arXiv subjects

Fabian Mies

Publications and source records attributed to Fabian Mies.

At least 19 recordsLinked to original sources

arXiv2026

Multifractional Stable Motion with Random Hurst Exponent

The fractional stable motion is a prototypical stochastic process exhibiting both heavy tails and long-range dependence, parameterized via a stability index $\alpha$ and a Hurst exponent $H$. We consider a nonstationary extension where the Hurst exponent is a function of time, and potentially random. The construction admits the standard linear fractional stable motion as tangent process, and we exactly determine its local H\"older exponent in terms of the pointwise values of the Hurst function. This is in contrast to other definitions of multifractional processes, where the Hurst function needs to have additional regularity in time.

math.PR↗

arXiv2026

Besov-Orlicz moduli of Brownian motion and polygonal partial sum processes

The sample paths of Brownian motion are known to admit the exact Besov-type smoothness exponent 1/2 when measured in the sub-Gaussian Orlicz norm. We extend these regularity results by deriving the exact limit of the sub-Gaussian Orlicz modulus for Brownian motion in Banach spaces, and we provide a rate of convergence towards this limiting value. The central technique is a new chaining bound for the Orlicz modulus of a stochastic process. The latter also applies to polyogonal partial sum processes of functional random variables and allows us to strengthen Donsker's invariance principle to all function spaces on the Besov-Orlicz scale up to the exact modulus with exponent 1/2. For the critical case, we establish the thresholded weak convergence of the Besov-Orlicz seminorm of the partial sum process. The analytical results find application in a nonparametric statistical testing problem, where Besov-Orlicz statistics are shown to detect a broader range of alternatives compared to Hölderian multiscale statistics.

math.PR↗

arXiv2026

Empirical Orlicz norms

The empirical Orlicz norm based on a random sample is defined as a natural estimator of the Orlicz norm of a univariate probability distribution. A law of large numbers is derived under minimal assumptions. The latter extends readily to a linear and a nonparametric regression model. Secondly, sufficient conditions for a central limit theorem with a standard rate of convergence are supplied. The conditions for the CLT exclude certain canonical examples, such as the empirical sub-Gaussian norm of normally distributed random variables. For the latter, we discover a nonstandard rate of $n^{1/4} \log(n)^{3/8}$, with a heavy-tailed, stable limit distribution. It is shown that in general, the empirical Orlicz norm does not admit any uniform rate of convergence for the class of distributions with bounded Orlicz norm.

math.ST↗

arXiv2025

Likelihood asymptotics of stationary Gaussian arrays

This paper develops an asymptotic likelihood theory for triangular arrays of stationary Gaussian time series depending on a multidimensional unknown parameter. We give sufficient conditions for the associated sequence of statistical models to be locally asymptotically normal in Le Cam's sense, which in particular implies the asymptotic efficiency of the maximum likelihood estimator. Unique features of the array setting covered by our theory include potentially nondiagonal rate matrices as well as spectral densities that satisfy different power-law bounds at different frequencies and may fail to be uniformly integrable. To illustrate our theory, we study efficient estimation for Gaussian processes sampled at high frequency and for a class of autoregressive models with moderate deviations from a unit root.

math.ST↗

arXiv2025

Rough Hurst function estimation

The fractional Brownian motion (fBm) is parameterized by the Hurst exponent $H\in(0,1)$, which determines the dependence structure and regularity of sample paths. Empirical findings suggest that the Hurst exponent may be non-constant in time, giving rise to the so-called multifractional Brownian motion (mBm). The Itô-mBm is an alternative to the classical mBm, and has been shown to admit more intuitive sample path properties in case the Hurst function is rough. In this paper, we show that the Itô-mBm also allows for a simplified statistical treatment compared to the classical mBm. In particular, estimation of the local Hurst parameter $H(t)$ with Hölder exponent $η>0$ achieves rates of convergence which are standard in nonparametric regression, whereas similar results for the classical mBm only hold for the smoother regime $η>1$. Furthermore, we derive an estimator of the integrated Hurst exponent $\int_0^t H(s)\, ds$ which achieves a parametric rate of convergence, and use it to construct goodness-of-fit tests.

math.ST↗

arXiv2025

At the edge of Donsker's Theorem: Asymptotics of multiscale scan statistics

For nonparametric inference about a function, multiscale testing procedures resolve the need for bandwidth selection and achieve asymptotically optimal detection performance against a broad range of alternatives. However, critical values strongly depend on the noise distribution, and we argue that existing methods are either statistically infeasible, or asymptotically sub-optimal. To address this methodological challenge, we show how to develop a feasible multiscale test via weak convergence arguments, by replacing the additive multiscale penalty with a multiplicative weighting. This new theoretical foundation preserves the optimal detection properties of multiscale tests and extends their applicability to nonstationary nonlinear time series via a tailored bootstrap scheme. Inference for signal discovery, goodness-of-fit testing of regression functions, and multiple changepoint detection is studied in detail, and we apply the new methodology to analyze the April 2025 power blackout on the Iberian peninsula. Our methodology is enabled by a novel functional central limit in Hölder spaces with critical modulus of continuity, where Donsker's theorem fails to hold due to lack of tightness. Probabilistically, we discover a novel form of thresholded weak convergence that holds only in the upper support of the distribution.

math.ST↗

arXiv2025

Simultaneous Nonparametric Confidence Bands for Load-Sharing Systems

Load-sharing systems arise in many different reliability applications, for instance, when modeling tensile strength of fibrous composites in textile industry or lifetimes of redundant technical systems in engineering. Sequential order statistics serve as a flexible model for the ordered component failure times of such systems and allow the residual lifetime distribution of the components to change after each component failure. In a proportional hazard rate setting, the model consists of some baseline distribution function and several model parameters describing successive adjustments of the hazard rates of the operating components. This work provides nonparametric confidence bands for the baseline distribution function, where the model parameters may be known or unknown. In case of known model parameters, we show how to construct exact confidence bands based on Kolmogorov-Smirnov type statistics, which are distribution-free with respect to the baseline distribution. If the model parameters are unknown, finite sample inference turns out to be infeasible, and asymptotic confidence bands for the baseline distribution function are derived. As a technical tool, we extend the existing asymptotic theory of semiparametric estimators based on the profile-likelihood approach.

stat.ME↗

arXiv2024

Strong Gaussian approximations with random multipliers

One reason why standard formulations of the central limit theorems are not applicable in high-dimensional and non-stationary regimes is the lack of a suitable limit object. Instead, suitable distributional approximations can be used, where the approximating object is not constant, but a sequence as well. We extend Gaussian approximation results for the partial sum process by allowing each summand to be multiplied by a data-dependent matrix. The results allow for serial dependence of the data, and for high-dimensionality of both the data and the multipliers. In the finite-dimensional and locally-stationary setting, we obtain a functional central limit theorem as a direct consequence. An application to sequential testing in non-stationary environments is described.

math.ST↗

arXiv2024

Rate-optimal estimation of mixed semimartingales

Consider the sum $Y=B+B(H)$ of a Brownian motion $B$ and an independent fractional Brownian motion $B(H)$ with Hurst parameter $H\in(0,1)$. Even though $B(H)$ is not a semimartingale, it was shown in [\textit{Bernoulli} \textbf{7} (2001) 913--934] that $Y$ is a semimartingale if $H>3/4$. Moreover, $Y$ is locally equivalent to $B$ in this case, so $H$ cannot be consistently estimated from local observations of $Y$. This paper pivots on another unexpected feature in this model: if $B$ and $B(H)$ become correlated, then $Y$ will never be a semimartingale, and $H$ can be identified, regardless of its value. This and other results will follow from a detailed statistical analysis of a more general class of processes called \emph{mixed semimartingales}, which are semiparametric extensions of $Y$ with stochastic volatility in both the martingale and the fractional component. In particular, we derive consistent estimators and feasible central limit theorems for all parameters and processes that can be identified from high-frequency observations. We further show that our estimators achieve optimal rates in a minimax sense.

math.ST↗

arXiv2024

Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning

Active learning of physical systems must commonly respect practical safety constraints, which restricts the exploration of the design space. Gaussian Processes (GPs) and their calibrated uncertainty estimations are widely used for this purpose. In many technical applications the design space is explored via continuous trajectories, along which the safety needs to be assessed. This is particularly challenging for strict safety requirements in GP methods, as it employs computationally expensive Monte-Carlo sampling of high quantiles. We address these challenges by providing provable safety bounds based on the adaptively sampled median of the supremum of the posterior GP. Our method significantly reduces the number of samples required for estimating high safety probabilities, resulting in faster evaluation without sacrificing accuracy and exploration speed. The effectiveness of our safe active learning approach is demonstrated through extensive simulations and validated using a real-world engine example.

cs.LG↗

arXiv2022

Functional estimation and change detection for nonstationary time series

Tests for structural breaks in time series should ideally be sensitive to breaks in the parameter of interest, while being robust to nuisance changes. Statistical analysis thus needs to allow for some form of nonstationarity under the null hypothesis of no change. In this paper, estimators for integrated parameters of locally stationary time series are constructed and a corresponding functional central limit theorem is established, enabling change-point inference for a broad class of parameters under mild assumptions. The proposed framework covers all parameters which may be expressed as nonlinear functions of moments, for example kurtosis, autocorrelation, and coefficients in a linear regression model. To perform feasible inference based on the derived limit distribution, a bootstrap variant is proposed and its consistency is established. The methodology is illustrated by means of a simulation study and by an application to high-frequency asset prices.

stat.ME↗

arXiv2022

Projection inference for high-dimensional covariance matrices with structured shrinkage targets

Analyzing large samples of high-dimensional data under dependence is a challenging statistical problem as long time series may have change points, most importantly in the mean and the marginal covariances, for which one needs valid tests. Inference for large covariance matrices is especially difficult due to noise accumulation, resulting in singular estimates and poor power of related tests. The singularity of the sample covariance matrix in high dimensions can be overcome by considering a linear combination with a regular, more structured target matrix. This approach is known as shrinkage, and the target matrix is typically of diagonal form. In this paper, we consider covariance shrinkage towards structured nonparametric estimators of the bandable or Toeplitz type, respectively, aiming at improved estimation accuracy and statistical power of tests even under nonstationarity. We derive feasible Gaussian approximation results for bilinear projections of the shrinkage estimators which are valid under nonstationarity and dependence. These approximations especially enable us to formulate a statistical test for structural breaks in the marginal covariance structure of high-dimensional time series without restrictions on the dimension, and which is robust against nonstationarity of nuisance parameters. We show via simulations that shrinkage helps to increase the power of the proposed tests. Moreover, we suggest a data-driven choice of the shrinkage weights, and assess its performance by means of a Monte Carlo study. The results indicate that the proposed shrinkage estimator is superior for non-Toeplitz covariance structures close to fractional Gaussian noise.

stat.ME↗

arXiv2022

Estimation of mixed fractional stable processes using high-frequency data

The linear fractional stable motion generalizes two prominent classes of stochastic processes, namely stable Lévy processes, and fractional Brownian motion. For this reason it may be regarded as a basic building block for continuous time models. We study a stylized model consisting of a superposition of independent linear fractional stable motions and our focus is on parameter estimation of the model. Applying an estimating equations approach, we construct estimators for the whole set of parameters and derive their asymptotic normality in a high-frequency regime. The conditions for consistency turn out to be sharp for two prominent special cases: (i) for Lévy processes, i.e. for the estimation of the successive Blumenthal-Getoor indices, and (ii) for the mixed fractional Brownian motion introduced by Cheridito. In the remaining cases, our results reveal a delicate interplay between the Hurst parameters and the indices of stability. Our asymptotic theory is based on new limit theorems for multiscale moving average processes.

math.ST↗

arXiv2022

Sequential Gaussian approximation for nonstationary time series in high dimensions

Gaussian couplings of partial sum processes are derived for the high-dimensional regime $d=o(n^{1/3})$. The coupling is derived for sums of independent random vectors and subsequently extended to nonstationary time series. Our inequalities depend explicitly on the dimension and on a measure of nonstationarity, and are thus also applicable to arrays of random vectors. To enable high-dimensional statistical inference, a feasible Gaussian approximation scheme is proposed. Applications to sequential testing and change-point detection are described.

math.PR↗

arXiv2021

An efficient jump-diffusion approximation of the Boltzmann equation

A jump-diffusion process along with a particle scheme is devised as an accurate and efficient particle solution to the Boltzmann equation. The proposed process (hereafter Gamma-Boltzmann model) is devised to match the evolution of all moments up to the heat fluxes while attaining the correct Prandtl number of 2/3 for monatomic gas with Maxwellian molecular potential. This approximation model is not subject to issues associated with the previously developed Fokker-Planck (FP) based models; such as having wrong Prandtl number, limited applicability, or requiring estimation of higher-order moments. An efficient particle solution to the proposed Gamma-Boltzmann model is devised and compared computationally to the direct simulation Monte Carlo and the cubic FP model [M. H. Gorji, M. Torrilhon, and P. Jenny, J. Fluid Mech. 680 (2011): 574-601] in several test cases including Couette flow and lid-driven cavity. The simulation results indicate that the Gamma-Boltzmann model yields a good approximation of the Boltzmann equation, provides a more accurate solution compared to the cubic FP in the limit of a low number of particles, and remains computationally feasible even in dense regimes.

physics.comp-ph↗

arXiv2021

Regularity of multifractional moving average processes with random Hurst exponent

A recently proposed alternative to multifractional Brownian motion (mBm) with random Hurst exponent is studied, which we refer to as Itô-mBm. It is shown that Itô-mBm is locally self-similar. In contrast to mBm, its pathwise regularity is almost unaffected by the roughness of the functional Hurst parameter. The pathwise properties are established via a new polynomial moment condition similar to the Kolmogorov-Chentsov theorem, allowing for random local Hölder exponents. Our results are applicable to a broad class of moving average processes where pathwise regularity and long memory properties may be decoupled, e.g. to a multifractional generalization of the Matérn process.

math.PR↗

arXiv2021

Confidence bands for exponential distribution functions under progressive type-II censoring

Based on a progressively type-II censored sample from the exponential distribution with unknown location and scale parameter, confidence bands are proposed for the underlying distribution function by using confidence regions for the parameters and Kolmogorov-Smirnov type statistics. Simple explicit representations for the boundaries and for the coverage probabilities of the confidence bands are analytically derived, and the performance of the bands is compared in terms of band width and area by means of a data example. As a by-product, a novel confidence region for the location-scale parameter is obtained. Extensions of the results to related models for ordered data, such as sequential order statistics, as well as to other underlying location-scale families of distributions are discussed.

math.ST↗

arXiv2020

Estimation of state-dependent jump activity and drift for Markovian semimartingales

The jump behavior of an infinitely active Itô semimartingale can be conveniently characterized by a jump activity index of Blumenthal-Getoor type, typically assumed to be constant in time. We study Markovian semimartingales with a non-constant, state-dependent jump activity index and a non-vanishing continuous diffusion component. A nonparametric estimator for the functional jump activity index is proposed and shown to be asymptotically normal under combined high-frequency and long-time-span asymptotics. Furthermore, we propose a nonparametric drift estimator which is robust to symmetric jumps of infinite variance and infinite variation, and which attains the same asymptotic variance as for a continuous diffusion process. Simulations demonstrate the finite sample behavior of our proposed estimators. The mathematical results are based on a novel uniform bound on the Markov generator of the jump diffusion.

math.ST↗