论文标题
在凸度约束模型中推断本地参数
Inference for local parameters in convexity constrained models
论文作者
论文摘要
我们考虑了基于从标准非参数回归模型的观测值，使用凸最小二乘估计器（LSE）$ \ wideHat {f} _n $，我们考虑了凸回归函数的局部参数的推断问题：[0,1] \ to \ mathbb {r} $。对于$ x_0 \ in（0,1）$，本地参数包括点函数函数值$ f_0（x_0）$，点式导数$ f_0'（x_0）$和unti-mode（即最小的最小化器）的$ f_0 $。估计错误的现有限制分布$（\ wideHat {f} _n（x_0） - f_0（x_0），\ wideHat {f} _n'（x_0） - f_0） - f_0'（x_0））$取决于未知的第二个衍生级$ f_0'（x_0）'（x_0）$，因此并不适用于unfection。为了避免这种僵局，我们表明以下本地归一化错误（LNE）享有关键限制行为：让$ [\ wideHat {u}（x_0）（x_0），\ wideHat {v}（x_0）] $是包含$ x_0 $ x_0 $ x_0 $ x_0 $ x_0 $的最大间隔。然后，在标准条件下，$$ \ binom {\ sqrt {n（\ wideHat {v}（x_0） - \ wideHat {u}（x_0））}（\ wideHat {f} {f} _n（x_0）-f_0（x_0（x_0）}} { \ sqrt {n（\ wideHat {v}（x_0） - \ wideHat {u}（x_0））^3}（\ wideHat {f} _n'（x_0）-f_0）-f_0）-f_0'（x_0））} \ rightquigaROWσ\ cdot \ binom {\ Mathbb {l}^{（0）} _ 2} {\ Mathbb {l}^{（1）_ 2}，$ n $是样本大小，$σ$，$σ$是错误的标准偏差，并且\ mathbb {l}^{（1）} _ 2 $是通用随机变量。这种渐近关键的LNE理论立即产生了一个简单的无调过程，用于构建具有渐近确切覆盖范围和最佳长度的顺式，$ f_0（x_0）$和$ f_0'（x_0）$。我们还为$ f_0 $的反模式构建了一个渐近的关键LNE，其限制分布甚至不取决于$σ$。这些渐近关键的LNE理论将进一步扩展到其他凸度/凹度约束模型（例如，对数 - 孔隙密度估计），该模型可用于特定于问题的估计器。
We consider the problem of inference for local parameters of a convex regression function $f_0: [0,1] \to \mathbb{R}$ based on observations from a standard nonparametric regression model, using the convex least squares estimator (LSE) $\widehat{f}_n$. For $x_0 \in (0,1)$, the local parameters include the pointwise function value $f_0(x_0)$, the pointwise derivative $f_0'(x_0)$, and the anti-mode (i.e., the smallest minimizer) of $f_0$. The existing limiting distribution of the estimation error $(\widehat{f}_n(x_0) - f_0(x_0), \widehat{f}_n'(x_0) - f_0'(x_0) )$ depends on the unknown second derivative $f_0''(x_0)$, and is therefore not directly applicable for inference. To circumvent this impasse, we show that the following locally normalized errors (LNEs) enjoy pivotal limiting behavior: Let $[\widehat{u}(x_0), \widehat{v}(x_0)]$ be the maximal interval containing $x_0$ where $\widehat{f}_n$ is linear. Then, under standard conditions, $$\binom{ \sqrt{n(\widehat{v}(x_0)-\widehat{u}(x_0))}(\widehat{f}_n(x_0)-f_0(x_0)) }{ \sqrt{n(\widehat{v}(x_0)-\widehat{u}(x_0))^3}(\widehat{f}_n'(x_0)-f_0'(x_0))} \rightsquigarrow σ\cdot \binom{\mathbb{L}^{(0)}_2}{\mathbb{L}^{(1)}_2},$$ where $n$ is the sample size, $σ$ is the standard deviation of the errors, and $\mathbb{L}^{(0)}_2, \mathbb{L}^{(1)}_2$ are universal random variables. This asymptotically pivotal LNE theory instantly yields a simple tuning-free procedure for constructing CIs with asymptotically exact coverage and optimal length for $f_0(x_0)$ and $f_0'(x_0)$. We also construct an asymptotically pivotal LNE for the anti-mode of $f_0$, and its limiting distribution does not even depend on $σ$. These asymptotically pivotal LNE theories are further extended to other convexity/concavity constrained models (e.g., log-concave density estimation) for which a limit distribution theory is available for problem-specific estimators.
