All Questions
Tagged with nonparametric kernel-smoothing
93
questions
0
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21
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Local linear kernel regression
It is know that the prediction for a given point $x$ is given by:
$$\hat{f}_h(x) = \hat{\beta}_0(x)$$
where
$$\hat{\beta}(x) = \arg\min_{\beta_0, \beta_1}\sum_{i=1}^nK\left(\frac{x - x_i}{h}\right)(...
- 145
1
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0
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40
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How to show $\sup_{x\in [a,b]}|f_n(x)-f(x)|=O_p(\sqrt{\frac{\log n}{nh}}+h^2)$ when the kernel $K(\cdot) $ is of bounded variation?
Consider the kernel estimate $f_n$ of a real univariate density defined by $$f_n(x)=\sum_{i=1}^{n}(nh)^{-1}K\left\{h^{-1}(x-X_i)\right\}$$
where $X_1,...,X_n$ are independent and identically ...
- 31
0
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36
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Implementing Convolution Function for Gaussian Kernel in Python for PDF Estimation
I am currently working on estimating a probability density function (PDF) nonparametrically using a Gaussian kernel. My goal is to determine the optimal bandwidth $h$ that minimizes the cross-...
- 273
1
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0
answers
32
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Strong consistency of kernel density estimator
I am studying the book Nonparametric and Semiparametric Models written by Wolfgang Hardle and have difficulty with the following exercise:
$\textbf{Exercise 3.13}$ Show that $\hat{f_h}^{(n)}(x) \...
- 111
2
votes
1
answer
56
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Gronwall's inequality
I am reading the article.
I am getting stuck with the first proof proposition 4 on page 32.
To be more specific, they understood the reason why they obtained $F(x) \le \frac{2K}{1-\frac{2R\epsilon}{\...
- 121
4
votes
1
answer
178
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Proving that the bias of the derivative of Parzen-Rosenblatt (kernel density) estimator is of order $O(h^2) $ and $O(h)$ when $h$ approaches $0$
I came across this property that I don't get and I couldn't find the proof anywhere:
Suppose we have a density $K$ of the standard normal distribution and $K'$ its derivative. Suppose that the density ...
- 241
2
votes
0
answers
133
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Propensity score non parametric estimation
In several papers, in the 'double machine learning' literature, the propensity score (a nuisance parameter) is estimated non parametrically. It is a bit unclear how this estimation is performed, as ...
- 111
1
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0
answers
127
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Gasser Müller estimator for estimating the derivative $m'(x)$ of a nonparametric regression function
I would like to compare the performance of the Gasser Müller estimator with other estimators for estimating the the derivative $m'(x)$ of the regression function $m(x)$.
Let's say we have the ...
- 111
1
vote
0
answers
38
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Maximum bias for NW estimator when $r(x)$ is Lipschitz (question 17, chapter 5 All of Non-Parametric Statistics)
The general condition is that $Y_i = r(X_i) + \epsilon_i$, and we want to estimate $r$ using Nadaraya–Watson kernel regression.
We additionally assume $r\colon [0,1] \to \mathbb{R}$ is lipschitz, so $|...
- 636
1
vote
0
answers
250
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Bias of kernel density estimator of pdf $f$, where $f$ has bounded first derivative $f'$
Let's say the kernel density estimator is given by
$$\hat f(x) = \frac{1}{nh_n} \sum_{i=1}^n K\left(\frac{X_i-x}{h_n}\right),$$ where $h_n \to 0$, $nh_n \to \infty$, $K$ a symmetric probability ...
- 636
0
votes
0
answers
40
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Kernel Density Estimator: Misunderstanding in Taylor Series and the bias of KDE [duplicate]
Let's say the kernel density estimator is given by
$\hat f(x) = \frac{1}{nh_n} \sum_{i=1}^n K(\frac{X_i-x}{h_n})$, where $h_n \to 0$, $nh_n \to \infty$, $K$ a symmetric probability distribution ...
- 636
1
vote
0
answers
31
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Closeness of two estimators of median under non parametric setup in a large sample situation
Median Regression under non-parametric set-up (Nadaraya Watson Estimate)
Data: $\{(Y_i,X_i):1\le i\le n\}$
Interested in estimating $\phi(x)=\text{median}(Y|X=x).$
Possible estimates are
Minimize the ...
- 385
2
votes
0
answers
211
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Question regarding Kernel Density Estimation bandwidth selection (Scott's rule)
I'm studying KDE and got trouble understanding Scott's rule or Silverman's rule for bandwidth selection.
I saw that the optimal bandwidth is the value that minimizes Mean Integrated Squared Error (...
- 43
3
votes
0
answers
479
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Pros and cons of Nadaraya–Watson estimator vs. RKHS method?
Recently I've been reading some materials about nonparametric methods. Two methods related to the word "kernel" rasied my interest-- Nadaraya–Watson estimator and RKHS method.
What's the ...
- 31
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votes
0
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50
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How to prove symmetry of a Uniform kernel?
I am trying to prove this kernel is valid,
$$
K(x) = \frac{1}{2}I(-1 < x < 1)
$$
So far I can integrate to 1, but how do I prove $$k(x) = k(-x)$$
Also, how do we satisfy that k(x) is $\ge$ 0 for ...
