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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., Comment: AISTATS 2024

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., Comment: This is an Accepted Manuscript of an article published by Taylor & Francis in the Journal of the American Statistical Association on September 27, 2021

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.

The linear fractional stable motion generalizes two prominent classes of stochastic processes, namely stable L\'evy 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 L\'evy 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.

Consider the sum $Y=B+B(H)$ of a Brownian motion $B$ and an independent fractional Brownian motion $B(H)$ with Hurst parameter $H\in(0,1)$. Surprisingly, even though $B(H)$ is not a semimartingale, Cheridito proved in [Bernoulli 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 a second surprise 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 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. The estimation of mixed semimartingales with correlation is motivated by applications to high-frequency financial data contaminated by rough noise.

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.

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.

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., Comment: AOM. This article has been accepted for publication in Journal of Statistical Computation and Simulation, published by Taylor & Francis

A recently proposed alternative to multifractional Brownian motion (mBm) with random Hurst exponent is studied, which we refer to as It\^o-mBm. It is shown that It\^o-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 H\"older 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 Mat\'ern process., Comment: accepted manuscript

As a specific proportional hazard rates model, sequential order statistics can be used to describe the lifetimes of load-sharing systems. Inference for these systems needs to account for small sample sizes, which are prevalent in reliability applications. By exploiting the probabilistic structure of sequential order statistics, we derive exact finite sample inference procedures to test for the load-sharing parameters and for the nonparametrically specified baseline distribution, treating the respective other part as a nuisance quantity. This improves upon previous approaches for the model, which either assume a fully parametric specification or rely on asymptotic results. Simulations show that the tests derived are able to detect deviations from the null hypothesis at small sample sizes. Critical values for a prominent case are tabulated., Comment: To appear in: IEEE Transactions on Reliability

The Blumenthal-Getoor (BG) index characterizes the jump measure of an infinitely active L\'evy process. It determines sample path properties and affects the behavior of various econometric procedures. If the process contains a diffusion term, existing estimators of the BG index based on high-frequency observations only achieve rates of convergence which are suboptimal by a polynomial factor. In this paper, a novel estimator for the BG index and the successive BG indices is presented, attaining the optimal rate of convergence. If an additional proportionality factor needs to be inferred, the proposed estimator is rate-optimal up to logarithmic factors. Furthermore, our method yields a new efficient volatility estimator which accounts for jumps of infinite variation. All parameters are estimated jointly by the generalized method of moments. A simulation study compares the finite sample behavior of the proposed estimators with competing methods from the financial econometrics literature.

The jump behavior of an infinitely active It\^o 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.

Real world networks are often subject to severe uncertainties which need to be addressed by any reliable prescriptive model. In the context of the maximum flow problem subject to arc failure, robust models have gained particular attention. For a path-based model, the resulting optimization problem is assumed to be difficult in the literature, yet the complexity status is widely unknown. We present a computational approach to solve the robust flow problem to optimality by simultaneous primal and dual separation, the practical efficacy of which is shown by a computational study. Furthermore, we introduce a novel model of robust flows which provides a compromise between stochastic and robust optimization by assigning probabilities to groups of scenarios. The new model can be solved by the same computational techniques as the robust model. A bound on the generalization error is proven for the case that the probabilities are determined empirically. The suggested model as well as the computational approach extend to linear optimization problems more general than robust flows.