A Fourier Analytical Approach to Estimation of Smooth Functions in Gaussian Shift Model
This provides a novel method for high-dimensional statistical estimation, potentially improving accuracy in real-world applications, though it builds on prior work on smoothness thresholds.
The paper tackles the problem of estimating smooth functions of unknown high-dimensional Gaussian means, developing a Fourier analytical estimator that achieves effective bias reduction and asymptotic normality when smoothness exceeds a threshold.
Let $\mathbf{x}_j = \mathbfθ + \mathbfε_j$, $j=1,\dots,n$ be i.i.d. copies of a Gaussian random vector $\mathbf{x}\sim\mathcal{N}(\mathbfθ,\mathbfΣ)$ with unknown mean $\mathbfθ \in \mathbb{R}^d$ and unknown covariance matrix $\mathbfΣ\in \mathbb{R}^{d\times d}$. The goal of this article is to study the estimation of $f(\mathbfθ)$ where $f$ is a given smooth function of which smoothness is characterized by a Besov-type norm. The problem of interest resides in the high dimensional regime where the intrinsic dimension can grow with the sample size $n$. Inspired by the classical work of A. N. Kolmogorov on unbiased estimation and Littlewood-Paley theory, we develop a new estimator based on a Fourier analytical approach that achieves effective bias reduction. Asymptotic normality and efficiency are proved when the smoothness index of $f$ is above certain threshold which was discovered recently by Koltchinskii et. al. (2018) for a Hölder type class. Numerical simulations are presented to validate our analysis. The simplicity of implementation and its superiority over the plug-in approach indicate the new estimator can be applied to a broad range of real world applications.