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在单变量随机设计模型中，Nadaraya-Watson估计量用于估计条件期望函数。假设一些正则性条件（如平滑性条件和带宽选择条件）满足，Nadaraya-Watson估计量在渐进意义下是正态分布的。

具体来说，Nadaraya-Watson估计量 (\hat{m}_h(x)) 的渐进分布形式可以表示为：

[ \sqrt{n h} \left( \hat{m}_h(x) - m(x) \right) \xrightarrow{d} N\left(0, \frac{\sigma^2(x)}{f(x) \int K^2(u) , du}\right) ]

其中：

• (n) 是样本量。
• (h) 是带宽参数。
• (m(x)) 是真实的条件期望函数。
• (\sigma^2(x)) 是误差项的条件方差。
• (f(x)) 是自变量的边际密度函数。
• (K(u)) 是核函数。

这个结果表明，经过适当的标准化后，Nadaraya-Watson估计量的误差项在渐进意义下服从均值为0的正态分布，其方差与核函数、带宽、样本量等因素有关。

在单变量随机设计模型和一些正则条件下，我们可以讨论Nadaraya-Watson估计量(\hat{m}_h(x))的渐近正态性。

Nadaraya-Watson估计量是一种用于非参数回归的核平滑方法。其基本思想是通过加权平均来估计回归函数的值，其中权重由核函数和带宽参数决定。

在讨论其渐近正态性时，我们通常需要假设一些正则条件，例如：

1. 核函数(K)是一个连续的对称函数，并且满足某些积分条件。
2. 带宽参数(h)随着样本量(n)的增加而趋于零，但不能太快趋于零。通常要求(h \to 0)且(nh \to \infty)。
3. 回归函数(m(x))在点(x)处具有某种光滑性，例如连续可微。

在这些条件下，可以证明Nadaraya-Watson估计量(\hat{m}_h(x))在渐近上服从正态分布。具体来说，当样本量(n)趋于无穷大时，估计量的分布可以近似为一个正态分布，其均值为真实的回归函数值(m(x))，方差则与核函数、带宽以及数据的分布有关。

这种渐近正态性为我们提供了一个理论基础，使得我们可以在大样本情况下对估计量进行推断，例如构建置信区间或进行假设检

The Nadaraya-Watson estimator is a popular nonparametric method for estimating the conditional expectation of a random variable. In the context of a univariate random design model, we can denote the estimator as (\hat{m}_h(x)), which estimates the conditional mean (m(x) = E[Y | X = x]) based on a sample of observations ((X_i, Y_i)) for (i = 1, \ldots, n).

Nadaraya-Watson Estimator

The Nadaraya-Watson estimator is defined as:

[ \hat{m}h(x) = \frac{\sum{i=1}^n K_h(X_i - x) Y_i}{\sum_{i=1}^n K_h(X_i - x)} ]

where:

• (K_h(u) = \frac{1}{h} K\left(\frac{u}{h}\right)) is a kernel function scaled by a bandwidth (h),
• (K(u)) is a kernel function (e.g., Gaussian, Epanechnikov) that integrates to 1,
• (h) is the bandwidth parameter that controls the smoothness of the estimator.

Asymptotic Normality

To discuss the asymptotic normality of the Nadaraya-Watson estimator, we need to consider the following regularity conditions:

1. Kernel Conditions: The kernel function (K(u)) is symmetric, integrates to 1, and has a bounded support. Additionally, (K(u)) should have a finite second moment.

2. Bandwidth Condition: The bandwidth (h) should satisfy (h \to 0) as (n \to \infty) and (nh \to \infty). This ensures that the estimator becomes more localized around the point (x) as the sample size increases.

3. Smoothness of the True Function: The true conditional mean function (m(x)) should be sufficiently smooth (e.g., continuous and differentiable) in the neighborhood of (x).

4. Independence and Identically Distributed (i.i.d.) Samples: The observations ((X_i, Y_i)) should be drawn from a joint distribution that is i.i.d.

Asymptotic Distribution

Under these conditions, the asymptotic distribution of the Nadaraya-Watson estimator can be derived. Specifically, as (n \to \infty), the estimator (\hat{m}_h(x)) converges in distribution to a normal distribution:

[ \sqrt{nh} \left( \hat{m}_h(x) - m(x) \right) \xrightarrow{d} N(0, \sigma^2(x)) ]

where:

• (\sigma^2(x)) is the asymptotic variance of the estimator, which can be expressed as:

[ \sigma^2(x) = \frac{1}{\int K^2(u) du} \cdot \text{Var}(Y | X = x) \cdot m'(x)^2 ]

This variance captures the variability of the estimator around the true conditional mean (m(x)) and depends on the choice of the kernel and the bandwidth.

Conclusion

In summary, the Nadaraya-Watson estimator (\hat{m}_h(x)) is asymptotically normal under certain regularity conditions. The convergence to a normal distribution allows for the construction of confidence intervals and hypothesis tests regarding the conditional mean function (m(x)). The choice of bandwidth (h) and kernel (K) plays a crucial role in the performance of the estimator, influencing both its bias and

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在单变量随机设计模型下，Nadaraya-Watson估计器是一种常用的非参数回归方法。为了讨论Nadaraya-Watson估计器(\hat{m}_h(x))的渐近正态性，我们需要考虑一些基本的假设和正则性条件。

首先，Nadaraya-Watson估计器的定义为：

[ \hat{m}h(x) = \frac{\sum{i=1}^{n} K_h(x - X_i) Y_i}{\sum_{i=1}^{n} K_h(x - X_i)} ]

其中，(K_h)是核函数，(h)是带宽参数，(X_i)是自变量，(Y_i)是因变量。

渐近正态性的条件

1. 核函数的选择：核函数(K)通常需要满足一些条件，例如是对称的、非负的，并且在某个区间内积分为1。此外，核函数的光滑性也会影响估计器的性质。

2. 带宽的选择：带宽(h)的选择对估计器的表现至关重要。通常要求(h)随着样本量(n)的增大而趋近于0，但要保证(nh)趋近于无穷大，以确保估计的稳定性。

3. 样本独立同分布：假设样本((X_i, Y_i))是独立同分布的，这样可以确保估计器的无偏性和一致性。

4. 光滑性条件：假设真实的回归函数(m(x))在某个邻域内是光滑的，这样可以保证估计器在该点附近的收敛性。

渐近正态性结果

在满足上述条件的情况下，可以证明Nadaraya-Watson估计器(\hat{m}_h(x))在样本量趋近于无穷大时，具有渐近正态性。具体来说，存在常数(C)和一个适当的标准差(\sigma_h(x))，使得：

[ \sqrt{n}(\hat{m}_h(x) - m(x)) \xrightarrow{d} N(0, \sigma_h^2(x)) ]

这里，(\xrightarrow{d})表示分布收敛，(N(0, \sigma_h^2(x)))表示均值为0、方差为(\sigma_h^2(x))的正态分布。

结论

综上所述，在适当的正则性条件下，Nadaraya-Watson估计器在样本量趋近于无穷大时，能够收敛到一个正态分布。这一结果为非参数回归分析提供了理论基础，使得我们可以在实际应用中对估计结果进行推断和置信区间的构建。

The Nadaraya-Watson estimator is a popular nonparametric method for estimating the conditional expectation of a random variable. In the context of a univariate random design model, the Nadaraya-Watson estimator for the conditional mean ( m(x) = E[Y | X = x] ) is defined as:

[ \hat{m}h(x) = \frac{\sum{i=1}^n Y_i K\left(\frac{X_i - x}{h}\right)}{\sum_{i=1}^n K\left(\frac{X_i - x}{h}\right)} ]

where:

• ( K(\cdot) ) is a kernel function (typically a symmetric and non-negative function that integrates to one),
• ( h ) is the bandwidth parameter,
• ( (X_i, Y_i) ) are the observed data points.

Asymptotic Normality

To discuss the asymptotic normality of the Nadaraya-Watson estimator ( \hat{m}_h(x) ), we need to consider the following regularity conditions:

1. Kernel Function: The kernel ( K(\cdot) ) is a bounded, symmetric function with ( \int K(u) du = 1 ) and ( \int u K(u) du = 0 ). Additionally, ( K(u) ) should have a finite second moment.

2. Bandwidth Condition: The bandwidth ( h ) should satisfy ( h \to 0 ) as ( n \to \infty ) and ( nh \to \infty ). This ensures that the estimator becomes more localized around the point ( x ) as the sample size increases.

3. Smoothness of the True Function: The true regression function ( m(x) ) should be sufficiently smooth (e.g., Lipschitz continuous) in a neighborhood of ( x ).

4. Independence and Identically Distributed (i.i.d.) Samples: The observations ( (X_i, Y_i) ) are assumed to be i.i.d. from some joint distribution.

Asymptotic Distribution

Under these conditions, the asymptotic distribution of the Nadaraya-Watson estimator can be derived. Specifically, as ( n \to \infty ), the estimator ( \hat{m}_h(x) ) converges in distribution to a normal distribution:

[ \sqrt{nh} \left( \hat{m}_h(x) - m(x) \right) \xrightarrow{d} N(0, \sigma^2(x)) ]

where ( \sigma^2(x) ) is the asymptotic variance of the estimator, which can be expressed as:

[ \sigma^2(x) = \frac{1}{\left( \int K(u) du \right)^2} \cdot \left( \int K^2(u) du \right) \cdot m'(x)^2 + \frac{1}{\int K(u) du} \cdot \text{Var}(Y | X = x) ]

Interpretation

• The term ( \sqrt{nh} ) indicates that the estimator's convergence rate is influenced by both the sample size ( n ) and the bandwidth ( h ).
• The variance ( \sigma^2(x) ) captures the variability of the estimator around the true mean function ( m(x) ) and depends on the choice of the kernel and the underlying distribution of the data.

Conclusion

In summary, under appropriate regularity conditions, the Nadaraya-Watson estimator exhibits asymptotic normality, allowing for inference about the conditional mean function ( m(x) ). This property is particularly useful in nonparametric statistics, as it provides a foundation for constructing confidence intervals and hypothesis tests based on the estimator.

要推导方差 ( \text{Var}(\hat{f}_h(x)) ) 的表达式，我们需要理解一些基本的概念和步骤。这里的 ( \hat{f}_h(x) ) 通常是指核密度估计（Kernel Density Estimation, KDE）中的估计值。我们将逐步推导这个方差的表达式。

核密度估计

核密度估计的形式为：

[ \hat{f}h(x) = \frac{1}{n} \sum{i=1}^{n} K_h(x - X_i) ]

其中 ( K_h(u) = \frac{1}{h} K\left(\frac{u}{h}\right) ) 是缩放后的核函数，( K ) 是一个对称的核函数，( h ) 是带宽参数，( X_i ) 是样本点。

方差的推导

1. 计算期望： [ \mathbb{E}[\hat{f}h(x)] = \mathbb{E}\left[\frac{1}{n} \sum{i=1}^{n} K_h(x - X_i)\right] = \frac{1}{n} \sum_{i=1}^{n} \mathbb{E}[K_h(x - X_i)] ] 在大样本情况下，假设 ( X_i ) 是独立同分布的，且 ( K_h(x - X_i) ) 的期望可以通过积分得到。

2. 计算方差： [ \text{Var}(\hat{f}h(x)) = \text{Var}\left(\frac{1}{n} \sum{i=1}^{n} K_h(x - X_i)\right) ] 由于 ( K_h(x - X_i) ) 是独立同分布的，我们可以使用方差的性质： [ \text{Var}(\hat{f}h(x)) = \frac{1}{n^2} \sum{i=1}^{n} \text{Var}(K_h(x - X_i)) = \frac{1}{n} \text{Var}(K_h(x - X_1)) ]

3. 计算 ( \text{Var}(K_h(x - X_1)) )： 通过计算 ( K_h(x - X_1) ) 的方差，我们可以得到： [ \text{Var}(K_h(x - X_1)) = \mathbb{E}[K_h(x - X_1)^2] - \left(\mathbb{E}[K_h(x - X_1)]\right)^2 ] 这里的 ( \mathbb{E}[K_h(x - X_1)^2] ) 可以通过积分得到。

4. 最终结果： 结合上述步骤，我们可以得到方差的最终形式： [ \text{Var}(\hat{f}_h(x)) = \frac{1}{n} \int K^2(u) du + O\left(\frac{1}{n} + nh^{d-2}\right) ] 其中 ( O\left(\frac{1}{n} + nh^{d-2}\right) ) 是高阶项，表示在带宽 ( h ) 较小或样本量 ( n ) 较大时的影响。

结论

通过以上步骤，我们推导出了核密度估计的方差表达式。这个过程涉及到期望和方差的计算，以及对核函数的性质的利用。希望这个推导过程对你理解方差的来源

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