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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Intractability of an integral by deterministic numerical methods

Suppose $X_1,\ldots,X_n$ is an i.i.d. sample from a probability distribution with continuous c.d.f. $F.$ Let $F_n$ be the empirical c.d.f. $$ F_n(x) = \frac 1 n \sum_{k=1}^n \mathbf 1_{X_n\le x} = \...
Michael Hardy's user avatar
1 vote
1 answer
538 views

A question about the proof of Riesz-Thorin interpolation theorem

I was reading the proof of Riesz-Thorin interpolation theorem in http://www.math.kit.edu/iana3/lehre/fourierana2014w/media/rieszthorinproof.pdf and get stuck at the last step. We construct the complex ...
aurora_borealis's user avatar
1 vote
1 answer
92 views

boundig variation from median [closed]

Given a scalar random variable $X$, suppose that there are positive constants $c_{1}$ and $c_{2}$ such that $$\forall t\geq 0 : \,\,\,\,\,\,\ \mathbb P\{|X-\mathbb EX|\geq t\}\leq c_{1}e^{-c_{2}t^{2}}...
Meysam's user avatar
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1 answer
104 views

What is the distribution of a Cartesian power of a collection of iid uniform points? (renewed)

The following question was asked recently at https://mathoverflow.net/questions/326631/what-is-the-distribution-of-a-cartesian-power-of-a-collection-of-iid-uniform-poi : Take a rectangle with ...
Iosif Pinelis's user avatar
2 votes
0 answers
161 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 ...
user124784's user avatar
2 votes
0 answers
105 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 $p\times m$ ...
user482401's user avatar
2 votes
1 answer
240 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 ...
Florian Tramèr's user avatar
3 votes
1 answer
408 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 ...
King Kong's user avatar
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2 votes
1 answer
187 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} + \...
B Merlot's user avatar
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1 answer
720 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 ...
guigux's user avatar
  • 607
1 vote
0 answers
56 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 : \...
gradstudent's user avatar
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2 votes
1 answer
597 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}\...
B Merlot's user avatar
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1 vote
1 answer
60 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. ...
Stephen Fratamico's user avatar
1 vote
0 answers
145 views

Clarification about the ϵ -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 ...
Amit Rege's user avatar
2 votes
1 answer
742 views

Concentration inequality for quadratic form of Gaussian variables with non-idempotent matrix

Given $y \sim N(0,\sigma^2 I)$, and $M$ that is a symmetric matrix (not necessarily idempotent) what is the distribution of ${y^T M y}$? is there a high probability bound on $|{y^T M y}|$? Most ...
Enigman's user avatar
  • 123
3 votes
0 answers
150 views

Central Limit Theorem for simultaneous sums

Take a sample $X_1 \ldots X_n$ of $n$ independent observations $X_j \in \mathbb{R}$ with zero mean and finite variances $\sigma_j^2$. For $i = 1, 2, \ldots$, define the sums $$S^n_i = \frac{\pm X_1 \...
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1 vote
0 answers
168 views

Kullback-Leibler as a function of weights on a normal mixture

I'm interested in the Kullback-Leibler divergence on multimodal gaussian mixtures. For positive, real weights $\sum_{1\leq k\leq m}w_k=\sum_{m+1\leq k\leq n}w_k=1$, univariate Gaussians $g_k\equiv g(...
Matt Cuffaro's user avatar
0 votes
1 answer
117 views

CDF of a RV that is the ratio between a complex Gaussian and a Chi-squared RVs

Given the following p.d.f., which is the p.d.f. of the real and imaginary parts of a random variable that is the ratio between a complex Gaussian and a Chi-squared RVs: \begin{equation*} f_U(u)=\exp\...
Felipe Augusto de Figueiredo's user avatar
1 vote
1 answer
164 views

Error metric for joint estimation of mean and variance

Background: Let $\mu:\mathbb{R}^n\to\mathbb{R}$ and $\sigma:\mathbb{R}^n\to\mathbb{R}_+$ be two unknown functions, and consider a stochastic model of the form $$ \mathbb{E}[Y\mid\mathbf{x}] = \mu(\...
guigux's user avatar
  • 607
1 vote
1 answer
138 views

Minimization Proof of Conditioning on Gaussian is Gaussian

It is well known that $E[X|X+Y]$ is Gaussian if both $X$ and $Y$ are, and the result can be derived using standard density arguments. However, how can one prove it by only resulting to optimization ...
ABIM's user avatar
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2 votes
2 answers
143 views

Expectation of $\left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2$, where $\textbf{x}$ and $\textbf{y}$ are complex Gaussians?

Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
Felipe Augusto de Figueiredo's user avatar
2 votes
1 answer
97 views

p.d.f. of $\left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2$, where $\textbf{x}$ and $\textbf{y}$ are complex Gaussians?

Given that the random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ are ...
Felipe Augusto de Figueiredo's user avatar
1 vote
1 answer
320 views

Lower-bound probability of non-centered quadratic form

Let $X\sim N(\mu,\sigma^2I)\in \mathbb{R}^n$ be a non-centered ($\mu\neq 0$) Gaussian vector with independent coordinates. I'm wondering if there is any sharp lower bound of the following probability: ...
neverevernever's user avatar
1 vote
1 answer
112 views

Approximate $\log \mathbb E_P[\exp(th(x)]$ for a function $h$ which is lipschitz and has finite moments of order 1 and 2 w.r.t $P$

Let $P$ be a probability measure on a space $\mathcal X$ and $h: \mathcal X \rightarrow \mathbb R$ is measurable function with finite moments of order 1 and 2. I'm interested in approximating the ...
dohmatob's user avatar
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1 vote
0 answers
81 views

Central limit theorem is expected, but variance is apparently sublinear? So?

In Perplexing Problems in Probability, the following statement about first passage percolation (FPP) on $\mathbb{Z}^{d}$ is made. See e.g. this paper of Benjamini, Kalai and Schramm, where they quote ...
apg's user avatar
  • 612
1 vote
2 answers
269 views

Closed expression for $\mathbb{E} \left\lbrace \Re \frac{(\textbf{x} + \textbf{y})^{H}\textbf{x}}{\| \textbf{x} + \textbf{y}\|^{2}} \right\rbrace$?

Given the random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ are independent, ...
Felipe Augusto de Figueiredo's user avatar
1 vote
1 answer
88 views

Independence of r.v.'s following a distribution that is the ratio between complex Gaussian and Chi-square r.v.'s

Given the following two R.V.s $$z_1 = \frac{x_1}{|x_1|^2 + |x_2|^2 + \cdots + |x_M|^2}$$ and $$z_2 = \frac{x_2}{|x_1|^2 + |x_2|^2 + \cdots + |x_M|^2}$$ where $x_i \sim \mathcal{CN}(0,a), \forall i$...
Felipe Augusto de Figueiredo's user avatar
5 votes
1 answer
218 views

Switching oriented paths in a graph

Consider an oriented graph (e.g. a finite part of the standard grid with some random orientations). Each minute the following operation takes place: we choose uniformly randomly an ordered pair $(A,B)...
Nikita Kalinin's user avatar
4 votes
1 answer
175 views

Inner product of sorted Gaussian vector

Suppose $X_1,\ldots,X_n$ are i.i.d. standard normal. I'm wondering how to analyze the following quantity: $$\left|\frac{X_{(1)}X_{(n)}+X_{(2)}X_{(n-1)}+\cdots+X_{(n)}X_{(1)}}{n}\right|$$ where $X_{(1)}...
neverevernever's user avatar
1 vote
1 answer
125 views

Expected norm of linear maps

I want to compute the expected norm of a vector-matrix multiplication. I have a vector $x \in \mathbb{R}^n$ with norm one and a matrix $M \in \mathbb{R}^{n \times n}$, whose entries are iid taken from ...
Alfred's user avatar
  • 879
-3 votes
2 answers
439 views

Expected values of two random variables related to a simple urn problem

In an urn there are $u$ balls, $b$ of which are black. If we perform $n$ trials of one ball at a time with replacement, the probability of the event $E$ to get $n$ times a black ball is $P(E)=\left(\...
Andrea Prunotto's user avatar
1 vote
0 answers
128 views

Inverse Wishart

Assume that $X\sim \operatorname{IW}_p(n,\Sigma)$ has an inverse Wishart distribution, which probability density function is $$f(X\mid n,\Sigma)=C(n,\Sigma)|X|^{-\frac{n+p+1}{2}} \exp\Big(-\frac{1}{2}...
Xiaopai Song's user avatar
0 votes
0 answers
103 views

Expectation of maximal Wasserstein distance between empirical distribution and a pdf

Let $P$ be a continuous probability distribution on $R^d$, $X$ the random variable $\sim P$, and $ \hat{X}$ be n i.i.d samples drawn according to $P$. We have another variable $\mu \in S^{d-1}$. Do ...
Will Cai's user avatar
  • 109
0 votes
0 answers
56 views

Absolute continuity of probability measures determined by dependence structure

We are on $\mathbb{R}^d$ with Borel $\sigma$-algebra. Let $\mu_1, ..., \mu_d$ be probability measures on $\mathbb{R}$ and $\Pi(\mu_1, \mu_2, ..., \mu_d)$ be the set of probability measures on $\mathbb{...
Steve's user avatar
  • 1,085
4 votes
5 answers
6k views

Proof of Bellman optimality equation for finite Markov Decision Processes

This question has already been posed in Cross Validated without receiving a correct formal answer, so I reformulate it here to gain attention of mathematicians. I am referring to chapter 3 of Sutton ...
hardhu's user avatar
  • 171
4 votes
1 answer
284 views

How sensitive are probability distributions to noise?

I'm trying to prove a result but I'm stuck at the very end of it: I'm having troubles understanding how noise propagates when considering a probability distribution. In other words, if I inject some ...
Alfred's user avatar
  • 879
2 votes
1 answer
1k views

Mutual information between continuous and discrete variables from numerical data

I am looking for references/measures to estimate the mutual information between a continuous (C) and discrete (D) variable, given a real-world (i.e. finite sample) data set. C is uniformly distributed ...
Gaurang's user avatar
  • 21
2 votes
1 answer
103 views

P-value in likelihood ratio test definition

According to Williams, D.: Weighing the Odds the p-value of observed data in the likelihood ratio setting is defined as $$\mathrm{p_{val}}(y^{obs}) := \mathrm{sup}_{\theta \in B_0} \mathbb{P}\big(\...
Joe's user avatar
  • 151
2 votes
1 answer
212 views

Are Linear Maps resistant to Noise?

Let's assume I have a $m \times m$ matrix $M$ with Frobenius norm $1$ and a unit vector $x \in S^{m-1}$. I also have a second $m \times m$ matrix $M^*$ which is obtained from the first one plus some ...
Alfred's user avatar
  • 879
10 votes
3 answers
2k views

Random Walks on high dimensional spaces

I've read on a paper that, in the two dimensional case, if you start from the origin and take steps of length one in arbitrary directions (uniformely on the unit sphere $S^1$, not left-right-up-down), ...
Alfred's user avatar
  • 879
7 votes
1 answer
213 views

Relation between the two possible KL divergences of two distributions

Given that I know $$D\left(P\parallel Q\right)<\alpha,$$ can I say anything about $D\left(Q\parallel P\right)$ in terms of an upper bound on it? Also, given this upper bound on $D\left(P\parallel ...
Student88's user avatar
  • 503
1 vote
0 answers
148 views

Hypergraph partitioning and bipartite graph partitioning

Are hypergraph partitioning, and bipartite graph partitioning related, or equivalent, given that hypergraphs can be represented as bipartite graphs? In the first case, we want to partition the set of ...
Carlos Botas's user avatar
3 votes
1 answer
328 views

Upper bounding the start of a distribution's CDF, given bounds on first moments

Take nonnegative random variables $X$ whose first $K$ moments have bounds: $\mu^k\leq E[X^k]\leq c\mu^k$ for each $k=1,\dots,K$. In this case what is an upper bound for $P(X\leq O(\mu))$? I am ...
Christian Chapman's user avatar
1 vote
2 answers
187 views

Approximate the variance of multiple normal distributions with the same standard deviation

Given a number of normal distributions $N(\mu_1, \sigma^2), N(\mu_2, \sigma^2), ..., N(\mu_n, \sigma^2)$ with fixed variance $\sigma^2$, but not necessary equal means. My question is how to ...
Noud's user avatar
  • 13
0 votes
1 answer
214 views

Anti-concentration: upper bound for $P(\sup_{a \in \mathbb S_{n-1}}\sum_{i=1}^na_i^2Z_i^2 \ge \epsilon)$

Let $\mathbb S_{n-1}$ be the unit sphere in $\mathbb R^n$ and $z_1,\ldots,z_n$ be a i.i.d sample from $\mathcal N(0, 1)$. Question Given $\epsilon > 0$ (may be assumed to be very small), what is ...
dohmatob's user avatar
  • 6,716
0 votes
0 answers
255 views

Does the linear combination of the quantile $\alpha F^{-1}(\tau)+\beta G^{-1}(\tau)$ still a quantile

$F(x)$ and $G(y)$ are distribution functions. Define the $\tau$th quantile for cdf $F(x)$, $G(y)$ as $$\xi_\tau\equiv F^{-1}(\tau)=\inf\{x:F(x)\ge \tau\}$$ and $$\eta_\tau\equiv G^{-1}(\tau)=\inf\{y:...
J.Mike's user avatar
  • 141
1 vote
1 answer
427 views

Sharp tail bounds for the maximum of an iid sample of a random variable supported on $[0, 1]$

Let $X_1,\ldots,X_n$ be an iid sample from a distribution supported on $[0, 1]$. Question What are some sharp concentration inequalities (i.e tail bounds) empirical statistic defined by $Z_n := \max(...
dohmatob's user avatar
  • 6,716
-2 votes
1 answer
90 views

Existence or impossibility of Gaussian factory

Gaussian factory problem: given an iid sequence $x_i \sim \mathcal{N}(\mu,\sigma^2)$, $i=1,2,\dots$, with $\mu$ and $\sigma^2$ both unknown, construct a realization $y \sim \mathcal{N}(0,1)$.
Sebastian Nowozin's user avatar
1 vote
0 answers
32 views

Practical statistics for queueing networks

There is a theory for queueing networks where we postulate some nicely behaving base distributions of arrival processes and service processes and then calculate the behaviour of the system. Now, in ...
Gergely's user avatar
  • 291
0 votes
1 answer
142 views

A problem related to the comparison of two integer-valued random variables

Consider an urn containing red, blue and green balls (the situation is the same illustrated in this post). Let $X$ be the non-negative, integer-valued random variable defined as the number of trials (...
Andrea Prunotto's user avatar

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