All Questions
Tagged with nonparametric density-estimation
18
questions with no upvoted or accepted answers
4
votes
0
answers
442
views
Derivation of k nearest neighbor classification rule
One way to derive the k-NN decision rule based on the k-NN density estimation goes as follows:
given $k$ the number of neighbors, $k_i$ the number of neighbors of class $i$ in the bucket, $N$ the ...
- 426
2
votes
0
answers
41
views
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 ...
- 2,680
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
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
1
vote
0
answers
251
views
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
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 ...
- 140
1
vote
0
answers
274
views
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
vote
0
answers
135
views
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
vote
0
answers
107
views
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
1
vote
0
answers
131
views
What is the resulting distribution of a data set that was originally normally distributed but has been quantized and had all negative values removed?
I am trying to benchmark a seasonal forecasting model and calculate not just the point forecasts but the forecast densities from the model.
To do this, I generated a simulated data set in the ...
- 1,381
1
vote
0
answers
42
views
Difficulties with orthogonal density estimation
I am working on an implementation of an orthogonal density estimator, using the basis
$$ \psi_0(t) = 1, \quad \psi_{2j}(t) = \sqrt{2}\text{cos}(2\pi j t), \quad \psi_{2j+1}(t) = \sqrt{2}\text{sin}(2\...
- 21
1
vote
0
answers
190
views
Optimal bandwidth selection in conditional density estimation
Consider the situation that we are estimating a $d$-dimensional density (with suitable regularity conditions) using kernel density estimation,
[Method1,conditional density estimation] We can proceed ...
- 2,480
1
vote
0
answers
53
views
Nonparametric density estimation, individual probablities
Consider the problem of doing nonparametric density estimation using kernel density estimator in the common form
$k(\frac{\textbf{x} - \mathbf{x_{j}}}{h})$,
$k(\textbf{u}) = \begin{cases}
1 & \...
- 121
0
votes
0
answers
85
views
Expected value (and variance) of a Dirichlet Process
Suppose I have a measure $G$ that follows a Dirichlet Process,
$$G \sim DP(H_0,\alpha)$$
where $H_0$ is some base measure. Is there a closed form solution for the expected value of $G$?
0
votes
0
answers
50
views
How to prove symmetry of a Uniform kernel?
I am trying to prove this kernel is valid,
$$
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 ...
