All Questions
Tagged with pr.probability st.statistics
1,134 questions
10
votes
2
answers
847
views
Minimum separation among $m$ random points on an $n$-dimensional unit sphere
Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be
$$
\rho = \min_{i,j\in{...
CommunityBot
- 1
3
votes
0
answers
82
views
Uniform mean-square-error estimates
Consider a standard statistical estimation problem with iid real observations $\{X_i\}_{i=1}^N$. For a collection of real functions $\mathcal{F}$, I want to get an estimate of the uniform rate of ...
- 111
9
votes
5
answers
1k
views
estimate the error term in CLT
Let $X_m = \frac{1}{\sqrt{m}}\sum_{k=1}^m Z_k$ where $Z_k$ are iid equally likely on $\{\pm 1\}$. Then $X_m$ convergens to $X \sim \mathcal{N}(0,1)$ in distribution by CLT.
Let $f$ be a smooth ...
- 128k
6
votes
1
answer
238
views
Positive semidefinite ordering for covariance matrices
Suppose that X and Z are matrices with the same number of rows. Let
$$ D = \left[\begin{array}{cc} X' X & X'Z \\ Z'X & Z'Z \end{array} \right]^{-1} - \left[\begin{array}{cc} (X' X)^{-1} & ...
- 128k
1
vote
0
answers
227
views
Uniform Law Of Iterated Logarithm for VC classes
Kenneth Alexander proved a uniform Law Of Iterated logarithm for Vapnik-Chervonenkis classes in the article Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm (Ann. ...
- 3,968
1
vote
1
answer
147
views
Proving that an integral related to order statistics is increasing in a certain parameter
Let $f$ and $F$ denote, respectively, the pdf and cdf of a probability distribution on $\mathbb R$. Take any natural $n\ge3$ and any real $a$ and $c$ such that $a\le c$.
Does it always follow that
$$...
- 128k
7
votes
2
answers
605
views
Uniform Concentration Bounds on Weighted Sum of i.i.d. Bernoulli Random Variables
Let $\delta_1,...,\delta_n$ be $n$ independent identically distributed Bernoulli random variables with $\mathbb{P}(\delta_1=1)=p$. We consider a set $\Omega = \{\mathbf{a}:=(a_1,...,a_n)~|~a_i\in [0,c/...
- 24.2k
2
votes
1
answer
150
views
Probability of collision of some family of hash functions
Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $ h(x) = \left\lfloor x + \sum^C_{i=1} U_i \right\rfloor$ for some ...
- 128k
3
votes
1
answer
378
views
What is the mathematics behind the random experiment which produces the data with this strange property?
I have a following scenario. there is a huge collection of data resulting from a random experiment $E$ (I do not say random variable yet, for reasons that you will need to explain in your answer). Let ...
- 698
1
vote
0
answers
88
views
Why is the classical secretary problem about ranks?
This relates here: https://math.stackexchange.com/questions/1820997/why-is-the-classical-secretary-problem-about-ranks
You want to stop optimal in a sequence of items presented sequentially, that is ...
CommunityBot
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4
votes
2
answers
255
views
The necessary sufficient condition for recurrence of a Markovian random walk
Suppose $\sigma_{1},\sigma_{2},...$are i.i.d random variables.$S_{0}=0$. Define $S_{n}=S_{0}+\sum_{i=1}^{n}\sigma_{i}$, then ${S_{n}}$ is a Markovian random walk.
I want to figure out the necessary ...
CommunityBot
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1
vote
0
answers
64
views
Posterior consistency of non linear model
This is possibly a reference request. Let $G$ : $\mathbb{R}^p \to \mathbb{R}^q$ be a continuous injective/bijective function. Let $\mu$(we may also assume this to be a non degenerate Gaussian) be ...
- 157
1
vote
1
answer
219
views
connection between the statistical properties of a scalar field and its columns
Consider a scalar field $s:[0,1]^3 \to \mathbb{R}$ and its "column" field
\begin{equation}
c: [0,1]^2 \to \mathbb{R}: (x,y) \mapsto \int_0^1 s(x,y,z) \,\mathrm{d}z.
\end{equation}.
What can be said ...
CommunityBot
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2
votes
1
answer
160
views
Do product distributions (or graph products) eventually cluster as more products are taken?
Say we have a joint distribution on a finite alphabet $\mathcal{X}\times \mathcal{Y}$. It could be a communication link where we want to send a random message $X$ over a channel, but it gets garbled ...
CommunityBot
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4
votes
1
answer
203
views
Can samples be compressed?
The Fisher information of a random variable $Y$ about a parameter $\theta$ upon which the probability of $Y$ depends is:
$\mathcal{I}_Y(\theta)= -E\left[\left.\strut \frac{\partial^2}{\partial \theta^...
- 5,971
2
votes
0
answers
1k
views
Moments of function of Poisson process
(I'm new to Poisson processes, so please edit if my terminology is incorrect.)
Edit: per comments, here is a (more) general version of the originally posted problem (which is now at the bottom, below ...
CommunityBot
- 1
1
vote
0
answers
87
views
Convergence of an rcll process along a random subsequence
I have a process $X_s$, for $s \ge 0$, taking values in a Polish space $T$ with an rcll version where I have shown, for every nonrandom increasing sequence $s_n$, that $X_{s_n} \to c$ in probability, ...
- 321
5
votes
1
answer
497
views
Size of KL-divergence neighbourhoods
I am new here. I was reading another
post
here and this got me wondering what can be said about the size of the following kl divergence neighborhoods.
Consider these two kl-divergence neighbourhood ...
CommunityBot
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2
votes
0
answers
54
views
Literature on transformed Gaussian matrices
I am considering real $n$-by-$m$ matrices of the following type:
$$
M=SM^\prime,\\
M^\prime_{ij}\sim^{iid}N(0,1).
$$
Here, $S$ is a fixed $n$-by-$n$ matrix and the entries of $M^\prime$ (same size ...
- 121
6
votes
1
answer
129
views
Choosing a sample based on where the density function is highest
Is there a name for the following process?
Say I have an absolutely continuous probability density function $f$ with compact support, and I take $k$ independent samples $x_1,\dots,x_k$ from $f$. ...
- 2,558
3
votes
1
answer
135
views
Is there a closed form expression for $E(X e^{-\mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$?
Is there any closed form expression for $E(X e^{- \mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$? If not, is there any tight upper bound for this quantity? Any idea how to proceed?
- 156
2
votes
0
answers
87
views
A question about probabilistic graphical models
Say one is given a probabilistic graphical model and a cut of the underlying graph. Do we know any statements about when and how can one or many of the marginals (of the sources) or the conditionals (...
- 2,246
1
vote
0
answers
69
views
Norm-averaging reference request
(Apology in advance for the broadness of this question) I recently came across a relatively simple application where I needed to "balance" the "spreaded-out-ness" of a function with the "peaked-ness" ...
- 123
5
votes
3
answers
4k
views
Integral over error function and normal distribution
Help me understand why
$\int_{-\infty}^{\infty}\frac{1}{2}[1+\operatorname{erf}(\frac{\theta-x}{\sqrt{2q^2}})]\frac{1}{\sqrt{2\pi\sigma^2}}{\exp(-\frac{(x-\mu)^2}{2\sigma^2})}dx \approx \frac{1}{2}[...
CommunityBot
- 1
3
votes
1
answer
187
views
Moment matching on the standard simplex
Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where
$$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j \...
- 1,127
4
votes
0
answers
573
views
An inequality involving conditional variance and its connection to information theory
Given absolutely continuous random variables $(X, Y)$ with joint distribution $P_{XY}$, we construct $Z:=\sqrt{\gamma} Y+N_\mathsf{G}$ where $N_\mathsf{G}\sim N(0, 1)$ and is independent of $(X,Y)$ ...
- 1,109
3
votes
1
answer
270
views
Learn a distribution from distributions on samples
There's many good ways to learn a distribution $p_X$ of an r.v. $X$ over $k$ symbols given many i.i.d. samples $X_1,\ldots, X_n$. The simplest is to use the sample relative frequencies $\hat{f}_X$ as ...
- 219
2
votes
1
answer
250
views
An Inequality Regarding the Squared Conditional Variance
Given absolutely continuous random variables $(X, Y)$ with joint distribution $P_{XY}$, we construct $Z:=\sqrt{\gamma} Y+N_\mathsf{G}$ where $N_\mathsf{G}\sim N(0, 1)$ and is independent of $(X,Y)$. ...
- 128k
1
vote
1
answer
494
views
Do there exist random variables that force transitivity of dependence? [closed]
In general, statistical dependence is not transitive. If $Y$ and $X_{1}$ are dependent, and $Y$ and $X_{2}$ are dependent, then $X_{1}$ and $X_{2}$ are NOT necessarily dependent.
However, in some ...
- 54.2k
4
votes
1
answer
151
views
Negative population variable importance
I asked this question on stats.stackexchange and even elsewhere, but it never received an answer.
I just state the probabilistic problem here. It is about the optimality of the conditional ...
CommunityBot
- 1
2
votes
1
answer
560
views
Covariance matrix as optimization problem solution?
I have seen the expectation of a random vector expressed as the solution to the optimization problem:
\begin{equation}
\mathbb{E}[X]=argmin_{v \in \mathbb{R}^n}\mathbb{E}[\|X-v\|_{l^2}^2](:= \int_{\...
- 5,405
1
vote
1
answer
115
views
Supremum of centered jointly generalized chi-square random variables
Let $\zeta_n$ be a sequence of centered jointly generalized chi-square random variables, i.e. $\zeta_n = \sum_{k=1}^{m_n} a_{k,n}(\xi_{k,n}^2 - E[\xi_{k,n}^2])$, and $\xi_{k,n}$ are centered jointly ...
CommunityBot
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8
votes
2
answers
2k
views
Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian
This is a question related to the statistical model behind independent component analysis (ICA).
We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...
CommunityBot
- 1
1
vote
1
answer
1k
views
Null hypothesis test for independent but not identically distributed samples
I'm trying to figure out the best statistical test to use for an edge case I've run into: trying to figure out the likelihood of the null hypothesis for a set of samples that each (potentially) come ...
1
vote
0
answers
64
views
Simulate a graph from a certain distribution
I am wondering if anyone can indicate whether the following is a solved problem. I don't care about time of the algorithm currently.
Consider a general probability distribution F on simple graphs ...
- 11
3
votes
1
answer
213
views
A lottery on coins in a convex set
You play the following game.
You get $4n$ gold coins and have to arrange them in the unit square in general position (no two coins have the same x or the same y coordinate). Call this set of coins $...
CommunityBot
- 1
11
votes
1
answer
283
views
Probability distribution derived from gamma function - does it have a name?
Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$.
Now, let's fix $\sigma$ and let t vary. Then consider the following expression:
$$|\Gamma(\sigma+it)|^2$$
For any choice of $\...
CommunityBot
- 1
4
votes
1
answer
161
views
Hellinger integral for the Student/Cauchy family
Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral is
$H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$.
Let now $p$ be ...
- 109k
9
votes
1
answer
385
views
A Generalized Version of Maximal Correlation and Hypercontractivity of Conditional Expectation Operator
Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as
$$\rho_2(X;Y):=\sup\frac{\mathbb{E}[f(X)g(Y)]}{||f||_2||...
CommunityBot
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6
votes
0
answers
578
views
Maximal Correlation versus Correlation Coefficient When one RV is Gaussian
Let a pair of random variables $(X,Y)$ be continuous random variables (i.e., they both have density with respect to Lebesgue measure) with joint distribution $P_{XY}$. The maximal correlation $\rho_m(...
- 1,109
1
vote
1
answer
124
views
"Convergence speed" results for the Langevin process
The Langevin process is defined by the following stochastic differential equation:
$$ \dot X = - \nabla \phi + \sqrt 2 dW_t $$
Its equilibrium distribution is the following:
$$ p_\infty (x) \propto ...
- 188k
10
votes
2
answers
925
views
Constructing an independent uniform random variable from two independent ones
Does there exist a continuous (differentiable) function $h:[0,1]\times [0,1] \to [0,1]$ such that if $\alpha,\beta\in [0,1]$ are independent and uniformly distributed on $[0,1]$, the random variable $...
- 4,466
4
votes
1
answer
229
views
How are the real-space RG transformations defined?
I'm reading Shang-keng Ma's book Modern theory of critical phenomena, and I'm a bit confused as to how the real-space RG transformations are defined. Ma basically says that these transformations are ...
CommunityBot
- 1
2
votes
0
answers
619
views
Laplace transform of a integral function of CIR/CEV process
The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...
- 323
0
votes
0
answers
322
views
Comparison of Parameter estimation using maximum likelihood and Maximum entropy
I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...
- 495
5
votes
1
answer
297
views
Random walk with continuously distributed steps on [-1,1]
A simple random walk $S_n = X_1 +\cdots +X_n$, where $P(X_i = 1) = p \not = 0.5$ and $P(X_i=-1)= q \triangleq 1-p$, admits the following probability
$$P(S_n \textrm{ reaches } a \textrm{ before} -b) =...
- 108
0
votes
2
answers
76
views
A way to possibly calculate one Binomial CDF function from another closely related one?
Let $y < z$ be two numbers between $0$ and $1$, is there a way to relate the CDF functions $F_{n,y}(s)$ and $F_{n,z}(s)$... or approximate one from another, without just saying $F_{n,z}(s) \le F_{n,...
- 37.7k
5
votes
1
answer
373
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assumptions on local rademacher complexities
A lot of the work on Local Rademacher complexities of Koltchinskii, and Bartlett for fast rates of convergence is based on Bousquet's version of Talagrand's inequality [1] (Theorem 2.11). However the ...
CommunityBot
- 1
1
vote
1
answer
202
views
An inequality for Maximal Correlation over a Markov Chain
Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...
- 14.2k
1
vote
0
answers
80
views
A variance-preserving Boolean function [closed]
Let a random variable $X$ be given with $P_X$ supported over $\mathcal{X}$. What are the necessary conditions for the existence of a boolean function $f:\mathcal{X}\to \{0,1\}$ such that $\mathsf{var}(...
- 1,109