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Marius Kroll

Publications and source records attributed to Marius Kroll.

7 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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arXiv2024

Detecting practically significant dependencies in metric spaces via distance correlations

We take a different look at the problem of testing the independence of two metric-space-valued random variables using the distance correlation. Instead of testing if the distance correlation vanishes exactly, we are interested in the hypothesis that it does not exceed a certain threshold. Our testing problem is motivated by the observation that in many cases it is more reasonable to test for a practically significant dependency since it is rare that a hypothesis of perfect independence is exactly satisfied. This point of view also reflects statistical practice, where one often classifies the strength of the association in categories such as `small', `medium' and `large' and the precise definitions depend on the specific application. To address these problems we develop a pivotal test for the hypothesis that the distance correlation between two random variables does not exceed a pre-specified threshold $\Delta$. We also determine a minimum value $\hat \Delta_\alpha$ from the data such that the hypothesis is rejected for all $\Delta \leq \hat \Delta_\alpha$ at controlled type I error $\alpha$. This quantity can be interpreted as a measure of evidence against the hypothesis that the distance correlation is less or equal than $\Delta$. The new test is applicable to processes taking values in separable metric spaces of strong negative type, covering Euclidean as well as functional data. We do not assume independent observations, and instead prove our results for absolutely regular sample generating processes, which includes many time series such as ARMA and GARCH models. Our approach is based on a new functional limit theorem for the sequential distance correlation process, and can also be used to construct confidence intervals for the distance correlation without the need for resampling.

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arXiv2024

Extension of Process Convergence With Application to Chatterjee's Rank Correlation

We give conditions under which weak convergence of a stochastic process indexed in the class of $d$-dimensional hyperrectangles is sufficient to ensure convergence in the larger class of functions of uniformly bounded Hardy-Krause variation. When applied to the empirical process, this can further be extended to derive weak convergence of V-processes indexed in the class of kernel functions which are coordinate-wise of uniformly bounded Hardy-Krause variation. Our proofs use a generalisation of the Koksma-Hlawka inequality for linear operators, allowing us to establish our results without any continuity assumptions on the functions involved. Our theory is complemented by two separate applications: First, we establish asymptotic normality of Chatterjee's rank correlation in the fully general setting. Second, we present new limit theorems for U- and V-processes of strongly mixing data.

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arXiv2023

A Simple Bootstrap for Chatterjee's Rank Correlation

We prove that an $m$ out of $n$ bootstrap procedure for Chatterjee's rank correlation is consistent whenever asymptotic normality of Chatterjee's rank correlation can be established. In particular, we prove that $m$ out of $n$ bootstrap works for continuous as well as for discrete data with independent coordinates; furthermore, simulations indicate that it also performs well for discrete data with dependent coordinates, and that it outperforms alternative estimation methods. Consistency of the bootstrap is proved in the Kolmogorov as well as in the Wasserstein distance.

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arXiv2021

A Bootstrap Test for Independence of Time Series Based on the Distance Covariance

We present a test for independence of two strictly stationary time series based on a bootstrap procedure for the distance covariance. Our test detects any kind of dependence between the two time series within an arbitrary maximum lag $L$. In simulation studies, our test outperforms alternative testing procedures. In proving the validity of the underlying bootstrap procedure, we generalise bounds for the Wasserstein distance between an empirical measure and its marginal distribution under the assumption of $\alpha$-mixing. Previous results of this kind only existed for i.i.d. processes.

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arXiv2020

Asymptotic Behaviour of the Empirical Distance Covariance for Dependent Data

We give two asymptotic results for the empirical distance covariance on separable metric spaces without any iid assumption on the samples. In particular, we show the almost sure convergence of the empirical distance covariance for any measure with finite first moments, provided that the samples form a strictly stationary and ergodic process. We further give a result concerning the asymptotic distribution of the empirical distance covariance under the assumption of absolute regularity of the samples and extend these results to certain types of pseudometric spaces. In the process, we derive a general theorem concerning the asymptotic distribution of degenerate V-statistics of order 2 under a strong mixing condition.

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