**2**

votes

**0**answers

38 views

### Where to read about this kind of “measure of irredundancy” of a set from a family of sets?

Studying a very practical problem from psychometrics, I encountered the following construction.
Let $(X,\mu)$ be a measure space; if preferred, you can presume $\mu$ is a probability measure. In any ...

**6**

votes

**3**answers

205 views

### Constructing a Bernoulli random variable for ratio of Bernoulli weights

$X$ and $Y$ are Bernoulli random variables with weights $0 < \alpha < 1$ and $0 < \beta < 1$. Is it possible to construct a sampler for the Bernoulli random variable with weight ...

**2**

votes

**2**answers

110 views

### Sampling from maximally skewed stable distribution

I am reading a paper which refers to a maximally skewed stable distribution $F(x;1,-1,\pi/2,0)$ . Is there an efficient way to sample from this distribution?
If $X$ has distribution ...

**2**

votes

**0**answers

157 views

### Hilbert Schmidt Operators and the Conditional Expectation Operator

Consider the function $\text{E}_W: L_2(\mathbb{R},P_X) \mapsto L_2(\mathbb{R},P_W)$ where $P_X$ and $P_W$ are two different probability measures. They are related in such a way that if $f_X$, $f_W$ ...

**5**

votes

**0**answers

190 views

### Inverse moment of the number of inversions of a permutation

Let $\pi$ be a permutation of $\{1,2,...,n\}$. A pair of elements ($\pi_i$,$\pi_j$) is called an inversion if $i$ $>$ $j$ and $\pi_i$ $<$ $\pi_j$. The total number of inversions in $\pi$ is ...

**3**

votes

**1**answer

126 views

### Estimating total variation distance from a given distribution

Given a known distribution supported on a finite set of $n$ elements with probabilities $p_1, \dots, p_n$ and an access to an unknown distribution $q$ is it known what is the number of samples from ...

**3**

votes

**0**answers

31 views

### Regression with correlation structure

I have a theoretical question about regression models.
Let's say I measured multiple responses from $n$ subjects and these responses are correlated with each other. For example, let's say I measured ...

**-1**

votes

**1**answer

92 views

### Express $cov(X^2,Z)$ in terms of means, variances, and covariance of $X$ and $Z$? [closed]

Suppose $X$ and $Z$ are random variables. Can the covariance of $X^2$ with $Z$ be expressed in terms of the means, population variances, and covariance of $X$ and $Z$ alone?
My attempts at solving ...

**2**

votes

**0**answers

39 views

### How to get the Expectation of the normalization of some log-normal-distributions?

Problem Definition:
Suppose that a random variable of multivariate Gaussian distribution $X \sim N(\Sigma,\mu)$, $\Sigma$ is the covariance matrix, and $\mu$ is the mean. For each $x_i$ from $X$, $x_i ...

**1**

vote

**1**answer

85 views

### Averaging function of sum of variables using central limit theorem

I'm trying to evaluate an integral of the following form
$$\int \prod_i \left[ dx_i \,P(x_i) \right] \; f \Big( \frac{1}{N} \! \sum_{i=1}^N x_i \Big)$$
and I know that the distribution of $x$ is ...

**5**

votes

**1**answer

377 views

### Central limit theorem for independent random variables, with a Gumbel limit

Consider independent random variables $Y_i$, $i>0$, such that $\mathbb{E}(Y_i)\approx \frac{1}{i}$ and $\text{Var}(Y_i)\approx \frac{1}{i^2}$, where $\approx$ means asymptotically equivalent up to ...

**1**

vote

**0**answers

55 views

### Efficient evaluation of multidimensional kernel density estimate

Edit I have copied this discussion to the stats community site here, since I feel it is more relevant. Please feel free to close this in due course.
I've seen a reasonable amount of literature about ...

**2**

votes

**2**answers

150 views

### Bounds for the fat tail after trimming the mean?

I am interested in the quantity $$f(X,t) = \int_t^\infty\negthinspace x\ p(x)\ dx,$$ where $p$ is a probability distribution for a positive variable $X$.
1) Does this quantity $f(X,t)$ have a name? ...

**1**

vote

**0**answers

61 views

### Whether r.v. with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)]$ is overdispersion?

When discrete r.v. $X$ is not Poisson distributed and ${\rm{Var}}X,EX < \infty $, I want to know whether r.v. $X$ with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)],({q_i} \in ...

**4**

votes

**0**answers

155 views

### Optimization problem involving Multivariate Normal

I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all
$n\geq3$, the function:
...

**-1**

votes

**1**answer

57 views

### Finiteness of “novel variance” from a kernel on a compact space [closed]

Let $c(i,i')$ be a kernel function on a reasonable index space $I$. Choose a dense sequence of points $\{i_1, i_2, \cdots \} \subseteq I$, and define the one-point kernel functions $k_n := c(\cdot, ...

**1**

vote

**3**answers

147 views

### How to estimate the entropy of a distribution on a power set?

Given a probability distribution $(X,p)$, its entropy is defined as $H=-\sum_{x\in X} p(x)\log p(x)$.
Given a sample of observations $x_n,n=1..N$, one can estimate $p(x)=\frac{\#\{i:x_i=x\}}{N}$ and ...

**1**

vote

**1**answer

94 views

### Can I test many p-values with KS or AD

(Sorry if this is a noob question. I'm a mathematician learning statistics.)
I would like to know if it's sound (or advisable) to test many p-values against the continuous uniform distribution using ...

**-4**

votes

**1**answer

91 views

### Is it possible to determine if these random numbers are not really random? [closed]

I've been given a big ordered list of integer numbers.
Looks like this :
10
-11
-3
-6
-10
-1
.....
.....
.....
Allegedly, these values are random from -12 to +12
However, there has been ...

**1**

vote

**2**answers

358 views

### Empirical estimator for total variation distance between two product distributions

Let $X = (X_1, X_2, \ldots , X_n)$ be an $n$-dimensional random variable, where each $X_i$ is a random variable on finite discrete set $S$. In addition, $X_i$ are independent of each other (but not ...

**5**

votes

**1**answer

203 views

### Measures which exhibit the “uncorrelated implies independent” property

Let $X$ be a topological linear space, and let $X^*$ be its dual space. Suppose that $X$ is complete and Hausdorff, and $X^*$ separates points. Let $Y$ be another such space, and let $f : X \to Y$ be ...

**0**

votes

**1**answer

86 views

### Understanding the derivation of a ML-estimator (statistics)

I'm trying to understand the derivation of a ML-estimator and more specifically the rewriting of the covariance matrix $\Sigma$. In this rewriting, a lemma is used to show that:
$$
\tag{1} ...

**1**

vote

**1**answer

213 views

### forward algorithm Hidden Markov Model

I am studying the the forward-backward algorithm used in Hidden Markov Models. I understand that that you are trying to propagate through a sequence (and the available states) to find the most ...

**0**

votes

**1**answer

102 views

### Estimating the variance of error in empirical approximation to a distribution

Let $X_1,X_2,\ldots,X_n$ be i.i.d. random variables in $\mathbb{R}$ with common cumulative distribution function (CDF) $F(x)$. The empirical approximation to $F(x)$ is defined as follows:
...

**1**

vote

**1**answer

92 views

### What is known about the distribution of the errors in empirical approximation of a CDF?

Let $X_1,X_2,\ldots,X_n$ be i.i.d. random variables in $\mathbb{R}$ with common cumulative distribution function (CDF) $F(x)$. The empirical approximation to $F(x)$ is defined as follows:
...

**2**

votes

**0**answers

83 views

### Question in Wainwright's paper about signed support recovery in lasso

Sharp thresholds for high dimensional and noisy sparsity recovery using $l_1$ constrained quadratic programming (Lasso)
This paper is about support recovery guarantees of the Lasso.
I have an issue ...

**13**

votes

**4**answers

885 views

### Are gaussians with different moments far in total variation distance?

If two Gaussians disagree on one moment, it seems like this should imply that they have a large variation distance--equivalently, if two Gaussians are close in variation distance it's hard for their ...

**2**

votes

**1**answer

77 views

### Why does differencing create wide-sense stationary time series?

In time series analysis, a common assumption made is that the series is wide-sense stationary, ex. that it has time invariant mean and covariance. However, as this is often not the case in real life, ...

**2**

votes

**0**answers

63 views

### Smallest distribution of points with genuinely different clusterings

An hierarchical clustering algorithm for (finite) sets of points in a given metric space is essentially determined by its linkage criterion, which defines the distance between arbitrary (finite) sets ...

**3**

votes

**2**answers

225 views

### expectation of log(x+a) when X follows a beta distribution

Is there a closed form expression for the expectation of $\log(x+a)$ (with $a>0$, the case $a=0$ is obvious) when X follows a beta distribution?

**2**

votes

**2**answers

118 views

### estimating variance of dependent normal distributed data

Let $X_{ij}$ with $1\leq i<j\leq n$ (that are $X_{12},\dots, X_{1n},\dots,X_{(n-1)n}$) be ${n \choose 2}$ identically normal distributed $N(0,\sigma^2)$ such that
$
\text{corr}(X_{ij},X_{rs})=\rho
...

**0**

votes

**2**answers

110 views

### Determine joint distribution from projections

Let $X=(X_1,\dots,X_d)$ be a random vector, and a.s. $X \in [0,1]^d$. Suppose that for every $a \in \mathbb{R}^d$, we know the probability distribution of the random variable $Y_a = <a,X>$. My ...

**2**

votes

**0**answers

122 views

### MLRP of random variables and order statistics

Suppose we have $N$ independent random variables $X_1, \cdots, X_N$ drawn from $f_1 > \cdots > f_N$ where $f_i > f_j$ indicates that $f_i$ and $f_j$ satisfy the monotone likelihood ratio ...

**11**

votes

**0**answers

343 views

### What kind of random matrices have rapidly decaying singular values?

I've been told that in machine learning it's common to compute the singular value decomposition of matrices in order to throw out all information in the matrix except that corresponding to, say, the ...

**1**

vote

**1**answer

91 views

### Ordinary least square and random projection

Let $X$ be a given $d \times T$ matrix, and let $M$ be an $n \times d$ random matrix (say i.i.d. centered coefficients). Define $Y=MX$ in $\mathbb{R}^n$ and $H=Y'(YY')^{-1}Y$, where $'$ denotes the ...

**5**

votes

**3**answers

647 views

### Deconvolution of sum of two random variables

Let $Z = X + c \cdot Y$ where $X$ and $Y$ are independent random variables drawn form the same distribution given by the pdf $g()$ and $0 < c < 1$
I have observations of $Z_i$'s and thus can ...

**3**

votes

**1**answer

231 views

### How to perform Importance Sampling with Prior Information

Let us define a random variable $X$ with density function $p(x)$. We wish to calculate $\mathbb{E}[f(X)] = \int f(x)p(x)dx$. We can compute the expectation by Monte Carlo simulations as
...

**2**

votes

**1**answer

276 views

### What are the Reasons for the Ambiguous Meaning of “Distribution” in Mathematics

The term "distribution" is commonly associated with statistics and, less commonly known, to generalized functions.
Questions:
what is known about the origin of the term in the two fields?
are the ...

**0**

votes

**1**answer

342 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 $Z$ ...

**1**

vote

**0**answers

313 views

### Incoherence of the row/column span

Due to V.Chandrasekaran., et al (p.11) : In general for any $k$-dimensional subspace of $A_{n×n}$ we have that:
$$\sqrt{(k/n)} \leq incoherence(A)\leq 1$$
where the lower bound is achieved (for ...

**1**

vote

**1**answer

342 views

### Sum of covariance matrix of products of dependent variables

Consider the sequences of random variables $\{X_i\}_{i=1}^n$ and $\{Y_i\}_{i=1}^n$, as well as the corresponding sequence of products, $\{X_i Y_i\}_{i=1}^n$. All $X_i$ share the same mean value, ...

**4**

votes

**2**answers

252 views

### Estimate on gaussian distribution

Let X be an $\mathbb R^d$-valued random variable with distribution $N_d(0,\Sigma)$. I'm looking for a function $f$ such that
$$P(|X_1|\leq M, |X_2|\leq M,\dots, |X_d|\leq M)\geq f(M),$$
and such that ...

**16**

votes

**5**answers

1k views

### Is a fair lottery possible?

I'm trying to come up with a scheme for a lottery where each individual has roughly the same chance of becoming the winner, regardless of the number of tickets one holds. So no individual should have ...

**14**

votes

**2**answers

413 views

### How to sample uniformly from singular matrices

I would like to uniformly sample from all singular $n$ by $n$ Bernoulli matrices (that is each entry is $1$ or $0$ with probability $1/2$). I could of course just sample from all $n$ by $n$ Bernoulli ...

**2**

votes

**0**answers

81 views

### sufficient statistics and isometries

Let $(M,g)$ be an infinite dimensional statistical manifold with the Fisher information metric $g$. Is it true that any isometry on this manifold must correspond to a sufficient statistic?

**4**

votes

**1**answer

193 views

### Consecutive Primes mod 3

Is anything known asymptotically about the binary "primes mod 3" sequence besides Dirichlet's result that 1 and 2 occur half of the time? For example, can you prove that it does not eventually cycle ...

**3**

votes

**0**answers

261 views

### Maximization of a total variation distance subject to another total variation distance in Markov chain

Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...

**8**

votes

**2**answers

421 views

### Inequality in information theory

I am reading the paper "chain independence and common information" (http://ttic.uchicago.edu/~yury/papers/independ.pdf). In this paper, an inequality is used several times (without proof) which looks ...

**3**

votes

**0**answers

186 views

### Quantile convergence

Let $X^1,\dots,X^n$ be a sample of (not necessarily iid) random variables and denote
$$F^n(x)=\frac{1}{n}\sum_{i=1}^n \mathbf 1_{X^i\leq x}$$
the empirical distribution function. Suppose that we know ...

**4**

votes

**1**answer

135 views

### Weak ergodicity of nonhomogenous products of 0-1 matrices

Here is a question which probably has a negative answer, but I couldn't find any literature directly on it.
Let $(A_n)$ be a sequence of rectangular 0-1 matrices (that is, the entries are restricted ...