Conference Agenda

Ranking problems are prevalent in modern statistical, machine learning, and computer science literature. This includes a variety of practical situations ranging from ranking experts/workers in crowd-sourced data, ranking players in a tournament or equivalently sorting objects based on pairwise comparisons. A main challenge in this field is to construct an estimator of the rank of the experts, based on incomplete and noisy data.
In this talk, we focus on understanding the problem of ranking both from an informational − namely, characterizing the fundamental statistical thresholds for optimal estimation − and a computational − namely, also characterising the fundamental limits of computationally efficient estimation − perspective. A core question for these problems is on whether statistical optimality is compatible with computational efficiency.
To do that, we first consider the simpler sub-problem of sub-matrix detection and estimation, which is useful to apprehend the more complex problem of ranking − and we will particularly focus on computational lower bounds. Based on results for this problem, we explain how they can be used to solve the more challenging problem of ranking.

Recent interest has emerged in community detection for dynamic networks which are observed along a trajectory of points in time. In this talk, we present a time-varying degree-corrected stochastic block model to fit a dynamic network which allows evolving heterogeneity in the degrees of nodes within a community over time. Considering the influence of the varying time window on the aggregation of network information from different time points, in the parameter estimation, we propose a smoothing-based method to recover time-varying degree parameters and communities. In particular we provide rates of consistency of our smoothed estimators for degree parameters and communities using a time-localised profile-likelihood approach. We illustrate our method by some comparative simulation studies and an application to a real data set.

Dynamic networks provide a powerful framework for characterizing time-varying functional connectivity in neuroimaging studies. In practice, such networks are typically collected from multiple subjects across time and exhibit both temporal dynamics and subject-specific heterogeneity. Brain functional connectivity networks also contain hub nodes, defined as highly connected regions that play critical roles in understanding brain functional connectivity. In this talk, we propose a mixed-effect dynamic stochastic block model with degree heterogeneity, which simultaneously disentangles the population connectivity structure from individual variability and recovers the trajectories of hub regions through time-varying degree parameters. We develop an efficient local approximate estimation procedure and evaluate its performance through extensive simulations and a case study of dynamic functional connectivity from the Human Connectome Project.

Statistical models that involve latent Markovian state processes have become immensely popular tools for analysing time series and other sequential data. However, the plethora of model formulations, the inconsistent use of terminology, and the various inferential approaches and software packages can be overwhelming to practitioners, especially when they are new to this area. Here we aim to provide guidance for both statisticians and practitioners working with latent Markov models by offering a unifying view on what otherwise are often considered separate model classes, from hidden Markov models over state-space models to Markov-modulated Poisson processes. In particular, we provide a roadmap for identifying a suitable latent Markov model formulation given the data to be analysed. Furthermore, we emphasise that it is key to applied work with any of these model classes to understand how recursive techniques exploiting the models' dependence structure can be used for inference. The R package LaMa adapts this unified view and provides an easy-to-use framework for fast numerical maximum likelihood estimation, allowing users to flexibly tailor a latent Markov model to their data using a Lego-type approach. Real-data examples from ecology, medicine and finance will be used to illustrate the modelling workflow.

We propose a robust multi-fidelity Gaussian process for integrating sparse, high-quality reference monitors with dense but noisy citizen-science sensors. The approach replaces the Gaussian log-likelihood in the high-fidelity channel with a global Huber loss applied to precision-weighted residuals, yielding bounded influence on all parameters, including the cross-fidelity coupling, while retaining the flexibility of co-kriging. We establish attenuation and unbounded influence of the Gaussian maximum likelihood estimator under low-fidelity contamination and derive explicit finite bounds for the proposed estimator that clarify how whitening and mean-shift sensitivity determine robustness. Monte Carlo experiments with controlled contamination show that the robust estimator maintains stable MAE and RMSE as anomaly magnitude and frequency increase, whereas the Gaussian MLE deteriorates rapidly. In an empirical study of PM2.5 concentrations in Hamburg, combining UBA monitors with openSenseMap data, the method consistently improves cross-validated predictive accuracy and yields coherent uncertainty maps without relying on auxiliary covariates. The framework remains computationally scalable through diagonal or low-rank whitening and is fully reproducible with publicly available code.

This paper presents a novel copula-based autoregressive framework for multi-layer arrays of integer-valued time series with tensor structure. Our framework generalizes recent advances in tensor time series models for real-valued data to a context that accounts for the unique properties of integer-valued data, such as discreteness and non-negativity. The model incorporates feedback effects for the counts’ temporal dynamics and introduces identification constraints. An asymptotic theory is developed for a Two-Stage Maximum Likelihood Estimator (2SMLE) for the model’s parameters. The estimator balances the challenges of parameter dimensionality, interdependence of the different count series, and computational stability. Together, this substantially pushes the frontier for modeling multi-dimensional, structured tensor time series of counts. An application to tensor crime counts demonstrates the practical usefulness of the proposed methodology.

We consider the setting where the state dynamics at each node in a network depend on interactions with its neighbors. We model this using the general framework of Network Stochastic Differential Equations (N-SDEs). The evolution at each node arises from three components: intrinsic dynamics (a momentum term), feedback from adjacent nodes (a network term), and a stochastic volatility component driven by Brownian motion. Our goals are twofold: (i) parameter estimation for N-SDE systems and (ii) recovery of the underlying graph.
The main motivation is to handle very high-dimensional time series by exploiting sparsity in the network structure. We study two settings. i) Known network structure: the graph is given, and we provide identifiability conditions for the parameters, accounting for the fact that the parameter dimension grows with the number of edges. ii) Unknown network structure: the graph must be learned from data; for this case, we propose an iterative procedure based on adaptive Lasso, developed for a particular class of N-SDE models.
We focus on oriented graphs, which supports applications to causal inference by allowing the investigation of directed cause–effect relationships in dynamical systems. Using simulations and real data, we illustrate the performance of the proposed estimators across several graph topologies in high-dimensional regimes. We establish non-asymptotic bounds for parametric estimation when the system dimension is large, in two observation schemes: (1) high-frequency data from an ergodic diffusion, and (2) continuous observation in a small-diffusion, not necessarily ergodic, setting.

Based on joint works with S.M. Iacus and N. Yoshida.

Exact testing of model assumptions is often of limited relevance, especially in high-dimensional settings. Structural assumptions on large-dimensional covariance matrices, such as sphericity, are rarely expected to hold exactly for real data, and practitioners are often primarily interested in whether such model assumptions are approximately satisfied. In this work, we propose a test for approximate sphericity of high-dimensional covariance matrices, where the tolerated level of deviation from sphericity can be chosen by the user. Our test statistic is based on estimators of the largest and smallest eigenvalues of the population covariance matrix in a high-dimensional regime, where the corresponding sample eigenvalues are not consistent. We derive theoretical guarantees showing that the test keeps the prescribed asymptotic level under the null hypothesis and is power consistent under the alternative. Our key theoretical contribution is a joint central limit theorem for the estimators of the extreme eigenvalues of the population covariance matrix, provided the corresponding eigenvalues exceed the critical phase transition threshold.

Functional principal component analysis (FPCA) is a fundamental tool for exploring variation in samples of random curves or surfaces. We propose a new approach to FPCA for functional data observed irregularly and sparsely over their domains, based on smoothing directly at the level of the eigenfunctions. Our formulation leads to an efficient optimization-based procedure whose computational and storage costs are comparable to those of standard multivariate PCA for regularly observed data. The method is flexible with respect to domain geometry and model class, accommodates structural constraints and penalties, and facilitates uncertainty quantification via resampling and asymptotic theory.

Depth plays a central role in modern deep learning, yet its probabilistic effects are subtle and are not fully captured by classical theories that primarily focus on the infinite-width limit. This talk explores how jointly scaling depth and width shapes the signal-propagation statistics of wide neural networks under two contrasting regimes: fully connected feedforward networks with independent weights across layers, and recurrent networks with shared weights. In feedforward networks, standard infinite-width analyses allow to stabilize forward and backward variance, ensuring well-behaved initialization. However, finite-width fluctuations accumulate with depth, breaking convergence to the Neural Tangent Kernel (NTK) regime. In contrast, in linear recurrent networks, finite-width effects already destabilize the forward-propagation variance, rendering conventional initialization schemes inadequate for long input sequences. Together, these results show that depth affects feedforward and recurrent architectures in qualitatively distinct ways that cannot be captured by infinite-width approximations.

Score-based generative modeling, implemented through probability flow ODEs, has shown impressive results in numerous practical settings. However, most convergence guarantees rely on restrictive regularity assumptions on the target distribution—such as strong log-concavity or bounded support. This work establishes non-asymptotic convergence bounds in the 2-Wasserstein distance for a general class of probability flow ODEs under considerably weaker assumptions: weak log-concavity and Lipschitz continuity of the score function. Our framework accommodates non-log-concave distributions, such as Gaussian mixtures, and explicitly accounts for initialization errors, score approximation errors, and effects of discretization via an exponential integrator scheme. Bridging a key theoretical challenge in diffusion-based generative modeling, our results extend convergence theory to more realistic data distributions and practical ODE solvers. We provide concrete guarantees for the efficiency and correctness of the sampling algorithm, complementing the empirical success of diffusion models with rigorous theory. Moreover, from a practical perspective, our explicit rates might be helpful in choosing hyperparameters, such as the step size in the discretization.

We introduce Random Quadratic Form (RQF): a stochastic differential equation which formally corresponds to the gradient flow of a random quadratic functional on a sphere. While one-point motion of the system is a Brownian motion on a sphere and thus has no preferred direction, the two-point motion exhibits nontrivial synchronizing behaviour. In this work we study synchronization of the RQF, namely we give both distributional and path-wise characterizations of the solutions by studying invariant measures and random attractors of the system.
The RQF model is motivated by the study of the role of linear layers in transformers and illustrates the synchronization by common noise phenomena arising in the simplified models of transformers. In particular, we provide an alternative (independent of self-attention) explanation of the clustering behaviour in deep transformers and show that tokens cluster even in the absence of the self-attention mechanism.

Distribution regression seeks to estimate the conditional distribution of a multivariate response given a continuous covariate. This approach offers a more complete characterization of dependence than traditional regression methods. Classical nonparametric techniques often assume that the conditional distribution has a well-defined density, an assumption that fails in many real-world settings. These include cases where data contain discrete elements or lie on complex low-dimensional structures within high-dimensional spaces. In this work, we establish minimax convergence rates for distribution regression under nonparametric assumptions, focusing on scenarios where both covariates and responses lie on low-dimensional manifolds. We derive lower bounds that capture the inherent difficulty of the problem and propose a new hybrid estimator that combines adversarial learning with simultaneous least squares to attain matching upper bounds. Our results reveal how the smoothness of the conditional distribution and the geometry of the underlying manifolds together determine the estimation accuracy.

Assume one wants to estimate the true parameter $\vartheta_0$, which is the {\it maximal} value {\it minimizing} a function $M(\vartheta)$ over $\vartheta$. Let $M_n(\vartheta)$ be a consistent estimator for $M(\vartheta)$ uniformly in $\vartheta$. Although uniform convergence holds, one cannot apply the argmin theorem in the non-unique minimum case. Using the {\it maximal} value {\it minimizing} the function $M_n(\vartheta)$ over $\vartheta$ generally does not give a consistent estimator. We consider a special case with real-valued parameter, and define a new consistent estimator. This method is then applied to estimate the gradual (smooth) change point $\vartheta_0$ of a nonparametric regression model $Y=m(X)+\varepsilon$ with real-valued covariates, and a continuous regression function $m$ with maximal value $\vartheta_0$, where $m$ is zero.

Flow Matching (introduced by Lipman et. al.) and associated models have recently attracted significant interest due to their simulation-free training via a straightforward least squares criterion and the extremely broad and consequently adaptable underlying ordinary differential equation framework. Despite being a generative model that aims to mimic an unknown distribution, its possible applications extend far beyond the core task of generating new samples. The cheap generation of new samples opens the door to efficient distribution estimation, an essential component of forecasting tasks such as weather prediction. In this talk, we first adapt the Flow Matching method to smooth conditional density estimation. We show that the resulting estimator is closely related to th Nadaraya-Watson estimator. Then, we bridge the gap between proper scoring rules, the established method of evaluating predictions, and the fundamental concept of risk in statistical learning. Building on this, we show that the Nadaraya-Watson estimator achieves a minimax optimal anisotropic rate of convergence with respect to the risk associated with the Fourier score. In the end, we transfer this result to the Flow Matching estimator and demonstrate its capability in practice.

Consider data points sampled independently from the uniform distribution on a known symmetric convex body in high-dimensional Euclidean space with unknown location parameter. In this setting, the set of maximum likelihood estimators (MLE set) is a convex body containing the true location parameter. The goal of this talk is to present non-asymptotic upper and lower bounds for the diameter of the MLE set.

In this talk I will discuss recent work on two-sample testing under a local differential privacy constraint where a permutation procedure is used to calibrate the tests. While permutation testing is a classical resampling technique, popular due to its ease of implementation and uniform Type I error control, its use under local privacy constraints is complicated by the fact that access to the data is limited. In this work we design appropriate mechanisms for private data collection, both interactive and non-interactive, that allow for permutation tests. Our analysis shows that these lead to minimax optimal separation rates in both discrete and continuous settings, with interactive procedures being significantly more powerful. This is recent joint work with Alexander Kent and Yi Yu (https://arxiv.org/abs/2505.24811).

In this talk, we consider the problem of monitoring changes in the covariance matrices of a sequence of multivariate normally distributed random vectors. Therefore, we introduce a Multivariate Exponentially Weighted Moving Average (MEWMA) control chart in which, at each time step, the empirical covariance matrix is computed and vectorized. The control limit and the corresponding Average Run Length (ARL) are determined not only by Monte Carlo simulation, but also by numerically solving an integral equation for the ARL. In order to set up this integral equation, the exact transition density of the monitoring statistic is approximated by its asymptotic transition density. This approximation exploits the fact that the asymptotic transition density is invariant under rotations of the sample covariance matrix. Finally, we provide an outlook on an application of the proposed control chart to data from a bridge monitoring project.

There are indeed many EWMA control charts for various parameters available. However, there is none for monitoring the linear correlation coefficient ρ. Despite it is known for a long time, the usage of he explicit distribution of the estimator of ρ while setting up a control chart seems to be non-existent. Here, we build an EWMA chart utilizing this estimator, namely the Pearson correlation, and calculate the most popular performance measure, the zero-state average run length (ARL), by means of various numerical methods. Less surprisingly, the two standard methods work poorly for certain chart designs. We solve these problems by utilizing piece-wise collocation. Moreover, we examine further configuration details and provide some guidelines. Two applications illustrate the usefulness of monitoring the ρ level.

Biodiversity underpins life on Earth, yet it is declining at an accelerating pace, sharpening the need for interventions that can slow, halt, or reverse these losses. Designing such interventions requires clear insight into the processes driving population declines in particular species—and into the relative importance of those processes—insight most directly generated by population dynamics models. Yet appropriate population dynamics models for quantifying declines and guiding conservation management of wild herbivore populations remain scarce, leaving a critical gap in both evidence and practice. To address this gap, we develop an integrated Bayesian state-space population dynamics model, using the Mara-Serengeti hartebeest population as a case study. The model extends and generalizes an earlier framework we developed and illustrated for the Mara-Serengeti topi (Mukhopadhyay et al. 2024), adding multiple features designed to improve realism, inference, and management relevance.

The model fuses ground demographic surveys with aerial monitoring data, explicitly representing population age–sex structure and key life-history traits and strategies. It links birth rates, age-specific survival rates, and sex ratios to meteorological covariates, prior population density, environmental seasonality, predation risk, and several environmental and anthropogenic covariates. Operating on a monthly time step, it enables fine-grained estimation of reproductive seasonality, phenology, synchrony, and birth prolificacy, as well as juvenile and adult recruitment dynamics. We evaluate performance using balanced bootstrap sampling and by comparing model predictions with empirical aerial estimates of population size. We perform detailed assessment of model robustness, including by checking for parameter redundancy, estimability and identifiability, performing sensitivity analysis of the priors and running multiple MCMC chains. Implemented as a hierarchical Bayesian model using MCMC methods for parameter estimation, prediction, and inference, the model reproduces several well-established features of the hartebeest population, including a steep and persistent decline, weakly seasonal births, and juvenile and adult recruitment patterns. The framework is general and flexible and easily adaptable for other species.

References Mukhopadhyay, S., Piepho, H. P., Bhattacharya, S., Dublin, H. T., & Ogutu, J. O. (2024). Hierarchical Bayesian integrated modeling of age-and sex-structured wildlife population dynamics. Journal of Agricultural, Biological and Environmental Statistics, 1-26.

Joseph O. Ogutu, Hans-Peter Piepho et al.

University of Hohenheim, Institute of Crop Science, Biostatistics Unit, Fruwirthstrasse 23, 70599 Stuttgart, Germany

The basic results concerning with the asymptotic theory of estimation and testing, Le Cam (1960) introduced so-called locally asymptotically normal (LAN) family of distributions. The convolution theorem for LAN case is obtained by Hájek (1970). The convolution result was extended by Le Cam (1972) to more general situations than that of LAN case. These results sometimes called the Hájek-Le Cam asymptotic minimax theorem. In this talk we derive the second order generalization of Hájek's convolution theorem. Furthermore, as a application of the second order Hájek's convolution theorem, we lead to the second order Hájek-Le Cam asymptotic minimax theorem. It automatically provides the conditions that the second order asymptotic efficient estimators should satisfy.

The change of the rank of the volatility function in Ito semimartingales poses a complicated signal-detection problem. In their paper from 2013 Jacod & Podolskij have derived a statistic to detect whether the rank of the volatility function is constant over the observation period. Based on their results we develop a statistic which allows us to detect local jumps in the rank which is based on random perturbation of the high-frequency observations on an Ito semimartingale. This statistic can be used to estimate the time points at which the rank jumps occur. We illustrate our results with some simulated data.

We consider the problem of estimating the density of the process associated with the small jumps of a pure jump Lévy process, possibly of infinite variation, from discrete observations of one trajectory. The interest of such a question lies on the observation that even when the Lévy measure is known, the density of the increments of the small jumps of the process cannot be computed in closed-form. We discuss results both from low and high-frequency observations. In a low frequency setting, assuming the Lévy density associated with the jumps larger than $\epsilon\in(0,1)$ in absolute value is known, a spectral estimator relying on the convolution structure of the problem achieves a parametric rate of convergence with respect to the integrated $L_2$ loss, up to a logarithmic factor. In a high-frequency setting, we remove the assumption on the knowledge of the Lévy measure of the large jumps and show that the rate of convergence depends both on the sampling scheme and on the behavior of the Lévy measure in a neighborhood of zero. We show that the rate we find is minimax up to a logarithmic factor. An adaptive penalized procedure is studied to select the cutoff parameter. These results are extended to encompass the case where a Brownian component is present in the Lévy process. Furthermore, we numerically illustrate the performances of our procedures.

We address the problem of estimating the drift parameter in a system of $N$ interacting particles driven by additive fractional Brownian motion of Hurst index \( H \geq 1/2 \). Considering continuous observation of the interacting particles over a fixed interval \([0, T]\), we examine the asymptotic regime as \( N \to \infty \). Our main tool is a random variable reminiscent of the least squares estimator but unobservable due to its reliance on the Skorohod integral. We demonstrate that this object is consistent and asymptotically normal by establishing a quantitative propagation of chaos for Malliavin derivatives, which holds for any \( H \in (0,1) \). Leveraging a connection between the divergence integral and the Young integral, we construct computable estimators of the drift parameter. These estimators are shown to be consistent and asymptotically Gaussian. Finally, a numerical study highlights the strong performance of the proposed estimators.

We introduce a new class of generative diffusion models that, unlike conventional denoising diffusion models, achieve a time-homogeneous structure for both the noising and denoising processes, allowing the number of steps to adaptively adjust based on the noise level. This is accomplished by conditioning the forward process using Doob’s h-transform, which terminates the process at a suitable sampling distribution at a random time. The model is particularly well suited for generating data with lower intrinsic dimensions, as the termination criterion simplifies to a first hitting rule. A key feature of the model is its adaptability to the target data, enabling a variety of downstream tasks using a pre-trained unconditional generative model. We highlight this point by demonstrating how our generative model may be used as an unsupervised learning algorithm: in high dimensions the model outputs with high probability the metric projection of a noisy observation $y$ of some latent data point $x$ onto the lower-dimensional support of the data – which we don't assume to be analytically accessible but to be only represented by the unlabeled training data set of the generative model.

In this talk the focus is on copula-based procedures for testing whether a finite collection of continuous random vectors is mutually independent. In particular, we look into the class of meta-elliptical copulas and test the hypothesis whether the copula correlation matrix is a block diagonal matrix. The test statistic is a Phi-dependence measure of a rank-based correlation matrix estimator, whose asymptotic distribution under the null is obtained for general (Phi) functions and general elliptical generators. In case of the Gaussian copula, we also develop asymptotics when optimal transport dependence measures are used for testing the null hypothesis of independent random vectors. Some numerical studies, including comparisons with existing methods, are reported on.

Irène Gijbels, Steven De Keyser

University of Leuven (KU Leuven), Belgium.

The pair-copula Bayesian Networks (PCBN) are graphical models composed of a directed acyclic graph (DAG) that represents (conditional) independence in a joint distribution. The nodes of the DAG are associated with marginal densities, and arcs are assigned with bivariate (conditional) copulas following a prescribed collection of parental orders. The choice of marginal densities and copulas is unconstrained. However, the simulation and inference of a PCBN model may necessitate possibly high-dimensional integration.
We present the full characterization of DAGs that do not require any integration for density evaluation or simulations. Furthermore, we propose an algorithm that can find all possible parental orders that do not lead to (expensive) integration. Finally, we show the asymptotic normality of estimators of PCBN models using stepwise estimating equations. Such estimators can be computed effectively if the PCBN does not require integration. A simulation study shows the good finite-sample properties of our estimators.

Missing values in multivariate dependent data are common in many applied settings and pose challenges for standard imputation methods, particularly when complex dependence structures are present. We introduce NPCoImp, a nonparametric copula-based approach for imputing multivariate missing data. The method relies on the empirical beta copula to estimate conditional distribution functions of missing variables given the observed ones, allowing the imputation process to account for the radial symmetry or asymmetry of the joint dependence structure. NPCoImp is highly flexible and can accommodate arbitrary missingness patterns in multivariate settings. We assess its performance through an extensive Monte Carlo simulation study, comparing it with classical imputation methods, the CoImp algorithm, and the machine-learning-based missForest approach. The results show that NPCoImp performs particularly well in preserving dependence structures across different sample sizes, missingness levels, and dependence strengths. The practical relevance of the method is illustrated through applications to real data from the agricultural sector.

We introduce a new dependence order, termed the conditional convex order, whose minimal and maximal elements characterize independence and perfect dependence. Moreover, it characterizes conditional independence, satisfies information monotonicity, and exhibits several invariance properties. Consequently, it is an ordering for the strength of functional dependence of a random variable Y on a random vector X. As we show, various recently studied dependence measures---including Chatterjee's rank correlation, Wasserstein correlations, and rearranged dependence measures---are increasing in this order and inherit their fundamental properties from it. We characterize the conditional convex order by the Schur order and by the concordance order, and we verify it in settings such as additive error models, the multivariate normal distribution, and various copula-based models. Our results offer a unified perspective on the behavior of dependence measures across statistical models.

We present test of symmetry of distribution and test of time symmetry for functional data. These test are Cramér - von Mises type tests based on empirical characteristic functionals. Specific variants of time symmetry including time symmetry of Wiener process are proposed. In general, the test statistics assume a relatively simple form if we use a Gaussian measure to construct the test. Then, we use bootstrap or permutation techniques to estimate the asymptotic critical values for the test statistics.

Event study plots are the centerpiece of Difference-in-Differences (DiD) analysis, but current plotting methods cannot provide honest causal inference when the parallel trends and/or no-anticipation assumption fails. We introduce a novel functional data approach to DiD that directly enables honest causal inference via event study plots. Our DiD estimator converges to a Gaussian process in the Banach space of continuous functions, enabling powerful simultaneous confidence bands. This theoretical contribution allows us to turn an event study plot into a rigorous, honest causal inference tool through equivalence and relevance testing: Honest reference bands can be validated using equivalence testing in the pre-treatment period, and honest causal effects can be tested using relevance testing in the post-treatment period. We demonstrate the performance of our method in simulations and two case studies.

Mercer's celebrated theorem is refined and extended for (weakly) differentiable symmetric kernels by associating not the common $L^2$-integral operator but a slightly more complex operator, that additionally takes into account information encoded in the (weak) derivatives of the kernel. The natural domain for this associated operator is the Sobolev Space $H^k(\Theta) = W^{k,2}(\Theta) \subset L^2(\Theta)$, where $\Theta \subset \R^d$ is some bounded domain and $k\in\N_0$ depends on the order of weak differentiability. The spectral decomposition of this operator then leads to a Mercer-type expansion of the kernel, which converges with respect to the $H^k$-norm and, if $k>d$, also uniformly \emph{without} requiring the kernel to be positive-definite. In case the kernel is also positive-definite and differentiable in the strong sense, a refinement of Mercer's theorem is obtained that additionally provides uniform convergence of the term-wise derivatives of the expansion to the respective derivatives of the kernel as well.\\
Finally, applied to the covariance kernel of a (weakly) differentiable stochastic process, these results provide a novel Karhunen-Loève-type representation that simultaneously approximates the process as well as its (weak) derivatives in a mean square optimal sense.

We revisit the classic situation in functional data analysis in which data items such as curves are observed at discrete (possibly sparse and irregular) arguments with observation noise. We focus on the reconstruction of individual curves, especially on prediction intervals and prediction bands for them. The standard approach is to proceed in two steps: First, one estimates the mean and covariance function of curves and observation noise variance function by smoothing techniques such as penalized splines. Second, under Gaussian assumptions, one derives the conditional distribution of a curve given its noisy discrete observations and constructs prediction sets with required properties (usually employing sampling from the predictive distribution). This approach is indeed well established, commonly used and theoretically valid but practically, it surprisingly fails in its key property: prediction sets constructed this way often do not have the required coverage. The actual coverage is lower than the nominal one. This has been little reported and studied in the literature. We investigate the cause of this issue and propose a remedy.

Tensor data consisting of multivariate outcomes over the items and across the subjects with longitudinal and cross-sectional dependence are considered. A completely distribution-free and tweaking-parameter-free detection procedure for changepoints at different locations is designed, which does not require training data. A CUSUM-type test statistic is employed, and its asymptotic properties are derived for a large number of available individual profiles. The considered test is shown to be consistent. The aim is to propose eigenbootstrap superstructure that overcomes the computational curse of dimensionality without any loss of information, while it preserves all the dependencies within and between the panels. The validity of this new and fast resampling algorithm is proved in this general setting. The empirical properties of the detection technique are investigated through a simulation study. The fully data-driven test is applied to real-world data from EEG and psychometrics.

Risk reserving is a fundamental task in non-life insurance and is performed on a regular basis. It is typically carried out using parametric estimation and prediction methods applied to aggregated data structured in so-called run-off triangles. In this talk, we present nonparametric, functional-based reserving alternatives that rely on the completion of MNAR functional segments in the underlying run-off triangles.

In addition to the theoretical and methodological framework, we focus on algorithmic details implemented in the recent R package ProfileLadder. The package offers a flexible and computationally efficient tools for pointwise and distributional reserve prediction and includes relevant visualization and diagnostic tools implemented via standard S3 methods. These nonparametric approaches provide modern, transparent, and extensible alternatives to classical reserving methods used by researchers, actuarial scientists, or insurance practitioners.

We identify shocks in a structural MGARCH model of asset returns using news-based proxy instruments. Structural parameters, including an orthogonal matrix, are estimated via Riemannian optimization. We study daily returns on the S&P500, the 10-year Treasury yield, and the USD index. The proxies identify an equity valuation shock, capturing shifts in expected dividend growth and risk premia, and a bond valuation shock, reflecting fundamental shocks in safe-haven asset pricing. The dynamic impact matrix is asymmetric, and sign changes in the bond valuation shock loading drive switches between negative and positive stock–bond co-movement. A decomposition of the COVID-19 episode shows that bond valuation shocks partially offset equity market stress and explain the temporary yield surge in mid-March 2020.

We propose a new estimator of low-frequency skewness that exploits high-frequency data through a direct functional mapping consisting of layers of convolutional neural networks followed by layers of MLPs. We show that the relevant high-frequency features converge to a continuous limit and that the latent skewness admits a continuous functional representation. This allows us to establish the unbiasedness of our NN estimator using classical universal approximation results and Rademacher complexity arguments. Monte Carlo experiments under stochastic volatility models, with and without jumps, show that the estimator reduces finite-sample bias relative to existing realized-skewness estimators and remains accurate under model misspecification. Empirically, our estimator exhibits temporal stability and delivers superior cross-sectional pricing performance in skewness-sorted portfolios. Another application finds no evidence that ESG-oriented firms exhibit lower crash risk. Overall, the results demonstrate how learning-based functionals can improve the estimation of nonlinear distributional characteristics from high-frequency data.

Based on discrete observations within the nonparametric Gaussian white noise model $dY_t = sigma(t)dW_t$, we develop a test to infer if the volatility function $sigma(cdot)$ is constant. In particular, at prescribed significance, we simultaneously identify those time intervals where a violation of the constancy hypothesis occurs without a priori knowledge of their number and size. The testing procedure is shown to be minimax-optimal and adaptive for infill asymptotics and these results entail that a deviation from the null hypothesis of constancy is best measured in terms of $sup_{tin [0,1]}|sigma(t)^2 /|sigma|_{L^2}^2 - 1|$. The derivation of the optimal constants requires to build hypotheses with height solving $F_n(x)=0$ for given functions $F_n$ and to understand the asymptotic behavior of their solution, which is done using the implicit function theorem.

We study the Langevin stochastic differential equation and its discrete approximations: the Euler–Maruyama scheme, commonly referred to as the Unadjusted Langevin Algorithm (ULA), and direct sampling from the continuous-time process. We show that the ULA process is geometrically ergodic in $\mathbb{R}^d$ under suitable conditions and derive a corresponding drift condition using a Foster–Lyapunov test function. We then analyze time-inhomogeneous approximations with diminishing step sizes and establish geometric recurrence for both chains—the ULA and the directly sampled chain.

Statistical analysis of stochastic processes increasingly relies on functional limit theorems for path-dependent estimators, particularly in the presence of jumps. Many estimators in econometrics and time series analysis, such as statistics used for cointegration testing, self-normalized inference, or high-frequency volatility estimation, can be expressed as functionals of stochastic integrals with random, data-dependent integrands, or as continuous-time limits thereof. Their asymptotic validity therefore hinges on weak convergence results that remain stable beyond the classical continuous-path regime. In particular, Skorokhod’s M1 topology becomes increasingly relevant, since it captures convergence in situations where large discontinuities are approximated by clusters of smaller jumps, a behavior that is typically not captured in the classical framework of the J1. Such phenomena arise naturally in econometrics and high-frequency data settings.

This talk develops a weak limit theory for stochastic integrals on the space of càdlàg paths under Skorokhod’s M1 topology. I present a new, self-contained approach based on good decompositions of semimartingale integrators, yielding tractable conditions under which Itô integration is continuous jointly in the integrator and integrand. The results unify classical J1 continuity theorems and provide new conclusions in M1. I also show that for families of local martingales, M1-tightness implies J1-tightness under a mild localised uniform integrability condition. I conclude with a discussion of applications, including anomalous diffusion models represented as stochastic integrals with respect to continuous-time random walks.

We are interested in the spectral density of a centered stationary Gaussian time series under local differential privacy constraints. Specifically, we propose new interactive privacy mechanisms for three tasks: recovering a single covariance coefficient, recovering the spectral density at a fixed frequency, and globally. Our approach achieves faster rates through a two-stage process: we apply first the Laplace mechanism to the truncated value and then use the former privatized sample to gain knowledge on the dependence mechanism in the time series. For spectral densities belonging to Hölder and Sobolev smoothness classes, we demonstrate that our algorithms improve upon the non-interactive mechanism of Kroll (2024) for small privacy parameter α, since the pointwise rates depend on nα² instead of nα⁴. Moreover, we show that the rate 1/(nα⁴) is optimal for estimating a covariance coefficient with non-interactive mechanisms. However, the L2 rate of our interactive estimator is slower than the pointwise rate. We show how to use these procedures to provide a bona-fide, locally differentially private estimator of the full covariance matrix.

Estimating the periodicity of a stationary time series via fitting a second order stationary autoregressive (AR(2)) model has been initiated by the seminal paper of Yule(1927). We investigate properties of this procedure when applied to general stationary processes possessing a spectral density with a dominant peak at some frequency λ₀ in (0,π).
We show that if the peak of the spectral density is sharp enough (in a way to be specified) then the AR(2) model, which best (in mean square sense) approximates the underlying process, correctly identifies the frequency λ₀.
To investigate consistency properties of the AR(2) based estimator of λ₀, a near to pole framework is adopted. Triangular arrays of stationary stochastic processes are considered that possess a spectral density the peak of which at λ₀ becomes more pronounced as the sample size n of the observed time series increases to infinity. It is shown in this set up, that the AR(2) based estimator achieves a rate of convergence which is larger than the parametric n^-1/2-rate and which can be arbitrarily close to n^-2/3, the best rate that can be achieved by this estimator.

Graphical models are ubiquitous for summarizing conditional relations in multivariate data. In many applications involving multivariate time series, it is of interest to learn an interaction graph that treats each individual time series as nodes of the graph, with the presence of an edge between two nodes signifying conditional dependence given the others. Typically, the partial covariance is used as a measure of conditional dependence. However, in many applications, the outcomes may not be Gaussian and/or could be a mixture of different outcomes. For such time series using the partial covariance as a measure of conditional dependence may be restrictive. In this article, we propose a broad class of time series models which are specifically designed to succinctly encode process-wide conditional independence in its parameters. For each univariate component in the time series, we model its conditional distribution with a distribution from the exponential family. We develop a notion of process-wide compatibility under which such conditional specifications can be stitched together to form a well-defined strictly stationary multivariate time series. We call this construction a conditionally exponential stationary graphical model (CEStGM). A central quantity underlying CEStGM is a positive kernel which we call the interaction kernel. Spectral properties of such positive kernel operators constitute a core technical foundation of this work. We establish process-wide local and global Markov properties of CEStGM exploiting a Hammersley-Clifford type decomposition of the interaction kernel. Further, we study various probabilistic properties of CEStGM and show that it is geometrically mixing. An approximate Gibbs sampler is also developed to simulate sample paths of CEStGM.

We suggest a dependence coefficient between a categorical variable and some general variable taking values in a metric space. In particular, this framework includes functional data.

We derive important theoretical properties and study the large sample behaviour of our suggested estimator. Moreover, we develop an independence test and prove that it is consistent against any violation of independence. The test is also applicable to the classical $K$-sample problem with possibly high- or infinite-dimensional distributions.

We obtain minimax-optimal convergence rates in the supremum norm,
including information-theoretic lower bounds,
for estimating the covariance kernel as well as principle component basis functions
of a stochastic process which is repeatedly observed at discrete,
synchronous design points.
We focus on the supremum norm instead of the simpler $L_2$ norm,
since it corresponds to the visualization of the estimation error and
forms the basis for the construction of uniform confidence bands.
For dense design, assuming Hölder-smooth sample paths we obtain
the $\sqrt n$-rate of convergence in the supremum norm without
additional logarithmic factors which typically occur in the results
in the literature.
Surprisingly, for the covariance kernel, in the transition from dense to
sparse design the rates do not reflect the two-dimensional nature
of the covariance kernel but correspond to those for univariate mean function
estimation.
Our estimation method can make use of higher-order smoothness of the
covariance kernel away from the diagonal, and does not require the same smoothness
on the diagonal itself. Hence, our results
cover covariance kernels of processes with rough, non-differentiable sample paths.
Moreover, the estimator does not use mean function estimation to form residuals,
and no smoothness assumptions on the mean have to be imposed.
In the dense case we also obtain central limit theorems in the supremum norm,
both for the covariance kernel and the principle component basis functions,
which can be used as the basis for the construction of uniform confidence sets.
Simulations and real-data applications illustrate the practical usefulness of the methods.

The two-sample problem for functional data is investigated for discrete, synchronous designs in each sample, in settings in which one sample is densely observed while the other is only relatively sparsely observed. This is motivated by comparing historical and more current daily temperature patterns, where more recent devices take measurements every 10 minutes, while historical measurements in the time period 1952 to 1972 are available only every hour. We use recently developed methods from transfer learning for functional data to estimate the difference of the mean functions at optimal rates in the supremum norm. Further, we derive a central limit theorem in the space of continuous functions and discuss the construction of uniform confidence bands using the multiplier bootstrap. We also show how our methods can be extended to functional time series.
In the application to daily temperature patterns we decompose the mean difference function into a daily average - the normalized integral of the mean difference function - as well as into the deviation from the average value. Using the developed inferential methodology we show that not only the daily average temperatures in each month have increased significantly, but also that the daily temperature patterns have changed for most months, with night temperature remaining relatively stable while daily temperatures increased beyond the daily average increase.