All Questions
Tagged with pr.probability st.statistics
1,135 questions
2
votes
1
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592
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Concentration rates for the posterior distribution
Sanov's theorem and Dvoretzky–Kiefer–Wolfowitz's inequality tell us how fast the empirical distribution concentrates around the true underlying probabilty distribution.
What is known about the ...
- 591
4
votes
0
answers
990
views
Probability distribution function for singular value sum of Gaussian random matrix
Let $\mathbf{X}$ be an $N \times N$ random matrix with IID Gaussian entries. They can be standard normal, but $N$ is not large: that is $N$ $<$ 6, typically. Call its singular value decomposition (...
4
votes
0
answers
76
views
How well does an estimator perform on another dataset?
Suppose $X \sim N(0, \Sigma)$ is a $d$-dimensional Gaussian random vector. And we have $2n$ $i.i.d$ sample $X_1, \ldots, X_{n}, \ldots, X_{2n}$.
Let $\hat{\Sigma}_1 = \frac{1}{n}\sum_{i=1}^nX_i X_i^\...
- 515
1
vote
0
answers
227
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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. ...
- 31
3
votes
1
answer
326
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Two matrix Fisher distributions on SO(3)?
After the uniform distribution (normalized Haar measure), the matrix Fisher distribution seems to be the most popular probability distribution on the Lie group SO(3). The density is proportional to ...
- 41
1
vote
1
answer
3k
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Convergence of Eigenvalues
Suppose we have a matrix $A_n = \frac{1}{n}\sum_{i=1}^nX_i X_i^T$, where $X_i$ is a $p$-dimensional random-vector. We also know that $E(XX^T) = \Sigma_{p \times p}$. Let us denote the $j$-th largest ...
- 63
4
votes
3
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433
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Expectation of $(c+e^{N(0,\sigma^2)})^{-n},\, n>0$
I would like to know if there's a way to compute or approximate the following expectation:
$$\mathbb{E}[(c+e^X)^{-n}]$$
where $X=N(0,\sigma^2)$ and $n,c>0$ (you can also assume that $n$ is a ...
- 81
8
votes
2
answers
1k
views
Order statistics (e.g., minimum) of infinite collection of chi-square variates?
Hi everyone,
This is my first time here, so please let me know if I can clarify my question in any way (incl. formatting, tags, etc.). (And hopefully I can edit later!) I tried to find references, ...
- 101
4
votes
0
answers
73
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Regularity Conditions for L1 convergence of maximum likelihood estimators
Let $X_1,\ldots, X_n$ be i.i.d. observations from a family of pdf or pmf $\{f_{\theta}: \theta \in \Theta \}$. We know that there are sufficient regularity conditions on the family $\{f_{\theta}: \...
- 41
-1
votes
1
answer
1k
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Rank of covariance matrix whose diagonal elements are same [closed]
Suppose A is a covariance matrix whose diagonal elements are same, i.e. $A_{1,1}=A_{2,2}=\cdots=A_{N,N}$, can we conclude that A is full rank?
Suppose the absolute values of the off-diagonal elements ...
4
votes
0
answers
980
views
Inverse Fourier Transform involving a Bessel Function, Exponential, and Power
I'm interested in this integral as a function of $r$ for various spectral densities $S(s)$:
$\frac{2 \pi}{r^{p/2}-1} \int_{0}^{\infty} S(s) J_{p/2-1}(2 \pi r s) s^{p/2} ds $, where $J_{p/2-1}$ is a ...
- 41
2
votes
1
answer
263
views
Probability distribution of uAv…
Consider the complex domain ℂ. If U and V are 2 unitary random matrices and A is a deterministic matrix.
What is the distribution of $u^HAv$ ( or $||u^HAv||^2$)
where : u is a column vector of U. v ...
- 233
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 ...
- 25
3
votes
1
answer
467
views
How to perform Importance Sampling with Prior Information
Let us define a random variable $X$ with density function $p(x)$. We wish to calculate $\mathbb{E}[f(X)] = \int f(x)p(x)dx$. We can compute the expectation by Monte Carlo simulations as
$$\mathbb{E}[...
- 33
5
votes
5
answers
3k
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Computing correlation between time series with missing data.
Suppose you have two simple Ar[1] series of the form $y_n=y_{n-1}+e_n$ and $x_n=x_{n-1}+m_n$, where $e_n$ and $m_n$ are normal white noise processes with no auto-correlation and $Corr(e_n,m_n)=p$. ...
- 109
1
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0
answers
87
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Convergence of an rcll process along a random subsequence
I have a process $X_s$, for $s \ge 0$, taking values in a Polish space $T$ with an rcll version where I have shown, for every nonrandom increasing sequence $s_n$, that $X_{s_n} \to c$ in probability, ...
- 321
4
votes
2
answers
653
views
Reading Material on Couplings
Does anybody have suggestions on what to read to learn more about couplings pertaining to statistics?
I'm working on a research project on Poisson approximations and am looking to perform a coupling ...
0
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2
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298
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Are all variables in a set of random variables independent if all pairs are independent?
If I have a sequence of random variables $X_1, X_2, \ldots, X_n$ (possibly infinite) such that all pairwise cdf's are factorized:
$$F(X_i, X_j) = F_i(X_i) F_j(X_j)$$
for all pairs $(X_i, X_j)$, does ...
- 923
2
votes
1
answer
169
views
higher-level independence of three or more correlated RVs
I'm hoping for some help in nailing down a vague idea about independence. It starts with finding the expectation of a product of three RVs (or more, but I'll stick to three for now). These are not ...
- 155
3
votes
2
answers
941
views
Probability distribution for two-state system that depends on residence time
I am a statistical physicist, and I've come across a problem that I don't know how to solve. I believe my issue lies with how to formulate it mathematically. I'd be very grateful for any assistance, ...
- 33
0
votes
1
answer
369
views
How to calculate eigenvalue density function of $XX^\dagger$ from the density function of X
Let X be a complex random matrix, which has the probability function (drawn from the ensemble) V($XX^\dagger$), where V(x) is some function which guaranties good behavior at infinity. Note the unitary ...
1
vote
1
answer
207
views
An efficient method to find the MLE of the combination of two point processes
I have a point process defined in two parts as follows. Consider first the main process which we call $A$ which is homogeneous Poisson process with conditional intensity
$$\lambda(t) = \mu$$
For ...
- 3,377
3
votes
2
answers
177
views
Sampling from maximally skewed stable distribution
I am reading a paper which refers to a maximally skewed stable distribution $F(x;1,-1,\pi/2,0)$ . Is there an efficient way to sample from this distribution?
If $X$ has distribution $F(x;\alpha,\...
- 3,377
0
votes
1
answer
915
views
Can you interpret this divergent integral?
In this ArXiv paper by Wilk and Wlodarczyk (published in Physical Review Letters), equation 16 has essentially the following definition of a function:
$$\text{f(x)=}\frac{c}{2Dx^2}\exp[\int^x_0 \frac{\...
- 33
4
votes
1
answer
193
views
Variance of central limit distribution for $P(x) \sim 1/x^{1+\alpha}$ for finite but large $N$?
Is it known what the next-to-leading order term is in the variance of the central limit distribution for the average of $N$ variables each of which is distributed according to $P(x) \sim 1/x^{1+\alpha}...
- 1,150
4
votes
1
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234
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Statistical models in terms of families of random variables
A statistical model is a function $P : \Theta \to \Delta(X)$, where $\Theta$ is a parameter space, and $\Delta(X)$ is the set of probability measures on a state space $X$.
Suppose that $\Theta$ and $...
- 8,512
1
vote
1
answer
405
views
Convergence to a k-dimensional Gaussian vector
Suppose I have a sequence of stochastic processes $X_{N}(t)$, $N=1,2,3,\ldots$ with mean zero and that I know for every fixed $t$, the random variable $X_{N}(t)$ converges in law to a Gaussian random ...
- 11
3
votes
0
answers
494
views
Maximization of a total variation distance subject to another total variation distance in Markov chain
Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...
- 1,109
3
votes
1
answer
372
views
Estimating total variation distance from a given distribution
Given a known distribution supported on a finite set of $n$ elements with probabilities $p_1, \dots, p_n$ and an access to an unknown distribution $q$ is it known what is the number of samples from $q$...
- 31
3
votes
2
answers
334
views
Scale random variables in a way they have equal probabilities of being minimal
I have several positive random variables $x_i,\ i=1,...,N$ taken from different unknown distributions (these distributions can be closely approximated by log-normal if needed). I can sample these ...
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
3
votes
1
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731
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name for $\underset{x}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n |x_i - x|$
Given a real-valued data set $ x_1, \dots, x_n $, what do you call the quantity
$$\underset{x}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n |x_i - x|$$
This seems like a pretty basic ...
6
votes
1
answer
1k
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Probability distributions: The maximum of a pair of iid draws, where the minimum is an order statistic of other minimums?
General question: What is the distribution for the maximum of 2 independent draws from cdf F(x), when we know that the minimum of those same two draws is the kth order statistic of the minimum of n ...
- 233
1
vote
2
answers
363
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A machine learning application question
I am familiar with basic probabilities, random processes but not so much of machine learning methods. This is the problem I am trying to solve.
I want to predict the nature of user activity on a ...
- 11
5
votes
2
answers
6k
views
Difference between Beta Process and Dirichlet process
I'm trying to understand the definition of a Beta process, as given in the paper:
www.ece.duke.edu/~lcarin/Paisley_BP-FA_ICML.pdf
The problem is that from the definition it follows that every ...
- 233
0
votes
1
answer
514
views
Relating percentiles to moments [closed]
There are at least two ways people look at statistical data:
A. For mathematicians, scientists, engineers, economists and such the most familiar distribution parameters would be analytical: mean, ...
- 2,205
1
vote
1
answer
3k
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Generating Bernoulli Correlated Random Variables with Space Decaying Correlations
Hi,
I have a set of N objects randomly distributed in a 2D physical space. Each object (i) generates a bernoulli random number (0 or 1) based on a marginal probability Pr(xi = 1) = p. These objects a ...
- 113
4
votes
3
answers
439
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Probability estimates for "beans & boxes"
From a discussion with some friends, this apparently easy problem has come out; I decided to post it here, because I believe that the answer is non-trivial and the maths beneath interesting. Partial ...
- 418
3
votes
1
answer
651
views
What conditions on a probability distribution defined by long-time averaging do I need to satisfy a central limit theorem?
For integer $n$, $1 \le n \le N$, consider the random variables
$X_n = \cos[t \omega_n]$
For any fixed $N$, we can take the mean
$Y_N = \frac{1}{N} \sum_{n=1}^N X_n$
and define a (cumulative) ...
- 846
8
votes
1
answer
2k
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Eigenvalue distributions of finite dimensional Wishart matrices
I am trying to obtain the eigenvalue distribution of a finite dimensional Wishart matrix. Let $A_{n\times n}\sim\mathbb{W}(\Sigma_{n\times n},m)$ where $\mathbb{W}(\Sigma_{n\times n},m)$ denotes the ...
0
votes
1
answer
227
views
two correlated processes
I apologize if this question is not placed in the right place. But I am having a hard time to figure it out. It would be greatly appreciated if some one could help me out.
Assume that there are two ...
- 1
3
votes
2
answers
231
views
Bounds for the fat tail after trimming the mean?
I am interested in the quantity $$f(X,t) = \int_t^\infty\negthinspace x\ p(x)\ dx,$$ where $p$ is a probability distribution for a positive variable $X$.
1) Does this quantity $f(X,t)$ have a name? ...
user44143
8
votes
0
answers
4k
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Taylor approximation of a function of a random variable
Suppose we have a random variable $X$ and a smooth function $g$. We want to calculate the expectation value $\mathbb{E}(g(X))$. To be able to write down at least an approximate solution, we perform a ...
- 91
3
votes
4
answers
2k
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statistical approach to multinomial distribution
Suppose a dice with $q$ faces is rolled $N$ times, where $N$ is very big.
We define a multinomial variable $X=(X_1,\ldots,X_q)$ which counts how many times any face is occurred ($X_i$ is the number ...
- 31
2
votes
0
answers
175
views
Implication of MGF inequality
Let X and Y be two random variables. Denote by $F_X(x)$ and $F_Y(y)$ their CDFs and by $M_X(t)$ and $M_Y(t)$ their MGFs.
It is known that X and Y have the same CDF iff they have the same MGF.
My ...
- 517
1
vote
1
answer
115
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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 ...
- 1,533
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
2
votes
3
answers
2k
views
Distribution of the sum of the $m$ smallest values in a sample of size $n$
Let $\mathbf X = [X_1, X_2, \ldots, X_n]^\mbox{T}$ be a vector random variable drawn from a known distribution with CDF $F(x)$. The CDF for the minimum value in $\mathbf X$ is clearly $P[\min_{i=1\...
- 183
1
vote
2
answers
789
views
Uniform law of large numbers for martingale difference
Let $\xi_{tn}(\theta),t=1,\dots,n$ be a real-valued martingale difference array indexed by a parameter $\theta \in \Theta \subset R$, where the set $\Theta$ is compact. Now, for all fixed $\theta \in \...
- 13
-2
votes
1
answer
347
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Forms of multivariate CLT [closed]
I am looking for a good reference for differnt kinds of multivariate central limit theorems. I was wondering how far the i.i.d. condition of the standard multivariate clt can be relaxed, as in can the ...
