**1**

vote

**2**answers

323 views

### Appropriate histogram comparison distance measure

I am working with hyperspectral image data in R, so I have subset an image to a region of 5000 pixels, each containing a vector 254 bands in length.
I would like to cluster this data in order to try ...

**4**

votes

**0**answers

165 views

### Pair of two-variable polynomial equations of high order

I have the following pair of equations to be solved for two variables $\rho$ and $D$ resulting from a certain Maximum Likelihood Estimation for a time series $X_n > 0$, $n=0, \ldots, N+1$ with $N ...

**1**

vote

**1**answer

80 views

### Estimating the relation between the covariance of a vector and a monotone function of the same vector

Let $\boldsymbol{X}\in\mathbb{R}^n$ be a random variable with positive entries ($X_i\geq a>0$). I want to characterize the relation between the second moment matrix $\boldsymbol{M}$, defined as
$$ ...

**1**

vote

**1**answer

147 views

### Mean -> Frechet mean, Standard deviation ->?

Given a finite set $A$ of points of a metric space $(X, d)$, I would like to
find its mean. A Frechet mean seems appropriate here: $\arg \min_{x \in X} \sum_{a \in A} d(x, a)^2$. I also would like ...

**4**

votes

**0**answers

356 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 ...

**3**

votes

**1**answer

140 views

### Markov Chains based on sampled transition probabilities [closed]

If I have a process that transitions between states with some set, unknown probability, I can sample to find the transition probability. This probability is a sample average, with a well understood ...

**-1**

votes

**1**answer

324 views

### 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 ...

**0**

votes

**2**answers

128 views

### Rewrite optimization objective

Hi,
I wanted to ask, under which conditions can one rewrite the optimization objective
$\min_x f(x)\;\;\;s.t.\;\;\;g(x) \leq s$
as
$\min_x g(x)\;\;\;s.t.\;\;\;f(x) \leq t$
I have particular ...

**1**

vote

**1**answer

187 views

### Understanding the rationale behind “batch means” estimation

Hello all,
I am implementing an MCMC algorithm for my work, and I've come upon something in the literature which I just can't understand.
Specifically, I am attempting to estimate the amount of ...

**4**

votes

**0**answers

148 views

### envelope function for a linear combination of gaussian distributions

Given a distribution $F$ defined as a linear combination of Gaussian distributions:
$F = \sum_{i=1}^n C_i*N(\mu_i,\sigma_i)$ with $\sum_{i=1}^n C_i = 1$
I want to find a Gaussian function $Q = ...

**1**

vote

**1**answer

114 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 ...

**4**

votes

**3**answers

498 views

### Incremental entropy computation

After a quick internet search I found no method for incremental entropy computation.
Question 1
Let $\{x_i\}_{i=1}^n$ and $\{x_i\}_{i=1+n}^{n+m}$ be two samples and let $S_i^j:=\sum_{k=i}^j x_k$. ...

**2**

votes

**0**answers

136 views

### Marginalizing multivariate normal over defined interval

Hello everyone,
I am trying to obtain an analytic expression for the following Gaussian integral
$$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} ...

**0**

votes

**3**answers

342 views

### Random infinite sequence : Can machines generate truly random sequences. [closed]

Test : "A True Random Sequence Source and a computer producing a certain sequence of numbers are kept in separate rooms and judges try to tell them apart by conducting a series of tests on the ...

**9**

votes

**0**answers

227 views

### Testing contrasts in statistics: Is this provably a hard problem, or not?

Scheffé's method for identifying statistically significant contrasts is widely known. A contrast among the means $\mu_i$, $i=1,\ldots,r$ of $r$ populations is a linear combination $\sum_{i=1}^r c_i ...

**2**

votes

**1**answer

103 views

### How to simulate random paths of a non-homogeneous continuous-time Markov process with discrete state space for a given infinitesimal generator matrix?

Let $X=(X_{t},t \in T)$ be a non-homogeneous, continuous time Markov process with a finite state space S={1,...,K}.
Let $\alpha_{i,j}(t)$ be the hazard rates of some $\varGamma$-distributed random ...

**4**

votes

**0**answers

66 views

### Importance sampling of finite path of stochastic difference equation

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...

**1**

vote

**0**answers

54 views

### Distribution for probability of an incorrect inference based on a comparison of only two samples?

I'm trying to demonstrate the problems of how taking a sample and assuming it reflects the population accurately can be problematic.
Imagine say an urn with some large number of balls, black and ...

**1**

vote

**2**answers

251 views

### 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 ...

**0**

votes

**1**answer

125 views

### What is the Bahadur-Anderson Algorithm?

What is the Bahadur-Anderson Algorithm, and which book could one read to learn it?

**1**

vote

**0**answers

222 views

### Joint distribution from multiple marginals

Consider an experiment consisting of a repeated trial with two random Bernoulli (=binary) variables, A and B. Each trial consists of multiple outcomes for both A and B. Each trial has the same number ...

**-1**

votes

**1**answer

187 views

### How to combine correlated signals !? [closed]

Hi everybody
There are 11 signals:
S_main : The original signal
S1 ~ S10 : 10 signals that are correlated to S_main with different correlation coefficients (coeff1 ~ coeff10)
Now here's the ...

**2**

votes

**0**answers

412 views

### Concentration of sum of independent random variables

Let $X_1, ..., X_n$ be i.i.d. sub-Gaussian random variables with mean $0$ and variance $1$. That is, we have $Pr[|X_i| > t] \leq \exp(1-t^2/K^2)$ for all $t>0$ and a parameter $K$.
Then we can ...

**1**

vote

**0**answers

378 views

### Prove that the sum of a certain infinite series is 1

Prove the (numerically-evident) proposition that
\begin{equation}
\Sigma_{i=0}^\infty f(i) = 1,
\end{equation}
where
\begin{equation}
f(i)= 2^{-4 i-6} q(i) \frac{\Gamma(3 i+\frac{5}{2}) \Gamma(5 ...

**2**

votes

**1**answer

422 views

### correlation for three variables? [closed]

suppose we have three variables here, x,y, z
now, what we know is that the correlation between x and z is 0.6, the correlation between y and z is 0.65.
Here is the question, is there any formula to ...

**3**

votes

**2**answers

343 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, ...

**0**

votes

**3**answers

491 views

### Probability that one RV will exceed many others

Assume the $1 \times N$ vector
$\mathbf X = [X_1, X_2, \ldots , X_N]$
contains i.i.d. normal samples such that $\mathbf X$ has a multivariate normal distribution. Now assume another random variable ...

**1**

vote

**0**answers

150 views

### Placing Bounds on Correlation/Covariance Through Correlation with an Intermediate Variable

I am trying to make the most of computations that have already been performed in previous steps of an algorithm. Throughout this problem statement I am only mentioning correlation, but I think it is ...

**1**

vote

**1**answer

121 views

### Transition time in finite voter model

I believe the following problem is related to something called the "voter model" in statistics. This is not my area of expertise so please forgive me if the answers turn out to be well known.
...

**1**

vote

**1**answer

177 views

### ROC curve with repeated measures

Hi, I have some repeated measures data, one measurement a day for three days in a row, and the measured variable looks normally distributed. I have two groups, the "really ill" and the "not ill after ...

**3**

votes

**1**answer

215 views

### Exact sampling from 2D Ising model where coupling is constant?

What progress has been made towards sampling from the 2D lattice Ising model with the following Hamiltonian:
$H=-J\sum_{\langle i,j \rangle}S_iS_j - \sum_i b_iS_i$
Where the first sum runs over all ...

**2**

votes

**1**answer

595 views

### Expectation of the trace of an inverse of a random matrix

Given a $N \times M$ matrix $X$ comprised of standard normal entries ($M > N$), I'm interested in approximating $E[trace((XX^T\frac{\gamma}{M} + I)^{-1}]$ in terms of $N, M$ and $\gamma$. ...

**2**

votes

**1**answer

234 views

### The first eigenvalue of a branching process matrix

Let $M$ be the real square matrix of a typed branching process, such that $M_{ij}$ is the expected value of offspring of type $j$ emanating from type $i$.
We know that if the first eigenvalue if $M$ ...

**0**

votes

**1**answer

56 views

### On Variance Break detection

Say that the variance of a constant mean scalar stochastic process can take finite number of values. The problem is to detect the the point of break in variance as observation data comes in.
I tried ...

**2**

votes

**1**answer

255 views

### Estimator for sum of independent and identically distributed (iid) variables

This interesting question was asked at http://math.stackexchange.com/questions/231455/estimator-for-sum-of-independent-and-identically-distributed-iid-variables a while ago but got no answers. The ...

**3**

votes

**0**answers

103 views

### A simplified MCMC / MH algorithm. Are there known convergence results?

Hi, I hope this isn't too basic. We were working on a simulation using a Monte Carlo Within Metropolis algorithm and noticed that the whole thing could be expressed in the form below and simplified ...

**3**

votes

**1**answer

499 views

### The average number of people that can sit on a bench of a given length.

Let me explain what I mean:
The width of the average person varies, perhaps with a normal distribution.
Given a specific variance, how many people (on average) can sit side-by-side on a bench of a ...

**0**

votes

**1**answer

375 views

### Expected value with a kronecker product and Gaussian distributional assumption

What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $? The kronecker product is where the confusion ...

**0**

votes

**1**answer

172 views

### How many ways we know to join two line segments with a smooth transitional function?

This topic was created to discuss how many ways we know to create piecewise linear functions with smooth transitions between the phases. An alternative is presents by Bacon & Watts (1971):
the ...

**0**

votes

**1**answer

381 views

### Has the controversy about *fiducial distribution* been settled? [closed]

Has the controversy about the correct meaning of Fisher's notion fiducial distribution meanwhile been settled? And are there newer applications than quoted in the following literature?
G.P. Klimov: ...

**10**

votes

**3**answers

566 views

### Rapid evaluation of multivariate normal integral

I'm implementing a model that requires me to numerically evaluate a multivariate normal integral of the following form
$$\int_{-\infty}^\infty \phi(z)\displaystyle\prod_{i=1}^N \Phi(a_iz+b_i) \, ...

**15**

votes

**1**answer

510 views

### The Chow & Robbins game ≈ 0.79295350640: improvements could come from simple statistics, or from a continuous version of the game

This question seeks help with improving a numerical estimate of the value of the Chow and Robbins game. Much about this game is unknown, such as whether its value is rational, but there are two routes ...

**0**

votes

**1**answer

203 views

### Expression for the square of the correlation of two Gaussian variables as an expectation value

Dear all,
I might just be blind, so forgive me if it is a trivial question. Given two normally distributed variables $x_1$ and $x_2$ (with zero mean), their correlation $c$ can be estimated from the ...

**2**

votes

**1**answer

105 views

### How to calculate average lifespan of a new population?

What's the best estimate one can make about the average lifespan of a new population?
For instance, let's say an alien kind of life came to earth and we're able to breed then. We then get 1000 aliens ...

**3**

votes

**1**answer

192 views

### Is the Binomial Expectation of a Multivariate Convex Function Convex in the Vector p?

Let $\mathbf{p}=(p_1,\dots,p_m)$ be a vector in $[0,1]^m$ and let $\mathbf{X}=(X_1,\dots,X_m)$ be a vector of independently-distributed binomial random variables such that $X_i\sim ...

**0**

votes

**0**answers

98 views

### Two Different Representations of Multivariate Bernstein Polynomials

In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following:
$$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in ...

**5**

votes

**3**answers

542 views

### Concentration of sum of pairwise squared Euclidean distances of random vectors

Let $X_1, \ldots, X_n $ be independent random vectors in $B(0, D) \subset R^d$ ($\ell_2$ ball of radius $D$ centered at the origin). I am trying to find the concentration of the following quantity ...

**2**

votes

**0**answers

202 views

### Is connected correlation/cumulant expansion additive?

Say X is a free field or a Gaussian random variable.
Then I want to analyse the connected correlation, $<(X + a (X^2 - \langle X^2 \rangle))^n>_c$
I think that for $n \geq 4$ there are no ...

**2**

votes

**1**answer

143 views

### Optimization problem

I'm trying to solve a very practical optimization problem and I think I hit a dead-end.
There are $N$ products ($N \sim 50$). Each product can have a price $p_i$ in range between 1 and 40 dollars. ...

**1**

vote

**1**answer

161 views

### Moments of the Kolmogorov distribution

Up to what order do the moments of the Kolmogorov distribution exist? References would be appreciated.