**3**

votes

**0**answers

124 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

**0**answers

35 views

### How to sample from the ratio between two distributions?

I want to sample a lot of $\theta$s from the density function below:
$$ r(\theta) = \frac{prior(\theta)}{Z}\frac{\int p(\theta,z_1)dz_1}{\int q(\theta,z_2)dz_2} $$
where $Z$ is the constant for ...

**0**

votes

**1**answer

233 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

157 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

824 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

99 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

117 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

53 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

107 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

140 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

70 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

325 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

74 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

178 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

161 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

83 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

172 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

331 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

79 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

63 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

302 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

689 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

91 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

251 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

47 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

111 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

139 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

270 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

401 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

249 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 ...

**1**

vote

**1**answer

51 views

### How are two tailed p values (especially) and one tailed p values useful given the following? [closed]

So I'm a self-learner which is always dangerous because I don't have anything to test if I am understanding things correctly, so I wanted to ask what is wrong/right with my assumptions.
When reading ...

**1**

vote

**0**answers

46 views

### Can anything be said of the correlation of X and Y / X? [closed]

I apologize in advance if I overstep my (relatively minimal) statistical knowledge.
I am looking at two random variables X and Y, and am unhappy with the correlation between the two. On a whim, I ...

**5**

votes

**1**answer

417 views

### Is there a mistake in Vapnik's “Basic Lemma”?

I have a concern about the "Basic Lemma" which Valdimir Vapnik states and proves in his 1998 book Statistical Learning Theory (ch. 14.3, pp. 574–76): It seems like a certain coefficient should have ...

**3**

votes

**1**answer

176 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 ...

**0**

votes

**0**answers

280 views

### Cholesky decomposition of a large covariance matrix

I have a tricky problem concerning a covariance matrix cholesky decomposition.
What I need is to obtain the cholesky decomposition of the estimated variance matrix of the set of samples stored in a ...

**0**

votes

**1**answer

81 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 ...

**0**

votes

**1**answer

205 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 ...

**2**

votes

**1**answer

70 views

### What is the problem with this model parameter estimation algorithm?

In a statistical model with parameters $\theta$ and unobserved laten variables $Z$, the model likelihood is
$$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$
The standard way to estimate $\theta$ ...

**0**

votes

**1**answer

47 views

### finite mixture of order statistics

Let $F(u)$ be a n-degree polynomial continuous distribution function in $[0,1]$, with $F(0)=0$, $F(1)=1$, that is $F(u)=\sum_{i=1}^{i=n} a_i u^i$. My question is: is that kind of distributions ...

**1**

vote

**0**answers

400 views

### random walk with reflecting barriers [closed]

Consider a random walk on the line 1,...,d. You start at point 1. At each step you flip a coin: heads means go left, tails means go right. If you're at 1 and get a heads, just stay where you are (same ...

**1**

vote

**0**answers

92 views

### Distribution of the Gram Matrices

Let $\mathbf{X}$ be an $m\times m$ random matrix full rank matrix, having the density function $f_{\mathbf{X}}(X)$. Also, let $\mathbf{W}$ be a deterministic $k\times m$ matrix of rank $k$ and ...

**1**

vote

**1**answer

165 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 ...

**1**

vote

**0**answers

237 views

### Inflated independent samples for Monte Carlo estimation

In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...

**3**

votes

**1**answer

264 views

### Mutual information decrease with coarse-graining

Let $X,A,Y,B,C,D$ be random binary variables. $D$ is independent from $X,A,C$ and $C$ is independent from $Y,B,D$.
Is it true that:
If $I(Y:B|D=0)\leq \epsilon$ then $I(X\oplus Y:A\oplus ...

**0**

votes

**1**answer

109 views

### Expected number of samples above certain value of a normally distributed variable with a given sample mean

Suppose $n$ values, $X_1,...,X_n,$ are generated by a random number generator with normal distribution $N(0,1).$ Suppose that the (sample) mean of $X_1,...,X_n$ is $\mu.$ What is known about the order ...

**0**

votes

**1**answer

429 views

### Compound Poisson process and central limit theorem [closed]

If I have a compound Poisson process
$$Y(t) = \sum_{i=1}{N(t)}D_{i}$$
where $ \{\,N(t) : t \geq 0\,\}$ is a Poisson process with rate $\lambda$, and $ \{\,D_i : i \geq 1\,\}$ are i.i.d random ...