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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3
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2answers
352 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
480 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. ...
1
vote
0answers
115 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. ...
1
vote
0answers
63 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: ...
16
votes
1answer
757 views

Gini Coefficient and Renyi Entropy

Gini coefficient (aka Gini Index) is a quantity used in economics to describe income inequality. It is 0 for uniformly distributed income, and approaches 1 when all income is in hands of one ...
3
votes
1answer
143 views

Concentration rates for the posterior distribution

Sanov's theorem and Dvoretzky–Kiefer–Wolfowitz's inequality tell us how fast the empirical distribution concentrates around the true underlying probabilty distribution. What is known about the ...
4
votes
2answers
212 views

Bounding the tail of an average using the the tail of individual members

Let $X_1,X_2,\ldots,X_n$ be an i.i.d. sequence of $n$ positive random variables with mean $E[X_1]=\mu_X<\infty$ and the second moment $E[X_1^2]=\infty$. I am interested in upper-bounding ...
4
votes
1answer
154 views

Statistical models in terms of families of random variables

A statistical model is a function $P : \Theta \to \Delta(X)$, where $\Theta$ is a parameter space, and $\Delta(X)$ is the set of probability measures on a state space $X$. Suppose that $\Theta$ and ...
0
votes
1answer
271 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 ...
-2
votes
1answer
119 views

Forms of multivariate CLT [closed]

I am looking for a good reference for differnt kinds of multivariate central limit theorems. I was wondering how far the i.i.d. condition of the standard multivariate clt can be relaxed, as in can the ...
4
votes
0answers
117 views

Pettis Integrability and Laws of Large Numbers

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_n : \Omega \to V$ be a sequence of Pettis ...
0
votes
1answer
96 views

Independence of Eigenvalues of Wishart

This question regards a previous post, but it is not immediately obvious the two are related, so I ask it anyways: are the eigenvalues of a Wishart matrix $\mathbf{S}$ $=$ ...
5
votes
3answers
398 views

Concentration inequalities for the maximum of the rescaled/normalized sum of iid random variables

I am interested in concentration inequalities for the maximum of the rescaled/normalized sum of iid random variables. Let $X_1,..., X_n$ be i.i.d random variables, $S_n$ their centered sum and $M_n$ ...
4
votes
0answers
253 views

Probability distribution function for singular value sum of Gaussian random matrix

Let $\mathbf{X}$ be an $N \times N$ random matrix with IID Gaussian entries. They can be standard normal, but $N$ is not large: that is $N$ $<$ 6, typically. Call its singular value decomposition ...
2
votes
1answer
504 views

Order statistics of independent NOT identically distributed random variables

I want to find the p.d.f of the n-th order statistics from a set of independent, but NOT identically distributed random variables $X_1, \dots, X_n$ (the p.d.f. of the $X_i$'s is at hand)
2
votes
1answer
289 views

Is this a closed set?

Let $\Theta$ and $X$ be two (Hausdorff) topological spaces. Let $\mathbb P : \Theta \to \Delta(X)$ be a "statistical model", i.e., a continuous function from parameter space $\Theta$ to the space of ...
3
votes
1answer
121 views

Variance of central limit distribution for $P(x) \sim 1/x^{1+\alpha}$ for finite but large $N$?

Is it known what the next-to-leading order term is in the variance of the central limit distribution for the average of $N$ variables each of which is distributed according to $P(x) \sim ...
2
votes
1answer
358 views

Central limit theorem for $P(x)\sim 1/x^3$ distribution

I have a random variable $x \in (0,\infty)$ with distribution $P(x)$ falling off slowly $P(x) \sim 1/x^3$ for large $x$. So the expectation value $\bar{x}$ is finite but the second moment $\bar{x^2}$ ...
2
votes
1answer
246 views

Maximum of a sequence of $n$ positive random variables where variance is an increasing function of $n$

Suppose I have a sequence of $n$ i.i.d. random variables $X_1,X_2,\ldots,X_n$. Each $X_i$ is positive and has variance $\sigma(n)$ that is an increasing function of the number of variables in the ...
9
votes
2answers
287 views

“Fractional sampling” from a probability distribution

My question concerns an operation on probability distributions which has arisen in some applied research. It is well-defined mathematically (at least in a limited context), but I don't know how to ...
1
vote
0answers
87 views

Distribute Monte Carlo samples among dimensions

Simplified problem: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am ...
0
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0answers
124 views

Behavior of the sum of the exponents of chi-squared random variables normalized by their maximum

Let $X_1,X_2,\ldots,X_n$ be a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom, and denote by $X_\max$ the maximum of this sequence. Furthermore, let $k=\omega(1)$ ...
-1
votes
1answer
63 views

Wishart random variables

I have a question about Wishart random variable. If X follows a Wishart distribution, then does X-Y follows a Wishart Distribution if Y is a Hermitian matrix? Thanks.
2
votes
1answer
445 views

Asymptotic behavior of max of chi-squared distribution

Suppose $X_{\max}$ is the maximum in a sequence $X_1,X_2,\ldots,X_n$ where each $X_i\sim\chi^2_k$ is an i.i.d. chi-squared random variable with $k$ degrees of freedom. Since chi squared distribution ...
0
votes
1answer
136 views

How to calculate eigenvalue density function of $XX^\dagger$ from the density function of X

Let X be a complex random matrix, which has the probability function (drawn from the ensemble) V($XX^\dagger$), where V(x) is some function which guaranties good behavior at infinity. Note the unitary ...
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vote
0answers
141 views

How far away is the maximum of $n$ i.i.d. chi-squared random variables from the rest of the sequence as $n$ gets large?

Suppose that I have a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom $X_1, X_2, \ldots, X_n$, and denote $X_{\max}=\max(X_1, X_2, \ldots, X_n)$. Let $k$ be increasing ...
5
votes
1answer
686 views

What is the maximum-entropy distribution given mean, variance, skewness, and kurtosis?

$X\in \mathbb{R}$. Which distribution $P(X)$ has the highest possible entropy given its expected value, variance, skewness, and kurtosis? Is it an exponential family distribution of the form $P(X) ...
18
votes
5answers
2k views

Inference using Topological Data Analysis: Is it worth it for a regular statistician to learn TDA?

After having read Gunnar Carlsson's http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/S0273-0979-09-01249-X.pdf I feel enthusiastic to use some topological data analysis (TDA) methods ...
0
votes
1answer
120 views

Relating percentiles to moments [closed]

There are at least two ways people look at statistical data: A. For mathematicians, scientists, engineers, economists and such the most familiar distribution parameters would be analytical: mean, ...
8
votes
2answers
784 views

Easier reference for material like Diaconis's “Group representations in probability and statistics”

I'm teaching a class on the representation theory of finite groups at the advanced undergrad level. One of the things I'd like to talk about, or possibly have a student do any independent project on ...
4
votes
1answer
136 views

diffusions corresponding to estimators

I am an undergraduate math student preparing my thesis. Currently I am reading L.D Brown's (1971) paper Admissible Estimators, Recurrent Diffusions, and Insoluble Boundary Value Problems. Here is a ...
1
vote
0answers
123 views

Sum of non-identical categorical random variables

Is there a named distribution for the sum of non-identical categorical random variables? When the categorical variables are i.i.d., the sum is a multinomial distribution. When the categorical ...
2
votes
1answer
174 views

Empirical estimator fot the total variation distance on a finite space

I have two probability measures $p$ and $p'$ on a finite set $X$ which I do not know precisely, but which I can sample from. I would like to estimate their total variation (omitting multiplier $2$): ...
1
vote
1answer
158 views

What is a likelihood kernel?

The paper, "The Multinomial-Poisson Transformation" by S. Baker (see http://www.math.ntnu.no/inla/r-inla.org/papers/multinomial-poisson.pdf) presents "likelihood kernels" for multinomial variables, ...
1
vote
0answers
102 views

Shrinkage (or Stein's phenomenon) in low dimensions, discrete contexts

I am trying to understand shrinkage, or the Stein phenomenon. As someone without a statistics background, the focus in most introductory presentations on normal distributions and squared error loss ...
1
vote
2answers
296 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
0answers
164 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
1answer
77 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
1answer
140 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
0answers
348 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
1answer
137 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
1answer
313 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
2answers
123 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
1answer
168 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
0answers
143 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
1answer
112 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
3answers
482 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
0answers
123 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
3answers
337 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
0answers
222 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 ...