All Questions
Tagged with nonparametric kernel-smoothing
93
questions
0
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0
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26
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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
$$\hat{\beta}(x) = \arg\min_{\beta_0, \beta_1}\sum_{i=1}^nK\left(\frac{x - x_i}{h}\right)(...
- 145
1
vote
0
answers
40
views
How to show $\sup_{x\in [a,b]}|f_n(x)-f(x)|=O_p(\sqrt{\frac{\log n}{nh}}+h^2)$ when the kernel $K(\cdot) $ is of bounded variation?
Consider the kernel estimate $f_n$ of a real univariate density defined by $$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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0
answers
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
vote
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
188
views
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
134
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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
vote
0
answers
128
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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
251
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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),$$ where $h_n \to 0$, $nh_n \to \infty$, $K$ a symmetric probability ...
- 636
0
votes
0
answers
40
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Kernel Density Estimator: Misunderstanding in Taylor Series and the bias of KDE [duplicate]
Let's say the kernel density estimator is given by
$\hat f(x) = \frac{1}{nh_n} \sum_{i=1}^n K(\frac{X_i-x}{h_n})$, where $h_n \to 0$, $nh_n \to \infty$, $K$ a symmetric probability distribution ...
- 636
1
vote
0
answers
31
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Closeness of two estimators of median under non parametric setup in a large sample situation
Median Regression under non-parametric set-up (Nadaraya Watson Estimate)
Data: $\{(Y_i,X_i):1\le i\le n\}$
Interested in estimating $\phi(x)=\text{median}(Y|X=x).$
Possible estimates are
Minimize the ...
- 385
2
votes
0
answers
215
views
Question regarding Kernel Density Estimation bandwidth selection (Scott's rule)
I'm studying KDE and got trouble understanding Scott's rule or Silverman's rule for bandwidth selection.
I saw that the optimal bandwidth is the value that minimizes Mean Integrated Squared Error (...
- 43
3
votes
0
answers
479
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Pros and cons of Nadaraya–Watson estimator vs. RKHS method?
Recently I've been reading some materials about nonparametric methods. Two methods related to the word "kernel" rasied my interest-- Nadaraya–Watson estimator and RKHS method.
What's the ...
- 31
0
votes
0
answers
50
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How to prove symmetry of a Uniform kernel?
I am trying to prove this kernel is valid,
$$
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 ...
2
votes
1
answer
737
views
Is Kernel-Regression parametric or non-parametric?
As the title says, is kernel regression a parametric or non-parametric method, and how can this be motivated/explained?
- 35
3
votes
1
answer
217
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How to calculate the expectation of the KDE using little-o?
This is possibly a duplicate of this question of mine, however, here I ask for clarification regarding an estimation that is done when calculating the expectation of the kernel density estimator (KDE) ...
- 249
6
votes
1
answer
643
views
Basic properties of the kernel density estimator
This is a question from a mathematical statistics textbook, used at the first and most basic mathematical statistics course for undergraduate students. This exercise follows the chapter on ...
- 249
1
vote
0
answers
96
views
Time Varying Coefficient Model with Uniform Kernel and Spline Estimator
I'm working on the BMACS data set data(BMACS) from library(npmlda).
I'm looking at the the time-varying coefficient model of post-CD4 versus smoking $X_1$, pre-HIV CD4 percent $X_2$ (centered) and age ...
- 25
1
vote
0
answers
274
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histogram vs. kernel in density estimation
Assume we have a problem of estimation of a density $f(x)$ over an interval $[0, 1]$. Can a regular histogram (i.e. with equal-sized bins) be viewed as some kind of a kernel?
- 676
2
votes
1
answer
115
views
Density plot with epanechnikov with exceedance data
I'm trying to replicate empirical density plot from the paper "Computing Maximum Likelihood Estimates for the Generalized Pareto Distribution".
The data is ...
- 8,445
0
votes
0
answers
31
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Biase of ASE estimation Kernel Regression
I'm trying to calculate the bias of the estimator $p(h)=n^{-1}\displaystyle\sum_{i=1}^{n}(Y_{j}-\hat{m}_{h}(X_{j})^{2}w(X_{j})$ of the averaged squared error. The result I find in the literature is ...
- 3
2
votes
0
answers
53
views
Linear model with partially time-varying coefficients
Suppose we have a linear model with time-varying coefficients
$$
y_i = x_i' \beta_{t_i} + \epsilon_i, \; i = 1, 2, \cdots, n
$$
where the design points are $t_i = \frac{i}{n}$, and $\beta(t): [0,1] \...
- 3,348
3
votes
1
answer
150
views
How to detect this change-point?
Let $Y\in\{0,1\}$ be a binary random variable. Let $P(x_1,x_2)=E(Y|x_1,x_2)$. By definition we have $0 \leq P(x_1,x_2) \leq 1$. Suppose $P(x_1,x_2)$ is strictly monotone in $x_1,x_2$ and $P(x_1,x_2)=1$...
- 2,966
2
votes
1
answer
2k
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Derivation of variance for kernel density estimator
My question refers to the book "Nonparametric Econometrics - Theory and Practice" by Li & Racine. Here, the variance for a kernel density estimator using the pointwise perspective (for fixed x) is ...
- 115
1
vote
1
answer
353
views
How Parzen window density estimate $f_n$ converges to f
I am trying to understand how Parzen window density estimate converges to actual density function f(x).[Actually i am trying to learn machine learning on my own using available free resources. Please ...
0
votes
0
answers
14
views
Doubt in kernel based method - unit hypercube(Parzan window estimate)
I recently started studying pattern recognition on my own. Please clarify me the following.
https://books.google.co.in/books?id=T0S0BgAAQBAJ&pg=PA53&lpg=PA53&dq=hypercube+of+side+h&...
2
votes
1
answer
839
views
Convergence of kernel density estimate as the sample size grows
Let $X\sim\text{Normal}(0,1)$ and let $f_X$ be its probability density function. I conducted some numerical experiments in the software Mathematica to estimate $f_X$ via a kernel method. Let $\hat{f}...
- 285
2
votes
1
answer
197
views
Kernel Estimation to Estimate Treatment Effect
I am trying to determine whether an estimator I came up with is just a non-parametric kernel estimator. I am performing a simulation study to estimate a treatment effect that I impose on my data.
My ...
- 497
3
votes
1
answer
100
views
how to understand this math formula for bandwidth calculation?
I am reading a paper that uses the following equation to calculate the optimal bandwidth, however, I am confused about the position of "4" and "3" in the equation. is this a typo? or what does it mean?...
- 445