**1**

vote

**0**answers

50 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**

votes

**1**answer

128 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 |) ...

**2**

votes

**2**answers

273 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

**3**answers

151 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

**0**answers

96 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

**0**answers

54 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: ...

**15**

votes

**1**answer

628 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

**1**answer

123 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

**2**answers

183 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

**1**answer

141 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

**1**answer

160 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

**1**answer

114 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 ...

**0**

votes

**0**answers

33 views

### Renormalised Pearson correlation for multivariate alpha-stable distributions

Assume we have a multivariate $\alpha$-stable distribution $\vec{X}$. Is there a way to define a renormalised version of the Pearson correlation coefficient for $X_i$ and $X_j$? For example, assuming ...

**3**

votes

**0**answers

107 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

**0**answers

44 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

**3**answers

252 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

**0**answers

184 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 ...

**1**

vote

**1**answer

188 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

**1**answer

269 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

**1**answer

110 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

**1**answer

324 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 $\bar{x^2}$ ...

**3**

votes

**1**answer

208 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 ...

**10**

votes

**2**answers

255 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 ...

**2**

votes

**0**answers

78 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**

votes

**0**answers

102 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**

vote

**0**answers

48 views

### Mathematical model building with dependent and independent variables [closed]

I have been working with data and building models on data. I have developed models in regression using cubic and power series. It works fine for variables with one dependent and one independent ...

**0**

votes

**1**answer

55 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.

**1**

vote

**0**answers

80 views

### Applying Anderson's theorem to Spherically symmetric distribution in Stein estimation

The question appears Example 3.1 of the paper "Stein Estimation for Spherically Symmetric Distributions: Recent Developments" ...

**2**

votes

**1**answer

254 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

**1**answer

120 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 ...

**1**

vote

**0**answers

105 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 ...

**4**

votes

**1**answer

433 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) ...

**16**

votes

**4**answers

889 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

**1**answer

82 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

**2**answers

450 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

**1**answer

133 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

**0**answers

71 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

**1**answer

141 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

**1**answer

99 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

**0**answers

93 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

**2**answers

147 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

**0**answers

155 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 ...

**0**

votes

**0**answers

61 views

### derivative of the logarithm of a complete homogeneous polynomials

I have the following complete homogeneous polynomial of degree $r$:
$p_(x_1, x_2,...x_n) = \sum_{i_1 + i_2 + ... +i_n = r, i_k\in {0,1,..r}} \phi_{i_1}(x_2)\phi_{i_2}(x_2)...\phi_{i_n}(x_n) $
where ...

**1**

vote

**1**answer

64 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

**1**answer

113 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

**0**answers

258 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

**1**answer

131 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

**1**answer

245 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

**2**answers

109 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 ...

**2**

votes

**1**answer

118 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 ...