**-1**

votes

**0**answers

6 views

### Mean and variance for unequal samples [migrated]

I have a sampling of variable sized plots. Each plot contains the number of trees present on the plot. Given:
$n=$ the number of plots
$s_i=$ the size of the $i^{th}$ plot
$y_i=$ the number of trees ...

**3**

votes

**4**answers

3k views

### Linear Regression Coefficients W/ X, Y swapped

Let's say I have a linear regression model of the form $ y = B_x x + I_x + \epsilon $, where $B_x$ is the beta coefficient of the $x$ term, $I_x$ is the intercept term and $\epsilon$ is additive, ...

**2**

votes

**3**answers

56 views

### Generate Bernoulli vector with given covariance matrix

I am from different background, so please forgive me if the answer is so well known.
Let $C=(c_{ij})$ be a given $n\times n$ matrix. Do we have a way to generate samples of random Bernoulli vectors ...

**1**

vote

**1**answer

344 views

### Size of KL-divergence neighbourhoods

I am new here. I was reading another
post
here and this got me wondering what can be said about the size of the following kl divergence neighborhoods.
Consider these two kl-divergence neighbourhood ...

**0**

votes

**0**answers

40 views

### How to find PDF of ordered random variables? [closed]

Assumpion: Let $X_1, X_2, \ldots, X_L$ be $L$ independent and identical random variables (RVs). Let $F_{X_i}(x_i)$ and $f_{X_i}(x_i)$ be CDF and PDF of $X_i$. Suppose that $F_{X_i}(x_i) = F_X(x_i)$ ...

**2**

votes

**1**answer

179 views

### Expected value (probability) maximization with binomial distribution

I need to solve an optimization problem that involves an expected value like
$$F(n,x) = \sum_{k=0}^n \binom{n}{k} p^k(1 - p)^{n - k} f(k,x).$$
Here $f(k,x)$ is actually a probability coming from a ...

**-1**

votes

**0**answers

31 views

### Statistical test for the independence of components in a N-dimension Gaussian random variable

Assuming that there is a N-dimenson Gaussian randon variable $\mathbf{X}=\left[\mathbf{x}_1,\mathbf{x}_2,...,\mathbf{x}_N\right]^T$ and we have K observations ($K\ll N$) of $\mathbf{X}$(i.e., we have ...

**4**

votes

**0**answers

139 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 ...

**0**

votes

**0**answers

28 views

### convergence of empirical distribution of random vectors

Given
(a) random matrices $A^{n} \in \mathbb R^{n\times n}$ with iid normal
entries $A_{ij}\sim \mathcal N(0, 1/n)$; and
(b) $X^{n} \in \mathbb R^{n}$ with its empirical distributions converging ...

**1**

vote

**4**answers

17k views

### How do I convert a uniform value in [0,1) to a standard normal (Gaussian) distribution value?

I have uniform value in [0,1). I'd like to transform it into a standard normal distribution value, in a deterministic fashion.
What I'm confused about with the Box-Muller transform is that it takes ...

**2**

votes

**1**answer

104 views

### Bounds on the probability of k-of-n events in terms of bounds on single and pairwise probabilities

Let $A_1,\dotsc,A_n$ be events in a probability space, and let $N = \sum_{i=1}^n \mathbf{1}_{A_i}$ be the random number of events that occur. For a fixed value $k \in \{1,\dotsc,n\}$, what can be ...

**2**

votes

**1**answer

102 views

### Where can I find a copy of Moussatat's 1976 thesis “On the Asymptotic Theory of Statistical Experiments and Some of Its Applications”?

It was apparently written at Berkeley under the direction of Le Cam, and it is cited in a number of contributions to mathematical statistics, for example in Strasser's (1985) book "Mathematical Theory ...

**0**

votes

**1**answer

163 views

### How are epidemic models simulated in case of mobility?

I am not a mathematician but out of curiosity I am trying to implement the SIS epidemic model when the nodes have mobility to understand how it will change the results. I understand how to perform ...

**8**

votes

**1**answer

670 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 ...

**0**

votes

**2**answers

64 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 ...

**0**

votes

**1**answer

207 views

### Are such averages known with representations of $S_n$?

Like is there a sense in which one can quantify that for two group elements (in different conjugacy classes) their characters are "close" for some fixed irreducible representation? (feel free to ...

**9**

votes

**2**answers

4k views

### Coin Pusher Game

While doing laundry at my local laundromat, I saw a coin pusher game. Below is a picture, and here is a video depicting how it works (disregard non-coins).
Essentially, one has a distribution of ...

**3**

votes

**2**answers

467 views

### An Upper Bound for the Average of Top Order Statistics

The following problem arises when we try to bound the expected offline optimal value of a simple online assignment problem with random values and unit weights, by its deterministic approximation.
The ...

**3**

votes

**1**answer

149 views

### A Generalized Version of Maximal Correlation and Hypercontractivity of Conditional Expectation Operator

Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as
...

**0**

votes

**0**answers

16 views

### Mixture model: optimization vs regression

Consider a sample $\mathcal D = \{T_n\}_{n=1}^N$ of independent random variables, s.t.:
$$
p(T_n) = p_n(T) = \sum _{m=1}^Mp_n(\mathcal C_m)p_n(T\mid \mathcal C_m) = \sum _{m=1}^Mw_{nm}q_m(T)
$$
I will ...

**0**

votes

**0**answers

40 views

### Stationarity and Regression

apologies this might turn out to be a bit on the simple side, but I've been thinking this through and haven't quite found the right approach.
Suppose I have a bunch of time series (say ...

**25**

votes

**12**answers

9k views

### why is it so cool to square numbers? (in terms of finding the standard deviation)

When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do
$$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$.
Why ...

**2**

votes

**1**answer

137 views

### Maximization of specific Likelihood function

N coins have probability $p_n = e^{-t_n/s}$ of heads, $t_n$ being specific for each coin. Coins 1 to m came up heads and m+1 to N came up tails. Now I'm trying to estimate $s$ using the Maximum ...

**1**

vote

**1**answer

52 views

### Rate-Distortion theory: What is the distribution of distortion on an optimal encoder?

If we wish to encode a gaussian source, $X\sim\mathcal{N}(0,\sigma^2)$ at rate $R$, then decode it to create an estimate $\hat{X}$, rate-distortion theory tells us that the lowest mean-squared-error ...

**6**

votes

**4**answers

90 views

### Uniform Sampling Subject to Linear Equalities and Non-Negativity Constraint

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints
$$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq ...

**0**

votes

**0**answers

17 views

### K nearest neighbors estimation with a kernel

If I have a bunch of data points $x_1,\dots,x_n$, I can build a density function $f(x)$ based on these data points by defining $f(x) = c/d_k(x)$ for an appropriate constant $c$, where $d_k(x)$ is the ...

**6**

votes

**4**answers

90 views

### What can be said about the concentration of measure of product of Gaussian variables?

I have a set of random variables $X_1,\ldots,X_n$, all Gaussian with mean 0 and variance 1, indepedent. Let $p(x_1,\ldots,x_n)$ be some polynomial that takes products and sums of $x_1,\ldots,x_n$.
...

**3**

votes

**0**answers

66 views

### Kullback Leibler “variance”: does that divergence have a name?

If you consider two probability distributions $p$ and $q$, one way to measure the distance between the two is the Kullback-Leibler divergence:
$$KL(p,q)=\int p \log (p/q) = E_p(\log p/q)$$
and this ...

**0**

votes

**1**answer

336 views

### Generating independent random variable from two correlated random variables

Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ ...

**6**

votes

**4**answers

3k views

### Constructing Bernoulli random variables with prescribed correlation

For which $n \times n$ correlation matrix $C$ can one construct Bernoulli random variables $(B_1, \ldots, B_n)$ with correlation $C$ ?
Following the approach described in this MO thread, one can ...

**0**

votes

**0**answers

71 views

### A question concerning distribution of $\mathbf{Y}/\|\mathbf{Y}\|_2$ where $\mathbf{Y}\sim \mathcal{N}(\boldsymbol{\mu},\mathbf{I})$

I know that when $\mathbf{Y}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$, $\mathbf{Y}/\|\mathbf{Y}\|_2$ is distributed uniformly on the unit sphere. But to my surprise, I failed to find a simple closed ...

**0**

votes

**0**answers

13 views

### Question on Asymptotic Normality of non-parameter estimands of a distribution

I'm currently taking an introductory statistics course and one of the topics we covered was Maximum Likelihood Estimates and their asymptotic normality (under reasonable conditions that were not ...

**0**

votes

**1**answer

54 views

### Finding the distribution of a random variable numerically with sample data? [closed]

Just a thought that I had recently. Suppose given discrete data points for a random variable, could one numerically generate the probability function values at these discrete values? I tried looking ...

**2**

votes

**0**answers

18 views

### Is there some kind of lower bound for estimation error of the estimation of (near) low-rank matrices in high-dimension?

I'm reading S.Negahban and M.J.Wainright's paper, ESTIMATION OF (NEAR) LOW-RANK MATRICES WITH NOISE AND HIGH-DIMENSIONAL SCALING. In the paper, they give a upper bound for estimation error of the ...

**4**

votes

**1**answer

135 views

### variance of compound binomial distributions

The below is motivated by a problem I'm observing in my experimental data
I have m boxes, where each box is supposed to contain k molecules of mRNA. The measurement process includes labeling all the ...

**0**

votes

**0**answers

61 views

### What is the concentration of measure for Gaussian random variables which are independent, but are transformed?

This might be a too easy question for Mathoverflow, but Googling led to similar questions and answers here (though not the one I was looking for).
The question is split into two:
I have a matrix $X ...

**1**

vote

**0**answers

47 views

### How do you use the bits you get back from Bits Back Coding?

Bits Back coding is a scheme to transmit an observation x.
You can read about it here [1]. To my understanding, it works like this:
The encoder samples a message z from a distribution Q(z|x) that it ...

**4**

votes

**3**answers

1k views

### Correspondence between Viterbi algorithm and Smith-Waterman

Viterbi is an algorithm for finding the maximum likelihood assignment to the hidden variables of an HMM, given the observed variables (we know the transition and emission probabilities of the HMM). ...

**2**

votes

**2**answers

805 views

### Approximation of a Normal Distribution function

I am reviewing and documenting a software application (part of a supply chain system) which implements an approximation of a Normal Distribution function; the original documentation mentions the ...

**3**

votes

**1**answer

261 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

**0**answers

41 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 ...

**4**

votes

**2**answers

293 views

### Expectation of Mahalanobis norm

Let $(g_i)_{i=1,...,d}$ sampled i.i.d. from a standard Gaussian, and $(\lambda_i)_{i=1,...,d}$ non-random s.t. $\max_i(\lambda_i)=1$ and $\lambda_i>0, \forall i$.
I am looking for the expectation ...

**1**

vote

**0**answers

69 views

### Bounding correlation between blocks of Gaussian stationary process

Let $X_n$ be a stationary Gaussian process with covariance function $\gamma(n)=\mathrm{Cov}[X(n),X(0)]$. Let $\mathbf{X}_p^q=(X_p,\ldots,X_q)$, $s_n^2=\mathrm{Var}(X_1+\ldots+X_n)$, and ...

**0**

votes

**2**answers

4k views

### Why 1.5IQR whiskers in boxplot? [closed]

Hi math people.
I'm in the process of analyzing some data that I collected through an experiment. The data are (somewhat) normally distributed and I represent the different data-sets using boxplot, ...

**2**

votes

**1**answer

92 views

### Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES.
And in the paper, they provide an inequation of the Schatten-p (quasi-)norm, ...

**2**

votes

**0**answers

81 views

### Implication of MGF inequality

Let X and Y be two random variables. Denote by $F_X(x)$ and $F_Y(y)$ their CDFs and by $M_X(t)$ and $M_Y(t)$ their MGFs.
It is known that X and Y have the same CDF iff they have the same MGF.
My ...

**2**

votes

**2**answers

107 views

### Do all positive distributions on $N$ variables factor pairwise?

The Hammersley-Clifford theorem says that any positive probability distribution satisfies one of the Markov properties with respect to an undirected graph G if and only if its density can be ...

**1**

vote

**0**answers

99 views

### Converse for Levy's continuity theorem

Levy's continuity theorem states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function ...

**3**

votes

**2**answers

141 views

### Statistical properties of principal components and their convergence rates.

Hello everyone, I'm interested in doing statistical tests on properties of principal components, but none of the literature I've found so far seems quite right for my purposes. Many articles present ...

**1**

vote

**1**answer

189 views

### Computing probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s

This question came up in my research: What is the probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s?
So far I only figured out that I can do Monte ...