$-based spectral methods. Most of the spectral concepts proposed in that literature suffer from one major drawback, though: their estimation requires the choice of a smoothing parameter, which has a considerable impact on estimation quality and poses challenges for statistical inference. In this paper, associated with the concept of copula-based spectrum, we introduce the notion of copula spectral distribution function or integrated copula spectrum. This integrated copula spectrum retains the advantages of copula-based spectra but can be estimated without the need for smoothing parameters. We provide such estimators, along with a thorough theoretical analysis, based on a functional central limit theorem, of their asymptotic properties. We leverage these results to test various hypotheses that cannot be addressed by classical spectral methods, such as the lack of time-reversibility or asymmetry in tail dynamics.

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arXiv2021

Learning extremal graphical structures in high dimensions

Extremal graphical models encode the conditional independence structure of multivariate extremes. Key statistics for learning extremal graphical structures are empirical extremal variograms, for which we prove non-asymptotic concentration bounds that hold under general domain of attraction conditions. For the popular class of H\"usler--Reiss models, we propose a majority voting algorithm for learning the underlying graph from data through $L^1$ regularized optimization. Our concentration bounds are used to derive explicit conditions that ensure the consistent recovery of any connected graph. The methodology is illustrated through a simulation study as well as the analysis of river discharge and currency exchange data.

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arXiv2021

Inference for Change Points in High Dimensional Data via Self-Normalization

This article considers change point testing and estimation for a sequence of high-dimensional data. In the case of testing for a mean shift for high-dimensional independent data, we propose a new test which is based on $U$-statistic in Chen and Qin (2010) and utilizes the self-normalization principle [Shao (2010), Shao and Zhang (2010)]. Our test targets dense alternatives in the high-dimensional setting and involves no tuning parameters. To extend to change point testing for high-dimensional time series, we introduce a trimming parameter and formulate a self-normalized test statistic with trimming to accommodate the weak temporal dependence. On the theory front, we derive the limiting distributions of self-normalized test statistics under both the null and alternatives for both independent and dependent high-dimensional data. At the core of our asymptotic theory, we obtain weak convergence of a sequential U-statistic based process for high-dimensional independent data, and weak convergence of sequential trimmed U-statistic based processes for high-dimensional linear processes, both of which are of independent interests. Additionally, we illustrate how our tests can be used in combination with wild binary segmentation to estimate the number and location of multiple change points. Numerical simulations demonstrate the competitiveness of our proposed testing and estimation procedures in comparison with several existing methods in the literature.

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arXiv2021

Rank-based Estimation under Asymptotic Dependence and Independence, with Applications to Spatial Extremes

Multivariate extreme value theory is concerned with modeling the joint tail behavior of several random variables. Existing work mostly focuses on asymptotic dependence, where the probability of observing a large value in one of the variables is of the same order as observing a large value in all variables simultaneously. However, there is growing evidence that asymptotic independence is equally important in real world applications. Available statistical methodology in the latter setting is scarce and not well understood theoretically. We revisit non-parametric estimation and introduce rank-based M-estimators for parametric models that simultaneously work under asymptotic dependence and asymptotic independence, without requiring prior knowledge on which of the two regimes applies. Asymptotic normality of the proposed estimators is established under weak regularity conditions. We further show how bivariate estimators can be leveraged to obtain parametric estimators in spatial tail models, and again provide a thorough theoretical justification for our approach.

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arXiv2020

An Analysis of Constant Step Size SGD in the Non-convex Regime: Asymptotic Normality and Bias

Structured non-convex learning problems, for which critical points have favorable statistical properties, arise frequently in statistical machine learning. Algorithmic convergence and statistical estimation rates are well-understood for such problems. However, quantifying the uncertainty associated with the underlying training algorithm is not well-studied in the non-convex setting. In order to address this shortcoming, in this work, we establish an asymptotic normality result for the constant step size stochastic gradient descent (SGD) algorithm--a widely used algorithm in practice. Specifically, based on the relationship between SGD and Markov Chains [DDB19], we show that the average of SGD iterates is asymptotically normally distributed around the expected value of their unique invariant distribution, as long as the non-convex and non-smooth objective function satisfies a dissipativity property. We also characterize the bias between this expected value and the critical points of the objective function under various local regularity conditions. Together, the above two results could be leveraged to construct confidence intervals for non-convex problems that are trained using the SGD algorithm.

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arXiv2020

Testing relevant hypotheses in functional time series via self-normalization

In this paper we develop methodology for testing relevant hypotheses about functional time series in a tuning-free way. Instead of testing for exact equality, for example for the equality of two mean functions from two independent time series, we propose to test the null hypothesis of no relevant deviation. In the two sample problem this means that an $L^2$-distance between the two mean functions is smaller than a pre-specified threshold. For such hypotheses self-normalization, which was introduced by Shao (2010) and Shao and Zhang (2010) and is commonly used to avoid the estimation of nuisance parameters, is not directly applicable. We develop new self-normalized procedures for testing relevant hypotheses in the one sample, two sample and change point problem and investigate their asymptotic properties. Finite sample properties of the proposed tests are illustrated by means of a simulation study and data examples. Our main focus is on functional time series, but extensions to other settings are also briefly discussed.

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