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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184 views

Lower bound on ratio of extreme order statistics

This question relates to bounds on expectations of order statistics, elaborated upon in the Book by Arnold and Balakrishnan (1989). Let $X_1,\ldots,X_n$ be i.i.d. continuous random variables ...
3 votes
1 answer
723 views

Navier-Stokes equations and machine learning

I am looking for a reference explaining how to solve the Navier-Stokes equations numerically using machine learning algorithms . Thank you in advance for your help .
2 votes
2 answers
135 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 ...
0 votes
1 answer
125 views

Analogues of Kac-Bernstein characterisation theorem for non-normal distributions

Let $X,Y$ be two independent random variables. The Kac-Bernstein theorem states that if $X+Y,X-Y$ are also independent, then $X,Y$ are Normal. Are there analogues of this theorem for non-normal, ...
0 votes
0 answers
58 views

Norms of Wigner matrices under power law decay

Suppose $\Sigma=\operatorname{diag}(h)$ where $h=(1^{-p},2^{-p},3^{-p},\ldots,d^{-p})$ and $p> 1$ $X$ is a matrix with $b$ rows sampled independently from $\operatorname{Normal}(0,\Sigma)$ Suppose $...
1 vote
2 answers
670 views

Upper bound about Gaussian tail bound

From the definition of sub-Gaussian distribution $X$ w.r.t. $\sigma$ i.e. $$\mathbb{P}(|X-\mathbb{E}(X)|\geq t) \leq 2 \exp(-\frac{t^2}{2\sigma^2}).$$ It's natural that when $X \sim \mathcal{N}(\mu, \...
1 vote
0 answers
109 views

When is the solution to a Fredholm integral equation a PDF?

I have two questions about inhomogenous Fredholm integral equations of the first kind: $$f(x) = \int_a^b K(x,t) g(t) dt$$ where $f, K$ are known and $g$ is not. If a unique solution for $g$ exists, ...
3 votes
0 answers
87 views

Question on an integral inequality

I am reading van de Vaart and Weller, Weak Convergence and Empirical Processes With Applications to Statistics. And I am stuck in the proof of Theorem 2.6.7 on page 141. For simplicity I restae the ...
0 votes
0 answers
71 views

Kernel density estimation is sub-gaussian

Let $X_1, ..., X_n$ be i.i.d. samples drawn from a pdf $f(x)$ on the real line. The kernel density estimator is defined as follows, $$\hat{f_n}(x) = \frac{1}{nh}\sum_1^n K(\frac{x-X_k}{h})$$ where $K:\...
14 votes
1 answer
3k views

How is the "conformal prediction" conformal?

The question is clarified by Prof.V.Vovk. See his answer below for discussion. Recently, early works of Gammerman, Vanpnik and Vovk[4] are rediscovered by Wasserman et.al[1] and proposed it as a ...
3 votes
3 answers
1k views

bracketing number vs covering number

Just want to double check if the lemma on page 9 of this slides is correct: http://www.math.leidenuniv.nl/~avdvaart/talks/09hilversum.pdf Lemma: $N(\epsilon,\cal F,||\cdot||)\leq N_{[]}(2\epsilon,\cal ...
2 votes
1 answer
79 views

p.d.f. of exponential family

I have a question about the p.d.f. of exponential family. Suppose $(X,\mathcal{F})$ is a measurable space and $\{F_{\theta},\theta\in \Theta\}$ is a distribution family on $(X,\mathcal{F})$. When $\{...
2 votes
1 answer
403 views

Weakly dependent central limit theorem

Say I have $N$ random variables $X_1,\cdots,X_i,\cdots,X_N$, with zero mean and finite variance. $X_i$ and $X_j$ are independent iif $|i-j|>m$, and positively correlated otherwise (say the ...
46 votes
13 answers
23k views

Why is it so cool to square numbers (in terms of finding the standard deviation)?

When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do $$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$. Why ...
0 votes
0 answers
114 views

Using projections to determine equidistribution

Suppose I have a collection of points on $\mathbb{S}^{n-1} \subset \mathbb{R}^n.$ I want to know that they are equidistributed (if you want to be more precise, you have a sequence of such collections, ...
3 votes
0 answers
86 views

Asymptotic approximation of Fisher information matrix for small Gaussian perturbation

Let $$ X=Y/a+b+\epsilon Z, $$ where $Y\sim\operatorname{Poisson}(\lambda)$ and $Z\sim\mathcal N(0,1)$ are independent. Also define $\theta=(\lambda,a,b,\epsilon)$. The Fisher information matrix $$ ...
3 votes
0 answers
86 views

Make inference on parameter $\lambda$ in Box-Cox transformation by MLE method

The original form of the Box-Cox transformation, as appeared in their 1964 paper, takes the following form: $$y(\lambda )=\begin{cases}\frac{y^{\lambda}-1}{\lambda}, & \lambda \neq 0\\ \log(y), &...
5 votes
0 answers
1k views

Multidimensional Berry–Esseen for probability density functions

This is a follow up to this recent question: Berry Esseen type result for probability density functions There exists a multidimensional version of the usual Berry–Esseen theorem (for cumulative ...
2 votes
1 answer
82 views

VC-based risk bounds for classifiers on finite set

Let $X$ be a finite set and let $\emptyset\neq \mathcal{H}\subseteq \{ 0,1 \}^{\mathcal{X}}$. Let $\{(X_n,L_n)\}_{n=1}^N$ be i.i.d. random variables on $X\times \{0,1\}$ with law $\mathbb{P}$. ...
1 vote
1 answer
108 views

Asymptotic expansion on the following integral of exponential function

I wish to obtain the asymptotic for the following integral $$ \int_{r: \|r\|\le 1} \exp(M\cdot a^Tr) \, dr, $$ where $a$ is a given vector in $\mathbb{R}^d$, $\|\cdot\|$ is a general norm function and ...
2 votes
1 answer
966 views

About maximal invariant statistics on a group family

I searched and I didn't find any answer (positive or negative): Suppose I have a group $G$, and a group family of probability measures . (i.e, there is a probability measure $P$ on $G$, and we define $...
1 vote
1 answer
117 views

Local maxima of the sum of Gaussian functions in *one dimension* are always strict local maxima - proof?

Motivated by this question asked earlier, I was wondering whether one can prove easily that the local maxima of the sum of Gaussians: $$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, \quad x_1 < x_2 < \...
0 votes
0 answers
100 views

Support function of the intersection of a hyper-ellipsoid and a Euclidean ball

Le $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_d$ be positive numbers. For any $x \in \mathbb R^d$ and $r \ge 0$, define $\gamma(x,r) := \sup_{z \in E(r)}x^\top z$, where $$ E(r) := E \cap B_2^d(r)...
2 votes
1 answer
154 views

Representer theorem for a loss / functional of the form $L(h) := \sum_{i=1}^n (|h(x_i)-y_i|+t\|h\|)^2$

Let $K:X \times X \to \mathbb R$ be a (positive-definite) kernel and let $H$ be the induced reproducing kernel Hilbert space (RKHS). Fix $(x_1,y_1),\ldots,(x_n,y_n) \in X \times \mathbb R$. For $t \ge ...
3 votes
2 answers
377 views

Precise asymptotics for moments of order statistics of normal distribution

Let $X_1, \cdots, X_n \sim N(0,1)$ be i.i.d. normal random variates. I am interested in understanding the first two moments of the quasi-range $X_{(n)}-X_{(n-1)}$ (i.e., the maximum value minus the ...
1 vote
1 answer
669 views

Quantifying aggregate vector strength/vector arithmetic

Say I have 5 vectors and I measure the similarity of each one to a fixed reference vector using cosine similarity. But now what I want to do is understand the aggregate or collective strength of these ...
2 votes
0 answers
54 views

Probability bounds of some ranked version of Dirichlet distribution

Recently I have come across a distribution defined on the open ranked simplex $\nabla^{n-1}_+ = \{\vec x \in \mathbb{R}^n:\sum_{k=1}^n x_k =1, x_1 \geq x_2 \geq \cdots \geq x_n > 0\}$, whose ...
9 votes
1 answer
612 views

Popular mistakes in probability

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...
-1 votes
1 answer
123 views

Under which conditions Mean Square Continuity implies Sample Continuity for Gaussian Processes?

First, let us give the setting. Let $(\Omega, \Sigma, \mathbf{P})$ be a probability space, let $T$ be some interval of time, and let $X: T \times \Omega \rightarrow S$ be a stochastic process. By Mean ...
0 votes
1 answer
134 views

Is the unconditional variance of a RV an upper bound for the variance of any conditional expectation of the RV?

Let $X$ and $Y$ be continuous random variables with finite first and second moments. Then, is it true that $Var[X]\geq Var[E(X|Y)]$?
3 votes
2 answers
229 views

Exact simulation of a large sample histogram

Say I want to create a histogram of $N$ random points from some simple compactly supported distribution on $\mathbb{R}$, where $N$ is very large, say $N = 10^{30}$. The histogram has $K$ disjoint bins,...
4 votes
1 answer
157 views

Existence of copula bound pointwise strictly smaller than the Fréchet-Hoeffding upper bound

Consider bivariate copulas $C_1$ and $C_2$ with $\max\{C_1(u,v), C_2(u,v)\}< M_2(u,v)$ for all $u,v \in(0,1)$, where $M_2(u,v) := \min\{u,v\}$ is the Fréchet-Hoeffding upper bound. Is there a ...
1 vote
1 answer
85 views

An inequality relating $\ell_1$ distance of input and output of a Markov krnel

Let $K$ be a Markov kernel from $\mathcal{X}$ to $\mathcal{Y}$, i.e., $K(\cdot|x)$ is a probability measure on $\mathcal{Y}$ for all $x\in \mathcal{X}$. Let $\mu$ and $\nu$ be two probability measures ...
2 votes
1 answer
312 views

Lower bound on sum of independent heavy-tailed random variables

I have a sum of $n$ i.i.d random variables $X_i$ such that $E[X_i] = 0$,$\mathrm{E}[|X_i|^{1 + \delta}]$ exists for some $0 < \delta < 1$ but $\mathrm{E}[|X_i|^{1 + \delta+ \epsilon}]$ does not ...
8 votes
3 answers
4k views

Sanov's Theorem and Chernoff bound

Sanov's Theorem (p.292, Thomas/Cover "Elements of Information Theory" (1991)) says that probability of a hypothesis $E$ according to distribution $Q$ is bounded above by $$(n+1)^k \exp (-n D(P^* \|Q)...
2 votes
1 answer
119 views

Justification of the use of residual plot

$\DeclareMathOperator\Cov{Cov}$Backround of my Question Let $Y$ be the response variable, $\mathbb{X}$ be the explanatory variables. The ultimate goal of prediction is finding a function $f^{*}$ that ...
0 votes
1 answer
57 views

Given positive $\epsilon$ and $c$, find a density $\phi$ such that $t\phi(\epsilon/t) \ge c \|\phi'\|_\infty$ for all positive $t$

A nice density (on $\mathbb R$) is function $\phi:\mathbb R \to \mathbb R$ such that (1) $\phi(x) \ge 0$ for all $x \in \mathbb R$, (2) $\int_{-\infty}^\infty \phi(x) \mathrm{d}x = 1$, (3) $\phi$ is ...
6 votes
1 answer
232 views

Violating an order statistic inequality?

[Edit: for posterity, I'm adding two small comments to the code explaining how to fix it, in light of Iosef Pinelis' answer below. Look for "Should be:" to find the corrections.] Suppose we ...
1 vote
1 answer
47 views

Represent multivariate data [closed]

I am not sure if this is the best place for my question. Please delete if it is not, but I would really appreciate some suggestions. I want to graphically represent multivariate data. I have 7 ...
1 vote
1 answer
1k views

Minimum number of support vectors? [closed]

I'm learning SVM and its written everywhere that the minimal number of support vectors is 2? But I couldn't find any formal proofs of that. Why cant there be less than 2 support vectors? Can somebody ...
2 votes
1 answer
159 views

Bayes risk of binary classification problem with conditionally independent covariates

In the setting of this problem, $\eta(\vec{x})$ is $P(Y=1|\vec{X}=\vec{x})$, $Y \in \{0,1\}$, and $X \in R^d$. Being the true probability know, the classification rule is simply $\eta(\vec{x})>0.5 \...
0 votes
0 answers
28 views

k-means errors for a block Gaussian vector

Consider a standard centered Gaussian vector $(X_1,...,X_n)$ with an approximate block structure, i.e. there is $q$ and a partition of $\{1,...,n\}$ in $q$ classes such that if $i,j$ are in the same ...
1 vote
2 answers
701 views

Finding decision boundary from empirical distribution

Based on measuring a certain characteristic, we want to classify measurements as coming from either of two populations. The true population distributions are unknown (and we don't want to take any ...
5 votes
2 answers
7k views

Upper bound total variation by Wasserstein distance for continuous distance

I am reading the survey of the relationships between metrics of distributions (see https://arxiv.org/pdf/math/0209021.pdf for the paper). The general results show that for general distributions, we ...
1 vote
0 answers
46 views

Sample complexity of estimating a doubly stochastic matrix

Let $P\in\mathbb{R}^{n\times n}$ be a doubly-stochastic matrix. That is: $$P(x,y)\geq 0,\quad \sum_xP(x,y)=1,\quad \sum_yP(x,y)=1.$$ I would like to know if lower and upper bounds on the sample ...
2 votes
2 answers
250 views

Nice way to parametrize a bunch of non-independent discrete random variables

I'm looking for a "nice" way to parametrize the joint distribution of multiple, possibly correlated discrete random variables on {0,1}. Even for N=2, there doesn't seem to be an obvious way to do it. ...
2 votes
0 answers
89 views

Training an energy-based model (EBM) using MCMC

I'm reading this paper about training energy-based models (EBMs) and don't understand the parameters that we are training for? The part that is relevant to the question is in pages 1-4. Here is the ...
3 votes
2 answers
2k views

Expected gradient vs. gradient of expectation

Suppose a function $f(x): \mathbb R^d \mapsto \mathbb R^D$, and its stochastic approximator, $g(x; W): \mathbb R^d \mapsto \mathbb R^D$. Here $W$ is some random variable. Then $g(x; W)$ is unbiased in ...
1 vote
1 answer
173 views

Rademacher complexity for a family of bounded, nondecreasing functions?

Let $\{\phi_k\}_{k=1}^K$ be a family of functions mapping from an interval $[a, b]$ to $[-1, 1]$. That is, $\phi_k \colon[ a,b] \to [-1, 1]$ are nondecreasing maps on some finite interval $[a, b] \...
2 votes
1 answer
99 views

Discrepancy of the Halton set

I am interested in low discrepancy sets for its applications in Monte Carlo integration - KH inequality tells us that the error will be lesser if the discrepancy of the sample is lesser. Every ...

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