For questions about mathematical-statistical models that involve at least one infinite-dimensional parameter and hence may also be referred to as "infinite-dimensional models." This field is closely related to functional analysis, measure theory, and topology on function spaces.
Questions tagged [nonparametric-statistics]
30 questions
4
votes
1 answer
U-statistics and independence testing
I wish you a happy new year.
I am reading this paper
I am struggling to understand a small part of section 5.1: Independence testing for multinomials at page 17. Specifically, I am having difficulty understanding the sentence:
In this case, the…
Pipnap
- 507
3
votes
0 answers
Conditional Density Estimation in RKHS
I would like to model the conditional density of two real-valued random variable and estimate it using the empirical conditional mean embedding. I am not sure which of these two are correct way of doing this in RKHS.
A
If we model the joint density…
domath
- 1,254
3
votes
0 answers
Some question about sub gaussian of orlicz norm.
I raised some questions when read Nickl's book "Mathematical Foundation of Infinite-Dimensional Statistical Models".
The first question is how to prove the following statement mentioned in page 38:
If $\xi$ which is sub-gaussian random variable is…
vincen
- 135
2
votes
1 answer
Given $D, X_1, X_2, Z$, how is $E(D Z)$ related with the nonparametric partial correlation $\eta^2_{D\sim Z\mid X_1, X_2}$?
Let $D, X_1, X_2, Z$ be some arbitrary random variables. If it is easier to think, we can assume they are mean zero.
I'm curious if there is any relationship between the correlation $E(D Z)$ and the nonparametric partial correlation $\eta^2_{D\sim…
Omega
- 743
- 3
- 12
2
votes
0 answers
Question on literature for contraction rates
I read in some lecture notes the following definition of contraction rate:
Definition (Posterior rate of contraction) The posterior distribution $\Pi_n\left(\cdot \mid X^{(n)}\right)$ is said to contract at rate $\epsilon_n \rightarrow 0$ at…
Grandes Jorasses
- 1,823
2
votes
0 answers
Sum of arrival times of Chinese Restaurant Process (CRP)
Suppose that a random sample $X_1, X_2, \ldots$ is drawn from a continuous spectrum of colors, or species, following a Chinese Restaurant Process distribution with parameter $|\alpha|$ (or equivalently $X_1,\ldots,X_n \mid P \sim P$ where $P$ is a…
Grandes Jorasses
- 1,823
2
votes
1 answer
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 $\hat{f_h}^{(n)}(x) \xrightarrow
{a.s.}f(x)$. Assume that $f$ possesses a…
graham
- 336
2
votes
1 answer
Coincise introduction to background for semiparametric statistics
I plan to study the theory behind Targeted Maximum Likelihood Estimation, Doubly Robust Estimation, and Semiparametric Theory. I have a background in bioinformatics: I took courses in basic linear agebra and proof-based calculus (and a little bit of…
wrong_path
- 369
2
votes
0 answers
Histogram asymptotic bias
I am reading All of Statistics from Casella. When trying to show the bias for a histogram estimator for some density distribution, he starts developing the formula for pj, that is, the probability some observations lies in that bin. Then he uses…
1
vote
0 answers
Is the closed unit ball in a RKHS closed in $ L^2$ norm?
Background and notations
Let $\mathcal{X}$ denote an input domain. Let $H$ denote the Reproducing Kernel Hilbert Space (RKHS) induced by the Radial basis function (RBF) kernel with bandwith 1, which is defined as the positive-definite…
Steve Shen
- 11
1
vote
1 answer
What is the cumulative kernel distribution using an Epanechnikov kernel?
good morning everyone.
I understand that kernel density estimation is a non-parametric technique used to estimate the probability density function of a random variable from a sample of data, with the kernel density defined as follows:
$$
\hat{f}(x)…
Samuel M
- 13
1
vote
1 answer
What hypothesis is tested by the Mann–Whitney U testing, and how does it differ from the Brunner-Munzel test?
It’s not clear to me what exactly is tested by the Mann–Whitney U test (also called Wilcoxon rank-sum test).
First, assume independence of the observations, and ordinality of the data.
My understanding is that under these conditions, the…
Guillaume F.
- 952
1
vote
0 answers
How to prove the one sample Hodges-Lehmann estimator is asymptotically normal and find its variance
In relation to the following URL question, I would like to consider a proof for the one-sample case.
https://stats.stackexchange.com/q/501493/401056
Definition
Consider the median of the average
$$
\theta_{HL} = \mathrm{med}_{i \leq j} \left(…
ytnb
- 666
- 2
- 9
1
vote
1 answer
Expectation of $L^2$ norm.
I am reading an article which are estimating the division kernel of a size - structured population.
I have some difficulties in unstanding the…
Pipnap
- 507
1
vote
0 answers
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'm trying to calculate the bias of this estimator of $f$ a $C^4$ mesurable function:
$$\hat{f'}_{h,n} = \cfrac{1}{nh^2}\sum_{j=1}^n K'\left(\cfrac{x-X_j}{h}\right) =\cfrac{1}{h^2}K'\left(\cfrac{x-X_1}{h}\right)$$
with $K$ being the standard normal…
wageeh
- 281