All Questions
Tagged with pr.probability st.statistics
1,135 questions
21
votes
3
answers
5k
views
James-Stein phenomenon: What does it mean that a James-Stein estimator beats least squares estimator?
Background James-Stein estimator and Stein's phenomenon, as described in Wikipedia are rather counterintuitive and amazing.
It is claimed that if one wants to estimate the mean $\Theta$ of
Gaussian ...
- 24.9k
1
vote
1
answer
84
views
Jeffreys' priors as coefficients of a linear estimator
I asked the following question in a forum more suitable for statistics, but I didn't get any answer; I hope, someone could shed light on my question:
I have three random variables, $X_1$, $X_2$, and $...
user386878
1
vote
1
answer
276
views
Upper bound for $P(X \geq x)$, where $X \sim \operatorname{Pois}(\lambda)$
I posted the following question in a comment on CDF of a log-concave discrete random variable. Since it is not related to my main question, I thought of reposting it as separate post.
Question:
Let $X ...
- 135
1
vote
1
answer
370
views
in Euclidean space defined by multivariate normal distribution, what fraction of points falls within n-ball (centered at origin) tangent to point p?
In a Euclidean space defined by the multivariate normal distribution, what fraction of all points falls within or are tangent to (as opposed to falling outside of) the n-sphere whose center is at the ...
- 21
1
vote
1
answer
106
views
What is the maximum possible coefficient of variation for data taking values within a specified range?
I have a question that seems very basic, and yet I have not managed to find an answer after probably several hours of Google-searching.
Fix $0<a<b<\infty$, and let $\mathcal{P}_{[a,b]}$ be ...
- 708
1
vote
1
answer
126
views
Probabilistic lower and upper-bounds for a certain random quartic form involving gaussian random matrices
Let $d,m \to \infty$ (integers) with $m/d \to \rho \in (0, \infty)$. Let $C$ be a $d \times d$ psd matrix with $trace(C)=\mathcal O(1)$, and let $w_1,\ldots,w_m$ be iid uniformly distributed on the ...
- 6,853
7
votes
2
answers
647
views
Moments of a positive random variable
Suppose one is handed a list of $K$ numbers, with a claim that these numbers are the first $K$ moments of a positive random variable $X$ (meaning there is 0 probability that $X<0$).
What is the ...
- 361
5
votes
3
answers
1k
views
Why a random variable is better described by its cumulants than by its characteristic function?
It is a classical and well known problem that a random variable $X$ is not uniquely determined by its moments $\mathbb{E}(X_n)$. The moment problem is the problem of determining the probability ...
- 137
5
votes
1
answer
363
views
Inverse marginal property of a collection of $\sigma$-algebras
In my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space"
I introduced the Inverse marginal property (IMP) for a collection of $\sigma$-algebras.
Let $(\...
- 935
0
votes
1
answer
160
views
Probability to cross an envelopp for 1D random walk?
Imagine we have an evolving sequence composed of 1 and -1 (ex: -1-11-111...) where the probability to get -1 or 1 is 1/2. n is the lengh of my sequence.
I can make an analogy with random walk: let ...
- 5
2
votes
1
answer
199
views
Do enough permutations of an initial set probably cover most permutations?
Fix $\alpha, \epsilon \in(0,1)$. Take $(S_n)_n$ to be any sequence of sets with each $S_n$ containing $ \lceil (n!)^\alpha\rceil$ permutations of $n$ elements. Also build another sequence of sets $(...
3
votes
1
answer
170
views
Donsker class and law of the iterated logarithm
Let $P$ be a probability measure on a measurable space $(E, \mathcal {E})$, and let
$\mathcal {F}$ be a countable collection of measurable functions $f : E \to \mathbb {R}$
which is a Donsker class ...
- 61
0
votes
1
answer
497
views
Laplace transform inversion
I have a probability distribution that is defined through it's Laplace transform by :
$$L(t) = \mathbb E(e^{-tX}) = e^{1 - \frac{1+t}{t}\ln(1+t)}$$
Using R and the invLT package, i have a numerical ...
- 686
1
vote
1
answer
520
views
The integral of a Gaussian process on a unit sphere
Suppose there exist a zero-mean Gaussian process $\mathbb{G} f_u$ indexed by $u \in \mathcal{S}^{p - 1}$ with known covariance $\mathrm{E} \big[ \mathbb{G} f_u \mathbb{G} f_v \big]$ when both $u$ and $...
- 331
0
votes
0
answers
202
views
$|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)|=O_P(\frac{1}{\sqrt{n}})$ under $E(|X_1|)<\infty$?
For i.i.d. random variables $X_1,\dots, X_n$ with $E(|X_1|)<\infty$. Does the following equation hold?
$$
\left|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)\right|=O_P\left(\frac{1}{\sqrt{n}}\right)
$$
I ...
- 193
3
votes
1
answer
902
views
Expectation of exponential of Gaussian random matrix
Let $X$ be an $(N, M)$ random Gaussian matrix where $M<N$. For a given vector $v$, I want to estimate the expectation of:
\begin{align}
E\left[ {{v^T}X{X^T}{v}} \right]
\end{align}
and
\begin{align}...
- 377
1
vote
0
answers
233
views
Variance-based localized Rademacher complexity for RKHS unit-ball
Let $\mathscr X$ be a compact subset of $\mathbb R^d$ (e.g the unit-sphere). Let $K: \mathscr X \times \mathscr X \to \mathbb R$ be a positive kernel function and let $\mathscr H_K$ be the induced ...
- 6,853
13
votes
1
answer
10k
views
KL divergence and mixture of Gaussians
Do we have an exact formula to compute the KL divergence between 2 mixtures of Gaussians (i.e convex combinations of a finite number of Gaussian distributions)?
If not exactly known, are there good ...
- 2,246
2
votes
1
answer
649
views
distribution on the inverse Wishart matrix eigenvalues summation
Let $\lambda_1>\lambda_2>....>\lambda_N$ be the ordered eigenvalues of Wishart matrix my objective is to find if it is possible the distribution of:
\begin{align}
s = \sum\limits_{i = 1}^...
- 377
1
vote
1
answer
88
views
tail probability of max of Gaussians
I'm trying to follow an argument in C. Giraud's "High Dimensional Statistics" (2nd Ed, p. 11 / $\S$ 1.2.3). The specific page is accessible via Google Books here but the formatting is awful....
- 13
2
votes
2
answers
206
views
non-homogeneous counting process
Consider a counting process $\{N(t), t\geq 0\}$ where the time distribution between any two consecutive events, say $k$ and $k+1$ has a Poisson rate $\lambda(k)$, which is an explicit function of $k$....
- 31
1
vote
1
answer
140
views
Does a sequence that verifies the assumptions of a square integrable martingale on some event need to be convergent on this event?
I came across this claim by reading some literature on stochastic approximation.
Let $(\Omega, \mathcal{A}, \mathbb{P}$) be a probability space, $(\mathcal{F}_n)$ a filtration on it. Let $(\epsilon_{n}...
- 115
0
votes
0
answers
769
views
sub-exponential type upper bound on the Poisson probability
I posted this question on Math Stack Exchange, though I'm not satisfied with the answer I received.
Question:
For a Poisson random variable $Z$ with the parameter $\lambda,\,$ what would be a good ...
- 11
0
votes
0
answers
91
views
Spectrally-weighted Stieltjes transform of random matrix $Z=XX^\top$ in terms of Stieltjes transform of $Z$ and the weighting function
Let $n$ and $d$ positive integers going to infinity such that $d/n \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ iid rows from $N(0,\Sigma)$, where $\Sigma = diag(\lambda_1,\ldots,\...
- 6,853
2
votes
0
answers
51
views
Spectral approximation of $(XX^\top/d)\circ(X\Sigma_dX^\top/d)$ where $X$ is an $n \times d$ random matrix with iid rows from $N(0,\Sigma_d)$
Let $X \in \mathbb R^{n \times d}$ be a random matrix with iid rows from $N(0,\Sigma_d)$ where $\Sigma_d$ is a $d \times d$ psd matrix verifying w.h.p,
$\mbox{trace}(\Sigma_d/d)= 1$.
$\|\Sigma_d\|_{...
- 6,853
4
votes
1
answer
362
views
Information monotonicity of divergence => function of $f$-divergence
It is well-known that $f$-divergences defined on $\mathcal P(\mathcal X)$ where $\mathcal X$ is a measure space with $\sigma$-algebra $\mathcal B$ satisfy the property of information monotonicity:
...
- 203
1
vote
0
answers
198
views
Weak convergence of Cesaro means of weakly converging infinite-dimensional distribution
Suppose we have sequences of random variables $\{X_{n,m},n \in \mathbb{N}\}$ where the distribution of $(X_{n,m})_{n\in\mathbb{N}}$ converges weakly to an infinite-dimensional normal distribution $\...
- 19
2
votes
1
answer
728
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:
$$\mathbb{E}_{v \sim N(0, I)}\left[\frac{vv^T}{||Mv||_2}\...
- 269
0
votes
1
answer
142
views
Covering number of the conditional distribution function
Suppose $Y$ is a random variable in $\mathbb{R}^d$, and we want to find the covering number
\begin{equation*}
\mathcal{F} = \big\{ F_{Y|W} (y | W) : y \in \mathbb{R}^d \big\}
\end{equation*}
where ...
- 331
2
votes
1
answer
177
views
Matrix-valued cumulant generating function for Wishart matrices
Suppose we have an axis-aligned Gaussian vector $v \sim \mathcal{N}(\mu, \sigma^2 I_{d \times d})$, and consider the Wishart matrix $W = vv^\top$.
Is there a simple closed form/"Lowener order ...
- 101
1
vote
1
answer
144
views
Bounds for the extreme singular-values of random matrix with thresholded entries
Let $n,d,k$ be large positive integers such that $\max(n/d,k/d) =: \lambda < 1$. Let $X$ be a random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $W$ be a $k \times d$ random ...
- 6,853
-3
votes
1
answer
123
views
Are the first 4 statistical moments independent? [closed]
Are the first 4 statistical moments independent? Is there a mathematical demonstration that can show independence one from each other? Can the value of one moment influence the value of another? If so,...
- 11
4
votes
1
answer
561
views
Intuition behind the noncentral chi square as Poisson mixing
It is known (cf. Wikipedia, Noncentral_chi_distribution) that the non-central chi-square distribution with k degrees of freedom is a Poisson weighted mixture of central chi-squared distributions).
...
- 43
1
vote
1
answer
337
views
Posterior expected value for squared Fourier coefficients of random Boolean function
Let $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ be a Boolean function. Let the Fourier coefficients of this function be given by
$$ \hat f(z) = \frac{1}{2^{n}} \sum_{x \in \{0, 1\}^{n}} f(x)(-1)^{x \cdot ...
4
votes
1
answer
239
views
Uniform inequality of the form $\text{Proba}(\sup_{v \in [-M,M]^k}|p^Tv-\hat{p}_n^Tv| \le \epsilon_n) \ge 1 - \delta$
Let $M > 0$, $k$ be a positive integer, and $\mathcal V:=[-M,M]^k$. Finally, let $p \in \Delta_k$, (where $\Delta_k$ is the $(k-1)$-dimensional probability simplex) and let $\hat{p}_n$ be an ...
- 6,853
2
votes
0
answers
172
views
Asymptotic lower and upper bounds for the eigenvalues of hadamard product $W \circ W$, where $W$ is a large Wishart matrix
Let $n$ and $d$ be large positive integers with $n,d \to \infty$ such that $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ random matrix with iid copies of log-concave isotropic ...
- 6,853
3
votes
1
answer
196
views
On estimating Covariance between a random variable and its non-linear transform
Let $X$ be a random variable taking values on the real line.
Let $R(X) = max\{0, X\}$. Is it true that the covariance $Cov[X, R(X)] \ge 0$ irrespective of the distribution of $X$? Many experiments, as ...
- 65
4
votes
1
answer
478
views
Order statistic - Rate of convergence of a p-quantile to the expectation
Fix some $k\in\mathbb N$ and some probability $p\in[0,1]$. Denote with $F_n$ the cdf of the k-th highest oder statistic (i.e. the distribution of the k-th highest draw) of $n$ draws from a uniform ...
- 43
1
vote
0
answers
147
views
Using maximum entropy principle for joint probability estimation
Let $X_1, \dots, X_n, Y$ be random variables, each taking values in $\{0,1\}$. Assume that we are interested in estimating, for each $v=(v_1,\dots,v_n)\in \{0,1\}^n$, the probability
$$
p(v) = P[Y=1|...
- 7,182
1
vote
0
answers
78
views
Canonical representation of the a probability distribution for Hammersley Clifford Theorem
I'm reading the following paper
http://www2.stat.duke.edu/~scs/Courses/Stat376/Papers/GibbsFieldEst/BesagJRSSB1974.pdf
On page 7 they give the result that
$$Q(\textbf{x}) = \sum_{1 \leq i \leq n} ...
- 903
2
votes
1
answer
185
views
Limiting distribution of "scatter matrix" $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ for iid $x_1,\ldots,x_n \in \mathbb R^p$
Let $x_1,\ldots,x_n$ be drawn iid from such "nice" distribution on $\mathbb R^p$ (but possibly very general!), and let $X$ be the $n$-by-$p$ matrix formed by vertically stacking the $x_i$'s.
...
- 6,853
0
votes
0
answers
171
views
A basic property of maximal correlation
Let $𝑋$ and $𝑌$ be random variables. Then the maximal correlation $\rho_{m}(X;Y)$ is defined as:
$$\rho_{m}(X;Y):=\max_{f,g}\mathbb{E}[f(X)g(Y)],$$
where the maximization is taken over real-valued ...
- 11
1
vote
1
answer
269
views
Can I prove that a polynomial representing the 4th moment of a weighted-sum of random variables is a sos?
I am looking at the 4th central moment of a weighted-sum of correlated random variables, which takes the form
$$\mu_4 = \sum_{i,j,k,l=1}^n w_i w_j w_k w_l \mu_{ijkl}$$
where $\mu_{ijkl}$ are the ...
- 173
4
votes
1
answer
320
views
The power of chi-square test
Under the null hypothesis, if we have
$$\sqrt{n} \vec{x} \, \rightarrow_d \, N(0, I_p),$$
the test statistic can be construct as:
$$\hat{\Psi} = n \vec{x}^{\top} \vec{x} \, \rightarrow_d \,\chi^2_p.$$
...
- 331
0
votes
1
answer
100
views
Independence between $X_{n-k:n}$ and $\sum\limits_i Y_{n-i:n}-Y_{n-k:n}$
If $(X_i,Y_i), i=1,\ldots,n,$ is i.i.d sample from the joint distribution $F$ and there is dependence between the two variables say $R$. Denote the order statistics for the two variables $X_{1:n},\...
- 1
3
votes
4
answers
375
views
Distinguishing between urn probability models
I have question about the urn model. Suppose I have an urn:
...
- 595
2
votes
1
answer
97
views
Local limit theorems for circular/spherical distributions
Here are some of the classical density functions for spherical distributions (on the $\mathcal{S}^{d-1}$ sphere, living in the Euclidean space $\mathbb{R}^d$):
$$\mathbf{x}\mapsto \frac{(\kappa/2)^{d/...
- 219
2
votes
2
answers
303
views
Expectation of the determinant of the inverse of non-central Wishart matrix
Let $A$ be $(n,n)$ central Wishart matrix with $k$ degrees of freedom.
my question is there is a way to estimate the expectation of:
\begin{align}
E[det(I+(I+A)^{-1})]
\end{align}
- 377
0
votes
0
answers
330
views
Lower-bound smallest eigenvalue of covariance matrix of $y = f(Ax)$, for $x$ uniform on unit-sphere
Let $A=(a_1,\ldots,a_)$ be a fixed $k \times d$ matrix (with $d$ large), and $x$ be a random vector uniformly distributed on the unit-sphere in $\mathbb R^d$. Let $f:\mathbb R \to \mathbb R$ be a ...
- 6,853
4
votes
0
answers
75
views
Marginalization of Wishart distribution
Consider the following Wishart distribution
$$
f({\bf W}) = \frac{ |{\bf W}|^{(n-p-1)/2} \exp\big[-\frac{1}{2}\text{tr}({\bf V}^{-1}{\bf W} ) \big] }{2^{np/2} |{\bf V}| \Gamma_p(\frac{n}{2})} \tag{1}
$...
