# 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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votes

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143 views

### Why does the assumption $|U_t| \le \frac1{p_{\min}}$ work in this paper?

I am reading a 2009 paper right now "Importance Weighted Active Learning" and on page 5, there is a theorem that uses the inequality $|U_t| \leq \frac{1}{p_{\min}}$. I am not sure how the paper found ...

**0**

votes

**0**answers

26 views

### Efficiency of importance sampling in terms of the size of the the support of sampling distribution

In importance sampling, one proposes to compute an integral $I:=\mathbb E_{x \sim P}[h(x)]$ by rewritting it as
$$
I=\mathbb E_{x \sim Q}\left[w(x)h(x)\right],\text{ with }w(x):=\frac{p(x)}{q(x)},
$$
...

**1**

vote

**0**answers

45 views

### Compare KS test and Wasserstein distance or Earth mover's distance

I have tried the following question in couple of exchange sites but I did not get any views or reply. I am asking here as I am kind of desperate for the answer, please be considerate. Any suggestion ...

**2**

votes

**0**answers

83 views

### Covering a sphere with ellipsoid-products in high dimension

For $\Sigma\geq 0$ a $k\times k$ matrix and large $n$, fix $E:= \{(x_i)_{i=1}^n: \sum_i x_i^\dagger \Sigma x_i \leq n\}$. Fix $(z_m)_m$ as $M$ points iid uniform on $\mathbb{S}^{nk-1}\subset \mathbb{...

**3**

votes

**2**answers

64 views

### Left tail of convex combinations of $\chi_1^2$

Suppose $a_1,...,a_n\geq0, \sum_{i=1}^na_i=1$ and $Z_1,...,Z_n$ are i.i.d. standard normal, what is a sharp upper bound of the following probability as $\delta\to0$ and what is the order?
$$\mathbb{P}(...

**2**

votes

**1**answer

191 views

### A modest generalization of the law of large numbers

Suppose I collect $2n$ independent samples of a probability density function $f$, which are separated into pairs $\{X_i^1, X_i^2\}$ for $1\leq i\leq n$. Suppose I now consider the set of all $2^n$ ...

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53 views

### Using mollifiers (or some other idea) to solve constrained minimax problem

Sorry in advance if this sounds like a more SE question.
Consider a continuously parametrized family of $L$-Lipschitz continuous $f_\theta: X \rightarrow \mathbb R_+$ on a metric space $X=(X,d)$. Let ...

**1**

vote

**1**answer

82 views

### What is the order of the left tail of a mixture of non-central chi-square?

Let $\mu\sim N(0,1)$, $Z\sim N(\mu,1)$. Then $Z$ can be viewed as a mixture of Gaussians. It can also be viewed as a Gaussian but there is a prior for the mean.
Let $X\sim\exp(\lambda)$ where the ...

**4**

votes

**0**answers

76 views

### Geometric meaning of the chi-square “measure of association”

In Statistics, there's a standard test of independence of two random variables taking values in finite sets $I,J$. It relies on the computation of $\chi$-square statistics,
$$
\chi^2:=\sum_{(i,j)\in ...

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vote

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27 views

### Supremum of expectations over a family distributions close to a base distribution

Let $p$ be a probability distribution on a space $X$ , $f:X \rightarrow \mathbb R_+$ be a measureable function, and $0 < \alpha \le 1 \le \beta < \infty$. Define a sub-collection $\mathcal Q \...

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votes

**1**answer

85 views

### how to derive stationary distribution of maximal entropy random walk

I was reading the paper 0810.4113v2, burda, which analyzed the stationary distribution maximal entropy random walk on the irregular lattice. I am confused on some of the steps.
Description:
The ...

**3**

votes

**1**answer

54 views

### How to value the extent of separation or mixing of point sets in plane?

As the image presented below, the reddish point set is totally separated from the blueish one and the greenish one, while the blueish point set is quite mixed with the greenish one.
A number of ...

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votes

**0**answers

57 views

### How to mathematically justify the “sampling” over only $100$ random matrices to estimate percolation thresholds?

As mentioned in the textbook "Introduction to Percolation Theory" (Chapter 4) by Stauffer et al., the variation of spanning cluster percolation probability $\Pi$ in a finite $L < \infty$ square ...

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37 views

### Reformulate Wasserstein constraint optimization on product space in terms of marginal

Let $X = (X,d_X)$ be a metric space and $Y$ be an abstract set (with at least two elements). Consider the metric on $X \times Y$ defined by
$$d((x,y),(x',y')) = \begin{cases}d_X(x,x'),&\mbox{ if }...

**3**

votes

**1**answer

131 views

### about an interesting moment generating function

Let $X_1,\ldots,X_n$ be iid Rademacher variables, i.e., $P(X_1=1)=P(X_1=-1)=1/2$. CLT says that $Y_n\equiv \sqrt{n}\bar{X}$ converges in distribution to $N(0,1)$ as $n\to\infty$. So $Y_n^2$ is ...

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23 views

### Approximate in $W_1$ sense, an empirical distribution with restriction of true distribution on a set

Let $\mu$ be a probability distribution on a metric space $X=(X,d)$ (to avoid unnecessary complications, assume the full filtration $2^X$) and let $x_1,x_2,\ldots,x_N$ be a sample of size $N$ drawn i....

**5**

votes

**1**answer

286 views

### KL divergence and mixture of Gaussians

Do we have an exact formula to compute the KL divergence between 2 mixtures of Gaussians (i.e convex combinations of a finite number of Gaussian distributions)?
If not exactly known, are there good ...

**5**

votes

**1**answer

243 views

### Approximation of Wasserstein distance between $p_\theta$ and $p_{\theta + d\theta}$

Given a parametric family of distributions $\{p_\theta\mid\theta \in \Theta\}$, one can show that under some regularity conditions, the following approximation is valid
$$\operatorname{KL}(p_\theta\...

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votes

**2**answers

77 views

### Draw samples from distribitions in the neighborhood of a fixed distribution

Disclaimer
Sorry in advance for vagueness. I'm still trying to get my ideas right on this one.
Setup
So, let $P$ be a distribution on a Euclidean space $X$ with an $\ell_p$ metric, and let $P_\...

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votes

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31 views

### Computation of singular multi-variate normal probabilities

Are there any statistical software packages that can compute probabilities from singular multi-variate normal distributions, such as described here?

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116 views

### Expected distance in hyperbolic space

In a hyperbolic space, $r$ and $\theta$ can represent a point in a polar coordinate system. If we suppose $\theta_1\sim \operatorname{Uniform}(t_1,t_2)$, $\theta_2\sim \operatorname{Uniform}(t_3,t_4)$,...

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131 views

### Hoeffding's inequality for random vectors

Let $x_1, \ldots, x_n$ be $n$ i.i.d. samples of a bounded random variable $X \in [a, b]$. We know from the Hoeffding's inequality that :
$$\mathbb{P} \left( \left| \frac{1}{n} \sum_{i=1}^n x_n - \...

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votes

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46 views

### How can least squares regression be modified to penalize errors more heavily for small values?

As usual, $f(x_i)$ is some linear combination of the variables, with the error total:
$$\sum_{i=1}^N (y_i - f(x_i))^2$$
I would like to penalize errors near $0$ more heavily. One conceivable ...

**3**

votes

**1**answer

89 views

### Statistical independence of eigenvectors of real symmetric Gaussian random matrices

What is known about the statistical independence of the eigenvectors of a real symmetric matrix with independent Gaussian entries with zero mean, and finite variance? The matrix elements are not ...

**2**

votes

**0**answers

104 views

### How high to build a dam in Amsterdam in order that the probability of a flooding within the next 100 years be less than 1%? [closed]

In the preface of the monography, Pr. D. Voiculescu Wrote:
"Free probability and operator algebras The well-known question about how high to build a dam in Amsterdam in order that the probability ...

**6**

votes

**0**answers

136 views

### Approximating a ray with an integer lattice point

Take $X$ uniform on the unit sphere in $\mathbb{R}^n.$ For $r>0$, take $S_r=\{x\in \mathbb{Z}^n: \sum_i x_i^2 \leq r^2\}.$
With $\|\cdot \|$ the 2-norm, what is the distribution (or at least the ...

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votes

**2**answers

175 views

### Expectation of minimum of correlated Gaussian

What is the order of the following expectation with respect to $n$?:
$$\mathbb{E}(\min_{1\leq i\leq n}|z_i|^2)$$
where
$$(z_1,...,z_n)^T\sim N(0,I+11^T), 1=(1,1,...,1)^T$$
I know that when $z_i$ are ...

**8**

votes

**2**answers

277 views

### Big ideas and big ways of thinking in statistics?

I'm moving to a new university for the fall semester, and I'll be teaching a statistics class for the first time. I'm familiar enough with doing statistics (my dissertation in math ed was a mixed-...

**3**

votes

**1**answer

303 views

### Expected value of the maximum of the periodogram

Let us suppose that $X_1,\ldots,X_n$ with $n\ge1$ are iid random variables such that $\operatorname EX_1=0$ and $\operatorname E|X_1|^s<\infty$ with some $s>2$ and define the DFT of $X_1,\ldots,...

**4**

votes

**1**answer

221 views

### A metric stronger than total variation

Let $P,Q$ be two distributions on a finite set $X$. Consider the following metric*
$$ d(P,Q) = \frac12\max_{\emptyset\neq A\subseteq X} \|P(\cdot\mid A)-Q(\cdot\mid A)\|_1. $$
Obviously, the total ...

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votes

**1**answer

77 views

### How to find the optimal convergence rate?

I have already asked that Question on Cross Validated:
Link
Suppose there is some data $X_{1},X_{2},\ldots,X_{n}$. We further suppose that there is some parameter $\theta$, for which we want to do ...

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votes

**3**answers

10k views

### What is the Katz-Sarnak philosophy?

It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some ...

**5**

votes

**1**answer

147 views

### A counterexample for the Mean Ergodic Theorem in $L_\infty$

The so-called Mean Ergodic Theorem goes back to von Neumann for Hilbert spaces. Later on, versions of this result in reflexive Banach spaces have also appeared (see, e.g., the book by Krengel, Ergodic ...

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votes

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225 views

### q-Means and the Mode of a Distribution

Let $f:\mathbb{R} \rightarrow [0,\infty)$ be a continuous probability density function on $\mathbb{R}$ such that
\begin{equation}
\int_{\mathbb{R}} |x| f(x)\, dx < \infty,
\end{equation}
and ...

**8**

votes

**2**answers

157 views

### Largest deviations for uniform order statistics

Let $n >0$.
Let $X_1,\ldots,X_n$ be i.i.d. uniform random variable on $[0,1].$ Denote by $X^{(1)}\leq X^{(2)} \leq \cdots \leq X^{(n)}$ their order statistics, and write $\Delta^{(i)} = \vert X^{(...

**0**

votes

**0**answers

56 views

### Is there any work on the distribution of the difference between samples and sample mean?

given $X_1,\cdots,X_n\overset{iid}\sim F$, where $F$ is a truncated normal, I wonder if there's something known about the distribution and specifically about the sub-Gaussianity of $X_i-\overline{X}$ ...

**6**

votes

**2**answers

92 views

### Order statistics of bounded variable : L2 concentration?

Let n >0
Let $X_1,...,X_n$ be i.i.d. random variable with a density (say $f(x)$) on [a,b]. Denote by $X_{(1)}\leq X_{(2)} \leq \ldots \leq X_{(n)}$ their order statistics.
I'm interested in ...

**2**

votes

**1**answer

128 views

### Fastest convergence of sum of uniform independent distributions to a Gaussian

The sum of uniform i.i.d. random variables follows the Irwin-Hall distribution. Through observation it seems that the convergence is faster in comparison to the sum of uniform independent but not ...

**5**

votes

**1**answer

222 views

### Isn't a Shapiro-Wilk normality test assuming its conclusion?

I am currently thinking about formalization of some statistics (in Coq). One thing I don't understand is the logic of e.g. the Shapiro-Wilk test for normality. To explain my problem, let's first look ...

**2**

votes

**1**answer

48 views

### Local distribution of sample covariance matrix when the number of observations/realisations is less than the matrix dimension

Given a true covariance matrix $M$ of dimension $p \times p$, we generate $n$ gaussian random vectors $X_1,..X_n \sim N(0,M)$. We then get a sample covariance matrix $M_s$ based on these $n$ ...

**4**

votes

**1**answer

130 views

### Bound for a conditional expectation

Let $a_i, i=1, \ldots, n$ be real numbers. Let $\epsilon_i, \, i=1, \ldots, n$ be a random variables that take values $\pm 1$ with equal probability and $r_i, i=1, \ldots, n$ be random variables ...

**4**

votes

**0**answers

126 views

### Probability of having complete conversion in a box of three different object with interaction rules

Say there a 3 types of Objects A,B,C which randomly interact in pairs to form new objects following the below rules:
$$A + B = AB$$
$$B + C = BC$$
$$C + A = CA$$
$$AB + C = ABC$$
$$BC + A = ABC$$
$$CA ...

**0**

votes

**0**answers

41 views

### n-D Gauss circle problem over a rectangle

I would like to approximate the amount of points in $\left(2^{-a\cdot n}\mathbb{Z}^n\right)\cap B^n(0,1)\cap C^n$ where $a>0,\ B^n(0,1)$ is the unit nball and $C^n$ is some rectangular domain ...

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votes

**3**answers

2k views

### On Mathematical Analysis of MathSciNet & MathOverflow

This question has two original motivations: mathematical and social.
The mathematical motivation is mainly based on what I have seen about Zipf's law here and there. The Zipf's law simply states ...

**4**

votes

**1**answer

319 views

### Minimize the variance of a Boltzmann distribution

N.B.: Sorry for cross-posting from https://stats.stackexchange.com/posts/347804/edit (I realized it was the wrong venue for the question, but couldn't find an easy way to transfer the question here).
...

**3**

votes

**1**answer

70 views

### Are there any statistical metrics that satisfy this kind of condition?

Let $f=N(\mu,\sigma^2)$ be a univariate normal distribution with mean $\mu$ and variance $\sigma^2$ and let $f_1 = N(\mu+\epsilon,\sigma^2)$ and $f_2=N(\mu,(\sigma+\epsilon)^2)$ be some small ...

**0**

votes

**1**answer

46 views

### Degree Distribution of Planar Minimum Spanning Trees

Question:
how are the vertex degrees of MSTs of the corresponding points that are uniformly and independently distributed in a square region of the euclidean plane and the edges being weighted ...

**2**

votes

**1**answer

107 views

### Monotonicity of the regularized incomplete gamma function

In the theory of chi-squared distribution in statistics, for the random variable $X$ following $\chi^2 (k)$, the probability that $X$ is lower than its expactation is given by
$$
P(X\le k) = \frac{\...

**2**

votes

**2**answers

72 views

### Why the autoregressive process to generate random time series?

in order to test some forecasting methods, I desire to generate random time series. I'm about to use the AR(1) model:
$X_k=\alpha X_{k-1}+ \epsilon_k$
With eventually: $\alpha>1$
How can I ...

**1**

vote

**0**answers

46 views

### Empirically random, quickly multiplicable matrices

I have encountered a need for fast computation of a transformation $Ax$ where $A\in \mathbb{C}^{K\times N},\ K\sim 10^7,\ N\sim 10^3$ is designed, and $x\in \mathbb{C}^N$ has iid $\mathcal{CN}(0,1)$ ...