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What is Learnable in Valiant's Theory of the Learnable? Learning Perturbations to Extrapolate Your LLM Byzantine-Robust Distributed Sparse Learning Revisited The Sample Complexity of Multiple Change Point Identification under Bandit Feedback A proximal gradient algorithm for composite log-concave sampling Model-based Bootstrap of Controlled Markov Chains Approximation of Maximally Monotone Operators : A Graph Convergence Perspective Posterior Contraction Rates for Sparse Kolmogorov-Arnold Networks in Anisotropic Besov Spaces MIST: Reliable Streaming Decision Trees for Online Class-Incremental Learning via McDiarmid Bound A Spectral Framework for Closed-Form Relative Density Estimation Fast Rates for Offline Contextual Bandits with Forward-KL Regularization under Single-Policy Concentrability Higher-Order Equilibrium Tracking for EM-Compressible Online Estimation Scaling Limits of Long-Context Transformers A Note on Non-Negative $L_1$-Approximating Polynomials Susceptibilities and Patterning: A Primer on Linear Response in Bayesian Learning 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Pass@k: A Bayesian Framework for Large Language Model Evaluation The Good, the Bad, and the Sampled: a No-Regret Approach to Safe Online Classification GOSPA and T-GOSPA quasi-metrics for evaluation of multi-object tracking algorithms A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows Multi-Armed Bandits With Machine Learning-Generated Surrogate Rewards Efficient compression of neural networks and datasets Out-of-Distribution Generalization of In-Context Learning: A Low-Dimensional Subspace Perspective Super-fast Rates of Convergence for Neural Network Classifiers under the Hard Margin Condition Sharp Gaussian approximations for Decentralized Federated Learning Learning Operators by Regularized Stochastic Gradient Descent with Operator-valued Kernels Smoothed Analysis of Learning from Positive Samples Statistical Impossibility and Possibility of Aligning LLMs with Human Preferences: From Condorcet Paradox to Nash Equilibrium Sharp Risk 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Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise
Axel Munk, Johannes Schmidt-Hieber · 2009-08-22 · via math.ST updates on arXiv.org

We consider the models Y_{i,n}=\int_0^{i/n} σ(s)dW_s+τ(i/n)ε_{i,n}, and \tilde Y_{i,n}=σ(i/n)W_{i/n}+τ(i/n)ε_{i,n}, i=1,...,n, where W_t denotes a standard Brownian motion and ε_{i,n} are centered i.i.d. random variables with E(ε_{i,n}^2)=1 and finite fourth moment. Furthermore, σand τare unknown deterministic functions and W_t and (ε_{1,n},...,ε_{n,n}) are assumed to be independent processes. Based on a spectral decomposition of the covariance structures we derive series estimators for σ^2 and τ^2 and investigate their rate of convergence of the MISE in dependence of their smoothness. To this end specific basis functions and their corresponding Sobolev ellipsoids are introduced and we show that our estimators are optimal in minimax sense. Our work is motivated by microstructure noise models. Our major finding is that the microstructure noise ε_{i,n} introduces an additionally degree of ill-posedness of 1/2; irrespectively of the tail behavior of ε_{i,n}. The method is illustrated by a small numerical study.

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