Classical statistical learning theory primarily concerns independent data. In comparison, it has been much less investigated for time dependent data, which are commonly encountered in economics, engineering, finance, geography, physics and other fields. In this talk, we focus on concentration inequalities for suprema of empirical processes which plays a fundamental role in the statistical learning theory. We derive a Gaussian approximation and an upper bound for the tail probability of the suprema under conditions on the size of the function class, the sample size, temporal dependence and the moment conditions of the underlying time series. Due to the dependence and heavy-tailness, our tail probability bound is substantially different from those classical exponential bounds obtained under the independence assumption in that it involves an extra polynomial decaying term. We allow both short- and long-range dependent processes, where the long-range dependence case has never been previously explored. We showed our tail probability inequality is sharp up to a multiplicative constant. These bounds work as theoretical guarantees for statistical learning applications under dependence.

The interest in this talk is in statistical (in)dependence between
a finite number of random vectors. Statistical independence between
random vectors holds if and only if the true underlying copula is the product
of the marginal copulas yielding zero dependence.
We discuss some recent approaches towards developing dependence measures
that completely characterize independence, such as
phi-divergence measures, and optimal transport measures.
We discuss statistical inference properties and provide illustrative examples.
In high-dimensional settings possible marginal independencies can be taken
into account by inducing (block) sparsity.

We examine the bivariate copulas that describe the serial dependencies in popular time series models from the ARCH/GARCH class. We show how these copulas can be approximated using a combination of standard bivariate copulas and uniformity-preserving transformations known as v-transforms. The insights can help us to construct stationary d-vine models to rival and often surpass the performance of GARCH processes in modelling and forecasting volatile financial return series.

We study the large-sample properties of sparse M-estimators in the presence of pseudo-observations. Our framework covers a broad class of semi-parametric copula models, for which the marginal distributions are unknown and replaced by their empirical counterparts. It is well known that the latter modification significantly alters the limiting laws compared to usual M-estimation. We establish the consistency and the asymptotic normality of our sparse penalized M-estimator and we prove the asymptotic oracle property with pseudo-observations, possibly in the case when the number of parameters is diverging. Our framework allows to manage copula-based loss functions that are potentially unbounded. Additionally, we state the weak limit of multivariate rank statistics for an arbitrary dimension and the weak convergence of empirical copula processes indexed by maps.
We apply our inference method to Canonical Maximum Likelihood losses with Gaussian copulas, mixtures of copulas or conditional copulas. The theoretical results are illustrated by two numerical experiments.

In this paper, we investigate the problem of detecting a change-point in a multiple time-series for both fixed and high-dimensions. For the fixed dimensional case, we detect change-points for each individual co-ordinates using a moving average technique and focus on testing synchronization of these change-points. The identification of synchronized change-points can often lead to finding an unanimous reason behind such changes. We provide an application of our study in speedy recovery of power grid system. Testing for Synchronization in High Dimension is also discussed.

We study the biomechanical joint angles from the hip, knee and ankle for runners who are experiencing fatigue. The data is cyclic in nature and densely collected by body worn sensors, which makes it ideal to work with in the functional data analysis (FDA) framework.We develop a new method for multiple change point detection for functional data, which improves the state of the art with respect to at least two novel aspects. First, the curves are compared with respect to their maximum absolute deviation, which leads to a better interpretation of local changes in the functional data compared to classical $L^2$-approaches. Secondly, as slight aberrations are to be often expected in a human movement data, our method will not detect arbitrarily small changes but hunts for relevant changes, where maximum absolute deviation between the curves exceeds a specified threshold, say $Delta >0$.

We recover multiple changes in a long functional time series of biomechanical knee angle data, which are larger than the desired threshold $Delta$, allowing us to identify changes purely due to fatigue. In this work, we analyse data from both controlled indoor as well as from an uncontrolled outdoor (marathon) setting.

We consider maximum deviations of sample autocovariances and autocorrelations from their theoretical counterparts over a number of lags that increases with the number of observations. The asymptotic distribution of such statistics e.g. for strictly stationary time series is of Gumbel type. However speed of convergence to the Gumbel distribution is rather slow. The well-known autoregressive (AR) sieve bootstrap is asymptotically valid for such maximum deviations but suffers from the same slow convergence rate. Braumann et al. 2021 showed that for linear time series the AR sieve bootstrap speed of convergence is of polynomial order. We use the idea of Gaussian approximation for high-dimensional time series to show that for the class of strictly stationary processes a wild-type bootstrap and a hybrid variant of the AR sieve bootstrap are asymptotically valid for our statistic of interest at a polynomial convergence rate. We close with results from a small simulation study that investigates finite sample properties of mentioned bootstrap proposals.

We present a hypothesis test to guide the search for the location of the parameter of the linear regression model in the high-dimensional setting. This parameter value may be unknown but often (part of) it is known in a different but similar context. Such external information can serve as a textit{signpost} in the vast parameter space. Our statistical hypothesis test evaluates the signpost's direction of the true parameter's location. Our test statistic measures the relevance of the signpost's direction. We derive the test statistic's limiting distribution and provide approximations to other cases. The signpost's significance is assessed by comparing the signpost's direction to that of randomly rotations of this direction. We present an Bayesian interpretation of the signpost test and its connection to the global test. In simulation we investigate the signpost test's type I error and power, with particular interest in the effect of regularization and high-dimensionality in finite samples on these properties, and under misspecification of the alternative hypothesis. We close with an application of the signpost test to a breast cancer study, which shows that the regression parameter estimate of a more prevalent subtype is informative for learning the same parameter in a less prevalent one.

Vine copulas are a flexible class of multivariate distributions beyond normality and allow for asymmetrical dependence structures in data. Their conditional distributions, important for prediction tasks, provide a flexible data representation. However, as the number of features in data increases, the computational complexity of estimating model parameters increases, and an overfitting problem arises. Hence, we propose methods to select relevant, irrelevant, and redundant variables for estimating conditional distributions of vine copulas in the presence of mixed continuous-ordinal features. We provide some empirical results to compare vine copula and machine learning models for prediction.

In this paper, we derive high-dimensional asymptotic properties of the Moore-Penrose inverse and the ridge-type inverse of the sample covariance matrix. In particular, the analytical expressions of the weighted sample trace moments are deduced for both generalized inverse matrices and are present by using the partial exponential Bell polynomials which can easily be computed in practice. The existent results are extended in several directions: (i) First, the population covariance matrix is not assumed to be a multiplier of the identity matrix; (ii) Second, the assumption of normality is not used in the derivation; (iii) Third, the asymptotic results are derived under the high-dimensional asymptotic regime. Our findings are used in constructing improved shrinkage estimators of the precision matrix, which asymptotically minimize the quadratic loss with probability one. Also, shrinkage estimators for the weights of the global minimum variance portfolio are obtained. Finally, the finite sample properties of the derived theoretical results are investigated via an extensive simulation study.

Nonparametric estimation of the conditional distributions of a real response Y, given a real covariate X, is possible if one imposes that these conditional distributions are non-decreasing with respect to the usual stochastic order or the stronger likelihood ratio order. We discuss briefly these both paradigms. Then we present various results connecting the likelihood ratio ordering of conditional distributions with total positivity of order 2 (TP2) of bivariate distributions. These considerations lead to a maximum empirical likelihood estimator. We also discuss some conjectures about projections of arbitrary bivariate distributions onto the space of TP2 distributions.

We consider survival data with a cure fraction, meaning that some subjects never experience the event of interest. For example, in oncology the event of interest is cancer relapse/death and the cured patients after treatment are immune to such event. It is common in this context to use a mixture cure model, consisting of two sub-models: one for the probability of being uncured (incidence) and one for the survival of the uncured subjects (latency). Various approaches, ranging from parametric to nonparametric, have been proposed to model the incidence component, with the logistic model being the standard choice. We consider a monotone single-index model for the incidence, which relaxes the parametric logistic assumption, while maintaining interpretability of the regression coefficients and avoiding the curse-of-dimensionality. A new estimation method is introduced that relies on the profile maximum likelihood principle, techniques from isotonic regression and kernel smoothing. We discuss some unique and challenging issues that arise when incorporating the monotone single-index model within the mixture cure model. The consistency of the proposed estimator is established, and its practical performance is investigated through a simulation study and an application to melanoma cancer data.

We consider a linear regression model in which the regression coefficients are estimated by minimising the empirical risk based on a convex loss function. The accuracy of the estimator depends on the choice of loss function; for instance, when the errors are non-Gaussian, ordinary least squares can be outperformed by estimators based on alternative loss functions. A natural question then is how to select a data-driven convex loss function that leads to optimal downstream estimation of the regression coefficients. We propose a nonparametric approach that approximates the derivative of the log-density of the noise distribution by a decreasing function, and explicitly identifies the convex loss function for which the asymptotic variance of the resulting M-estimator is minimal. We show that this optimisation problem is equivalent to a version of score matching, which corresponds to a log-concave projection of the noise distribution not in the usual Kullback–Leibler sense, but instead with respect to the so-called Fisher divergence.

Consider these two problems. The first one: astronomers are interested in the study of the way stars are distributed in the universe. In this setting, globular clusters - spherical aggregations of stars held together by gravity - are a topic of particular focus. But how to determine the distribution of stars within these three-dimensional structures when given only a 2D photo of the clusters? The second one: having material that presents a globular microstructure and that cannot be scanned but only sectioned, how can we determine the distribution of particles' size inside the material when given only a limited number of 2D cross sections?

Despite their seemingly distant nature, both of these problems share a common thread: the inherent structure of a nonlinear inverse problem, originally conceptualized by Wicksell.

In this talk, we explore the properties of nonparametric Bayesian estimators based on the Dirichlet process prior for F in this problem, which is the first time the Dirichlet process is studied in an inverse setting. In particular, we illustrate contraction rates results and uncertainty quantification (Bernstein-von Mises type of results) for our methodology and we compare it to the state-of-the-art nonparametric shape-constrained estimator.

Based on Godambe's theory of estimating functions, we propose a class of cumulative sum, CUSUM, statistics to detect breaks in the dynamics of time series under weak assumptions. First, we assume a parametric form for the conditional mean, but make no specific assumption about the data-generating process (DGP) or even about the other conditional moments. The CUSUM statistics we consider depend on a sequence of weights that influence their asymptotic accuracy.
Data-driven procedures are proposed for the optimal choice of the sequence of weights, in the sense of Godambe. We also propose modified versions of the tests that allow to detect breaks in the dynamics even when the conditional mean is misspecified.
Our results are illustrated using Monte Carlo experiments and real financial data.

Training deep learning neural networks often requires massive amounts of computational ressources. We propose to sequentially monitor predictions to trigger retraining only if the predictions are no longer valid. The approach is based on projected second moments monitoring, a problem also arising in other areas such as finance. Open-end as well as closed-end monitoring is studied under mild assumptions on the training sample and the observations of the monitoring period. The results allow for high-dimensional non-stationary time series data and thus, especially, non-i.i.d. training data. Asymptotics is based on Gaussian approximations of projected partial sums allowing for an estimated projection vector. Estimation is studied both for classical non-$l0$-sparsity as well as under sparsity. For the case that the optimal projection depends on the unknown covariance matrix, hard- and soft-thresholded estimators are studied. The method is analyzed by simulations and supported by synthetic data experiments.

Time series containing a non-negligible portion of possibly dependent zeros, whereas the remaining observations are positive, are considered. They are regarded as GARCH processes consisting of non-negative values. The aim lies in the estimation of the omnibus model parameters taking into account the semi-continuous distribution. The hurdle distribution, together with dependent zeros, causes the classical GARCH estimation techniques to fail. Two different likelihood-based approaches are derived, namely the maximum likelihood estimator and a new quasi-likelihood estimator. Both estimators are proved to be strongly consistent and asymptotically normal. Predictions with bootstrap add-ons are proposed. The empirical properties are illustrated in a simulation study, which demonstrates the computational efficiency of the methods employed. The developed techniques are presented through an actuarial problem concerning sparse insurance claims.

A damage function measures quantitatively how aggregated economies respond to climate change and it has been used as a powerful tool to provide trajectories of future economic development. However, the specification of the damage function remains highly contentious. In this paper we extend the conventional damage function by introducing interactive terms between temperature and precipitation. Our new specification allows for heterogeneous responses to climate change in different climate conditions, making possible the response to temperature change dependent on precipitation levels, and vice versa. The results show that all temperature, precipitation, as well as their interaction are statistically significant factors affecting economic growth.The most sensitive economy to climate change is the combination of cold temperature with excessive precipitation, in which case, either reduced rainfall or a warming trend could benefit economic growth considerably such as in Canada and Northern Europe countries. Compared to cold climate economies, economies with moderate to warm climates are more resilient to precipitation change, which could possibly be attributed to their adaptation to climates characterizing high variability in precipitation.

This paper aims at capturing time-varying spillover effects with panel data. We consider panel models where the outcome of a unit not only depends on its characteristics (private effects) and also on the characteristics of other units (spillover effects). The private effects can be unit-specific or homogeneous. We allow the linkage structure, i.e., which units affect which, to be latent. Moreover, the structure and the spillover effects may both change at an unknown breakpoint. To estimate the breakpoint, linkage structure, spillover effects, and private effects, we solve a penalized least squares optimization and employ double machine learning procedures to improve the convergence rate and statistical inferences. We establish the super consistency of the breakpoint estimator, which allows us to make inferences on other parameters as if the breakpoint was known. We illustrate the theory via simulated and empirical data.

We propose a new unified approach to identifying and estimating spatio-temporal dependence structures in large panels. The model accommodates global cross-sectional dependence due to global dynamic factors as well as local cross-sectional dependence, which may arise from local network structures. Model selection, filtering of the dynamic factors, and estimation are carried out iteratively using a new algorithm that combines the Expectation-Maximization algorithm with proximal minimization, allowing us to efficiently maximize an l1- and l2-penalized state space likelihood function. A Monte Carlo simulation study illustrates the good performance of the algorithm in terms of determining the presence and magnitude of common factors and local spillover effects. In an empirical application, we investigate monthly US interest rate data on 12 maturities over almost 35 years. We find that there are heterogeneous local spillover effects among neighboring maturities. Taking this local dependence into account improves out-of-sample forecasting performance.

During its first two years, the SARS-CoV-2 pandemic manifested as multiple waves shaped by complex interactions between variants of concern, non-pharmaceutical interventions, and the immunological landscape of the population. Understanding how the age-specific epidemiology of SARS-CoV-2 evolved throughout the pandemic is crucial for informing policy. In this paper, we develop an inference-based modeling approach to reconstruct the
burden of true infections and hospital admissions in children, adolescents,
and adults over the seven waves of four variants (wild-type,
Alpha, Delta, Omicron BA.1) during the first two years of the pandemic,
using the Netherlands as the motivating example. We find that
reported cases are a considerable underestimate and a generally poor
predictor of true infection burden, especially because case reporting
differs by age. The contribution of children and adolescents to
total infection and hospitalization burden increased with successive
variants and was largest during the Omicron BA.1 period. However,
the ratio of hospitalizations to infections decreased with each subsequent
variant in all age categories. During the Delta and Omicron BA.1 periods, primary infections were
common in children but relatively rare in adults who experienced
either re-infections or breakthrough infections. Our approach can be
used to understand age-specific epidemiology through successive
waves in other countries where random community surveys are absent but basic surveillance and
statistics data are available

We introduce a quasi-Bayesian technique designed to relax the rigidity of moment conditions,
thereby accommodating model misspecification. We establish new Bernstein–von mises (BvM)
type theorems for the quasi-posterior distributions. We also demonstrate the practical usefulness
of this method through simulation exercises and empirical application to linear instrumental
variable (IV) models and nonlinear quantile regression models. Our results show that this approach still provides informative inference even when there is a significant degree of misspecification.

The impact of missing data on quantitative research can be serious,
leading to biased estimates of parameters, loss of information, increased
standard errors, weakened generalizability etc.
In this paper, we discuss and apply to a real-world data set a regression-
based data imputation method when data is a sequence (time series) of
multidimensional measurements

Computational models are central and data based modeling has become increasingly important over the last years. Sufficient amount of data has to be collected to build accurate data based models. Data collection can be time and labor intensive and sometimes even safety critical. Safe Active Learning is a sequential experimental design that selects highly informative measurements and learns to cope with safety considerations on the fly. This presentation will show the mathematical core of the algorithm as well as its impact on automation of data collection.

This contribution is based on previous publications.

We present a new method for combining (or aggregating or ensembling) multivariate probabilistic forecasts, considering dependencies between quantiles and marginals through a smoothing procedure that allows for online learning. We discuss two smoothing methods: dimensionality reduction using Basis matrices and penalized smoothing. The new online learning algorithm generalizes the standard CRPS learning framework into multivariate dimensions. It is based on Bernstein Online Aggregation (BOA) and yields optimal asymptotic learning properties. The procedure uses horizontal aggregation, i.e., aggregation across quantiles. We provide an in-depth discussion on possible extensions of the algorithm and several nested cases related to the existing literature on online forecast combination. We apply the proposed methodology to forecasting day-ahead electricity prices, which are 24-dimensional distributional forecasts. The proposed method yields significant improvements over uniform combination in terms of continuous ranked probability score (CRPS). We discuss the temporal evolution of the weights and hyperparameters and present the results of reduced versions of the preferred model. A fast C++ implementation of the proposed algorithm will be made available in connection with this contribution as an open-source R-Package on CRAN.

References: Berrisch, J., & Ziel, F. (2023). CRPS learning. Journal of Econometrics, 237(2), 105221.

Semi-structured regression (SSR) jointly learn interpretable structured effects of tabular data and additional unstructured effects modeled through a (deep) neural network. These two parts are embedded and trained together in one unifying network architecture. The structured part allows for an interpretation as in common statistical regression models whereas the remaining variation in the data can be explained by the more powerful (unstructured) neural network part. SSR also open up new ways of modeling multi-modal data, e.g., datasets with both tabular and image information, while preserving statistical model characteristics typically relied on in various fields such as medical research or psychology. In order to correctly interpret a SSR, an orthogonalization operation is used. There also exist various ways to perform statistical inference for SSR. By embedding statistical models in neural networks, SSR can be implemented in a generic software framework that leverages the full modularity of current deep learning platforms while also enabling many different statistical model specifications. This further allows classic statistical models themselves to be more flexible and scalable.

A method to forecast the demand and flexibility level of consumers of electricity is presented. The price-response model is defined by an optimization program whose defining parameters are represented by time series of prices, and minimum and maximum load flexibility levels. These parameters are, in turn, estimated from observational data by exploiting an approach based on duality theory. The proposed methodology is data-driven and exploits information from covariates via Kernel regression functions, such as price, and weather variables, to account the non-linearity for changes in the parameter estimates. The resulting estimation problem is a tractable mixed-integer linear program. Furthermore, we include a regularization term that is statistically adjusted by cross-validation and the estimated model is used to forecast the demand of customers and the flexibility level in a real dataset.

Empirical evidence shows that calibrating exponential Lévy models by options with different maturities leads to conflicting information. In other words, the stationarity implicitly assumed in the exponential Lévy model is not satisfied. An identifiable time-inhomogeneous Lévy model is proposed that does not assume stationarity and that can integrate option prices from different maturities and different strike prices without leading to conflicting information. In the time-inhomogeneous Lévy model, the convergence rates are derived, and confidence intervals are shown for the estimators of the volatility, the drift, the intensity and the Lévy density. Previously, confidence intervals have been constructed for time-homogeneous Lévy models in an idealized Gaussian white noise model. In the idealized Gaussian white noise model, it is assumed that the observations are Gaussian and given continuously across the strike prices. This simplifies the analysis significantly. The confidence intervals are constructed in a discrete observation setting for time-inhomogeneous Lévy models, and the only assumption on the errors is that they are sub-Gaussian. In particular, all bounded errors with arbitrary distributions are covered. Additional results on the convergence rates extend existing results from time-homogeneous to time-inhomogeneous Lévy models.

In the one-dimensional case, the self-similarity of the stochastic process {X_t, t ≥ 0}
is inferred by assessing the statistical significance of the minimal diameter of the rescaled distributions of X_t. By construction, for i.i.d. random variables, the one-dimensional minimal diameter reduces to the widely recognized Kolmogorov-Smirnov (KS) statistic. However, when independence is not strictly enforced, the distribution of the KS statistic undergoes significant alterations, contingent upon the intensity and direction of dependence. An illustrative instance is fractional Brownian motion (fBm), for which the critical values of the KS test cease to offer
reliable rejection thresholds, displaying a type I or type II error depending on whether the value of the Hurst exponent of the fBm is greater or less than 1/2. We observe that, in case of fBm, the KS statistic distributes as the absolute maximum of a fractional Brownian bridge (fBb) on the time interval [0,T] associated to the number of data of the rescaled empirical distribution. Consistently with this premise, employing the Maximum Likelihood Estimation (MLE) approach, we find that the Generalized Extreme Value (GEV) distribution captures the distributions of minimal diameters, providing more accurate critical values depending on the value of H and on the sample length.

We investigate the Local Asymptotic Property for fractional Brownian models based on discrete observations contaminated by a Gaussian noise. We consider both situations of low and high-frequency observations in a unified setup and we show that the convergence rate $n^{1/2} (nu_n Delta_n^{-H})^{-1/(2H+1)}$ is optimal for estimating the Hurst index $H$, where $nu_n$ is the noise intensity.

Simulation of bridge processes is a widely used tool for statistics for stochastic processes. Such processes arise when the original process is conditioned to be in a given state at a given time. A common tool to study bridge processes is Doob's h-transform. However, a problem with the transformation is that it relies on the, typically intractable, transition density of the process. We instead consider the technique of conditioning by guiding, which circumvents this problem by using the same transformation but with a different h-function whilst maintaining absolute continuity with respect to the true bridge process.

The talk will focus on conditioning manifold-valued semimartingales. We describe semimartingales on manifolds through the so-called "rolling without slipping" (Eels-Elworthy-Malliavin) construction: Mapping an R^d-valued semimartingale to the frame bundle of the manifold and then projectiong it back to the manifold.

In the talk I'll briefly discuss the construction of manifold-valued semimartingales and then move to the simulation of bridge processes through guiding.

The Compound Poisson Process (CPP) is a mathematical model used to describe phenomena in medicine, reliability, risk and catastrophe bonds. In this work, we developed four methods to offer the confidence intervals of the homogeneous CPP. We presented the exact method for constructing the confidence intervals of CPP by using the Renewal Reward Theorem and the properties of the parameter estimators. Using the delta method, the other two approaches of CPP's confidence intervals were estimated in both the original and logarithm scales. Furthermore, a special case of CPP with exponential distribution was discussed in detail.

We presented our four proposed methods with the numerical simulation and compared these methods to the referential method. The simulated data was synthesised by combining a Poisson process with several non-negative distributions, including discrete and continuous. The delta method in the logarithm scale achieved the highest coverage compared to the other methods. Furthermore, this method outperformed the other methods when the sample size was small. This method has been proven to be effective and was recommended to be used in practice to achieve a more precise confidence interval.

Multivariate functional data can be intrinsically multivariate like movement trajectories in 2D or complementary such as precipitation, temperature and wind speeds over time at a given weather station. We propose a multivariate functional additive mixed model (multiFAMM) and show its application to both data situations using examples from sports science (movement trajectories of snooker players) and phonetic science (acoustic signals and articulation of consonants). The approach includes linear and nonlinear covariate effects and models the dependency structure between the dimensions of the responses using multivariate functional principal component analysis. Multivariate functional random intercepts capture both the auto-correlation within a given function and cross-correlations between the multivariate functional dimensions. They also allow us to model between-function correlations as induced by, for example, repeated measurements or crossed study designs. Modelling the dependency structure between the dimensions can generate additional insight into the properties of the multivariate functional process, improves the estimation of random effects, and yields corrected confidence bands for covariate effects. Extensive simulation studies indicate that a multivariate modelling approach is more parsimonious than fitting independent univariate models to the data while maintaining or improving model fit.

Functional data, as observations of complexe processes, pose challenges when it comes to modeling spatial data in the form of curves, shapes, images, and more complex structures. This talk delves into Principal Component Analysis (PCA) for multivariate functional spatial data. We explore the synergy between spatial, and functional dimensions, unraveling hidden structures and patterns within complex functional spatial datasets.

The talk will give an overview of multivariate functional spatial data, highlighting its ubiquity in diverse fields such as environmental monitoring, geostatistics, and biomedical research. We will then delve into the theoretical underpinnings of Multivariate Functional Principal Component Analysis, emphasizing its adaptability to spatial datasets.

Practical applications of functional PCA in uncovering spatial dependencies, capturing variability, and dimensionality reduction will be illustrated. Case studies will showcase the effectiveness of the proposed methodology.

Additionally, we will touch upon challenges and considerations when applying PCA to multivariate functional spatial data, including handling large datasets, addressing computational complexities, and interpreting results in the context of real-world applications.

We discuss a system of spatial autoregressive models for stock returns, in which general dependencies, dependencies between industrial branches and country-specific dependencies are included. The model parameters are estimated with the method of moments, the estimators are consistent and asymptotically normal. It is shown that the model allows for convincing VaR-forecasting due to its sparsity. Furthermore, we present a fluctuation test for structural breaks in the model parameters and a specification test, which is based on the magnitude of the estimation residuals. (The talk is based on several publications with different coauthors.)

This work introduces the novel Space-Time Integer AutoRegressive and Moving Average (STINARMA) class of statistical models, aimed to cope with the temporal and spatial dependency of integer-valued processes. These models are inspired by key ideas of the continuous Space-Time ARMA (STARMA) and the Integer-valued ARMA (INARMA) models. To ensure the discrete nature of the process, the binomial thinning operator (BTO) replaces the multiplication of the continuous STARMA. Furthermore, the Gaussian distributed innovations are replaced by discrete random variables. The space-time information is introduced through a weight matrix embedded into the matrix-BTO. The focus of this work is on the full STINARMA(p{f1,...,fp}, q{m1,...,mq}) model, for which the univariate INARMA formulation appears as a special STINARMA case for fp=mq=0. First- and second-order moments as well as space-time autocorrelation functions are derived to characterise the process. The STINARMA(1{1},1{1}) with Poisson innovations is studied in detail and theoretical estimation results are derived via the method of moments (MM) and conditional maximum likelihood (CML). The finite-sample performance of MM is evaluated via simulation. Finally, the STINARMA(1{1},1{1}) model is applied to analyse real count data consisting of the daily number of hospital admissions, over time, in three different Portuguese locations.

We illustrate the main results of a non-stationary spatio-temporal precipitation model interpolation process of three different precipitation scenarios distributed throughout Austria for the years 1973-1092 and 2013-2022. We model mean and maximum precipitation
as well as the length of dry spells with a Gamma, blended generalized extreme value
and negative Binomial distribution. A generalized additive model accounts for influencing covariates as elevation and the coordinates of the monitoring stations which is then rewritten in a Bayesian hierarchical form. The spatial dependencies within the data are modelled through a non-stationary Matérn covariance function and the temporal ones through AR(1) dynamics. The stochastic partial differential equation (SPDE) approach is used to tackle the "big n" problem. Inference is performed through integrated nested Laplace approximation (INLA) which comes along with a user friendly R-INLA package. The model outputs are visualised and give insights into changes in precipitation patterns over two different time periods.

Consider a duration time $T$, a possibly endogenous covariate $Z$ and a vector of exogenous covariates $X$ such that $T=varphi(Z,X,U)$ is increasing in $U$ with $U sim U[0,1]$. Moreover, let $T$ be right-censored by a censoring time $C$ such that only their minimum, denoted by the follow-up time $Y=min{T,C}$, is observed. In this paper, we construct a test statistic for the hypothesis that $Z$ is exogenous w.r.t. $T$, where $T$, given $Z$ and $X$, is assumed to follow a proportional hazards model. Note that this is equivalent to testing whether $U$ is independent of $Z$. Our test makes use of an instrumental variable $W$ that is independent of $U$, since it can be shown that $Z$ is exogenous w.r.t. $T$ if and only if $V_T = F_{T mid Z,X}(T mid Z,X)$ is independent of $W$. We prove some asymptotic properties of the proposed test, provide possible bootstrap approximations for the critical value and show that we have a good finite sample performance via simulations. Lastly, we give an empirical example using The National Job Training Partnership Act (JTPA) Study.

The INAR (integer-valued autoregressive) and INGARCH (integer-valued GARCH)
classes are among the most commonly employed approaches for count time series
modelling, but have been studied in largely distinct strands of literature. In this
paper, a general model class unifying a large number of INAR and INGARCH mod-
els is introduced and its stochastic properties are studied. The starting point is
given by a novel thinning-based representation of compound-Poisson INGARCH pro-
cesses, which is subsequently generalized. Particular attention is given to a gener-
alization of the INAR(p) model which parallels the extension of the INARCH(p) to
the INGARCH(p, q) model. Models from the class have a natural interpretation
as stochastic epidemic processes. Different instances of the class, including both es-
tablished and newly introduced models, are compared in a real-data application on
infectious disease counts from German routine surveillance.

Resampling-based approaches are very helpful to construct prediction intervals when typically questionable distribution assumptions have to be avoided. Recent literature introduces different procedures to construct bootstrap prediction intervals for (continuous) autoregressive models dealing with the conditional nature of predictive inference in a time series setup. We adapt these findings to count data time series. In contrast to continuous time series, the construction of prediction intervals with a pre-determined level is not reasonable. This is because, due to the integer nature of count data, it is generally not possible to have (not even asymptotically) valid prediction intervals with desired coverage. Instead, we propose to consider pre-defined sets and to estimate the conditional probability that a future observation falls in these sets. Then, the accuracy of the predictive inference procedure and other quality criteria can be evaluated by its capability to mimic the distribution of the estimated conditional probabilities. In simulations, we consider different bootstrap-based procedures to account for the various sources of randomness and variability. Additionally, we deal with the aspect of possible model misspecification for the (point) prediction and propose a robustification of our predictive inference procedure by using non- and semiparametric estimation.

Autoregressive (AR) modeling of real-valued time series data has been used for decades, but it is still popular in practice, because they are flexible and enable an explicit estimation of the model parameters. When dealing with integer-valued time series, autoregressive modeling is not straightforward as classical AR models do not respect the integer-valued range. For count data time series, integer-valued AR (INAR) models based on binomial thinning are popular in practice, because they are of autoregressive nature and respect the integer-valued range. INAR models are still easy to estimate, but their parameter range is restricted in comparison to AR models. While the literature that investigates both model classes separately is huge, it lacks a unified joint approach for real-valued and integer-valued time series.

In this paper, we propose a joint vector-autoregressive model for real- and integer-valued time series. By construction, it respects the corresponding time series ranges and allows for an unrestricted range of the AR model parameters, which coincides with that of a classical vector-autoregressive model. We provide explicit estimators of all model parameters, derive their asymptotic properties and discuss suitable bootstrap approaches. The estimation performance is evaluated by simulations and applicability is illustrated on a real data set.

We derive mixing properties for a class of count time series satisfying a certain contraction condition. Using specific coupling techniques, we can deduce absolute regularity at a geometric rate for non-stationary Poisson-GARCH(1,1) processes with a possibly explosive trend and seasonal patterns.

Moreover, we propose a new model for nonstationary integer-valued time series which is particularly suitable for data with a strong trend. In contrast to popular Poisson-INGARCH models, but in line with classical GARCH models, we propose to pick the conditional distributions from nearly scale invariant families where the mean absolute value and the standard deviation are of the same order of magnitude. Again we provide sufficient conditions for absolute regularity of these count process with exponentially decaying coefficients. Finally, we illustrate the statistical use of our results in a statistical application.

Structural Health Monitoring (SHM) is increasingly applied in civil engineering. One of its primary purposes is detecting and assessing changes in structure conditions to reduce potential maintenance downtime. Recent advancements, especially in sensor technology, facilitate data measurements, collection, and process automation, leading to large data streams. We propose a function-on-function regression framework for modeling the sensor data and adjusting for confounder-induced variation. Our approach is particularly suited for long-term monitoring when several months or years of training data are available. It combines highly flexible yet interpretable semi-parametric modeling with functional principal component analysis and uses the corresponding out-of-sample phase-II scores for monitoring. The method proposed can also be described as a combination of an `input-output' and an `output-only' method.

Self-starting control charts as introduced by Quesenberry and Hawkins are of practical interest to monitor processes for which we do not know parameters based on historical data. We will show a case study with the Royal Dutch Navy from the national PrimaVera project to illustrate this.

Since self-starting control charts update the estimates of the unknown parameters at every observation, one might think that even if the observations are independent, the resulting control chart statistics are no longer independent, which would cause technical problems in assessing the performance of the control chart procedure. However, there are claims in the literature that surprisingly, independence of the statistics of self-starting control charts is preserved. We will indicate that there are problems with the correctness of these proofs. We will show how to adapt one of the existing proofs so that we not only get a clear, correct proof for the one-dimensional case, but also a proof for the more general case of regression control charts.

This is based on joint work with bachelor student Gijs Pennings, while the case study was performed by master student Esmée Stijns together with Wieder Tiddens and Tiedo Tinga from the Dutch Royal Navy.

The online quality monitoring of a process with low volume data is a very challenging task and the attention is most often placed in detecting when some of the underline (unknown) process parameter(s) experience a persistent shift. Self-starting methods, both in the frequentist and the Bayesian domain aim to offer a solution. Adopting the latter perspective, we propose a general closed-form Bayesian scheme, whose application in regular practice is straightforward. The testing procedure is build on a memory-based control chart that relies on the cumulative ratios of sequentially updated predictive distributions. The derivation of control chart's decision-making threshold, based on false alarm tolerance, along with closed form conjugate analysis, accompany the testing. The theoretic framework can accommodate any likelihood from the regular exponential family, while the appropriate prior setting allows the use of different sources of information, when available. An extensive simulation study evaluates the performance against competitors and examines the robustness to different prior settings and model type misspecifications, while continuous and discrete real datasets, illustrate its implementation.

Generative AI applications such as ChatGPT, GitHub Copilot, Bard, Midjourney, and others have created worldwide buzz and excitement due to their ease of use, broad utility, and perceived capabilities. This talk will introduce two projects both first attempts to understand the impact of ChatGPT on analytics.

In the first part, I will introduce ChatSQC, an innovative chatbot system that combines the power of OpenAI’s Large Language Models (LLM) with a specific knowledge base in Statistical Quality Control (SQC). Our research focuses on enhancing LLMs using specific SQC references, shedding light on how data preprocessing parameters and LLM selection impact the quality of generated responses.

In the second part, I will share ongoing work focused on defining quality metrics to evaluate Generative AI’s analytics capabilities. Currently, Generative AI systems are evaluated mainly in designing and training the LLM models that generate output in various forms depending on the user’s request. The models are not, however, universally evaluated based on the quality of the output in terms of the output’s fitness for use by the user. We therefore define user oriented quality metrics and evaluate, from a user perspective, the LLMs generated output in a variety of analytics tasks.

We introduce an approach to high-dimensional extremes. Based on concepts from high-dimensional statistics and modern convex programming, we can obtain fine-grained models within seconds on a standard laptop. We will illustrate these properties with finite-sample theories and empirical analyses.

The surge of massive image databases has spurred the development of scalable machine learning methods particularly convolutional neural network (CNNs), for filtering and processing such data. Current theoretical advancements in CNNs primarily focus on standard nonparametric denoising problems. However, in image classification datasets, the variability arises not from additive noise but from variations in object shape and other characteristics of the same object across different images. To address this problem, we consider a simple supervised classification problem for object detection in grayscale images. While from a function estimation point of view, every pixel is a variable and large images lead to high-dimensional function recovery tasks suffering from the curse of dimensionality, increasing the number of pixels in our image deformation model enhances image resolution and simplifies object classification problem easier. We introduce and theoretically analyze two procedures: one based on support alignment, demonstrating perfect classification under minimal separation conditions, and another fitting CNNs to the data showcasing a misclassification error depending on the sample size and number of pixels. Both methods are empirically validated using the MNIST handwritten digits database.

This is joint work with Johannes Schmidt-Hieber (Twente).

We investigate the statistical behavior of gradient descent iterates with dropout in the linear regression model. In particular, non-asymptotic bounds for expectations and covariance matrices of the iterates are derived. In contrast with the widely cited connection between dropout and ell_2-regularization in expectation, the results indicate a much more subtle relationship, owing to interactions between the gradient descent dynamics and the additional randomness induced by dropout.

For more details see [2306.10529] on arXiv.

We present a significance test for any given variable in nonparametric regression with many variables {via estimating derivatives of a nonparametric function}. The test is based on the moment generating function of the partial derivative of an estimator of the regression function, where the estimator is a deep neural network whose structure is allowed to become more complex as the sample size grows.
This test finds applications in model specification and variable screening for high-dimensional data. To render our test applicable to high-dimensional inputs, whose dimensions can also increase with sample size, we make the assumption that the observed high-dimensional predictors can effectively serve as proxies for certain latent, lower-dimensional predictors that are actually involved in the regression function. Additionally, we finely adjust the regression function estimator, enabling us to achieve the desired asymptotic normality under the null hypothesis, as well as consistency for any fixed scenarios and certain local alternatives.

We will consider the problem of nonparametric inference in multi-dimensional diffusion models from low-frequency data. Due to the computational intractability of the likelihood, implementation of likelihood-based procedures in such settings is a notoriously difficult task. Exploiting the underlying (parabolic) PDE structure of the transition densities, we derive computable formulas for the likelihood function and its gradients. We then construct a Metropolis-Hastings Crank-Nicolson-type algorithm for Bayesian inference with Gaussian priors, as well as gradient-based methods for computing the MLE and Langevin-type MCMC. The performance of the algorithms is illustrated via numerical experiments.

Uncertainty quantification is a key issue when considering the application of deep neural network methods in science and engineering. To this end, numerous Bayesian neural network approaches have been introduced. The main challenge is to construct an algorithm which is applicable to the large sample sizes and parameter dimensions of modern applications on the one hand and which admits statistical guarantees on the other hand. A stochastic Metropolis-Hastings step saves computational costs by calculating the acceptance probabilities only on random (mini-)batches, but reduces the effective sample size leading to less accurate estimates. We demonstrate that this drawback can be fixed with a simple correction term. Focusing on deep neural network regression, we prove a PAC-Bayes oracle inequality which yields optimal contraction rates and we analyze the diameter and show high coverage probability of the resulting credible sets. The method is illustrated with a simulation example.

Mixture models are a method of creating flexible models to study reality. These models contain three components: a finite-dimensional parameter, the mixing distribution and the kernel. As Bayesians, we would put priors on the finite-dimensional parameter and the mixing distribution. A natural choice for priors on the mixing distribution is a species sampling process prior. We study the frequentist properties of such priors when the parameter of interest is the finite-dimensional parameter.
Three fundamental questions arise: 1) will the posterior ever converge near the truth? 2) How fast will the posterior contract to the truth? 3) Can we use credible sets as valid confidence sets? By answering these questions, we get an insight into designing priors that are optimal for learning about our problem.
To answer these questions, we prove a semiparametric Bernstein-von Mises theorem. We then provide the tools to verify the assumptions of this result. The new tools allow us to check the change of measure condition of semiparametric BvMs. We then use these tools to prove the Bernstein-von Mises theorem for mixture of finite mixtures and Dirichlet process mixtures for the frailty and errors-in-variables model.

In many high-dimensional problems, the final 'summary statistic' is amenable to a univariate CLT, but with a complicated unknown covariance structure due to underlying dependencies. To take up a general perspective, we consider a stationary, weakly dependent sequence of random variables. Given only mild conditions, allowing for polynomial decay of the autocovariance function, we show a Berry-Esseen bound of optimal order $n^{-1/2}$ for studentized (self-normalized) partial sums, both for the Kolmogorov and Wasserstein (and $L^p$) distance. The results show that, in general, (minimax) optimal estimators of the long-run variance lead to suboptimal bounds in the central limit theorem, that is, the rate $n^{-1/2}$ cannot be reached, refuting a popular belief in the literature. This can be salvaged by simple methods: We reveal that in order to maintain the optimal speed of convergence $n^{-1/2}$, simple over-smoothing within a certain range is necessary and sufficient. The setup contains many prominent dynamical systems and time series models, including random walks on the general linear group, products of positive random matrices, functionals of Garch models of any order, functionals of dynamical systems arising from SDEs, iterated random functions and many more.

Many fields of modern sciences are faced with high-dimensional data sets. In this talk, we investigate the spectral properties of a large sample correlation matrix R. Results for the spectral distribution, extreme eigenvalues and functionals of the eigenvalues of R are presented in both light- and heavy-tailed cases. The findings are applied to independence testing and to the volume of random simplices.

This paper focuses on optimizing high-dimensional problems with linear constraints under uncertain conditions, often caused by noisy data. We address challenges in large-scale, data-driven applications where the parameter matrix is only approximately known due to noise and limited data samples. Our approach is a linear shrinkage method that blends random matrix theory and robust optimization principles. It aims to minimize the Frobenius distance between the estimated and the true parameter matrix, especially when dealing with a large and comparable number of constraints and variables. This data-driven method excels in simulations, showing superior noise resilience and more stable performance in both achieving objectives and adhering to constraints compared to traditional robust optimization. Our findings highlight the effectiveness of our method in improving the robustness and reliability of optimization in high-dimensional, data-driven scenarios.

The paper "A Test on the Location of Tangency Portfolio for Small Sample Size and Singular Covariance Matrix" explores the tangency portfolio's positioning within feasible portfolios under specific conditions: small sample sizes and singular covariance matrices for asset returns. A new test is presented to determine this location, deriving the exact distribution of the test statistic under both null and alternative hypotheses and the high-dimensional asymptotic distribution as both portfolio dimension and sample size increase. The numerical study compares the asymptotic test with an exact finite sample test, showing effective performance. This research is pivotal for understanding tangency portfolio characteristics in challenging conditions, like singularity and limited data, enhancing financial decision-making strategies.

Learning from positive and unlabeled data is actively researched machine learning task.
The goal is to train a binary classification model based on a training dataset containing part of positives which are labeled, and unlabeled instances. Unlabeled set includes remaining part of positives and all negative observations. Unlike in many prior works, we consider a realistic setting for which probability of label assignment, i.e. propensity score, is instance-dependent.
In proposed approach we investigate minimizer of an empirical counterpart of a joint risk which depends on both posterior probability of inclusion in a positive class as well as on a propensity score. The non-convex empirical risk is alternately optimised with respect to parameters of both functions. In the theoretical analysis we establish risk consistency of the minimisers using recently derived methods from the theory of empirical processes. Besides the important development here is a proposed novel implementation of an optimisation algorithm, for which sequential approximation of a set of positive observations among unlabeled ones is crucial. Experiments conducted on 20 data sets for various labeling scenarios show that the proposed method works on par or more effectively than state-of-the-art methods based on propensity function estimation.

In deep learning, the task is to estimate the functional relationship between input and output using deep neural networks. In image classification, the input data consists of observed images and the output data represents classes of the corresponding images that describe what kind of objects are present in the images. The most successful methods, especially in the area of image classification can be attributed to deep learning approaches and, in particular, to convolutional neural networks (CNNs). Recently, Kohler, Krzyzak and Walter have shown that CNN image classifiers that minimize empirical risk are able to achieve dimension reduction, however, in practice, it is not possible to compute the empirical risk minimizer. Instead, gradient descent methods are used to obtain a small empirical risk. Furthermore, the network topologies used in practice are over-parameterized, i.e., they have many more trainable parameters than training samples. The goal of this work is to derive the rate of convergence results for over-parameterized CNN image classifiers, which are trained by gradient descent. Thus this work should provide a better theoretical understanding of the empirical success of CNN image classifiers.

Spike trains data find a growing list of applications in computational neuroscience,

imaging, streaming data and finance. Machine learning strategies for spike

trains are based on various neural network and probabilistic models. The probabilistic

approach is relying on parametric or nonparametric specifications of the underlying spike

generation model. In this paper we

consider the two-class statistical classification problem for a class of spike train data

characterized by nonparametrically specified intensity functions. We derive the optimal

Bayes rule and next form the plug-in nonparametric kernel classifier. Asymptotical

properties of the rules are established including the limit with respect to the

increasing recording time interval and the size of a training set.

In particular the convergence of the kernel classifier to the Bayes rule is proved.

The obtained results are supported by a finite sample simulation studies.

In the first part of my talk I will talk about Adversarial Classification. In multiple domains such as malware detection, automated driving systems, or fraud detection, classification algorithms are susceptible to being attacked by malicious agents willing to perturb the value of instance covariates in search of certain goals. Such problems pertain to the field of adversarial machine learning and have been mainly dealt with, perhaps implicitly, through game-theoretic ideas with strong underlying common knowledge assumptions. These are not realistic in numerous application domains in relation to security. We present an alternative statistical framework that accounts for the lack of knowledge about the attacker’s behavior using adversarial risk analysis concepts.

In the second part I will discuss about an adversarial risk analysis framework for the software release problem. A major issue in software engineering is the decision of when to release a software product to the market. This problem is complex due to, among other things, the uncertainty surrounding the software quality and its faults, the various costs involved, and the presence of competitors. A general adversarial risk analysis framework is proposed to support a software developer in deciding when to release a product and showcased with an example.

In this paper, several regression-type models for multivariate ordinal time series are developed. The regression equations are inspired by existing GARCH-type models for univariate discrete-valued time series and include feedback terms in addition to the usual lagged observations. The corresponding terms from other individuals are represented by weighted averages which are calculated based on a proximity matrix. The marginal conditional distributions are either binomial (employing the simplifying rank-count formulation) or multinomial. The approach can generalized to obtain VARMA-type models to allow for more specific dependence between individuals. Additionally, different copulas are considered to model possible cross-correlation explicitly. The main data example concerns the daily air quality (ordinal) in three cities in North China. Here, a spatial dimension is present, which can be exploited in the definition of the proximity matrix and the copulas.

We analyze data occurring in a regular two-dimensional grid for spatial dependence based on spatial ordinal patterns (SOPs). After having derived the asymptotic distribution of the SOP frequencies under the null hypothesis of spatial independence, we use the concept of the type of SOPs to define the statistics to test for spatial dependence. The proposed tests are not only implemented for real-valued random variables, but a solution for discrete-valued spatial processes in the plane is provided as well. The performances of the spatial-dependence tests are comprehensively analyzed by simulations, considering various data-generating processes. The results show that SOP-based dependence tests have good size properties and constitute an important and valuable complement to the spatial autocorrelation function. To be more specific, SOP-based tests can detect spatial dependence in non-linear processes, and they are robust with respect to outliers and zero inflation. To illustrate their application in practice, two real-world data examples from agricultural sciences are analyzed.

The classification of movement in space is one of the key tasks in environmental science. Various geospatial data such as rainfall or other weather data, data on animal movement or landslide data require a quantitative analysis of the probable movement in space to obtain information on potential risks, ecological developments or changes in future. Usually, machine-learning tools are applied for this task. Yet, machine-learning approaches also have some drawbacks, e.g. the often required large training sets and the fact that the algorithms are often hard to interpret. We propose a classification approach for spatial data based on ordinal patterns. Ordinal patterns have the advantage that they are easily applicable, even to small data sets, are robust in the presence of certain changes in the time series and deliver interpretative results. They, therefore, do not only offer an alternative to machine-learning in the case of small data sets but might also be used in pre-processing for a meaningful feature selection. In this talk, we introduce the basic concept of multivariate ordinal patterns and the corresponding limit theorem. We focus on the discrete case, that is, on movements on a two dimensional grid. The approach is applied to rainfall radar data.

In time series analysis, ordinal patterns describe the spatial ordering of consecutive observations in temporally-ordered data.These have been well-studied for univariate time series. Since a definition of ordinal patterns presupposes a total ordering of observations, there is, however, no straightforward extension of this notion to multivariate data. Nevertheless, applications often require an analysis of data in $mathbb{R}^d$, $d>1$. A lack of canonical ordering of $mathbb{R}^d$ can be overcome by the concept of statistical depth, i.e. by measuring how deep a data point lies in a given reference distribution. The corresponding center-outward ordering of observations in multivariate time series data naturally leads to the definition of ordinal patterns for multivariate data (depth patterns). Given the definition of depth patterns, we are interested in the probability of observing a specific pattern in a time series. For this, we consider the relative frequency of depth patterns as natural estimators for their occurrence probabilities. Depending on the choice of reference distribution and the relation between reference and data distribution, we distinguish different settings that are considered separately. Within these settings we study statistical properties of ordinal pattern probabilities, establishing consistency and asymptotic normality in specific cases under the assumption of weakly dependent time series data.

A rank-invariant clustering of variables is introduced that is based on the predictive strength between groups of variables, i.e., two groups are assigned a high similarity if the variables in the first group contain high predictive information about the behaviour of the variables in the other group and/or vice versa. The method presented here is model-free, dependence-based and does not require any distributional assumptions. Various general invariance and continuity properties are investigated, with special attention to those that are beneficial for the agglomerative hierarchical clustering procedure. A fully non-parametric estimator is considered whose excellent performance is demonstrated in several simulation studies and by means of real-data examples.

Zero-inflated data naturally appears in many applications, such as health care, weather forecasting, and insurance. Analyzing zero-inflated data is challenging as the high amount of observations in zero invalidates standard statistical techniques. For example, assessing the level of dependence between two zero-inflated random variables becomes difficult due to limitations when applying standard rank-based measures of association, such as Kendall’s tau or Spearman’s rho. Recent work tackles this issue and suggests an estimator of Kendall’s tau for zero-inflated continuous distributions. However, such an estimator does not show satisfactory performances for zero-inflated count data. We fill this gap and propose an adjusted estimator for zero-inflated discrete distributions. We derive the estimator analytically and show that it outperforms existing estimators in various simulated scenarios. Finally, we facilitate the interpretability of the proposed estimator by deriving its achievable bounds.

Kendall's tau and conditional Kendall's tau matrices are multivariate (conditional) dependence measures between the components of a random vector. For large dimensions, available estimators are computationally expensive and can be improved by averaging. Under structural assumptions on the underlying Kendall's tau and conditional Kendall's tau matrices, we introduce new estimators that have a significantly reduced computational cost while keeping a similar error level. In the unconditional setting we assume that, up to reordering, the underlying Kendall's tau matrix is block-structured with constant values in each off-diagonal blocks. The estimators take advantage of this block structure by averaging over (part of) the pairwise estimates in each off-diagonal blocks. Derived explicit variance expressions show their improved efficiency. In the conditional setting, the conditional Kendall's tau matrix is assumed to have a block structure given some covariate. Conditional Kendall's tau matrix estimators are constructed as in the unconditional case by averaging. We establish their joint asymptotic normality, and show that the asymptotic variance is reduced compared to the naive estimators. We perform a simulation study displaying improved performance for all estimators. The estimators are used to compute VaR of a large stock portfolio; backtesting illustrates the obtained improvements compared to the previous estimators.

Optimal transportation theory and the related $p$-Wasserstein distance ($W_p$, $pgeq 1$) are widely-applied in statistics and machine learning. In spite of their popularity, inference based on these tools has some issues. For instance, it is sensitive to outliers and it may not be even defined when the underlying model has infinite moments. To cope with these problems, first we consider a robust version of the primal transportation problem and show that it defines the {robust Wasserstein distance}, $W^{(lambda)}$, depending on a tuning parameter $lambda > 0$. Second, we illustrate the link between $W_1$ and $W^{(lambda)}$ and study its key measure theoretic aspects. Third, we derive some concentration inequalities for $W^{(lambda)}$. Fourth, we use $W^{(lambda)}$ to define minimum distance estimators, we provide their statistical guarantees and we illustrate how to apply the derived concentration inequalities for a data driven selection of $lambda$. Fifth, we provide the {dual} form of the robust optimal transportation problem and we apply it to machine learning problems (generative adversarial networks and domain adaptation). Numerical exercises %(on simulated and real data) provide evidence of the benefits yielded by our novel methods.

This talk presents some novel contributions to persistence homology based statistical inference for Topological Data Analysis (TDA). Along the way, we are also discussing, on a more general level, statistical challenges underlying the construction of such inference methods. The presented novel inference methods consist of bootstrap based confidence regions for (persistent) Betti numbers and Euler characteristic curves. In contrast to most of the other existing inference methods for TDA, our methods are based on one data set of size n, and large sample guarantees are thus established for n tending to infinity. On a technical level, the presented results depend critically on the notion of stabilization that has been developed in geometric probability theory.

Biostatistics increasingly face the urgent challenge of efficiently dealing with extensive experimental data. In particular, high-throughput assays are transforming the study of biology as they generate a complex and diverse collection of high-dimensional data sets. Through compelling statistical analysis, these extensive data sets lead to inaccessible discoveries and knowledge. Building such systematic knowledge is a cumulative process requiring analyses integrating multiple sources, studies, and technologies. The increased availability of studies on related clinical populations poses two important multi-study statistical questions: 1) To what extent is biological signal reproducibly shared across different studies? 2) How can we detect and quantify local signals that may be masked by global solid signals? We will answer these questions by introducing a novel class of methodologies for the joint analysis of different studies. The goal is to identify and estimate separately common factors reproduced across multiple studies and study-specific factors. We present different medical and biological applications. In all the cases, we clarify the benefits of a joint analysis compared to the standard methods. Our method could accelerate the pace at which we can combine unsupervised analysis across different studies and understand the cross-study reproducibility of signals in multivariate data.

In order to estimate the proportion of 'immune' or 'cured' subjects who will never experience failure, a sufficiently long follow-up period is required. Several statistical tests have been proposed in the literature for assessing the assumption of sufficient follow-up. However, they have not been satisfactory for practical purposes due to their conservative behaviour or underlying parametric assumptions. A novel method is proposed for testing sufficient follow-up under general nonparametric assumptions. This approach differs from existing methods. The hypotheses are formulated in a broader context, eliminating the requirement for event times of interest to have compact support. Instead, the notion of sufficient follow-up is characterized by the quantiles of the distribution. The underlying assumption for the proposed method is that the event times have a non-increasing density function, which can also be relaxed to an unimodal density. The test is based on a shape-constrained density estimator such as the Grenander or the kernel-smoothed Grenander estimator. The performance of the test is investigated through a simulation study, and the method is illustrated on data from cancer clinical trials.

The shape parameter in the Crow-AMSAA model is, in practice, more interpretable than the scale parameter. Hence, incorporating initial knowledge in the estimation seems more necessary for it than for the scale parameter. In this talk we will show the application of the profile likelihood estimation method to Crow-AMSAA and the combination of this method with Bayesian estimation. We show the possibility of using double-truncated gamma distribution. As part of the numerical investigation, we present an analysis of the sensitivity of posterior inference to the incorrect selection of the hyperparameters of the prior distributions.

New crop varieties are extensively tested in multi-environment trials in order to obtain a solid basis for recommendations to farmers. When the target population of environments is large, a division into sub-regions is often advantageous. If the same set of genotypes is tested in each of the sub-regions, a linear mixed model (LMM) may be fitted with random genotype-within-sub-region effects.

The first analytical results to optimizing allocation of trials to sub-regions have been obtained in Prus and Piepho (2021). In that paper the genotype effects are assumed to be uncorrelated. However, this assumption is not always suitable for practical situations. In praxis, genetic markers are often used in plant breeding for determining genetic relationships of genotypes, which helps to model their correlation.

In this work a more general LMM with correlated genotype effects is considered. An analytical solution and a computational approach are proposed for optimal allocation of trials.

Prus, M. and Piepho, H.-P. (2021). Optimizing the allocation of trials to sub-regions in multi-environment crop variety testing. Journal of Agricultural, Biological and Environmental Statistics, 26, 267–288.

A society is said to exhibit more cumulative deprivation (affluence) when more individuals score low (high) in all dimensions of well-being. To measure these concepts, we adopt a position-based approach which builds on copulas.

First, we summarize the information about cumulative deprivation using the cdf of the maximal position across all dimensions, which is determined by the diagonal section of the copula. Second, we measure cumulative affluence through the cdf of the minimal position, which is closely related to the diagonal section of the survival copula.

We depict these functions into a single hairpin-type diagram along with two benchmarks: independence and perfect alignment. From this diagram, we define an index of cumulative deprivation (affluence) as the normalized area between the diagonal section of the copula (survival copula) and that representing independence. We discuss the properties of these indices and show its relationship with multivariate Spearman’s footrule. Several examples with well-known multivariate copulas illustrate our results.

We derive nonparametric estimators of the indices above and we apply them to analyse how cumulated deprivation and affluence have evolved worldwide from 2007 to 2021. The application involves yearly data on the three dimensions included in the Human Development Index: health, education and income.

Motivated by a recently established result saying that within the family of bivariate Archimedean copulas standard pointwise convergence implies the generally stronger weak conditional convergence,
i.e. convergence of almost all conditional distributions, this result is extended to the class of multivariate Archimedean copulas.
Working with the fact that generators of Archimedean copulas are $d$-monotone functions, pointwise convergence within the family of multivariate Archimedean copulas is characterized in terms of convergence of the corresponding generators,
derivatives of the generators, marginal copulas as well as marginal densities.
Furthermore, weak conditional convergence is a consequence of any of the afore-mentioned properties.
Utilizing that generators of Archimedean copulas can be represented via Williamson transforms of one dimensional probability measures,
it is established that weak convergence of the probability measures is equivalent to uniform convergence of the Archimedean copulas.
Using Markov kernels, Archimedean copulas inherit absolute continuity, singularity and discreteness from the afore-mentioned probability measures,
leading to the surprising result that absolutely continuous, singular, as well as discrete copulas are dense in the class of Archimedean copulas with respect to the uniform metric.

In the talk we consider strategies for model selection of copulas within parametric families and for preventing overfitting. The parameter is estimated by an approximate minimizer of the estimated Cramér-von Mises divergence supplemented by a regularization term. We provide results on strong consistency and on asymptotic normality of the estimator. Moreover, we consider statistical tests in order to compare the approximation quality of several model classes.

The empirical success of Generative Adversarial Networks (GANs) caused an increasing interest in theoretical research. The statistical literature is mainly focused on Wasserstein GANs and generalizations thereof, which especially allow for good dimension reduction properties.Statistical results for Vanilla GANs, the original optimization problem, are still rather limited and require assumptions such as smooth activation functions and equal dimensions of the latent space and the ambient space. To bridge this gap, we draw a connection from Vanilla GANs to the Wasserstein distance. By doing so, problems caused by the Jensen-Shannon divergence can be avoided and existing results for Wasserstein GANs can be extended to Vanilla GANs. In particular, we obtain an oracle inequality for Vanilla GANs in Wasserstein distance. The assumptions of this oracle inequality are designed to be satisfied by network architectures commonly used in practice, such as feedforward ReLU networks. Using Hölder-continuous ReLU networks we conclude a rate of convergence for estimating probability distributions.

We investigate the statistical behavior of Stochastic Gradient Descent (SGD) with constant step size under the framework of iterated random functions. Unlike previous studies establishing the convergence of SGD in probability measure, e.g., Wasserstein distance, our approach provides the convergence in Euclidean distance by showing the Geometric Moment Contraction (GMC) of SGD. This new convergence can address the non-stationarity of SGD due to fixed initial points and can provide a more refined asymptotic analysis of SGD. Specifically, we prove a quenched central limit theorem and a quenched invariance principle for averaged SGD (ASGD) regardless of the initial points. Furthermore, we provide a novel perspective to understand the impact of step sizes in SGD by studying its derivative with respect to the step size. The existence of stationary solutions for the first and second derivative processes are shown under mild conditions. Subsequently, we utilize multiple step sizes and show an enhanced Richardson-Romberg extrapolation with improved bias representation, which brings ASGD estimates closer to the global optimum. Finally, we propose a new online inference method and a bias-reduced variant for the extrapolated ASGD. Empirical confidence intervals are constructed and the coverage probabilities are shown to be asymptotically correct by numerical experiments.

Optimization problems with continuous data appear in, e.g., robust machine learning, functional data analysis, and variational inference. Here, the target function is an integral over a family of (continuously) indexed target functions—integrated concerning a probability measure. Such problems can often be solved by stochastic optimization methods: performing optimization steps for the indexed target function with randomly switched indices. In this talk, we will discuss a continuous-time variant of the stochastic gradient descent algorithm for optimization problems with continuous data. The stochastic gradient process consists of a gradient flow minimizing an indexed target function coupled with a continuous-time process determining the index. Index processes are, e.g., reflected diffusions, and pure jump processes in compact spaces. Thus, we study multiple sampling patterns for the continuous data space and allow for data simulated or streamed at runtime of the algorithm. We analyze the approximation properties of the stochastic gradient process and study its long-time behavior and ergodicity under constant and decreasing learning rates.

Deep Reinforcement Learning (DRL) has shown remarkable success in solving complex tasks across various research fields. However, transferring DRL agents to the real world is still challenging due to the significant discrepancies between simulation and reality. To address this issue, we propose a robust DRL framework that leverages platform-dependent perception modules to extract task-relevant information and train a lane-following and overtaking agent in simulation. This framework facilitates the seamless transfer of the DRL agent to new simulated environments and the real world with minimal effort. We evaluate the performance of the agent in various driving scenarios in both simulation and the real world, and compare it to human players and the PID baseline in simulation. Our proposed framework significantly reduces the gaps between different platforms and the Sim2Real gap, enabling the trained agent to achieve similar performance in both simulation and the real world, driving the vehicle effectively.

We consider the classical factor model of Jöreskog (1969) within a change point detection framework with the aim of discovering intervals of local homogeneity of the model. Our tests for structural breaks in the variance (homogeneity in variance) as well as both in the mean and the variance (complete homogeneity) are based on a maximum statistic of sequential generalized likelihood ratios. We approximate the small-sample distribution by means of a multiplier bootstrap. To handle the high-dimensional parameter problem, we suggest a novel bias correction for the multiplier bootstrap. Simulations show that the tests perform very well in terms of size and power. In our empirical application, we study structural breaks for moderately sized equity portfolios

Vine copulas have become increasingly popular in modeling multivariate data. Their key building blocks are bivariate copulas selected from a limited number of parametric families. Apart from the Student's t copula none of these families is particularly good at modeling heteroskedastic data. In this regard, certain mixture copulas have been shown to perform better. We develop a feasible algorithm to determine the optimal bivariate mixture model. We demonstrate that the direct maximum likelihood optimization is comparative to an EM-type algorithm and can even outperform it. Furthermore, we highlight the advantages of copula mixture models that mix with the independence copula. Since the search for the best mixture model is time-consuming, we also present a simple a priori check to determine whether a mixture copula might yield significant improvements.

In the talk, we discuss a complex problem of recognizing, detecting, and estimating stochastically relevant (significant) changepoints within in a time series of specific functional profiles (the option market implied volatility smiles ) while the main focus is on changes caused by various exogenous effects (meaning that the observed changes are not due the market itself and its dynamics but rather because of some human-made interactions).

The standard implied volatility tool (commonly used for the market analysis by practitioners) is shown to be insufficient for a proper detection and analysis of this type of risk because exogenous changes are typically dominated by endogenous effects coming from a specific trading mechanism or a natural market dynamics.

We propose a whole methodological approch (statistical theory, computational algorithms, and practical recommendations) based on "artificial options" that always have a constant (over time) maturity and formal statistical tests for detecting significant changepoints are proposed under different theoretical, computational, as well as applicational scenarios.

I will discuss a simple example of an interacting particle process which consists of individuals that are either susceptible, infected or recovered. Transitioning from susceptible to infected depends on the number of infected neighbours, causing interaction. I will assume only some individuals at some times can be observed. Can we infer the states of all individuals at all times? The proposed solution is valid in more general settings and if time permits I will comment on that.

This paper delves into a nonparametric estimation approach for the interaction function within diffusion-type particle system models. We introduce two estimation methods based upon an empirical risk minimization. Our study encompasses an analysis of the stochastic and approximation errors associated with both procedures, along with an examination of certain minimax lower bounds.

We propose statistical methods to infer a semi-martingale efficient log-price process in a boundary model with one-sided microstructure noise for high-frequency limit order prices. Two main challenges are to discriminate price jumps from continuous log-price movements and volatility estimation. We develop test and estimation methods and establish asymptotic results in a high-frequency regime. Convergence rates and detection boundaries are shown to hinge on characteristics of the imposed noise distribution. We address the estimation of these noise characteristics and adaptive inference on the semimartingale. For illustration, we shed light on the related asymptotic properties for point estimation of boundary parameters.

We consider a nonparametric Bayesian approach to estimation of the volatility function of a stochastic differential equation driven by a gamma process. The volatility function is modelled a priori as piecewise constant, and we specify a gamma prior on its values. This leads to a straightforward MCMC procedure for posterior inference. We give theoretical performance guarantees (minimax optimal contraction rates for the posterior) for the Bayesian estimate in terms of the regularity of the unknown volatility function. We illustrate the method on synthetic and real data examples.

The talk is based on the joint work with D. Belomestny, M. Schauer, and P. Spreij.

https://doi.org/10.3150/21-BEJ1413

https://doi.org/10.1016/j.indag.2023.03.004

In this talk we treat statistical inference for an intrinsic wavelet estimator of curves of symmetric positive definite (SPD) matrices in a log-Euclidean manifold. Examples for these arise in Diffusion Tensor Imaging or related medical imaging problems as well as in computer vision and for neuroscience problems.

Our proposed linear wavelet (kernel) estimator preserves positive-definiteness and enjoys permutation-equivariance, which is particularly relevant for covariance matrices. Our second-generation wavelet estimator is based on average-interpolation and allows the same powerful properties, including fast algorithms, known from nonparametric curve estimation with wavelets in standard Euclidean set-ups.

At the heart of this talk is the proposition of confidence sets based on our wavelet estimator in a non-Euclidean geometry. We derive asymptotic normality of this estimator, including explicit expressions of its asymptotic variance. This opens the door for constructing asymptotic confidence regions which we compare with our proposed bootstrap scheme for inference. Numerical simulations confirm the appropriateness of our suggested inference schemes.

This is joint work with D. Rademacher (U Heidelberg) and J. Krebs (KU Eichstätt) [J. Stat. Plan. Inf. (2023)]. We also refer to work in progress with G. Bailly (UCLouvain) on the more adaptive non-linear wavelet threshold estimator version of this approach.

In this talk, we tackle the problem of estimating time-varying covariance matrices (TVCM; i.e. covariance matrices with entries being time-dependent curves) characterized by either pronounced local peaks or inhomogeneous smoothness. To address this challenge, wavelet denoising estimators are particularly appropriate. Specifically, we model TVCM using a signal-noise model within the Riemannian manifold of symmetric positive definite matrices (endowed with the log-Euclidean metric) and use the intrinsic wavelet transform, designed for curves in Riemannian manifolds. Within this non-Euclidean framework, the proposed estimators preserve positive definiteness.

Although linear wavelet estimators for smooth TVCM achieve good results in various scenarios, they are less suitable if the underlying curve features singularities. Consequently, our estimator is designed around a nonlinear thresholding scheme, tailored to the characteristics of the noise in covariance matrix regression models. The effectiveness of this novel nonlinear scheme is assessed by deriving mean-squared error consistency and by numerical simulations, and its practical application is demonstrated using electroencephalography data. Joint work with R. von Sachs (UCLouvain).

Comparative evaluation of forecasts of statistical functionals relies on comparing averaged losses of competing forecasts after the realization of the quantity $Y$, on which the functional is based, has been observed. Motivated by high-frequency finance, in this paper we investigate how proxies $tilde Y$ for $Y$ - say volatility proxies - which are observed together with $Y$ can be utilized to improve forecast comparisons.
We extend previous results on robustness of loss functions for the mean to general moments and ratios of moments, and show in terms of the variance of differences of losses that using proxies will increase the power in comparative forecast tests. These results apply both to testing conditional as well as unconditional dominance. Finally, we numerically illustrate the theoretical results, both for simulated high-frequency data as well as for high-frequency log returns of several cryptocurrencies.

The study of the function-indexed empirical process and its weak convergence under dependence has a substantial history. A great share of the literature deals with empirical processes constructed from stationary time series, and recently, some results concerning nonstationary settings have also been published. In contrast, much less is known about the sequential empirical process indexed in function classes, and the literature so far seems to focus on the stationary case. In an attempt to partially close this gap, its weak convergence in a nonstationary setting is studied, and it is shown to be asymptotically equicontinuous provided suitable uniform moment bounds are available for the increments of the empirical process. Restrictions on the dependence are imposed solely in terms of these bounds and their assumed properties in order to achieve a certain degree of generality. Limitations, possible extensions and statistical applications are discussed.

Recently a lot of progress has been made regarding the theoretical understanding of machine learning methods. One of the very promising directions is the statistical approach, which interprets machine learning as a collection of statistical methods and builds on existing techniques in mathematical statistics to derive theoretical error bounds and to understand phenomena such as overparametrization. The talk surveys this field and describes future challenges.