Mathematics

Questions tagged [nonparametric-statistics]

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.

27 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
  • 549
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
  • 1,264
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
  • 8
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
  • 753
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
  • 1,821
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
  • 1,821
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…
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…
Javier Moreno Sepena
  • 21
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 kernel…
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.
  • 972
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
  • 682
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 line…
Pipnap
  • 549
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
1
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