arXiv subjects

Axel Bücher

Publications and source records attributed to Axel Bücher.

At least 19 recordsLinked to original sources

arXiv2026

Simultaneous Change-Point Inference for High-Dimensional Functional Time Series

We develop a framework for simultaneous change-point inference of high-dimensional functional time series. The observations are modeled as temporally dependent vectors whose coordinates take values in possibly different separable Hilbert spaces, thereby covering a broad class of functional data. Heterogeneous mean changes may occur at coordinate-specific locations, and the contemporaneous dependence across coordinates is left unrestricted. Our procedure is based on coordinatewise cumulative-sum statistics and a residual block multiplier bootstrap that provides a common critical value for the global test and all coordinatewise decisions. We establish a nonasymptotic Gaussian approximation, quantitative nonasymptotic bounds for strong family-wise error control under arbitrary mixtures of changed and unchanged coordinates, covariance-adaptive detection guarantees, and simultaneous high-probability bounds for change-point localization. The bounds accommodate high-dimensional regimes in which the number of functional coordinates grows exponentially in a power of the sample size. We investigate finite-sample performance in simulations and illustrate the method using river discharge curves and high-frequency financial log returns.

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arXiv2026

Copulas for Geostatistical Data: Foundations, Modeling Principles and Statistical Inference

Spatial statistics commonly describes spatial dependence through second-order quantities such as covariance functions and variograms, often within Gaussian random-field models and under structural assumptions such as stationarity, isotropy, or distance-based decay. Copulas offer a complementary framework that separates marginal distributions from dependence and permits a broad range of non-Gaussian dependence structures. Because the finite-dimensional distributions of a spatial random field can always be decomposed into margins and copulas through Sklar's theorem, copulas provide a natural language for studying spatial dependence beyond second-order summaries. Yet the relevant literature has developed along several largely separate strands across spatial statistics, copula modeling, stochastic processes, and application domains, often with different terminology and modeling objectives. This review brings these strands together: We revisit classical concepts from spatial statistics through a copula lens, discuss copula-based tools for describing spatial dependence, and systematically review constructions of spatial copula models. Particular emphasis is placed on Kolmogorov consistency and on the distinction between models defined for a fixed set of locations and genuinely process-level constructions. We also discuss statistical inference, extensions to spatio-temporal settings, and emerging directions involving flexible marginal and dependence models. By clarifying the relationships among existing approaches and their respective strengths and limitations, the review provides a unified perspective on the interface between copula modeling and spatial statistics.

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arXiv2026

Evidence Synthesis in Probabilistic Extreme Event Attribution: From Attribution Measures to Model Parameters

Probabilistic extreme event attribution aims to quantify how anthropogenic climate change has altered the likelihood or intensity of a class of extreme events. Existing studies commonly combine evidence from observational products and climate-model ensembles by first estimating attribution measures, such as probability ratios or intensity changes, for each data source and then synthesizing the resulting estimates. We critically assess this approach, identify potential shortcomings of a respective benchmark procedure from the literature, and propose both targeted modifications and a new parameter-level synthesis method. The latter combines estimates of the underlying nonstationary distributional regression parameters, thereby enabling inference across multiple event thresholds and counterfactual climate conditions. In controlled simulation studies, the proposed modifications substantially improve upon the benchmark procedure, while parameter-level synthesis provides competitive overall performance. The practical usefulness is illustrated through a case study of the heavy precipitation associated with Storm Boris in September 2024.

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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

Dimension Reduction in Multivariate Extremes via Latent Linear Factor Models

We propose a new and interpretable class of high-dimensional tail dependence models based on latent linear factor structures. Specifically, extremal dependence of an observable vector is assumed to be driven by a lower-dimensional latent $K$-factor model, where $K \ll d$, thereby inducing an explicit form of dimension reduction. Geometrically, this is reflected in the support of the associated spectral dependence measure, whose intrinsic dimension is at most $K-1$. The loading structure may additionally exhibit sparsity, meaning that each component is influenced by only a small number of latent factors, which further enhances interpretability and scalability. Under mild structural assumptions, we establish identifiability of the model parameters and provide a constructive recovery procedure based on a margin-free tail pairwise dependence matrix, which also yields practical rank-based estimation methods. The framework combines naturally with marginal tail models and is particularly well suited to high-dimensional settings. We illustrate its applicability in a spatial wind energy application, where the latent factor structure enables tractable estimation of the risk that a large proportion of turbines simultaneously fall below their cut-in wind speed thresholds.

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arXiv2025

Consistency of M-estimators for non-identically distributed data: the case of fixed-design distributional regression

This paper explores strong and weak consistency of M-estimators for non-identically distributed data, extending prior work. Emphasis is given to scenarios where data is viewed as a triangular array, which encompasses distributional regression models with non-random covariates. Primitive conditions are established for specific applications, such as estimation based on minimizing empirical proper scoring rules or conditional maximum likelihood. A key motivation is addressing challenges in extreme value statistics, where parameter-dependent supports can cause criterion functions to attain the value $-\infty$, hindering the application of existing theorems.

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arXiv2025

Structured linear factor models for tail dependence

A common object to describe the extremal dependence of a $d$-variate random vector $X$ is the stable tail dependence function $L$. Various parametric models have emerged, with a popular subclass consisting of those stable tail dependence functions that arise for linear and max-linear factor models with heavy tailed factors. The stable tail dependence function is then parameterized by a $d \times K$ matrix $A$, where $K$ is the number of factors and where $A$ can be interpreted as a factor loading matrix. We study estimation of $L$ under an additional assumption on $A$ called the `pure variable assumption'. Both $K \in \{1, \dots, d\}$ and $A \in [0, \infty)^{d \times K}$ are treated as unknown, which constitutes an unconventional parameter space that does not fit into common estimation frameworks. We suggest two algorithms that allow to estimate $K$ and $A$, and provide finite sample guarantees for both algorithms. Remarkably, the guarantees allow for the case where the dimension $d$ is larger than the sample size $n$. The results are illustrated with numerical experiments and two case studies.

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arXiv2025

Extreme Value Analysis based on Blockwise Top-Two Order Statistics

Extreme value analysis for time series is often based on the block maxima method, in particular for environmental applications. In the classical univariate case, the latter is based on fitting an extreme-value distribution to the sample of (annual) block maxima. Mathematically, the target parameters of the extreme-value distribution also show up in limit results for other high order statistics, which suggests estimation based on blockwise large order statistics. It is shown that a naive approach based on maximizing an independence log-likelihood yields an estimator that is inconsistent in general. A consistent, bias-corrected estimator is proposed, and is analyzed theoretically and in finite-sample simulation studies. The new estimator is shown to be more efficient than traditional counterparts, for instance for estimating large return levels or return periods.

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arXiv2024

On the lack of weak continuity of Chatterjee's correlation coefficient

Chatterjee's correlation coefficient has recently been proposed as a new association measure for bivariate random vectors that satisfies a number of desirable properties. Among these properties is the feature that the coefficient equals one if and only if one of the variables is a measurable function of the other. As already observed in Mikusinski, Sherwood and Taylor (Stochastica, 13(1):61-74, 1992), this property implies that Chatterjee's coefficient is not continuous with respect to weak convergence. We discuss a number of negative consequences for statistical inference. In particular, we show that asymptotic tests for stochastic independence based on Chatterjee's empirical correlation coefficient, or boosted versions thereof, have trivial power against certain alternatives for which the population coefficient is one.

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arXiv2024

On the maximal correlation coefficient for the bivariate Marshall Olkin distribution

We prove a formula for the maximal correlation coefficient of the bivariate Marshall Olkin distribution that was conjectured in Lin, Lai, and Govindaraju (2016, Stat. Methodol., 29:1-9). The formula is applied to obtain a new proof for a variance inequality in extreme value statistics that links the disjoint and the sliding block maxima method.

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arXiv2024

Bootstrapping Estimators based on the Block Maxima Method

The block maxima method is a standard approach for analyzing the extremal behavior of a potentially multivariate time series. It has recently been found that the classical approach based on disjoint block maxima may be universally improved by considering sliding block maxima instead. However, the asymptotic variance formula for estimators based on sliding block maxima involves an integral over the covariance of a certain family of multivariate extreme value distributions, which makes its estimation, and inference in general, an intricate problem. As an alternative, one may rely on bootstrap approximations: we show that naive block-bootstrap approaches from time series analysis are inconsistent even in i.i.d.\ situations, and provide a consistent alternative based on resampling circular block maxima. As a by-product, we show consistency of the classical resampling bootstrap for disjoint block maxima, and that estimators based on circular block maxima have the same asymptotic variance as their sliding block maxima counterparts. The finite sample properties are illustrated by Monte Carlo experiments, and the methods are demonstrated by a case study of precipitation extremes.

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arXiv2024

The empirical copula process in high dimensions: Stute's representation and applications

The empirical copula process, a fundamental tool for copula inference, is studied in the high dimensional regime where the dimension is allowed to grow to infinity exponentially in the sample size. Under natural, weak smoothness assumptions on the underlying copula, it is shown that Stute's representation is valid in the following sense: all low-dimensional margins of fixed dimension of the empirical copula process can be approximated by a functional of the low-dimensional margins of the standard empirical process, with the almost sure error term being uniform in the margins. The result has numerous potential applications, and is exemplary applied to the problem of testing pairwise stochastic independence in high dimensions, leading to various extensions of recent results in the literature: for certain test statistics based on pairwise association measures, type-I error control is obtained for models beyond mutual independence. Moreover, bootstrap-based critical values are shown to yield strong control of the familywise error rate for a large class of data generating processes.

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arXiv2023

Limit theorems for non-degenerate U-statistics of block maxima for time series

The block maxima method is a classical and widely applied statistical method for time series extremes. It has recently been found that respective estimators whose asymptotics are driven by empirical means can be improved by using sliding rather than disjoint block maxima. Similar results are derived for general non-degenerate U-statistics of arbitrary order, in the multivariate time series case. Details are worked out for selected examples: the empirical variance, the probability weighted moment estimator and Kendall's tau statistic. The results are also extended to the case where the underlying sample is piecewise stationary. The finite-sample properties are illustrated by a Monte Carlo simulation study.

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arXiv2023

Regional Pooling in Extreme Event Attribution Studies: an Approach Based on Multiple Statistical Testing

Statistical methods are proposed to select homogeneous locations when analyzing spatial block maxima data, such as in extreme event attribution studies. The methods are based on classical hypothesis testing using Wald-type test statistics, with critical values obtained from suitable parametric bootstrap procedures and corrected for multiplicity. A large-scale Monte Carlo simulation study finds that the methods are able to accurately identify homogeneous locations, and that pooling the selected locations improves the accuracy of subsequent statistical analyses. The approach is illustrated with a case study on precipitation extremes in Western Europe. The methods are implemented in an R package that allows easy application in future extreme event attribution studies.

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arXiv2022

Statistics for Heteroscedastic Time Series Extremes

Einmahl, de Haan and Zhou (2016, Journal of the Royal Statistical Society: Series B, 78(1), 31-51) recently introduced a stochastic model that allows for heteroscedasticity of extremes. The model is extended to the situation where the observations are serially dependent, which is crucial for many practical applications. We prove a local limit theorem for a kernel estimator for the scedasis function, and a functional limit theorem for an estimator for the integrated scedasis function. We further prove consistency of a bootstrap scheme that allows to test for the null hypothesis that the extremes are homoscedastic. Finally, we propose an estimator for the extremal index governing the dynamics of the extremes and prove its consistency. All results are illustrated by Monte Carlo simulations. An important intermediate result concerns the sequential tail empirical process under serial dependence.

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arXiv2022

Testing for independence in high dimensions based on empirical copulas

Testing for pairwise independence for the case where the number of variables may be of the same size or even larger than the sample size has received increasing attention in the recent years. We contribute to this branch of the literature by considering tests that allow to detect higher-order dependencies. The proposed methods are based on connecting the problem to copulas and making use of the Moebius transformation of the empirical copula process; an approach that has already been used successfully for the case where the number of variables is fixed. Based on a martingale central limit theorem, it is shown that respective test statistics converge to the standard normal distribution, allowing for straightforward definition of critical values. The results are illustrated by a Monte Carlo simulation study.

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arXiv2021

On the Disjoint and Sliding Block Maxima method for piecewise stationary time series

Modeling univariate block maxima by the generalized extreme value distribution constitutes one of the most widely applied approaches in extreme value statistics. It has recently been found that, for an underlying stationary time series, respective estimators may be improved by calculating block maxima in an overlapping way. A proof of concept is provided that the latter finding also holds in situations that involve certain piecewise stationarities. A weak convergence result for an empirical process of central interest is provided, and, as a case-in-point, further details are worked out explicitly for the probability weighted moment estimator. Irrespective of the serial dependence, the estimation variance is shown to be smaller for the new estimator, while the bias was found to be the same or vary comparably little in extensive simulation experiments. The results are illustrated by Monte Carlo simulation experiments and are applied to a common situation involving temperature extremes in a changing climate.

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arXiv2020

Statistical analysis for stationary time series at extreme levels: new estimators for the limiting cluster size distribution

A measure of primal importance for capturing the serial dependence of a stationary time series at extreme levels is provided by the limiting cluster size distribution. New estimators based on a blocks declustering scheme are proposed and analyzed both theoretically and by means of a large-scale simulation study. A sliding blocks version of the estimators is shown to outperform a disjoint blocks version. In contrast to some competitors from the literature, the estimators only depend on one unknown parameter to be chosen by the statistician.

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