On Estimation of $L_{r}$-Norms in Gaussian White Noise Models

Nonparametric estimation of functionals has long been a core topic in statistics, with linear and quadratic functionals well-understood in Gaussian models. However, non-smooth functionals like L_r norms pose unique challenges due to their non-differentiability and irregularity. Early work by Lepski (1999) and subsequent studies addressed smooth functionals, but the non-smooth case remained less explored. Recent advances in polynomial approximation and Hermite polynomial techniques have enabled progress in high-dimensional settings, yet a complete theory for L_r norms, especially for non-even r, was lacking. This work builds on foundational theories in minimax estimation, polynomial approximation, and Gaussian process analysis, aiming to fill this gap.

The core problem is to construct estimators for L_r norms of an unknown function in Gaussian white noise models that are both rate-optimal and adaptive over Nikolskii-Besov spaces. The difficulty stems from the non-smoothness of |u|^r at zero, which invalidates classical plug-in estimators. Existing methods either require prior smoothness knowledge or fail to achieve optimal rates for non-even r. The challenge is to develop a unified framework that handles both smooth and non-smooth regimes, ensuring minimax optimality and adaptivity simultaneously, especially for high-dimensional data where computational efficiency is also critical.

The paper introduces a hybrid approach combining polynomial approximation of |u|^r with Hermite polynomial-based unbiased estimators, tailored for non-smooth functionals. This approach allows precise bias-variance trade-offs and leverages the properties of Hermite polynomials to handle Gaussian noise effectively. The authors develop a novel adaptive bandwidth selection method based on Lepski’s technique, enabling the estimator to automatically adjust to unknown smoothness levels. Additionally, they rigorously derive asymptotic bounds, proving the estimator’s minimax optimality and revealing the fundamental difference between even and non-even r in terms of adaptivity constraints.

• �� Define the function space as Nikolskii-Besov Bs p,∞(L), characterizing smoothness via r-th order differences.
• �� Approximate the target functional |f|^r using polynomial approximation, with coefficients derived from best polynomial fits.
• �� Construct unbiased estimators for polynomial terms using Hermite polynomials, exploiting their orthogonality and Gaussian properties.
• �� Implement sample splitting to obtain independent estimates, reducing bias and variance.
• �� Use kernel projection to localize the estimation, controlling bias via bandwidth h.
• �� Derive asymptotic bounds for the mean squared error, matching upper and lower bounds through information-theoretic arguments.
• �� Develop an adaptive bandwidth selection procedure based on Lepski’s method, ensuring minimax rates without prior smoothness knowledge.

Simulations involve Gaussian noise with known variance, testing the estimator’s performance across various smoothness levels and r values. Real data applications include signal strength estimation in noisy environments. Hyperparameters such as polynomial degree and bandwidth are tuned via theoretical guidelines. Comparisons with classical plug-in and thresholding methods demonstrate superior bias control and convergence rates, especially in non-smooth regimes. Robustness is assessed under different noise intensities, confirming the estimator’s stability and optimality.

The proposed estimators achieve the minimax rate of \((n \log n)^{-s/(2s+1)}\) for both r=1 and non-even r>1, with the upper bounds matching the derived lower bounds. For adaptive estimation, the non-even r case requires no penalty, while even r incurs a polynomial penalty, aligning with classical theory. These results hold uniformly over Nikolskii-Besov spaces, confirming the estimators’ robustness and optimality. The theoretical bounds are validated through asymptotic limit theorems, polynomial approximation bounds, and Hermite polynomial properties, demonstrating the effectiveness of the approach in high-dimensional noisy settings.

Immediate applications include high-dimensional signal denoising, image reconstruction, and financial risk assessment where non-smooth metrics are relevant. The methodology can be integrated into machine learning pipelines for feature selection and regularization involving irregular functionals. Long-term, the framework could influence the design of adaptive algorithms in neural networks, nonparametric Bayesian inference, and complex data environments, enabling efficient estimation of irregular functionals with minimal prior assumptions.

The approach assumes known noise variance, limiting direct applicability in unknown noise scenarios. Computational complexity increases with polynomial degree and data dimension, posing challenges for large-scale problems. The asymptotic guarantees may not fully capture finite-sample behavior, especially under model misspecification or heavy-tailed noise. Extending the framework to multivariate or non-Gaussian settings remains an open challenge.