# Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

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### How is Nadayara Watson KDE proof

I was looking at Wikipedia article on Nadayara-Watson Kernel regression section, in the proof part they state But I'm having trouble understand why: Turns to just yi. Sorry I'm missing something so ...
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### Clarification of line in proof of consistency theorem (Vapnik)

In Vapnik's Statistical Learning Theory (1998 edition) on pages 89-92, he proves a "key theorem of learning theory" that states the conditions for when: "the following two statements ...
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### why in real gases we have potential energy? [closed]

I read that in ideal gases, we dont have any potential energy. But I also read what in real gases we have potential energy(positive Ep). Can someone explain why in real gases we have potential energy?
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### What is a convincing and useful real application of the Central Limit Theorem?

I understand what the central limit theorem is about (ie. that the shape of the distribution of the sample means of any population (each calculated by taking 30+ random measurements from the ...
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### How are the kernel density estimators and the tree structure related in Tree Parzen Estimators?

I'm still a bit confused about a few details of TPEs. So this should be a follow-up to this discussion: What is the "tree" structure in Tree Parzen Estimators? See the original TPE paper ...
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### Shapiro-Wilk vs Anderson Darling test for normality

What makes the Shapiro-Wilk test have the best power for a given significance when compared to Anderson-Darling test and other tests.
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### What is a confidence distribution?

I am going through this paper. I am unable to understand the definition of a confidence distribution. A statistic C is an exact confidence distribution for a real parameter ρ if: ρ → C(ρ; y) is a ...
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### Drawing graph of variance using R [closed]

I am a self -learner and try to learn statistics with R ,but i encounter with a problem i could not handle it such that I want to produce a graph of the variance of a binomial distribution with a ...
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### Random effects one-way layout analysis

I am analyzing the model $X_{ij} = \mu + \alpha_i + \epsilon_{ij}$ where $\alpha_i \sim N(0, \tau^2)$ (i.i.d), $\epsilon_{ij} \sim N(0, \sigma^2)$ (i.i.d), and they're each independent of each other. ...
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### Maxmium likelihood for the random effects one-way layout

Question: I am hoping someone can help me figure this out, or at least tell me whether I'm on the right track or not. My attempt: The likelihood of $\mathbf X$ will be the product of the multivariate ...
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### How is it possible to have $P(A|B \cup C)$ lower than both $P(A|B)$ and $P(A|C)$?

Let's say we investigate disease probability given two symptoms. Thus we have 3 variables: A) has disease B) has symptom 1 C) has symptom 2 We have following data: Now we want to see which symptom ...
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### Getting pearson corelation coefficient greater than 1

I was learning about the metrics which measure the relationship between the variables. I wrote a python code that can generate and calculate covariance and person correlation coefficient. ...
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### Expectation of inverse of sum of iid random variables [closed]

Description: There is a indicator function called $I_{i}^{k}$ as follows: \begin{align*} \begin{split} I_{i}^{k}= \left\{ \begin{array}{lr} 1, \; {\rm if \; the \; event \; happened\; at\; time \...
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### PCA: is linearity important?

I have a 6 dimensions dataset where I want to apply PCA to remove one dimension. I did a small analysis to check for relationships in my data and concluded that there is very low linear correlation ...
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### Estimating Population mean given a sample, and the population size

For my study, I have population size known (in order of millions), and a sample data (about 30 sample size). I can estimate population mean using 95% confidence interval by t distribution or normal ...
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### Should I make equally sized samples for the Mann-Whitney U test if originally I have unequal sample sizes

I have 2 groups with unequal sizes (control 70% and test 30%) and need to find out if there is a significant difference between these groups. I've read that MW test works fine with unequal sized ...
### Sufficient statistic $\sum_{j=1}^{n} |x_{j}|$ for laplace distribution
Let be $X_{1},\ldots , X_{n}$ random variables independent and identically distributed with density function: $$f_{\theta}(x)=\dfrac{1}{2}e^{-|x-\mu|}, \quad x,\mu \in \mathbb{R}$$ Find the joint ...
When proving the delta method of distributions in my textbook we make the following assumption: Let $X_{n}$ be a sequence of random variables. and: \${\sqrt{n}[X_n- c]\,\xrightarrow{D}\,\mathcal{N}(0,1)...