**2**

votes

**2**answers

138 views

### What is the sum capacity of a scalar gaussian broadcast channel?

"On the Achievable Throughput of a Multiantenna Gaussian Broadcast Channel" by Giuseppe Carie and Shlomo Shamai talks, in part, about the following type of link (paraphrasing):
A transmitter with ...

**5**

votes

**1**answer

108 views

### 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 ...

**5**

votes

**0**answers

110 views

### What's the variance in the Six Degrees model?

Recall the six degrees of Kevin Bacon game. You can even play the game at The Oracle of Bacon, and their search works via Breadth First Search.
I interpret the punchline as saying that if I start ...

**4**

votes

**0**answers

324 views

### An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$
where $dH$ is the unit invariant Haar measure on the group of unitary ...

**0**

votes

**0**answers

79 views

### Continuous self-information

Let $I(X,Y)$ be the mutual information between two continuous random variables $X$ and $Y$.
We have $I(X,Y) = H(X)-H(X|Y)$, and setting $X=Y$ leads to $I(X,X) = H(X)-H(X|X)$. If $X$ was discrete, ...

**0**

votes

**1**answer

52 views

### distances-based dispersion measuring approach

Is there any known approach or method to measure the dispersion of a set depending on the distances between its points (i.e.: without calculating the average or the mean) ?
thanks.

**1**

vote

**0**answers

172 views

### Bound the expectation of trace norm of random Hermitian matrix

Suppose $H_i$ are traceless $d\times d$ Hermitians, $X_i$ are Standard normal distribution for $1\leq i\leq d^2$.
We would like to bound the following expectation on the trace norm
...

**1**

vote

**1**answer

221 views

### 1-wasserstein distance v.s. total variation distance

Suppose that $\mu_1$ and $\mu_2$ are two distributions defined on $\mathbb{R}^n$ and $\gamma$ is a symmetric distribution (around $0$) on $\mathbb{R}^n$ with compact support. Let $\gamma_x$ denote the ...

**2**

votes

**1**answer

429 views

### Statistical distance between discrete and continuous distributions

Are there any statistical distance functions that are capable of comparing a continuous and a discrete distribution? From reading this list
http://en.wikipedia.org/wiki/Statistical_distance
the only ...

**0**

votes

**0**answers

76 views

### Integral over conditioning variable of a Gaussian

The marginal of a multivariate Gaussian can be computed in closed form, i.e.,
$p(x) = \int_y \mathcal{N}((x,y);\mu,\Sigma)\ dy$
is simple. But what I need is
$L(x) = \int_y \mathcal{N}((x\mid y); ...

**3**

votes

**1**answer

152 views

### Does bounding moments make distributions close in total variation distance?

Let $W\sim\mathcal{N}(0,\sigma^2)$ be a "reference" Gaussian random variable.
Suppose I have a set of distributions, $\mathcal{W}$, where $W_a\in\mathcal{W}$ if it satisfies the following criteria:
...

**1**

vote

**0**answers

78 views

### Lower bound on difference between polynomials at moderate distance

Fix $r > 0$ and $k, n \in \mathbb{N}$. Also consider a function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$. Let $x_{1},\ldots, x_{n+1}$ be points chosen uniformly from $[-r,r]^{d}$. For $1 \leq i ...

**3**

votes

**2**answers

274 views

### PDF of the product of normal and Cauchy distributions

I am having trouble in finding out the resulting PDF of the product of normal and Cauchy distributions. It turns out that we have a general formula for calculating the PDF of product of two random ...

**0**

votes

**2**answers

117 views

### What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them?
Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all ...

**8**

votes

**4**answers

793 views

### What does it mean when we say we have computed a number to a certain accuracy using a probabilistic algorithm?

My intention is to ask a general question about probabilistic (Monte Carlo) algorithms. But to keep things simple, I will focus on a few specific examples.
Let me start the discussion with ...

**1**

vote

**0**answers

112 views

### limit distribution of multinomial distribution with increasing categories

If $\bf{X} \sim \text{multi}(n,p)$ with $k$ categories, we know
$$ \sqrt{n}\left( \frac{\bf{X}}{n} - \bf{p} \right) \rightarrow^D N(0,\Sigma),$$
where $\bf{X}=(X_1,\ldots,X_k)^T$ and ...

**1**

vote

**0**answers

37 views

### Inverse of the covariance of the estimate of a covariance

I have a covariance matrix, $V_{ij}$, which (for reasons that aren't important) I'm going to call the visibilities. I have an estimator for the visibilities $\hat V_{ij}$, and I've derived that the ...

**3**

votes

**0**answers

134 views

### Sum of the entries of the inverse covariance matrix

Let $T \in\left(0,1\right)$, $n\in\mathbb{N}$ and $e_n = [1,\ldots,1]\in\mathbb{R}^n$. Consider the covariance matrix $\mathfrak{A}_n = ...

**0**

votes

**1**answer

242 views

### Continuity of a Functional

A certain functional $T$ is defined as:
$$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$
where $M$ is a probability measure with support $[\alpha,1-\alpha]$,for $\alpha>0$.
The result that above functional is ...

**3**

votes

**0**answers

162 views

### Donsker's Theorem for triangular arrays

I should mention that I already posed this question on Math Stack Exchange, but didn't receive much feedback.
Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given ...

**10**

votes

**0**answers

831 views

### Table with the most seated customers in Chinese restaurant process

Suppose we have some initial configuration of people seated at some tables. We start taking new customers and seat them following Chinese restaurant process. Is there some known work on finding the ...

**5**

votes

**0**answers

106 views

### In what sense is the Bayesian posterior mean a “convex combination”?

I asked this on math.stackexchange with no response, I'm hoping someone here might have something.
Suppose I want to estimate $x \in \mathbb{R}^n$ from two signals with zero mean, normally ...

**3**

votes

**1**answer

119 views

### Characterizing space that preserves positive-definiteness property

Given a symmetric positive-definite matrix $\Sigma$, consider the space $\mathcal{D}$ of diagonal matrices such that $\forall D\in\mathcal{D}$, the matrix $\Sigma-D\Sigma^{-1}D$ is positive definite. ...

**1**

vote

**0**answers

57 views

### Finding a general form of the density function when we have a four dimensional random variable

Consider a subject having time of the specific event $T_i$, which is a single sample from a
distribution $F_i$ with density $f_i$ and support
$[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...

**1**

vote

**1**answer

109 views

### Rademacher complexity of a Lipschitz class: Are the boundedness constraints necessary?

Consider the following function class: $F={f:R^d\rightarrow [a,b], f(x)=\sigma(w^Tx)}$ where $\sigma(.)$ is Lipschitz, and $w\in R^d$ is a parameter vector. The problem I'm working on is a machine ...

**0**

votes

**1**answer

93 views

### Expectation of exp(-1/(ax^2)) when x is a standard normal variable and a>0 is a parameter [closed]

I would like to know if the mean value of $\exp(-1/(ax^2)) $ when $x \sim N(0,1)$ and $a>0$ is a parameter is known.

**4**

votes

**1**answer

143 views

### Earth mover/Wasserstein distance between a pdf and an empirical distribution

This question is inspired by this much older question:
Convergence of an empirical distribution w.r.t. the Hellinger distance
Let $P$ be a continuous probability distribution on a compact subset of ...

**1**

vote

**2**answers

71 views

### Is it possible to find an asymptotic distribution for the LRT without the ML estimators being consistent?

I'm reading a comment(last page) to a paper, and the author states that sometimes, even though the estimators (found by ML or maximum quasilikelihood) may not be consistent, the test may be ...

**1**

vote

**0**answers

36 views

### Asymptotic results for functions of order statistics

There are $n$ ($n \ge 3$) iid random variables $\{ {c_i}\} _{i = 1}^n$ on the interval $[\underline c,\bar c]$ ($\underline c>0$). The cdf $F(\cdot)$ and pdf $f(\cdot)$ are unkown to us, but we ...

**3**

votes

**2**answers

332 views

### Consistent price index

This question came out of a discussion with a colleague from economics about price indices. Here is MattF's formulation of the question which differs somehow from the original problem.
Let ...

**1**

vote

**0**answers

84 views

### Is there an efficient algorithm for sampling from the negative hypergeometric distribution? [closed]

I'm writing a small statistics library currently. One of the algorithms I'm implementing has two variants: one that samples the hypergeometric distribution and one that samples the negative ...

**3**

votes

**1**answer

187 views

### Proof for power-law tail of Poisson-Dirichlet distribution (Pitman-Yor process & Zipf's law)

I'm trying to understand the motivation of using Pitman-Yor (PY) processes in language modeling, in particular Teh's hierarchical LM based on PY processes. A motivation frequently stated in research ...

**3**

votes

**2**answers

200 views

### Is a function of complete statistics again complete?

suppose $T$ is a complete stats for a parameter $\theta$.
Is any function $f(T)$ again complete?
It sounds weird but the definition seems to confirm that $f(T)$ is indeed complete..

**1**

vote

**0**answers

39 views

### Bounds on product of CDF or Beta function

I have functions of the form
\begin{align}
I_i = \int_0^\infty F_0(x)^aF_1(x)^b(1-F_0(x))^c(1-F_1(x))^ddF_i(x),~~~~i = 0,1.
\end{align}
$F_0(x)$ and $F_1(x)$ are CDFs corresponding to the random ...

**2**

votes

**0**answers

89 views

### Mean and variance of a general multivariate skew normal distribution

I have a problem about a general multivariate skew normal distribution. There is a $p\times 1$ vector, $\mathbf{y}=(\mathbf{y}_1',\mathbf{y}_2',\ldots,\mathbf{y}_n')',p>n$, which has the density as
...

**6**

votes

**1**answer

173 views

### Closure of random rotations

Are matrix Fisher random variables closed under multiplication?
For those unfamiliar with the jargon, let me unpack the terms above and repose my question.
This is a question about probability ...

**5**

votes

**3**answers

342 views

### A. Markov's papers?

A. Markov published several papers on his chains, starting in 1906, so it is written, in the journal:
(1) Извѣстія Физико-математического общества при Казанском университете
I am surprised by the ...

**2**

votes

**1**answer

83 views

### Linear least squares with unordered response variable

In the classical linear regression model one considers the equation
$$ y = X \beta + \epsilon.$$
I was wondering whether there are also results when the ordering of the response variable $y$ is not ...

**1**

vote

**1**answer

64 views

### How to extend Dirichlet distribution to Dirichlet process

For a Dirichlet process, there are two parameter $\alpha$ and $H$, and the Dirichlet process $X$ is defined as
$$(X(B_1),\cdots,X(B_n))\sim Dir(\alpha H(B_1),\cdots,\alpha H(B_n))$$
...

**2**

votes

**1**answer

65 views

### Unbiased sample from a product

Let $X = (x_1,\ldots,x_n)$ be an i.i.d sample from distribution $F%$ and let $y = \prod_{i=1}^n x_i$
Can we derive a randomized, unbiased. estimator $\hat{y}$ of $y$ that on average considers only a ...

**5**

votes

**2**answers

303 views

### Random Vornoi Diagrams (particular measures)

This is my second question about Random Voronoi diagrams, in my first question was given some excellent advice but i was not clear in explaining what i was looking for.
I'm interested to know ...

**8**

votes

**2**answers

702 views

### Random Voronoi Diagrams

I'm interested in what research has already been done with regards to the statistics of random voronoi diagrams. I have had a look on google scholar and results are a little inconclusive. I'm ...

**1**

vote

**1**answer

94 views

### Distance between two distribution of image

I am looking for a common distance method to compare two distribution (ex: histogram of image). Please suggest to me some common method to do it. I found some method ex: Bhattacharyya distance , K-L ...

**2**

votes

**2**answers

253 views

### Distribution of a random walk on a directed line

Is there a closed formula for the distribution of $x_t$ in the following random process, describing a random walk on a directed line?
$x_0 = n$
$x_t$ is a uniformly random integer between 1 and ...

**1**

vote

**1**answer

54 views

### Multivariate Rayleigh [closed]

What is the closed form formula (pdf) for a multivariate Rayleigh distribution. Is it -
$x^T \Sigma^{-1} x \times \exp(\frac{-x^T \Sigma^{-1} x}{2})$
How do you prove it is from the exponential ...

**6**

votes

**0**answers

116 views

### Convergence of Maximum Likelihood Estimator

I apologize for the basic question. If $\{p_\theta(x): \theta\in K\subseteq\mathbb{R}\}$ is a smooth family of distributions, then the MLE $\hat{\theta}_n,$ under suitable regularity conditions ...

**3**

votes

**1**answer

141 views

### 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 ...

**1**

vote

**1**answer

308 views

### Markov chain Monte Carlo: why is non-reversible MC MC not as popular?

I am new to methods for simulating Markov chains in order to sample from the target, unknown distribution. After a couple days of reading, I found out that even though people have realized that ...

**5**

votes

**1**answer

455 views

### How to check if a symmetric random variables is the difference of two iid symmetric random variables

I have the continuous symmetric random variable $X$ in $\mathbb{R}$. If I know its distribution function $F(x)$ what are the conditions on $F(x)$ so that $X=Y_1 - Y_2$ where $Y_i$ are also iid ...

**2**

votes

**2**answers

252 views

### Gaussian expectation of an exponentiated outer product

Given a normal random column vector $\mathbf{x} \sim N(\mu, \Sigma)$, I need the expectation,
$$ E\left[ \exp(\mathbf{xx}^\top)\right]$$
where $\exp(\cdot)$ is element-wise exponential function (not ...