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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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25 views

Choice of residual function for least squares error minimization

Good morning, I have the a set of data $(\sigma,D,\alpha_0)_i$, $i=1...n$ data. I want to determine two parameters $K_{IC}$, $C_f$ in the basic equation given as $K_{IC} = \sigma \sqrt{D} k_0(\...
3
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1answer
82 views

Largest eigenvalues of a (random) correlation matrix?

I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some ...
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0answers
62 views

Asymptotic distribution of $n\mathbb E_{\hat{P}_n}[g(Z;\theta)]^T\operatorname{Cov}_{\hat{P}_n}[g(Z;\theta)]^{-1}\mathbb E_{\hat{P}_n}[g(Z;\theta)]$

Setup This question is a followup on this question. I'm interested in the asymptotic distribution of certain quadratic forms. So, let $Z$ be a $p$-dimensional random vector with (unknown) ...
2
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1answer
59 views

How does a statistical divergence change under a Lipschitz push-forward map?

Let $\mu, \nu$ be two probability measures on a space $X$ (assume Polish space). $T: X \rightarrow Y$ is a Lipschitz-map that acts as a push-forward on these measures; let $\mu^\prime = T_{\#\mu}$ and ...
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0answers
185 views

What happens in the martingale CLT if I norm by the conditional variance instead?

TLDR: I'm a statistician (bear with me!) trying to use the martingale CLT but I only can estimate the conditional variance instead of the unconditional one. Can I do anything to get a CLT with norming ...
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1answer
44 views

References for Hellinger distance/affinity involving mixture distributions

For two continuous probability distributions $F,G$ and their densities, $f,g$, the (squared) Hellinger distance/affinity is given by $d^2_H(F,G)=1-\int_{\mathbb{R}} \sqrt{fg}~dx$. Suppose that $f,g$ ...
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1answer
24 views

Cumulative Order Statistics of Independent Non Identical Distributions

I understand that the p.d.f of order statistics for Independent Non Identical Distributions are given by the Bapat-Beg theorem as previously explained in another question. As explained in the article, ...
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25 views

Rate of convergence of centered Hotelling's statistic to Chi-squared distribution

Consider the Hotelling's statistic $H_n := n\mu_n\Sigma_n^{-1}\mu_n$, where $\mu_n$ (resp. $\Sigma_n$) is the empirical mean (resp. empirical covariance matrix) of a zero-mean random $d$-dimensional ...
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0answers
64 views

Open problems in Monte Carlo Simulation [closed]

I want to know some open problems in Monte Carlo Simulation, which is being studied or in a stalemate. Could you please give me some advice? Thanks a alot
3
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2answers
209 views

Effect of perturbing the atoms of a measure on the Wasserstein distance

Let $(X,d)$ be a metric space, $x_1,\ldots,x_N\in X$ and $x_1',\ldots,x_N'\in X$ be atoms, and $G=\sum_{i=1}^Np_i\delta_{x_i}$, $G'=\sum_{i=1}^Np_i'\delta_{x_i}$, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...
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1answer
132 views

Asymptotic distribution of $\mathbb E_{\hat{P}_n}[Z]^T\operatorname{Cov}_{\hat{P}_n}[Z]^{-1}\mathbb E_{\hat{P}_n}[Z]$

Under very general conditions on the random $p$-dimensional vector $Z$, what can be said about the asymptotic distribution of the (random) scalar quantity $R_n := \mathbb E_{\hat{P}_n}[Z]^T\...
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0answers
14 views

Large Deviations Rate of Convergence and Robbins Monro

I am looking for a result/paper (if there is any) on the large deviations rate of convergence of the Robbins-Monro (RM) algorithm. Specifically, given $X_k \rightarrow X$ a.s. in the RM algorithm, I ...
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2answers
108 views

lower bound the probability of at least L collisions

Lets say we get a list $M$ containing $|M|=\sqrt{L\cdot N}$ randomly and independtly drawn elements from a set of size $N$. And lets denote the $i$-th element of the list $M$ by $M[i]$. If we now ask ...
2
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1answer
101 views

Why the VC dimension of triangles in 2D space is not greater than 7?

I understand that there are sets of 7 points on a circle that can be fully shattered using triangles.But, it is not clear to me why it cannot shatter 8 points. Is there any intuitive way of arriving ...
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1answer
85 views

Is the normal product distribution sub-gaussian?

Consider the normal product distribution, which is the distribution of the product of two or more independent normal variables. Particulary, focus in the case where the multiplied normal variables are ...
3
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1answer
142 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 ...
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0answers
25 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 $$ I=\mathbb E_{x \sim Q}\left[w(x)h(x)\right],\text{ with }w(x):=\frac{p(x)}{q(x)}, $$ ...
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0answers
42 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
81 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{...
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25 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 ...
3
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2answers
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}(...
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1answer
189 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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0answers
49 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
80 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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0answers
72 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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0answers
25 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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1answer
81 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
53 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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55 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 }...
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1answer
127 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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21 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
161 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
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1answer
231 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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2answers
72 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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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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114 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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98 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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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 ...
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1answer
87 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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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 ...
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133 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
173 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
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2answers
270 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
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1answer
112 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
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1answer
212 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 ...
4
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1answer
74 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 ...
27
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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 ...
5
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1answer
145 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 ...
8
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2answers
223 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 ...