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Asymptotic Normality of Nadaraya-Watson Estimator in Univariate Models

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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GPT-4o Mini
Asymptotic Normality of Nadaraya-Watson Estimator

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。

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GPT-4o Mini
Asymptotic Normality of the Nadaraya-Watson Estimator

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.

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GPT-4o Mini
Derivation of Variance in Kernel Density Estimation

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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FAQ
What is GPT-4o Mini?

GPT-4o Mini is a streamlined version of the GPT-4o model, designed to be more efficient with fewer computational resources while still offering strong language processing capabilities.

How does GPT-4o Mini differ from the full GPT-4o model?

GPT-4o Mini has fewer parameters compared to GPT-4o, which makes it faster and less resource-intensive. While GPT-4o Mini is efficient and cost-effective, GPT-4o provides deeper contextual understanding and handles more complex queries with greater accuracy.

How can I use GPT-4o Mini?

You can start using GPT-4o Mini directly on our website GPT4oMini.app for text generation and conversations. No registration or software download is needed.

Is GPT4oMini.app free or paid?

Currently, GPT4oMini.app is available for free. We plan to introduce a paid version in the future, which will offer additional features and access to more advanced models, including the full GPT-4o.

What features will the paid version include?

The paid version will provide access to more GPT models beyond GPT-4o Mini, including the full GPT-4o, and support for context-aware conversations with enhanced capabilities.

What are the main advantages of using GPT-4o Mini over GPT-4o?

The main advantages of using GPT-4o Mini include faster response times and lower resource consumption, making it more suitable for applications with high request volumes or limited resources. It is also more cost-effective compared to GPT-4o, which may be beneficial for budget-conscious projects.

Can GPT-4o Mini handle long conversations or complex queries effectively?

While GPT-4o Mini is efficient and capable of handling many tasks well, it may not perform as effectively as GPT-4o for very long conversations or highly complex queries. GPT-4o offers more advanced contextual understanding and can better manage intricate discussions.

How fast are GPT-4o Mini’s response times compared to GPT-4o?

GPT-4o Mini generally offers faster response times due to its optimized architecture. In contrast, GPT-4o may have longer response times but provides more detailed and nuanced responses.

How can I provide feedback or report issues?

If you encounter any issues or have suggestions, please submit feedback through the email [email protected]. Your input is valuable for improving our services.

Does GPT-4o Mini support multiple languages like GPT-4o?

Yes, GPT-4o Mini supports multiple languages, but the quality may vary compared to GPT-4o. GPT-4o typically offers higher accuracy and better performance across a wider range of languages.