All Questions
Tagged with pr.probability st.statistics
1,134 questions
6
votes
0
answers
388
views
Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance
Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...
- 8,071
2
votes
1
answer
387
views
Weak convergence of sum of log normal random variables
Let $S_t$ be the Geometric Brownian Motion, we know that
$$dS_t=rS_tdt+\sigma S_tdW_t, t\in [0,T], S_0>0, r>0,\sigma>0$$
and the distribution of $S_t$ is known explicitly. Please see the ...
- 489
2
votes
1
answer
652
views
Concentration inequality for subgaussian^4
Let $X_1,...,X_N$ be IID, mean-zero random variables whose tail is bounded by a subgaussian-tailed variable to the fourth moment, i.e., for some $t \ge t_0 > 0$
$$
P(|X_i| > t) \le C\exp\left( -...
- 719
2
votes
1
answer
269
views
Distribution-free statistics on compact Lie groups
(Cross-listed from the math stackexchange)
Let $(X_i)_{i=1}^n$ be iid random variables with joint cdf $F$. Recall that the empirical distribution function is:
$$
F_n(x) = \frac{1}{n} \sum_{i=1}^n \...
CommunityBot
- 1
1
vote
1
answer
54
views
Realization property implies expectation property
Is there a theorem that says if every realization $X(t)$ of a random process $X_{\omega}(t)$ satisfies some property, then the expectation $\mathbb{E}X(t)$ also satisfies the same property?
What ...
CommunityBot
- 1
3
votes
1
answer
345
views
Second moment of cos(x,y) for Normal x,y?
I'm trying to figure out second moment of the following quantity
$$y = \frac{\langle x_1, x_2 \rangle}{\left\|x_1\right\|\left\|x_2\right\|}$$
Where $x_1$, $x_2$ are sampled independently from $\...
CommunityBot
- 1
3
votes
0
answers
841
views
Is uniform distribution on unit sphere subgaussian?
Is uniform distribution on unit sphere subgaussian?
To be specific, let $X = (X_1,\dots,X_d) \sim \mbox{Unif}(\mathcal{S}^{d-1})$. What is the Orlicz-$\psi_2$ norm of $X$?
- 719
0
votes
1
answer
171
views
Distance of distributions of random variables, without PDF
Consider an interval $I$ with a smooth probability measure $d\mu (x) = c(x) dx$ and two known real measurable functions $f_1(x)$,$f_2(x)$. Both functions define a distribution on $X = {\rm Im} \, [f_1]...
- 3,574
-1
votes
6
answers
2k
views
Chances to win an election
Let's say that tomorrow national president election is held. A poll asks 1100 persons which of the two candidates, A or B, will he or she will vote. 750 say will vote A, and 250 say will vote B. What ...
CommunityBot
- 1
4
votes
2
answers
475
views
midpoint between two normal distributions for the Rao-Fisher metric
Given two multivariate gaussian distributions $G_0 \sim N(\mu_0,\Omega_0)$ and $G_1 \sim N(\mu_1,\Omega_1)$, is there a closed-form formula for the gaussian distribution equidistant from them that is ...
- 3,574
1
vote
0
answers
809
views
Generalized Chi squared distribution
What is the distribution of $Y Y^\top$ if $Y \sim N(\mu,\Sigma)$ is a generic multivariate gaussian vector?
- 719
11
votes
1
answer
1k
views
What are some of the surprising results of finite sample statistical estimation?
I'm trying to familiarize myself with the latest results in finite sample statistics. It seems to me that these results can be classified into two categories:
Unsurprising results confirm that the ...
CommunityBot
- 1
4
votes
1
answer
347
views
Concentration of functional of Gaussian random variable
Suppose I have two Gaussian distributions
$p(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_p|^{1/2}}\exp(-\frac{1}{2}x^\top \Sigma_p^{-1} x)$ and $q(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_q|^{1/2}}\exp(-\frac{1}{2}x^\...
CommunityBot
- 1
5
votes
1
answer
365
views
power laws emerging from the sandpile model
Is there a rigorous proof that the abelian sandpile model generates a power law distribution of avalanche lengths?
- 5,063
2
votes
0
answers
171
views
Distribution for the extreme values of a cumulative sum of normal variables
Suppose I have a sample $X$ of $n$ iid Normal random variables $(X_1,X_2,..,X_n)$.
Now, define the sample mean $u=\frac1n\sum_{i=1}^n X_i$ and let $Y_i=X_i-u$.
Let $Z_k=\sum_{i=1}^k {Y_i}$. Note ...
- 6,711
3
votes
3
answers
292
views
A question in central limit theorem
Suppose $\{X_n,n\ge1\}$ are independent r.v., $E(X_n)=0$, $\operatorname{Var} \left(X_n\right)=\sigma_n^2<\infty$. Set $S_n=\sum_{i=1}^nX_i$ and $s_n^2=\sum_{i=1}^n\sigma_i^2$, assume
$$\frac{S_n}{...
- 211
6
votes
1
answer
291
views
Is there a name for this quantity between two distributions?
Let $f$ be a probability density on a compact domain $D$, and say that $x_1,\dots,x_n$ are samples from $f$. If we wanted to compute the Wasserstein distance between $f$ and the empirical ...
CommunityBot
- 1
2
votes
0
answers
74
views
Best estimator for a 3 coin problem
Let $X,Y,Z$ be three unfair coins. We consider the coin $\Gamma=XI(Z=1)+YI(Z=0).$ We are given the sample $S=\{(\Gamma_i,Z_i)| i \leq n \}$ Let $k=\sum_{i \leq n}I(Z_i=1)$. For every pair of $(\...
- 103
2
votes
1
answer
1k
views
forward algorithm Hidden Markov Model
I am studying the the forward-backward algorithm used in Hidden Markov Models. I understand that you are trying to propagate through a sequence (and the available states) to find the most probable ...
- 6,239
8
votes
1
answer
1k
views
How to generate Voronoi diagram with polygons of equal area?
I would like to generate some random set of points so that their Voronoi diagram consist of equal-area polygons. Is it possible to impose some constraints on the points in order to have the same areas ...
- 151k
2
votes
1
answer
327
views
Recursive parameter estimation for partially observed Ito SDEs
I'm trying to get my head around online (recursive) maximum-likelihood parameter estimation in the language of stochastic processes and in the context of stochastic filtering, i.e. where we have a ...
- 1,675
4
votes
3
answers
273
views
Concentration inequalities for random sets
$\newcommand{\abs}[1]{\left|#1\right|}$
There is a population $O$ with a countable (finite or infinite) number of subjects. The population is colored randomly: for each subject, an unbiased coin-toss ...
- 6,541
3
votes
0
answers
75
views
Covariance of censored/clipped Gaussians
I am interested in the covariance of two clipped (or censored) Gaussian variables.
More precisely, let $g_1 \sim N(0,\sigma_1^2)$ and $g_2 \sim N(0,\sigma_2^2)$ be two (dependent) Gaussians with $\...
CommunityBot
- 1
0
votes
0
answers
124
views
Which sub-sequence selection rules preserve the iid property?
Let $\xi_1,\ldots,\xi_n$ be an iid sequence of random variables. If we take a sub-sequence $\xi_{i_1},\ldots,\xi_{i_k}$ with constant indices $1\leq i_1 <\ldots <i_k\leq n$, then the sub-...
- 251
1
vote
1
answer
375
views
convergence of Bayesian posterior with non iid data
Let $(\epsilon_t)_t$ be a sequence of iid random variables, distributed according to the density $f:\mathbb{R}\to (0,\infty)$ and
$$
x_t = q( \theta^\star, x_1,x_2, \ldots, x_{t-1}) + \epsilon_t \,.
...
CommunityBot
- 1
0
votes
0
answers
102
views
Probability of random variable being lesser than the other
Say there are two independent random variables, $X$ and $Y$, and we have samples $\{x_1,\dots x_n\},\{y_1,\dots y_n\}$. I am interested in bounding the probability of the event $C = \mathbb{1}_{X<Y}...
- 197
6
votes
0
answers
554
views
a variation on Hanson-Wright inequality
The classic Hanson-Wright inequality states that for a Gaussian random vector $\mathbf{x}\in\mathbb{R}^n$ distributed as $\mathcal{N}(\mathbf{0},\mathbf{I})$ and $\mathbf{A}\in\mathbb{R}^{n\times n}$ ...
- 859
1
vote
2
answers
1k
views
Gibbs sampler with linear constraints
My problem concerns the estimation of truncated multivariate normal distributions under constraints.
Let $X_1$ and $X_2$ two random variables following normal distributions $\mathcal{N_1}(m_1,\...
CommunityBot
- 1
0
votes
1
answer
905
views
Parameter estimation distribution for hypergeometric distribution
Let the hypergeometric distribution is given by $h(k\mid N;M;n)$, where
$k$ is the number of observed successes,
$N$ is the population size,
$M$ is the number of success states in the population and
$...
4
votes
0
answers
147
views
The asymptotic behavior of the ratio between the largest two of $n$ i.i.d. chi-square random variables
My question is about the asymptotic behavior of the ratio between the largest and second largest values of $n$ independent chi-square random variables.
Let $X_1, \ldots, X_n$ be $n$ independent and ...
- 24.8k
4
votes
2
answers
314
views
Convexity of truncated expectation
Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$.
Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...
CommunityBot
- 1
-1
votes
2
answers
512
views
Deriving the joint distribution of multivariate normal transformed into Bernoulli
Given a covariance matrix $\sum_{ij}$ and a mean vector $\mu$ I have sampled $N$ multivariate normal vectors $Z = (z_1,...z_n)$ My goal is to create a vector of Bernoulli random variables $Y = (y_1,......
- 37.7k
1
vote
1
answer
259
views
Conditional Expectation Relative to "Random Time" - Consistency of the Substitution Rule
I am thinking of the following situation:
On a probability space $\left( \Omega, \mathscr{F}, \cal{P} \right)$ with arbitrary structure, suppose we are given a random function (as it is called in the ...
- 75
4
votes
1
answer
463
views
Variance and expectation of timed-change squared Bessel process
Let $X_t$ be a squared Bessel process satisfying the SDE:
$$
dX_t=\left(1-\frac{\beta}{(1-\beta)(1-\rho^2)} \right) dt +2\sqrt{X_t}dW^{(1)}_t
$$
and $v_t=v_0e^{-\alpha^2 t/2+\alpha W^{(2)}_t}$ be a ...
- 6,045
2
votes
0
answers
101
views
Best describing a stochastic process in terms of others
Intuitive Question
Suppose I'm given a set of $k$ time-series $\{X_t^1,\dots X_t^k\}$. Is there a way to determine how much of each series is dependent on the others.
Formal Question
More ...
- 5,405
2
votes
0
answers
80
views
Maximal and second max of chi-squared (normal squared) distribution
Suppose $X_i$ are i.i.d.~$\log \chi^2$ where $\chi\sim N(0,1)$ distribution, in which case
$$
F(x) = \mathbb{P}\left( \log \chi^2 \le x \right)
= 2\Phi(e^{x/2}) - 1,
$$
and
$$
f(x) = e^{x/2}\phi(e^{x/...
CommunityBot
- 1
1
vote
1
answer
142
views
Subclass of semimartingales for which all characteristics can be estimated?
I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great.
An Ito semimartingale is a martingale for which the ...
- 21
1
vote
1
answer
510
views
Total variation distance between multinomial laws
Can someone help me with the following problem:
Let $P_n$ and $Q_n$ two multinomial laws with parameters $(p,n)$ and $(q,n)$, where $p$ and $q$ are two probability measures on some measurable space ...
- 1
4
votes
0
answers
73
views
Regularity Conditions for L1 convergence of maximum likelihood estimators
Let $X_1,\ldots, X_n$ be i.i.d. observations from a family of pdf or pmf $\{f_{\theta}: \theta \in \Theta \}$. We know that there are sufficient regularity conditions on the family $\{f_{\theta}: \...
- 41
3
votes
1
answer
942
views
Cumulative integral of the Marchenko-Pastur density for Wishart eigenvalues
I can't find any work on the cumulative density function of the well-known Marchenko-Pastur density for the eigenvalues of a standard Wishart Matrix as its dimension goes to $\infty$, i.e.
$$F(\beta)=...
CommunityBot
- 1
18
votes
1
answer
3k
views
Distribution of maximum of random walk conditioned to stay positive
I have an $n$ step random walk which starts at zero $X_0 = 0 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$, but the walk is conditioned on the hypothesis that it ...
CommunityBot
- 1
2
votes
1
answer
79
views
Asymptotic rate of multivariate normal sample mean
I want to establish an asymptotic rate for the quantity $\|\bar{x} - \mu\|_2^2$. Here, $\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$ where $x_i$ are iid ${\rm N}_p(\mu, \Sigma)$ for $i=1,\dots, n$. Here, I ...
- 133
11
votes
0
answers
536
views
Bounding the probability that a random variable is maximal
Question: Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_N$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$.
I am looking ...
- 2,680
3
votes
4
answers
22k
views
Football Squares
Dear Colleagues,
This is a math question for people who know the rules of (American) football.
Every year my barber runs a “football squares” game. He finds 100 customers, each put in 20 dollars, ...
- 1
0
votes
1
answer
254
views
Solution for 2 variable recurrence for a problem similar to gambler's ruin [closed]
Let $p_A,p_B$ with $p_A+p_B=1$, $p_A \geq p_B$ be the probabilities of a biased coin flip. Player $A$ gets 1 point if the coin flip gives heads, $B$ gets 1 if tails. The player who reaches $N$ points ...
- 1,897
3
votes
0
answers
699
views
How does Jensen Shannon divergence and KL divergence correlate?
I am wondering if there is way to derive the correlation between Jensen Shannon divergence and KL divergence for two distributions: P and Q, in order to show that if JSD(P,Q) decreases, KLD(P,Q) ...
CommunityBot
- 1
1
vote
1
answer
534
views
PDF and CDF using Gauss-Legendre quadrature
Consider the unit interval $I$ with a continuous probability measure $\mu$, and consider a smooth random variable $f:I\to \mathbb{R}$. We can define its cumulative distribution function and ...
CommunityBot
- 1
1
vote
0
answers
80
views
Estimation with an unbalanced loss function
I'm interested in estimating the mean $\mu$ of a distribution given i.i.d samples. The empirical mean is a totally acceptable estimator (and can even be shown to be asymptotically/close to optimal ...
- 591
1
vote
1
answer
313
views
Finding the joint distribution from Poisson conditionals
Suppose that for two discrete random variables $X_1$ and $X_2$, we know their conditional distributions. Namely
$$X_1~|~X_2 = x_2 \sim \mathrm{Poisson}(\lambda_1 + ax_2),$$
$$X_2~|~X_1 = x_1 \sim \...
- 54.2k
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