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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.

-4
votes
0answers
41 views

Statistics flipping a coin [on hold]

if I were to flip a coin 10 times and it results in 4 heads and 6 tails, I have no reason to assume there is something 'wrong' with my coin. Now if I were to flip it 1 000 000 times and it results ...
0
votes
1answer
53 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\...
0
votes
1answer
26 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|\mathbf{x}] = \mu(\...
1
vote
1answer
61 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 ...
2
votes
2answers
93 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 $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
2
votes
1answer
64 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 ...
1
vote
0answers
42 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: ...
1
vote
1answer
48 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 ...
1
vote
0answers
53 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 ...
2
votes
2answers
188 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, ...
1
vote
0answers
27 views

Distribution of a normalized multivariate log-gaussian process

Let $p, q \in \mathbb{N}$ and $Z$ a gaussian process over $D \subset \mathbb{R}^{p+q}$ : $Z(\cdot) \sim \mathcal{GP}(\mu(\cdot), k(\cdot,\cdot))$. For $(x_1, \ldots, x_p, y_1, \ldots, y_q)^T \in D$, ...
2
votes
1answer
63 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 $x_i \sim \mathcal{CN}(0,a), \forall i$...
5
votes
1answer
162 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)...
4
votes
1answer
101 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)}...
1
vote
1answer
54 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 ...
0
votes
0answers
34 views

Marginal Distribution of Partition Matrix

Suppose that $X\sim IW_{p}(n,I_p)$ has an inverse Wishart distribution, which probability density function is $$f(X\mid n)\propto |X|^{-\frac{n+p+1}{2}}exp\Big(-\frac{1}{2}tr( X^{-1})\Big),~~\qquad (...
-3
votes
2answers
259 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(\...
1
vote
0answers
75 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}...
0
votes
0answers
66 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 ...
0
votes
0answers
39 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{...
2
votes
3answers
181 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 ...
4
votes
1answer
239 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 ...
2
votes
1answer
50 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 ...
0
votes
0answers
35 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(\...
2
votes
1answer
186 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 ...
8
votes
2answers
524 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), ...
5
votes
1answer
170 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 ...
1
vote
0answers
35 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 ...
3
votes
0answers
78 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 ...
1
vote
2answers
43 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 ...
0
votes
1answer
50 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 ...
0
votes
0answers
19 views

Anti concentration of $\frac{1}{n}zz^T + \text{diag}(z_1^2,\ldots,z_n^2)$ for sub-gaussian i.i.d $z_1,\ldots,z_n$ and $z:=(z_1,\ldots,z_n)$

Let $z_1,\ldots,z_n$ be an i.i.d sample from a sub-gaussian distribution. Define the $n$-by-$n$ p.s.d matrix $C_n := \frac{1}{n}zz^T + \text{diag}(z_1^2,\ldots,z_n^2)$, where $z:=(z_1,\ldots,z_n)$. ...
0
votes
0answers
18 views

Concentration of $X^T\eta\eta^TX \in \mathbb R^d$ for i.i.d $(x_i,\eta_i)$ and sub-gaussian $\eta_i$

Suppose $(x_1,\eta_1),\ldots,(x_n,\eta_n)$ are $n$ i.i.d points in $\mathbb R^{d+1}$ such that $\eta_1,\ldots,\eta_n$ are $\sigma$-subgaussian. Let $X \in \mathbb R^{n \times d}$ be the vertical ...
0
votes
0answers
60 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:...
2
votes
0answers
40 views

Enumerating lattice points in a product of balls, in limit with dimension

Fix $(L_n)_n$ to be a sequence of lattices, each $L_n\subset \mathbb{R}^n$, where both the effective-inradius and effective-outradius go to 1 (i.e. the Voronoi region of the lattices approach a ball ...
1
vote
1answer
67 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(...
-2
votes
1answer
75 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 $y \sim \mathcal{N}(0,1)$.
0
votes
0answers
18 views

Non-asymptotic tail-bounds for Hotelling $T^2$ statistic

Let $X_1,\ldots,X_n$ be an i.i.d sample from a distribution on $\mathbb R^p$ with mean $\mu = 0 \in \mathbb R^p$ and $p$-by-$p$ covariance matrix $\Sigma$ of rank $r \le p$. Consider the centered ...
1
vote
0answers
19 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 ...
0
votes
1answer
109 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 (...
1
vote
1answer
96 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
0answers
72 views

How to formally connect the log-Euclidean and Riemannian metrics for Symmetric Positive Definite matrices?

I'm a statistician working on a research project dealing with metrics on SPD matrices, specifically the log-Euclidean $d_{LE}(X, Y) = \|\log(X) - \log(Y)\|$ and the Riemannian metric $d_{R}(X, Y) = \|\...
2
votes
0answers
56 views

Proof of a technical fact in the book of Schapire and Freund on boosting

Disclaimer: I asked this question on math.stackexchange.com two weeks ago but it has not been answered yet so I figured that I might as well try to also post it here. I am currently looking at ...
0
votes
0answers
45 views

Does lattice mod preserve direction?

For high enough dimension $n$ there are lattices $L_n$ in $\mathbb{R}^n$ whose Voronoi partition's base regions encompass all but a negligible proportion of a $(1-\varepsilon)$-ball, and also nearly ...
0
votes
1answer
63 views

Reconstructing Euclidian space from distance matrix

The setup. Let's say that we have a set of objects $O_i$ for which we have a dissimilarity measure $M(O_1,O_2)$. With this we can build a distance matrix $D_{ij}$. Let's also assume that we have NO ...
4
votes
3answers
115 views

Expected distance of nearest matching pair in the game of pairs

Recently I was playing several rounds of the game of pairs with my children. I was surprised that almost every time, one matching pair was adjacent (either next to each other in a row, or vertically). ...
1
vote
1answer
116 views

Expected values of two non-negative, integer-valued random variables related to an urn problem

Consider an urn containing $c$ distinguishable balls, $\alpha$ of which are red, $\beta$ of which are blue, and $\gamma$ of which are green, and $\alpha+\beta+\gamma=c$. We assume $\alpha,\beta,\gamma&...
-1
votes
1answer
45 views

Characterisation of a superset of the simplex

Does there exist a nice description of the following set: \begin{equation} A:=\left\lbrace x\in\mathbb{R}^{n}\ \colon\ 0< x_{i}-\bar{x}+\frac{1}{n}< 1\ \text{for} \ i=1,\dots,n\right\rbrace, \...
3
votes
1answer
83 views

Non-asymptotic tail bounds for $D_{\text{Hellinger}}(P\|\hat{P}_N)$

Let P be a distribution on a finite set of size $k$ and let $\hat{P}_N=(N_1/N,\ldots,N_k/N)$ be the empirical distribution (frequencies) from a samples of size $N$. Consider the Hellinger distance ...
1
vote
0answers
44 views

Gaussian as a product of two independent random variables [duplicate]

Ideally what I am looking for two random variables, $X$ and $Y$ (if one is positive then that's even better) such that $Z=X\cdot Y\sim\mathcal{N}(0,1)$ where $X,Y$ are some distributions I can ...