arXiv subjects

Nina Dörnemann

Publications and source records attributed to Nina Dörnemann.

16 recordsLinked to original sources

arXiv2026

Monitoring for a Phase Transition in a Time Series of Wigner Matrices

We develop methodology and theory for the detection of a phase transition in a time-series of high-dimensional random matrices. In the model we study, at each time point $ t = 1,2,\ldots $, we observe a deformed Wigner matrix $ \mathbf{M}_t $, where the unobservable deformation represents a latent signal. This signal is detectable only in the supercritical regime, and our objective is to detect the transition to this regime in real time, as new matrix--valued observations arrive. Our approach is based on a partial sum process of extremal eigenvalues of $\mathbf{M}_t$, and its theoretical analysis combines state-of-the-art tools from random-matrix-theory and Gaussian approximations. The resulting detector is self-normalized, which ensures appropriate scaling for convergence and a pivotal limit, without any additional parameter estimation. Simulations show excellent performance for varying dimensions. Applications to pollution monitoring and social interactions in primates illustrate the usefulness of our approach.

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arXiv2026

Tracy-Widom, Gaussian, and Bootstrap: Approximations for Leading Eigenvalues in High-Dimensional PCA

Under certain conditions, the largest eigenvalue of a sample covariance matrix undergoes a well-known phase transition when the sample size $n$ and data dimension $p$ diverge proportionally. In the subcritical regime, this eigenvalue has fluctuations of order $n^{-2/3}$ that can be approximated by a Tracy-Widom distribution, while in the supercritical regime, it has fluctuations of order $n^{-1/2}$ that can be approximated with a Gaussian distribution. However, the statistical problem of determining which regime underlies a given dataset is far from resolved. We develop a new testing framework and procedure to address this problem. In particular, we demonstrate that the procedure has an asymptotically controlled level, and that it is power consistent for certain alternatives. Also, this testing procedure enables the design a new bootstrap method for approximating the distributions of functionals of the leading sample eigenvalues within the subcritical regime -- which is the first such method that is supported by theoretical guarantees.

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arXiv2026

Goodness-of-Fit Testing for Groupwise Spherical Error Structures

The analysis of large data panels is important in econometrics and beyond. Prediction and inference methods for such data typically rely on simplifying model assumptions for the covariance structure of errors. One convenient assumption is what we call groupwise sphericity: that errors are uncorrelated across individuals and have constant variance within certain groups. While theoretically useful, in large panels groupwise sphericity is often too restrictive to apply in practice. We therefore develop new quantitative inference tools to test whether deviations from this model assumption are practically relevant. Our approach covers both large-dimensional data matrices and regression panels, in a regime where the cross-sectional dimension is proportional to the sample size. The theory is based on the analysis of extreme eigenvalues of the empirical covariance matrix and uses recent advances in random matrix theory. Numerical experiments demonstrate accurate size control and good power in finite samples.

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arXiv2026

Eigenvalue Statistics for Change-Point Detection in Covariance Matrices

Testing for change points in sequences of covariance matrices is an important and equally challenging problem in statistical methodology with applications in various fields. Motivated by the observation that even in cases where the ratio between dimension and sample size is as small as $0.05$, tests based on a fixed-dimension asymptotics do not keep their preassigned level, we propose to derive critical values of test statistics using an asymptotic regime where the dimension diverges at the same rate as the sample size. This paper introduces a novel and well-founded statistical methodology for detecting change points in a sequence of moderately dimensional covariance matrices. Our approach utilizes a min-type statistic based on a sequential process of likelihood ratio statistics. This is used to construct a test for the hypothesis of the existence of a change point with a corresponding estimator for its location. We provide theoretical guarantees by thoroughly analyzing the asymptotic properties of the sequential process of likelihood ratio statistics. In particular, we prove weak convergence towards a Gaussian process under the null hypothesis of no change. To identify the challenging dependency structure between consecutive test statistics, we employ tools from random matrix theory and stochastic processes.

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arXiv2026

Change Point Detection and Localization in High-Dimensional Time Series

We present new inference tools for change point detection in high-dimensional time series. We discuss two distinct statistical applications: First, sequential change point testing in an incoming data-stream. Second, retrospective localization of multiple changes, with confidence intervals at a globally controlled error level. Test statistics are built on the maximum norm to generate power against sparse and asynchronous changes. Both problems are tackled by related multiscale statistics that search for changes in the data at many different levels of resolution. For fixed dimension, our statistical approaches can be validated using traditional Hölderian invariance principles. In this paper, we present the high-dimensional analogue: Hölder-Gauss-approximations, which can be (roughly) interpreted as the Gaussian approximation for a Hölder-norm of the high-dimensional partial sum process. Such approximations are of interest beyond change point detection and can be used for other problems such as for stationarity testing in high dimensions. We evaluate finite-sample performance in a simulation study and give an application to air contamination due to wildfires in California, which occurs asynchronously across a panel of measuring stations.

stat.ME↗

arXiv2026

Ties, Tails and Spectra: On Rank-Based Dependency Measures in High Dimensions

This work is concerned with the limiting spectral distribution of rank-based dependency measures in high dimensions. We provide distribution-free results for multivariate empirical versions of Kendall's $τ$ and Spearman's $ρ$ in a setting where the dimension $p$ grows at most proportionally to the sample size $n$. Although rank-based measures are known to be well suited for discrete and heavy-tailed data, previous works in the field focused mostly on the continuous and light-tailed case. We close this gap by imposing mild assumptions and allowing for general types of distributions. Interestingly, our analysis reveals that a non-trivial adjustment of classical Kendall's $τ$ is needed to obtain a pivotal limiting distribution in the presence of tied data. The proof for Spearman's $ρ$ is facilitated by a result regarding the limiting eigenvalue distribution of a general class of random matrices with rows on the Euclidean unit sphere, which is of independent interest. For instance, this finding can be used to derive the limiting spectral distribution of sample correlation matrices, which, in contrast to most existing works, accommodates heavy-tailed data.

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arXiv2026

Does PCA Work for Rough Functional Data?

Functional data analysis is concerned with the analysis of infinite-dimensional data functions. Functional principal component analysis (FPCA) is a key method to obtain finite-dimensional summaries. Consistency of FPCA has been theoretically established for sufficiently regular data functions. However, empirical evidence shows that FPCA can become severely inconsistent when the underlying functions are too rough. This paper provides the first theoretical explanation for this phenomenon. We propose a model that explicitly captures the roughness of functional data and allows us to quantify the resulting bias of FPCA, depending on the functional roughness. The model undergoes a phase transition marking the point at which FPCA becomes entirely uninformative. Based on these probabilistic results, we discuss diagnostic tests for informative principal components. As an additional contribution, we derive results on spectral statistics that may serve as a foundation for goodness-of-fit tests for rough functional data. Mathematically, our approach combines recent advances in random matrix theory and generic chaining with tools from FDA. We illustrate the effects of roughness on FPCA using simulations, as well as climate and environmental datasets.

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arXiv2025

Two-Sample Covariance Inference in High-Dimensional Elliptical Models

We propose a two-sample test for large-dimensional covariance matrices in generalized elliptical models. The test statistic is based on a U-statistic estimator of the squared Frobenius norm of the difference between the two population covariance matrices. This statistic was originally introduced by Li and Chen (2012, AoS) for the independent component model. As a key theoretical contribution, we establish a new central limit theorem for the U-statistics under elliptical data, valid under both the null and alternative hypotheses. This result enables asymptotic control of the test level and facilitates a power analysis. To the best of our knowledge, the proposed test is the first such method to be supported by theoretical guarantees for elliptical data. Our approach imposes only mild assumptions on the covariance matrices and does neither require sparsity nor explicit growth conditions on the dimension-to-sample-size ratio. We illustrate our theoretical findings through applications to both synthetic and real-world data.

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arXiv2025

A New Two-Sample Test for Covariance Matrices in High Dimensions: U-Statistics Meet Leading Eigenvalues

We propose a two-sample test for covariance matrices in the high-dimensional regime, where the dimension diverges proportionally to the sample size. Our hybrid test combines a Frobenius-norm-based statistic as considered in Li and Chen (2012) with the leading eigenvalue approach proposed in Zhang et al. (2022), making it sensitive to both dense and sparse alternatives. The two statistics are combined via Fisher's method, leveraging our key theoretical result: a joint central limit theorem showing the asymptotic independence of the leading eigenvalues of the sample covariance matrix and an estimator of the Frobenius norm of the difference of the two population covariance matrices, under suitable signal conditions. The level of the test can be controlled asymptotically, and we show consistency against certain types of both sparse and dense alternatives. A comprehensive numerical study confirms the favorable performance of our method compared to existing approaches.

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arXiv2024

Likelihood ratio tests under model misspecification in high dimensions

We investigate the likelihood ratio test for a large block-diagonal covariance matrix with an increasing number of blocks under the null hypothesis. While so far the likelihood ratio statistic has only been studied for normal populations, we establish that its asymptotic behavior is invariant under a much larger class of distributions. This implies robustness against model misspecification, which is common in high-dimensional regimes. Demonstrating the flexibility of our approach, we additionally establish asymptotic normality of the log-likelihood ratio test statistic for the equality of many large sample covariance matrices under model uncertainty. For this statistic, a subtle adjustment to the centering term is needed compared to normal case. A simulation study and an analysis of a data set from psychology emphasize the usefulness of our findings.

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arXiv2024

Detecting Spectral Breaks in Spiked Covariance Models

In this paper, the key objects of interest are the sequential covariance matrices $\mathbf{S}_{n,t}$ and their largest eigenvalues. Here, the matrix $\mathbf{S}_{n,t}$ is computed as the empirical covariance associated with observations $\{\mathbf{x}_1,\ldots,\mathbf{x}_{ \lfloor nt \rfloor } \}$, for $t\in [0,1]$. The observations $\mathbf{x}_1,\ldots,\mathbf{x}_n$ are assumed to be i.i.d. $p$-dimensional vectors with zero mean, and a covariance matrix that is a fixed-rank perturbation of the identity matrix. Treating $\{ \mathbf{S}_{n,t}\}_{t \in [0,1]}$ as a matrix-valued stochastic process indexed by $t$, we study the behavior of the largest eigenvalues of $\mathbf{S}_{n,t}$, as $t$ varies, with $n$ and $p$ increasing simultaneously, so that $p/n \to y \in (0,1)$. As a key contribution of this work, we establish the weak convergence of the stochastic process corresponding to the sample spiked eigenvalues, if their population counterparts exceed the critical phase-transition threshold. Our analysis of the limiting process is fully comprehensive revealing, in general, non-Gaussian limiting processes. As an application, we consider a class of change-point problems, where the interest is in detecting structural breaks in the covariance caused by a change in magnitude of the spiked eigenvalues. For this purpose, we propose two different maximal statistics corresponding to centered spiked eigenvalues of the sequential covariances. We show the existence of limiting null distributions for these statistics, and prove consistency of the test under fixed alternatives. Moreover, we compare the behavior of the proposed tests through a simulation study.

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arXiv2023

A CLT for the difference of eigenvalue statistics of sample covariance matrices

In the case where the dimension of the data grows at the same rate as the sample size we prove a central limit theorem for the difference of a linear spectral statistic of the sample covariance and a linear spectral statistic of the matrix that is obtained from the sample covariance matrix by deleting a column and the corresponding row. Unlike previous works, we do neither require that the population covariance matrix is diagonal nor that moments of all order exist. Our proof methodology incorporates subtle enhancements to existing strategies, which meet the challenges introduced by determining the mean and covariance structure for the difference of two such eigenvalue statistics. Moreover, we also establish the asymptotic independence of the difference-type spectral statistic and the usual linear spectral statistic of sample covariance matrices.

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arXiv2022

Fluctuations of the diagonal entries of a large sample precision matrix

For a given $p\times n$ data matrix $\textbf{X}_n$ with i.i.d. centered entries and a population covariance matrix $\bfΣ$, the corresponding sample precision matrix $\hat{\bfΣ}^{-1}$ is defined as the inverse of the sample covariance matrix $\hat{\bfΣ} = (1/n) \bfΣ^{1/2} \textbf{X}_n\textbf{X}_n^\top \bfΣ^{1/2}$. We determine the joint distribution of a vector of diagonal entries of the matrix $\hat{\bfΣ}^{-1}$ in the situation, where $p_n=p< n$ and $p/n \to y \in [0,1)$ for $n\to\infty$ and $\bfΣ$ is a diagonal matrix. Remarkably, our results cover both the case where the dimension is negligible in comparison to the sample size and the case where it is of the same magnitude. Our approach is based on a QR-decomposition of the data matrix, yielding a connection to random quadratic forms and allowing the application of a central limit theorem for martingale difference schemes. Moreover, we discuss an interesting connection to linear spectral statistics of the sample covariance matrix. More precisely, the logarithmic diagonal entry of the sample precision matrix can be interpreted as a difference of two highly dependent linear spectral statistics of $\hat{\bfΣ}$ and a submatrix of $\hat{\bfΣ}$. This difference of spectral statistics fluctuates on a much smaller scale than each single statistic.

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arXiv2022

Limiting spectral distribution for large sample correlation matrices

In this paper, we consider the empirical spectral distribution of the sample correlation matrix and investigate its asymptotic behavior under mild assumptions on the data's distribution, when dimension and sample size increase at the same rate. First, we give a characterization for the limiting spectral distribution to follow a Marchenko-Pastur law assuming that the underlying data matrix consists of i.i.d. entries. Subsequently, we provide the limiting spectral distribution of the sample correlation matrix when allowing for a dependence structure within the columns of the data matrix. In contrast to previous works, the fourth moment of the data may be infinite, resulting in a fundamental structural difference. More precisely, the standard argument of approximating the sample correlation matrix by its sample covariance companion breaks down and novel techniques for tackling the challenging dependency structure of the sample correlation matrix are introduced.

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arXiv2021

Linear spectral statistics of sequential sample covariance matrices

Independent $p$-dimensional vectors with independent complex or real valued entries such that $\mathbb{E} [\mathbf{x}_i] = \mathbf{0}$, ${\rm Var } (\mathbf{x}_i) = \mathbf{I}_p$, $i=1, \ldots,n$, let $\mathbf{T }_n$ be a $p \times p$ Hermitian nonnegative definite matrix and $f $ be a given function. We prove that an approriately standardized version of the stochastic process $ \big ( {\operatorname{tr}} ( f(\mathbf{B}_{n,t}) ) \big )_{t \in [t_0, 1]} $ corresponding to a linear spectral statistic of the sequential empirical covariance estimator $$ \big ( \mathbf{B}_{n,t} )_{t\in [ t_0 , 1]} = \Big ( \frac{1}{n} \sum_{i=1}^{\lfloor n t \rfloor} \mathbf{T }^{1/2}_n \mathbf{x}_i \mathbf{x}_i ^\star \mathbf{T }^{1/2}_n \Big)_{t\in [ t_0 , 1]} $$ converges weakly to a non-standard Gaussian process for $n,p\to\infty$. As an application we use these results to develop a novel approach for monitoring the sphericity assumption in a high-dimensional framework, even if the dimension of the underlying data is larger than the sample size.

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arXiv2019

Likelihood ratio tests for many groups in high dimensions

In this paper we investigate the asymptotic distribution of likelihood ratio tests in models with several groups, when the number of groups converges with the dimension and sample size to infinity. We derive central limit theorems for the logarithm of various test statistics and compare our results with the approximations obtained from a central limit theorem using a two step procedure: first consider the number of groups fixed and assume that the sample size and dimension converge to infinity, secondly investigating the resulting distribution if the number of groups converges to infinity.

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