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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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23 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 ...
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0answers
23 views

Please Help Solve This Correlation Problem [on hold]

The correlation between x and y is equal to 0.8 (Pxy = 0.8). A new variable z is defined as: z = 6x. What is Pzy? Mathematically prove your answer.
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0answers
48 views

Covering a sphere with ellipsoids in high dimension

For $k\times k \ \Sigma\geq 0$, $n$ large, fix $E:= \{(x_1,\dots, x_n): \frac{1}{n}\sum_i x_i^\dagger \Sigma x_i \leq 1\}$. Fix $(z_m)_m$ as $M$ points iid uniform on $\mathbb{S}^{nk-1}\subset \...
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0answers
18 views

Conditional distribution of maximum of the multivariate normal distribution

Suppose $X=[X_1,...,X_n]^T$ follows an $n$-dimensional multivariate normal distribution $\mathcal{N}_n(\mu_1,\Sigma_1)$ and $Y=[Y_1,...,Y_n]^T$ follows $\mathcal{N}_n(\mu_2,\Sigma_2)$, and $Y$ is ...
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0answers
25 views

Generating correlated random variables

Suppose I have a vector $X$ of random independent time series (uniform distribution) and I'd like to transform them into dependent time series having a covariance $Q$. So, first I standardize $X$ to ...
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2answers
56 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}(...
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1answer
178 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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46 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 ...
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1answer
59 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 ...
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64 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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22 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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26 views

Measurements to deterministic value [migrated]

I have a number of measurements of two variables: the number of products, the weight. Sometimes the weight is missing and sometimes the number of products is missing. I want to use the given data to ...
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1answer
73 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 ...
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1answer
50 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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0answers
54 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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0answers
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 }...
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1answer
121 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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20 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....
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1answer
85 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 ...
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1answer
186 views

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

Given a parametric family of distributions $\{p_\theta|\theta \in \Theta\}$, one can show that under some regularity conditions, the following approximation is valid $$\operatorname{KL}(p_\theta\| p_{...
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2answers
64 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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0answers
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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113 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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64 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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0answers
29 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 approach ...
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1answer
85 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 ...
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0answers
103 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 ...
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0answers
132 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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2answers
170 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 ...
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2answers
261 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-...
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1answer
96 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,...
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1answer
163 views

A metric stronger than total variation

Let $P,Q$ be two distributions on a finite set $X$. Consider the following metric* ​$$ d(P,Q) = \frac12\max_{\emptyset\neq A\subseteq X} ||P​(\cdot|A)-Q(\cdot|A)||_1. $$ Obviously, the total variation ...
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1answer
73 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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3answers
2k 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 ...
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1answer
142 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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2answers
219 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 ...
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2answers
126 views

Largest deviations for uniform order statistics

Let n >0 Let $X_1,...,X_n$ be i.i.d. uniform random variable on [0,1]. Denote by $X^{(1)}\leq X^{(2)} \leq \ldots \leq X^{(n)}$ their order statistics, and write $\Delta^{(i)} = \vert X^{(i)} - EX^{(...
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55 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 $X_i-\overline{X}$ ...
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2answers
88 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
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1answer
95 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
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1answer
210 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 ...
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1answer
47 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 $n$ ...
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1answer
117 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 ...
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1answer
72 views

Estimating histogram of discrete function with large domain and small range

I have a function $f$ which maps a LARGE range of positive integers into a small set of positive integers. For example it maps the integers between 0 and $2^{2^{10}}$ into the range [2,250]. It ...
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124 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 ...
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0answers
40 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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3answers
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 ...
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1answer
255 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
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1answer
67 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 ...
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1answer
43 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 ...