提出新自适应带宽选择法，实现局部多项式回归的最优逼近。
A new approach to locally adaptive polynomial regression
- 基于ℓ₂范数投影构建带宽选择准则，区别于传统方差估计方法。
- 在任意点上近似最优适应局部霍尔德指数，风险有非渐近界保证。
- 仅需一个全局调参即可实现全域自适应，适合理论研究者参考。
自适应带宽选择是非参数回归中的核心挑战。本文提出一种受ℓ₀-正则化回归最优性准则启发的新带宽选择方法。虽与Lepski方法精神相似，即选取满足可接受性条件的最大区间，但本方法基于区间投影的ℓ₂-范数准则，而非显式点估计与方差估计。我们为基于该带宽选择的局部多项式回归方法建立了非渐近风险界，证明其能在所有定义域点上近似最优地适应底层回归函数的局部霍尔德指数。此外，我们表明在每种情形下均存在单一理想全局调参，使上述局部自适应性质成立。方法的最优风险源于一类新的‘带宽选择方程’的解，该方程本身具有独立研究价值。我们认为本方法为经典且持续相关的局部自适应非参数回归问题提供了新视角。
原文摘要 · Abstract (English)
Adaptive bandwidth selection is a fundamental challenge in nonparametric regression. This paper introduces a new bandwidth selection procedure inspired by the optimality criteria for $\ell_0$-penalized regression. Although similar in spirit to Lepski's method and its variants in selecting the largest interval satisfying an admissibility criterion, our approach stems from a distinct philosophy, utilizing criteria based on $\ell_2$-norms of interval projections rather than explicit point and variance estimates. We obtain non-asymptotic risk bounds for the local polynomial regression methods based on our bandwidth selection procedure which adapt (near-)optimally to the local Hölder exponent of the underlying regression function simultaneously at all points in its domain. Furthermore, we show that there is a single ideal choice of a global tuning parameter in each case under which the above-mentioned local adaptivity holds. The optimal risks of our methods derive from the properties of solutions to a new ``bandwidth selection equation'' which is of independent interest. We believe that the principles underlying our approach provide a new perspective to the classical yet ever relevant problem of locally adaptive nonparametric regression.
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