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Session Overview |
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Poster Exhibition
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Optimal Control of McKean-Vlasov Stochastic Partial Differential Equations Technische Universität Berlin, Germany
We consider the control of stochastic partial differential equations (SPDE) of McKean-Vlasov type via deterministic controls using the variational approach to SPDE. Based on a recent novel approach to the Lions derivative for Banach space valued functions by Stannat and Vogler (https://arxiv.org/abs/2407.14884), we prove the Gateaux differentiability of the control to state map and, using adjoint calculus, we derive explicit representations of the gradient of the cost functional and a Pontryagin maximum principle.
Further, we prove the existence of optimal controls using a martingale approach and compactness methods.
Our setting uses monotone coefficients and allows the drift and diffusion coefficients to depend on the state, the distribution of the state and the control.
Limit Theorem for Trace of the Squared Sample Correlation Matrices in High Dimensions Stockholm University, Sweden
From a sample of $n$ observations of a $p$-dimensional random vector, we construct the sample correlation matrix. In this work, we establish the asymptotic behavior of the trace of the squared correlation matrix as both $p$ and $n$ grow large.
Known results confirm that for p and n growing at the same rat and finite fourth moments of the vector components, the trace of the squared correlation matrix satisfies a central limit theorem (CLT). Additionally, we identify more general conditions under which this asymptotic normality holds, revealing how the relationship between $p$, $n$, and the distribution of the random vector influences the limiting behavior. These findings extend existing results and provide a broader framework for independence testing in high dimensions.
Multiple Contrast Test for Youden-Indices in Factorial Diagnostic Trials 1Institute for Mathematical Stochastics, Faculty of Mathematics, Otto-von-Guericke-University Magdeburg, Germany; 2Institute of Biometry and Clinical Epidemiology, Charité-Universitätsmedizin Berlin, Germany
Diagnostic tests are important for clinical decision-making. However, many possible biomarkers and other factors could often be used, and their examination in a diagnostic trial is needed to find the optimal method to answer the clinical question. A well-known method for a single biomarker is using the cutoff value corresponding to the maximized Youden index. Bantis et al. have developed several methods to compare two biomarkers to calculate statistical tests and confidence intervals for the difference of two maximized Youden indices [bantis2021].
As there are often more than two diagnostic methods or biomarkers and possibly more than one rater or clinician involved in the trial, its design has a factorial structure with rather complex dependencies. To select in this cases the biomarker with the highest accuracy, multiple statistical tests need to be calculate. We propose to use as multiple contrast test the max T test based on ideas from Bretz et al [bretz2001], which controls the family wise error rate in the strong sense and uses the correlations between the test statistics.
In our work, we extend the theoretical background and implemented R code to execute for arbitrarily many biomarkers and general factorial designs the multiple contrast tests and calculate the corresponding simultaneous confidence intervals. Different methods for the calculation of the Youden indices have been implemented, which are based on a normal, a power normal and an arbitrary distribution assumption of the sample data.
We will discuss methods’ properties such as their control of the type-1 error rate and power obtained from extensive simulation studies.
References
[bantis2021] Bantis LE, Nakas CT, Reiser B. Statistical inference for the difference between two maximized Youden indices obtained from correlated biomarkers. Biometrical Journal. 2021; 63:1241–1253.
[bretz2001] Bretz F, Genz A, Hothorn LA. On the numerical availability of multiple comparison procedures. Biom J. 2001;43(5):645-656.
Mixed moving average field guided learning with unbounded losses TU Chemnitz, Germany
Influenced mixed moving average fields are versatile models for spatio-temporal data. However, apart from the Gaussian distributed case, their predictive distribution is not generally known. To allow the use of such models in forecasting tasks also in the non-Gaussian setup and for data with short and long memory, ensemble forecasts for data with such an underlying model have been established using a generalized Bayesian algorithm and a bounded loss function in [1]. We extend this approach to the unbounded loss function setup by determining a novel PAC Bayesian bound for $\theta$-lex weakly dependent data.
Stable convergence in law in approximation of stochastic integrals with respect to diffusions University of Zagreb, Faculty of Science, Croatia
We assume that the one-dimensional diffusion $X$ satisfies a stochastic differential equation of the form:
$dX_t=\mu(X_t)dt+\nu(X_t)dW_t$, $X_0=x_0$, $t\geq 0$. Let $(X_{i\Delta_n},0\leq i\leq n)$ be discrete observations along fixed time interval $[0,T]$. We prove that the random vectors which $j$-th component is $\frac{1}{\sqrt{\Delta_n}}\sum_{i=1}^n\int_{t_{i-1}}^{t_i}g_j(X_s)(f_j(X_s)-f_j(X_{t_{i-1}}))dW_s$, for $j=1,\dots,d$, converge stably in law to mixed normal random vector with covariance matrix which depends on path $(X_t,0\leq t\leq T)$, when $n\to\infty$. We use this result to prove stable convergence in law for $\frac{1}{\sqrt{\Delta_n}}(\int_0^Tf(X_s)dX_s-\sum_{i=1}^nf(X_{t_{i-1}})(X_{t_i}-X_{t_{i-1}}))$.
Perpetual American standard and lookback options in models with progressively enlarged filtrations 1LSE, United Kingdom; 2University of New South Wales, Australia
We derive closed-form solutions to optimal stopping problems related to the pricing of perpetual American standard and lookback put and call options in extensions of the Black-Merton-Scholes model under progressively enlarged filtrations. It is assumed that the information available from the market is modelled by Brownian filtrations progressively enlarged with the random times at which the underlying process attains its global maximum or minimum, that is, the last hitting times for the underlying risky asset price of its running maximum or minimum over the infinite time interval, which are supposed to be progressively observed by the holders of the contracts. We show that the optimal exercise times are the first times at which the asset price process reaches certain lower or upper stochastic boundaries depending on the current values of its running maximum or minimum depending on the occurrence of the random times of the global maximum or minimum of the risky asset price process. The proof is based on the reduction of the original necessarily three-dimensional optimal stopping problems to the associated free-boundary problems and their solutions by means of the smooth-fit and either normal-reflection or normal-entrance conditions for the value functions at the optimal exercise boundaries and the edges of the state spaces of the processes, respectively.
A comparison of multiple imputation algorithms University hospital of Essen, Germany
Missing data is a major problem in medicine and other branches of science. There are a lot of competing algorithms that deal with missing data. In this poster we take the point of view that the decision to use multiple imputation by chained equations has already been made and it only remains to choose the exact algorithm within this class. We will see that predictive mean matching is superior to all other algorithms.
Efficient Estimation of a Gaussian Mean with Local Differential Privacy Institute of Science and Technology Austria (ISTA), Austria
In this paper, we study the problem of estimating the unknown mean $\theta$ of a unit variance Gaussian distribution in a locally differentially private (LDP) way. In the high-privacy regime ($\epsilon\le 1$), we identify an optimal privacy mechanism that minimizes the variance of the estimator asymptotically. Our main technical contribution is the maximization of the Fisher-Information of the sanitized data with respect to the local privacy mechanism $Q$. We find that the exact solution $Q_{\theta,\epsilon}$ of this maximization is the sign mechanism that applies randomized response to the sign of $X_i-\theta$, where $X_1,\dots, X_n$ are the confidential iid original samples. However, since this optimal local mechanism depends on the unknown mean $\theta$, we employ a two-stage LDP parameter estimation procedure which requires splitting agents into two groups. The first $n_1$ observations are used to consistently but not necessarily efficiently estimate the parameter $\theta$ by $\tilde{\theta}_{n_1}$. Then this estimate is updated by applying the sign mechanism with $\tilde{\theta}_{n_1}$ instead of $\theta$ to the remaining $n-n_1$ observations, to obtain an LDP and efficient estimator of the unknown mean.
STATISTICAL GUARANTEES FOR APPROXIMATE STATIONARY POINTS OF SHALLOW NEURAL NETWORKS 1UHH, Germany; 2RUB, Germany
Since statistical guarantees for neural networks are usually restricted to global optima of intricate objective functions, it is unclear whether these theories explain the performances of actual outputs of neural network pipelines.
The goal of this paper is, therefore, to bring statistical theory closer to practice.
We develop statistical guarantees for shallow linear neural networks that coincide up to logarithmic factors with the global optima but apply to stationary points and the points nearby.
These results support the common notion that neural networks do not necessarily need to be optimized globally from a mathematical perspective.
We then extend our statistical guarantees to shallow ReLU neural networks, assuming the first layer weight matrices are nearly identical for the stationary network and the target.
More generally, despite being limited to shallow neural networks for now,
our theories make an important step forward in describing the practical properties of neural networks in mathematical terms.
Adaptive Kernel Density Estimation in L2-norm using artificial data 1Georg-August-Universität Göttingen; 2Universidad de Valparaiso; 3Université Paris Dauphine
This paper deals with the study of Kernel Density Estimator using artificial data. We investigate theoretical and practical properties of such estimators and propose a data-driven procedure such that the estimator is adaptive to the regularity of the underlying density of the data.
Estimating the density of a sample is one of the most useful steps in any data analysis. Among the nonparametric methods used for this goal, the Kernel Density Estimator is perhaps the most widely used. These estimators depend on a kernel $K$ and a bandwidth $h$. We study an adaptive and data driven method to select the bandwidth $h$, as introduced in \cite{goldenshluger2011bandwidth}, in the context of artificial data. More precisely, we are interested in estimating the density $f_Y$ of a random variable $Y$ that satisfies $Y = m(X)$ where $X$ is another random variable and $m : \mathbb{R} \mapsto \mathbb{R}$ is an unknown function to be estimated. We observe two independent and identically distributed samples generated from theses variables. The first one is quite difficult to obtain and rather small: ${(X_1, Y_1), \dots, (X_n, Y_n)}$, that satisfies $Y_i = m(X_i)$. The second one (independent of the first one) is simpler to obtain ${X_{n+1}, \dots, X_N}$ and can be as large as the statistician needs.
To estimate $f_Y$, we can use two approaches. In the classical approach, we use the sample $Y_1 , \dots, Y_n$. In the artificial data approach, we estimate the unknown function $m$ by $\hat{m}$ using $(X_1 , Y_1 ), \dots, (X_n , Y_n)$, then we construct the artificial data $\hat{Y}_{n+1} = \hat{m}(X_{n+1}), \dots ,\hat{Y}_{n+N} = \hat{m}(X_{n+N})$, and finally we estimate $f_Y$ using these artificial data.
We show that the kernel estimators using artificial data achieves a faster convergence rates when compared to the same estimator in the classical approach. Moreover, we propose a Goldenshluger-Lepski method to select the bandwidth in the artificial data approach and prove that its converges achieves the optimal rate of $\varphi_n(\beta)= n^{-4\beta/2\beta+3} (\log n)^{8\beta/2\beta+3}$. Finally, we perform a simulation study and compare the results via the Mean Integrated Square Error criterion.
Measuring Dependence between Events 1Heidelberg Institute for Theoretical Studies, Germany; 2Ruprecht Karl University of Heidelberg, Germany; 3Goethe University Frankfurt, Germany
Measuring dependence between two events, or equivalently between two binary random variables, amounts to expressing the dependence structure inherent in a $2\times 2$ contingency table in a real number between $-1$ and 1. Countless such dependence measures exist but there is little theoretical guidance on how they compare and on their advantages and shortcomings. Thus, practitioners might be overwhelmed by the problem of choosing a suitable dependence measure. We provide a set of natural desirable properties that a \emph{proper} dependence measure should fulfill. We show that Yule's $\mathsf{Q}$ and the little-known Cole coefficient are proper, while the most widely-used measures, the phi coefficient and all contingency coefficients, are improper. They have a severe attainability problem, that is, even under perfect dependence they can be very far away from $-1$ and $1$, and often differ substantially from the proper measures in that they understate strength of dependence. The structural reason is that these are measures for equality of events rather than of dependence. We derive the (in some instances non-standard) limiting distributions of the measures and illustrate how asymptotically valid confidence intervals can be constructed. In a case study on drug consumption we demonstrate that misleading conclusions may arise from the use of improper dependence measures.
Testing Monotonicity of Regression in Sublinear Time 1Georg August University of Göttingen, Germany; 2Cluster of Excellence "Multiscale Bioimaging: from Molecular Machines to Networks of Excitable Cells'' (MBExC), University of Göttingen, Germany
Modern data sets have grown in size and complexity, exposing the scalability limitations of classical statistical methods, where computational efficiency is as crucial as statistical accuracy. In the context of testing monotonicity in regression functions, we propose FOMT (Fast and Optimal Monotonicity Test), a novel methodology designed to overcome these challenges. FOMT employs a sparse collection of local tests, strategically generated at random, to detect violations of monotonicity scattered throughout the domain of the regression function. This sparsity enables significant computational efficiency, achieving sublinear runtime in most cases, and quasilinear runtime (i.e., linear up to a log factor) in the worst case. In contrast, existing statistically optimal tests typically require at least quadratic runtime. FOMT's statistical accuracy is achieved through the precise calibration of these local tests and their effective combination, ensuring both sensitivity to violations and control over false positives. More precisely, we show that FOMT separates the null and alternative hypotheses at minimax optimal rates over Hölder function classes of smoothness order in (0,2]. Further, for cases with unknown smoothness, we introduce an adaptive version of FOMT, based on the Lepskii principle, which attains statistical optimality and meanwhile maintains the same computational complexity as if the intrinsic smoothness were known. Extensive simulations confirm the competitiveness and effectiveness of both FOMT and its adaptive variant.
Some results on statistical classification University of Trier, Germany
We present
1. an explicit representation of the stepwise Bayes classification functions introduced by Wald and Wolfowitz (1951),
2. a simple formula for the minimax classification of two binomials,
3. a minimax risk upper bound in terms of the spectral radius of a Hellinger matrix,
4. a phonetic application of the minimax bound, showing that people can be distinguished by their hesitation behavior with small error probabilities.
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