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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.

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P-value for testing a median to be $M \geq M_0$

I'm trying to solve a question from an introductory textbook on statistics. I am to use the Sign Test and determine if there is significant evidence that the median of a dataset $M$ is "at least&...
fatCat9999's user avatar
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Nonparametric likelihood function $\mathcal{L}_n(f) = \prod_{i=1}^nf(X_i)$ doesnt attain maximum in set of all densities

Let $X_1, \dots, X_n$ be i.i.d random variables with distribution function $F$, and $\mathcal{L}_n(f) = \prod_{i=1}^nf(x_i)$ it's likelihood function. Let $\mathcal{F}$ be the family of all possible ...
nicoyanovsky's user avatar
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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 ...
Grandes Jorasses's user avatar
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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 ...
Grandes Jorasses's user avatar
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Distribution of a transformation of the Dirichlet process

Consider a Dirichlet process $P \sim \operatorname{DP}(\alpha)$ on $(X,\mathcal{X})$ and a measurable $\psi : X \mapsto \mathbb{R}$. We know then that: $$ P \circ \psi^{-1} \sim \operatorname{DP}(\...
Grandes Jorasses's user avatar
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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 $$ \...
ytnb's user avatar
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Does there exist a test for statistical significance to compare two distributions that is independent of sample size?

Essentially, I am generating datasets in which I can make the sample sizes as large as I want. Therefore, any statistical test that I do between the generated sample distributions are somewhat ...
Decebalus's user avatar
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Calculate the integral with a normalized kernel

$K$ is a baseline kernel function that is nonnegative, symmetric and supported on [-1,1]; $h$ is a bandwidth in local smoothing .Let $p_j$ denote the marginal density of $X_j$.We define the estimator ...
Lee's user avatar
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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) \...
graham's user avatar
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find the UMVUE of a parametric function

Assume $(X_1, . . . , X_n)$ is an i.i.d. sample from $P = \{f : f$ is a pdf and $E_f|X|< ∞\}$. Use the conditioning approach to find an UMVUE of $τ (f) = (E_f (X))^2$. Can someone provide a hint ...
Tapi's user avatar
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Conditional mean square convergence and consistency

Let $\{(Y_i,X_i)\}$ be i.i.d. random pairs that satisfies $$Y_i=m(X_i)+e_i, \;\; \mathsf{E}(e_i|X_i)=0, \ i\in\{1,\dotsc,n\}.$$ Let $\hat m(x)$ be an estimator of $m(x)$ at point $x$ and $\boldsymbol{...
Celine Harumi's user avatar
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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}{\...
Pipnap's user avatar
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Expected value non-parametric?

I have some non-parametric data and I would like to determine the expected value. I can't find any guidance on this in my old stat books (I find formulas for the EV of various distributions, but ...
CBRF23's user avatar
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The property of Hölder continuous density function

I am reading the article but I don't understand some lines in page 10. First, assumptions on the density $h$ are needed. Of course, since $h$ is the density of a symmetric probability distribution on ...
Pipnap's user avatar
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170 views

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 $$\mathbb{E}[A_3^2]=\left\|\mathbb{E}[\hat{h}_{\ell}]...
Pipnap's user avatar
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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}...
wageeh's user avatar
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Finding Sample Size (n) using the binomial equation for non-parametric test design

Im using the non-parametric binomial equation, found in the link here https://reliawiki.org/index.php/Reliability_Test_Design It is written below: 1−C=∑(i=0 to f)(n,i) ((1−R)^i)*R^(n−i) Where C is the ...
Cameron Roberson's user avatar
2 votes
1 answer
155 views

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 ...
wrong_path's user avatar
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The relationship between Spearman coefficient and Pearson Coefficient

The Spearman coefficient is defined as following:$r_s = 1- \frac{6\Sigma d^2}{n(n^2 -1)}=1$ and the Person Coefficient is given by $r_p=\frac{\sum ^n _{i=1}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum ^...
mathbeginner's user avatar
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2 votes
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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 ...
Javier Moreno Sepena's user avatar
1 vote
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111 views

Convergence of sup norm estimate of functions

I am working in the context of Gaussian white noise model (where we observe $n$ trajectories sampled via $d X(t)=f(t) d t+\sigma d W(t)$) and also in the non-parametric regression model (where we ...
BabaUtah's user avatar
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187 views

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 ...
vincen's user avatar
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