**-1**

votes

**0**answers

32 views

### Maximum chi-square distance between norm vectors [on hold]

What is the maximum possible chi-square distance between two normalized vectors? The representation of chi-square distance is below.
$d(x,y) = \sum_i \frac{(x_i-y_i)^2}{x_i+y_i}$

**2**

votes

**2**answers

87 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

46 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 ...

**0**

votes

**0**answers

30 views

### Variance of the estimator of a geometric distribution [closed]

Suppose I have $N$ geometric random variables, $x_1, x_2, ..., x_N$, which the following probability mass function
$ f(x) = (1 - p)^{x-1} p$
The maximum likelihood estimation of $p$ will be
...

**4**

votes

**0**answers

116 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:
...

**0**

votes

**0**answers

13 views

### How to generalize uncertainty coefficient to set-valued classes?

This question is the reason I asked How to estimate the entropy of a distribution on a power set?
Proficiency (AKA uncertainty coefficient) is an information-theoretic measure of predictor quality, ...

**0**

votes

**1**answer

44 views

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

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

89 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

46 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

70 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

**3**answers

106 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

163 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

69 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

**0**answers

45 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

55 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

65 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:
...

**1**

vote

**0**answers

52 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

453 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

50 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

44 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

84 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

63 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

86 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

96 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 ...

**9**

votes

**0**answers

190 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

50 views

### ordinary least square and random projection

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

**5**

votes

**3**answers

307 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

144 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
...

**0**

votes

**0**answers

70 views

### Fitting distribution to spatial data

I am studying a physical process generating data which projects nicely into two dimensions with non-negative values. Each process has a (projected) track of $x$-$y$ points -- see the image below.
...

**2**

votes

**1**answer

239 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 ...

**-1**

votes

**1**answer

183 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

146 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

206 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

222 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 ...

**15**

votes

**6**answers

984 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

366 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

61 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

143 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

148 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

302 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

94 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

88 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 ...

**0**

votes

**0**answers

57 views

### Monte carlo Method to estimate a proportion

I'd like to use Monte Carlo method to estimate a proportion and I'd like to be sure my idea is correct mathematically speaking.
Let a pool full of red and blue balls.
I'd like to estimate the ...

**5**

votes

**1**answer

95 views

### Deviation bound for the maximum of the norm of Wiener process

Let $W(t)$ be an $n$-dimensional Wiener process. Denote by $\chi_n^2$ a chi-squared random variable with $n$ degrees of freedom. I have recently found the following inequality given without proof:
$$
...

**5**

votes

**3**answers

214 views

### How do you call the problem of approximating a continuous distribution with a simple discrete distribution?

The following problem came up on the Mathematica forum as "Generating a list of integers that roughly satisfy a distribution": Given $n$, find $n$ integers (possibly with duplicates) whose ...

**0**

votes

**1**answer

113 views

### Expected rank of players in a Bradley-Terry round-robin tournament

Let $[n]$=$\{1,\dots,n\}$ be a set of players in a round-robin tournament. Each player $i$ has an associated skill parameter, $\lambda_{i}$, and the probability that player $i$ defeats player $j$ is ...

**-6**

votes

**1**answer

100 views

### How do you calculate a 95% confidence interval for the difference of 2 values? [closed]

Need to know how to calculate a 95% confidence interval of the difference between 2 values.
Information:
...

**1**

vote

**0**answers

84 views

### What is the range of a positive random variable after whitening?

Let ${\bf x}\in\mathbb R^N$ be a positive multivariate random variable, i.e.
$$x_i\in [0,\infty).$$
What is the range after whitening, i.e. the range of ${\bf y} = \sqrt{C}^{-1}{\bf x}$ with the ...

**0**

votes

**0**answers

62 views

### Eigen value distribution of autocorrelated Wishart matrix

Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on ...

**1**

vote

**2**answers

198 views

### Proof of Von Neumann's debiasing algorithm

Assume you have a source of random binary information that has a bias but no correlation between consecutive bits. John von Neumann describes an algorithm to debias the random source and output a ...