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arXiv 1409.0733math.STstat.TH
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核光滑积分逼近

Integral approximation by kernel smoothing

Bernard Delyon, François Portier

备注：
Published at http://dx.doi.org/10.3150/15-BEJ725 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm). arXiv admin note: text overlap with arXiv:1312.4497
发表信息：
Bernoulli 2016, Vol. 22, No. 4, 2177-2208
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摘要：

令$(X_1,\ldots,X_n)$为i.i.d. $\mathbb{R}^d$、$d\geq 1$中的随机变量序列。我们证明了，对于任何函数$φ:\mathbb{R}^d\rightarrow\mathbb{R}$，在正则性条件下，\[n^ {1/2}\Biggl(n^{-1}\sum_{i=1}^n\frac{φ(X_i)}{\widehat{f}^(X_i)}- \int φ(x)\,dx\Biggr)\stackrel{\mathbb{P}}{\longrightarrow}0,\]，其中$\widehat{f}$是$X_1$密度的经典核估计。这个结果是惊人的，因为它加快了传统的速度，在根$n$，来自中心极限定理时，$\widehat{f}=f$。虽然本文突出了一些应用，我们主要解决与后一个结果相关的理论问题。我们推导出概率收敛速度的上界。这些界限取决于函数$φ$和$f$的正则性、维度$d$和核估计器$\widehat{f}$的带宽。此外，他们被证明是准确的，因为他们被用来作为重正化序列在两个中心极限定理，每个反映不同程度的光滑$φ$。作为随机设计回归建模的一个应用，我们给出了回归函数线性泛函估计的渐近正态性。作为上述结果的结果，渐近方差不依赖于回归函数。最后，我们讨论了积分近似带宽的选择，并通过模拟强调了我们的程序的良好行为。

英文摘要：

Let $(X_1,\ldots,X_n)$ be an i.i.d. sequence of random variables in $\mathbb{R}^d$, $d\geq 1$. We show that, for any function $φ:\mathbb{R}^d\rightarrow\mathbb{R}$, under regularity conditions, \[n^ {1/2}\Biggl(n^{-1}\sum_{i=1}^n\frac{φ(X_i)}{\widehat{f}^(X_i)}- \int φ(x)\,dx\Biggr)\stackrel{\mathbb{P}}{\longrightarrow}0,\] where $\widehat{f}$ is the classical kernel estimator of the density of $X_1$. This result is striking because it speeds up traditional rates, in root $n$, derived from the central limit theorem when $\widehat{f}=f$. Although this paper highlights some applications, we mainly address theoretical issues related to the later result. We derive upper bounds for the rate of convergence in probability. These bounds depend on the regularity of the functions $φ$ and $f$, the dimension $d$ and the bandwidth of the kernel estimator $\widehat{f}$. Moreover, they are shown to be accurate since they are used as renormalizing sequences in two central limit theorems each reflecting different degrees of smoothness of $φ$. As an application to regression modelling with random design, we provide the asymptotic normality of the estimation of the linear functionals of a regression function. As a consequence of the above result, the asymptotic variance does not depend on the regression function. Finally, we debate the choice of the bandwidth for integral approximation and we highlight the good behavior of our procedure through simulations.

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