Asymptotic equivalence of density estimation and Gaussian white noise

研究问题: 独立同分布的密度观测实验，能否在严格的统计决策意义下近似为高斯白噪声实验，从而使非参数风险分析能够迁移？
方法与思路: 作者用 Le Cam 亏损距离刻画实验等价，对光滑且一致远离零的密度建立白噪声近似，证明使用与 Hungarian 构造相关的经验过程技术。
材料陈述的贡献: 摘要称 Hölder 指数大于二分之一且密度一致远离零时，密度观测与平方根密度漂移的白噪声实验全局渐近等价，支持有界损失下的决策迁移。
阅读关注点: 应核对参数空间的一致性限制、等价距离的定义与证明构造，以及从白噪声模型转移风险界时损失函数和估计程序的要求。
阅读边界: 仅据版本未注明的索引摘要；不能把结论扩展到低光滑度、允许密度为零或任意无界损失的情形，也不能据此判断有限样本等价。

GPT-6.1 Sol · 高 · 2026-10-05

原文摘要

Signal recovery in Gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation, having all the essential traits in a pure form. The equivalence has mostly been stated informally, but an approximation in the sense of Le Cam's deficiency distance $\Delta$ would make it precise. The models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. In nonparametrics, a first result of this kind has recently been established for Gaussian regression. We consider the analogous problem for the experiment given by n i.i.d. observations having density f on the unit interval. Our basic result concerns the parameter space of densities which are in a Hölder ball with exponent $\alpha > 1/2$ and which are uniformly bounded away from zero. We show that an i. i. d. sample of size n with density f is globally asymptotically equivalent to a white noise experiment with drift $f^{1/2}$ and variance $1/4 n^{-1}$ . This represents a nonparametric analog of Le Cam's heteroscedastic Gaussian approximation in the finite dimensional case. The proof utilizes empirical process techniques related to the Hungarian construction. White noise models on f and log f are also considered, allowing for various "automatic" asymptotic risk bounds in the i.i.d. model from white noise.Signal recovery in Gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation, having all the essential traits in a pure form. The equivalence has mostly been stated informally, but an approximation in the sense of Le Cam's deficiency distance $\Delta$ would make it precise. The models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. In nonparametrics, a first result of this kind has recently been established for Gaussian regression. We consider the analogous problem for the experiment given by n i.i.d. observations having density f on the unit interval. Our basic result concerns the parameter space of densities which are in a Hölder ball with exponent $\alpha > 1/2$ and which are uniformly bounded away from zero. We show that an i. i. d. sample of size n with density f is globally asymptotically equivalent to a white noise experiment with drift $f^{1/2}$ and variance $1/4 n^{-1}$. This represents a nonparametric analog of Le Cam's heteroscedastic Gaussian approximation in the finite dimensional case. The proof utilizes empirical process techniques related to the Hungarian construction. White noise models on f and log f are also considered, allowing for various "automatic" asymptotic risk bounds in the i.i.d. model from white noise.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）

笔记与分组

分类依据与纠错

Asymptotic equivalence of density estimation and Gaussian white noise

代码、数据与补充材料

密度估计与高斯白噪声的渐近等价性

原文摘要

同方向的基础文献

相关领域论文

Testing Cumulative Treatment Effects in Continuous Time with ODE Dynamics and Noisy Observations

On Local Inference for Functionals of Ill-Posed Inverse Problems

Inverse Probability Weighting-based Mediation Analysis for Microbiome Data

You've Got to be Efficient: Ambiguity, Misspecification and Variational Preferences

Sharp Asymptotic Theory of Maximum Likelihood Estimation for Gaussian Processes with an RBF Kernel

PICPIs: Prediction-Interval-Conditional Prediction Intervals