Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies.

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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
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0answers
152 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: ...
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1answer
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, ...
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3answers
141 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
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1answer
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 ...
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1answer
88 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 ...
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2answers
301 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 ...
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1answer
193 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 ...
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1answer
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
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1answer
188 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
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1answer
97 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
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1answer
88 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
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0answers
79 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 ...
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4answers
815 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
1answer
72 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
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0answers
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
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2answers
175 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
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2answers
107 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
2answers
105 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
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0answers
119 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 ...
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0answers
311 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
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1answer
84 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
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3answers
583 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
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1answer
212 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
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1answer
271 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
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1answer
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 $Z$ ...
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0answers
264 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
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1answer
304 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
2answers
251 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
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5answers
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
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2answers
406 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
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0answers
79 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
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1answer
185 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
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0answers
256 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 ...
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2answers
413 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
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0answers
164 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
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1answer
133 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
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0answers
97 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
1answer
113 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
3answers
272 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
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1answer
130 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 ...
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0answers
90 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 ...
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0answers
89 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 ...
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2answers
470 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 ...
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0answers
64 views

Stochastic process inference from partial observations

Consider a set $U$. My signal is a piece-wise constant "function" $Sig: t \mapsto s$, i.e. the signal at time $t$ equals to some subset $s \subset U$. One can see $Sig(t)$ as a stochastic process. ...
3
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1answer
156 views

Equivalent method for maximum likelihood estimation of covariance parameters

My goal is to estimate the parameters of a covariance matrix $\Omega$, by maximizing the following log-likelihood function: $$\log L(\vec\tau, \rho, \sigma \mid W, X) = -m\ln(\left | \Omega \right |) ...
3
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2answers
408 views

What is the maximum entropy distribution on the natural numbers?

On the reals $\mathbb{R}$, the maximum entropy distribution with a given mean and variance is the Gaussian distribution. Let $\mu, \sigma > 0$. What is the maximum entropy distribution on the ...
4
votes
3answers
762 views

Online estimation of covariance matrix

I am trying to dynamically estimate the (low-dimensional) covariance matrix ${\mathbb E}[{\bf x}_t{\bf x}_t^\top]$ of a stream of data points ${\bf x}_t\in{\mathbb R}^N$ online, without any memory. ...
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0answers
127 views

Uniform Law Of Iterated Logarithm for VC classes

Kenneth Alexander proved a uniform Law Of Iterated logarithm for Vapnik-Chervonenkis classes in the article Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm (Ann. ...
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68 views

Small ball probabilities for functions of correlated normals

Let $f : \mathbb{R}^k \rightarrow \mathbb{R}$ and let $X$ be distributed k-dimensional normal with mean $0$ (with "arbitrary" covariance matrix). I am looking for references with bounds of the form: ...