# Questions tagged [st.statistics]

Applied and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments.

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14 views

### Minimum population size given the average [on hold]

I had a thought experiment that I was interested in formalising. I don't have a maths background so apologies.
How can you calculate the possible population sizes for a given average of whole numbers?...

**-2**

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12 views

### Confidence interval in Chi squared testing [migrated]

How to calculate confidence interval when performing a Chi-Square significance testing?
For reference where it's used - http://www.evanmiller.org/ab-testing/chi-squared.html
Can we apply the same ...

**1**

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45 views

### Infinitesimal matrix rotation towards orthogonality

TLDR; I am trying to prove the existence of an infinitesimal rotation which always moves a matrix "closer" to being orthogonal.
Setting
In this setting, we have a matrix $W \in \mathbb{R}^{n \times ...

**0**

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66 views

### Davis-Kahan $\sin(\theta)$ Theorem assuming population eigengap

The Davis-Kahan Theorem allows to bound the angle between the subspace spanned by some population principal directions and a sample estimate of it. There is a version (http://www.cs.columbia.edu/~...

**3**

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59 views

### Expectation of a Random Matrix that Contains Wishart Form

I am interested in calculating the expectation of the following random matrix:
$$A=WX(X^TWX)^{-1},$$
where $W \sim W_p(n,I)$ is a $p\times p$ random Wishart matrix, and $X$ is a fixed $p\times m$ ...

**2**

votes

**1**answer

49 views

### Asymptotic rate for the expected value of the square root of sample average

I have iid random variables $X_1, \dots, X_n$ with $X_i \geq 0$, $E[X_i]=1$ and $V[X_i] = \sigma^2$.
Let $S_n = \frac{\sum_{i=1}^n X_i}{n}$.
I'd like to say that $E[\sqrt{S_n}] = 1-O(1/n)$.
My first ...

**2**

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50 views

### Order statistics of correlated bivariate Gaussian

Suppose $(X_1,Y_1),...,(X_n,Y_n)$ are i.i.d. bivariate Gaussian with mean zero. Each coordinate has variance 1 and correlation between coordinates is $\rho\in[-1,1]$.
I'm interested in the following ...

**-1**

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19 views

### Uniform law of large number in a Markov decision process setting?

Consider
$$
R =\sup_{f\in\mathcal{F}}
\left[
\frac{1}{n}\sum_{i=1}^n f(X_i) - \mathbb{E}[f(X)]
\right]
$$
If $X_i$'s are i.i.d., then uniform law of large number shows that, if $\...

**0**

votes

**1**answer

68 views

### Strictly Proper Scoring Rules and f-Divergences

Let $S$ be a scoring rule for probability functions. Define
$EXP_{S}(Q|P) = \sum \limits_{w} P(w)S(Q, w)$.
Say that $S$ is striclty proper if and only if $P$ always minimises $EXP_{S}(Q|P)$ as a ...

**2**

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**1**answer

53 views

### Bounding integral arising from expectation of a random variable satisfying Bernstein's inequality

Let $X$ be a random variable s.t. for $v, b > 0$ and $C \geq 1$:
$$ P(X \geq t) \leq C\exp\left(-\frac{t^2}{2(v^2 + bt)} \right) $$
I am trying to show that $\mathbb{E}X \leq 2v(\sqrt{\pi} + \...

**3**

votes

**1**answer

82 views

### concentration inequality for a weighted sum of independent but not identical binary variables

Let $\alpha\in[0,1]$ be a fixed constant, and let
$w,x\in[0,1]^n$ be two vectors such that $\sum_i w_i x_i=\alpha$.
Define $Y = \sum_i w_i X_i$, where $X_i \sim \operatorname{Bernoulli}(x_i)$, so it ...

**1**

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47 views

### About a class of expectations

Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \...

**3**

votes

**1**answer

66 views

### Gaussian expectation of outer product divided by norm (check)

I am trying to get compute at least the directional component of the following expectation, where $M$ is a symmetric, invertible, PD matrix:
$$\mathbb{E}_{v \sim N(0, I)}\left[\frac{vv^T}{||Mv||_2}\...

**1**

vote

**1**answer

48 views

### Performing Statistical Analysis on a Data Set With a lot of Null Responses

I am currently trying to perform some statistical analysis on some data to see if there is any meaningful conclusion for a research project I am working on; however, I have come across a problem. ...

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82 views

### Clarification about the ? -net argument

I have been reading the paper Do GANs learn the distribution? Some theory and empirics.
In Corollary D.1, they reference the paper Generalization and Equilibrium in Generative Adversarial Nets which ...