All Questions
Tagged with nonparametric density-estimation
39
questions
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 ...
- 9,427
1
vote
0
answers
43
views
Why is histogram density estimation nonparametric?
My understanding of histogram density estimation:
For $k$ predefined equal-width bins $(b_0, b_1], (b_1, b_2], ..., (b_{k-1}, b_k]$ and $n$ observations $x_1,...,x_n \in (b_0,b_k]$, we estimate ...
- 11
5
votes
1
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698
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Expected value and variance of KDE
I need to find the expected value and variance of KDE given that $$(i) E[u] = 0 \to \int u\phi(u)du=0\\
(ii)V[u] = \sigma^2 \to \int u^2\phi(u)du=\sigma^2$$ where $\phi$ is the kernel function.
I've ...
0
votes
0
answers
85
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Expected value (and variance) of a Dirichlet Process
Suppose I have a measure $G$ that follows a Dirichlet Process,
$$G \sim DP(H_0,\alpha)$$
where $H_0$ is some base measure. Is there a closed form solution for the expected value of $G$?
5
votes
2
answers
549
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Is density estimation the same as parameter estimation?
I was studying parameter estimation from Sheldon Ross' probability and statistics book. Here the task of parameter estimation is described as follows:
Is this task the same of density estimation in ...
- 68.6k
9
votes
2
answers
3k
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Estimating the gradient of log density given samples
I am interested in estimating the gradient of the log probability distribution $\nabla\log p(x)$ when $p(x)$ is not analytically available but is only accessed via samples $x_i \sim p(x)$.
There ...
- 25.2k
1
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0
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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
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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
0
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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 ...
4
votes
3
answers
251
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Fast multivariate unimodal density estimator
I have a sample $\boldsymbol{x}_i$ for $i$ in $1,\dots, n$, from a $d$ dimensional density $f(\boldsymbol{x})$ and I would like to estimate this unknown density. In addition I know that $f(\boldsymbol{...
1
vote
0
answers
102
views
Optimal rate of convergence of nonparametric density estimators
Suppose that $X_1, X_2, \dots, X_n$ forms an independent and identically distributed sample from some $d$-dimensional probability distribution with unknown probability density function $f$. Let $x$ be ...
- 81.5k
1
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0
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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
1
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0
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135
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Extraction of modes from a multi-modal density function
I am trying to extract modes from a multi-modal density function and not just peaks. For example, in the two density functions below (images), I would like to extract the curves contained in the black ...
- 323
1
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0
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107
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Convex hull version of density estimation (or lines of constant density)
Background:
So I had a thought, tried it out, and liked what it did. I'm sure someone else has done this. It feels very convenient. It also gives an interesting take on robust nonparametric density ...
- 9,580
2
votes
0
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41
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Unexpected zero on posterior density of Dirichlet process mixture
I was reading this notebook from the PyMC3 documentation about Dirichlet Process Mixtures and, on the last figure, the estimated density reaches almost zero for a particular value, despite the ...
- 81.5k