# Questions tagged [st.statistics]

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

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9k views

### Research situation in the field of Information Geometry

I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field.
I have read (4) and parts of (3). ...

**20**

votes

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797 views

### Random Distance Matrices

My question is motivated by the following recent paper:
http://arxiv.org/abs/1110.6333
Assume you have a metric space $(X,d)$ equipped with a Borel probability measure $\mu$. We can further assume ...

**10**

votes

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268 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 \...

**10**

votes

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476 views

### Bounding the probability that a random variable is maximal

Question: Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_N$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$.
I am looking ...

**10**

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**0**answers

369 views

### Question from an economist: solving a model of traders' behavior with expectations about the future values of the variable they are currently optimizing

Motivation
I am an economist writing a paper for an academic finance journal. My paper is about the behavior of currency traders, who choose the price at which they will sell currency today, based on ...

**9**

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1k views

### Has the Lie group preserving a probability distribution been used in Bayesian statistics?

For a (possibly signed) nondegenerate probability measure $\pi$ on $\{1,\dots,n\}$ define
$$\langle \pi \rangle := \{R \in \operatorname{STO}(n): \pi R = \pi \}.$$
Here $\operatorname{STO}(n)$ denotes ...

**8**

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256 views

### First Table of Random Numbers

What was the first table of random numbers of any sort?
The best I can do is Tippett and Pearson's Random Sampling Numbers of 1927.
Can anybody identify an earlier table?
Thanks for any insight....

**7**

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192 views

### Is the Dimer Model a TQFT?

The answer to my question is "yes". Technically, it's a spin-TQFT but now I am trying to make sense of that answer.
Dimers on surface graphs and spin structures. I David Cimasoni, Nicolai ...

**7**

votes

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142 views

### Can one “smooth over” k-wise independence to get actual independence?

I came across the following toy problem and was curious if there was a simple solution or counterexample. Suppose you have a distribution $p$ on $m$ random variables $X_1, \ldots, X_m$, each with ...

**7**

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462 views

### Convergence of Maximum Likelihood Estimator

I apologize for the basic question. If $\{p_\theta(x): \theta\in K\subseteq\mathbb{R}\}$ is a smooth family of distributions, then the MLE $\hat{\theta}_n,$ under suitable regularity conditions ...

**6**

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136 views

### Approximating a ray with an integer lattice point

Take $X$ uniform on the unit sphere in $\mathbb{R}^n.$ For $r>0$, take $S_r=\{x\in \mathbb{Z}^n: \sum_i x_i^2 \leq r^2\}.$
With $\|\cdot \|$ the 2-norm, what is the distribution (or at least the ...

**6**

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367 views

### Maximal Correlation versus Correlation Coefficient When one RV is Gaussian

Let a pair of random variables $(X,Y)$ be continuous random variables (i.e., they both have density with respect to Lebesgue measure) with joint distribution $P_{XY}$. The maximal correlation $\rho_m(...

**6**

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146 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 ...

**6**

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198 views

### Given that a conditional measure is Gaussian, how bad can the original measure be?

Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...

**6**

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2k views

### Taylor approximation of a function of a random variable

Suppose we have a random variable $X$ and a smooth function $g$. We want to calculate the expectation value $\mathbb{E}(g(X))$. To be able to write down at least an approximate solution, we perform a ...

**5**

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350 views

### Closedness of a set of measures, where conditional marginals are in closed $\varepsilon$-ball w.r.t. Wasserstein distance

Let $(E,d)$ be a bounded polish space (separable, complete metric space satisfying $\sup_{x,y\in E} d(x,y) < \infty$). By $\mathcal{P}(E)$ we denote the space of Borel probability measures on $E$ ...

**5**

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340 views

### Hierarchical Random Walk (also known as Hierarchical Hidden Markov Model)

Let us consider the following hierarchical (recursive) random walk model, which is also known as the hierarchical hidden Markov model in computer science (https://en.wikipedia.org/wiki/...

**5**

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124 views

### What's the variance in the Six Degrees model?

Recall the six degrees of Kevin Bacon game. You can even play the game at The Oracle of Bacon, and their search works via Breadth First Search.
I interpret the punchline as saying that if I start ...

**5**

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266 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 ...

**5**

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139 views

### Positive estimator

Suppose that one knows how to generate (independent) random samples $X_1, X_2, \ldots$ distributed as the random varable $X$ with $\mathbb{E}[X]=\mu \in \mathbb{R}$. It is then easy to construct an ...

**5**

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432 views

### Another generalized coupon collector's problem

Suppose there are $L$ types of coupons, the probabilities that they appear are $a_1,a_2,\ldots,a_L$ respectively, $\sum_i^La_i=1$. Each of them is associated with a constrain number $d_i,i=1,\ldots,L$....

**5**

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1k views

### Multidimensional Berry-Esseen for probability density functions

This is a follow up to this recent question: Berry Esseen type result for probability density functions
There exists a multidimensional version of the usual Berry-Esseen theorem (for cumulative ...

**5**

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384 views

### Missing mass estimate

Let $S$ be a finite set with probability distribution $P$. Define the random variable $m_i$ to be the "missing mass" after seeing $i$ iid samples from $S$ under $P$. That is, $m_i$ is the total mass ...

**4**

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189 views

### What happens in the martingale CLT if I norm by the conditional variance instead?

TLDR: I'm a statistician (bear with me!) trying to use the martingale CLT but I only can estimate the conditional variance instead of the unconditional one. Can I do anything to get a CLT with norming ...

**4**

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76 views

### Geometric meaning of the chi-square “measure of association”

In Statistics, there's a standard test of independence of two random variables taking values in finite sets $I,J$. It relies on the computation of $\chi$-square statistics,
$$
\chi^2:=\sum_{(i,j)\in ...

**4**

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126 views

### Probability of having complete conversion in a box of three different object with interaction rules

Say there a 3 types of Objects A,B,C which randomly interact in pairs to form new objects following the below rules:
$$A + B = AB$$
$$B + C = BC$$
$$C + A = CA$$
$$AB + C = ABC$$
$$BC + A = ABC$$
$$CA ...

**4**

votes

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106 views

### Total Variation distance of polynomials of Bernoulli R.V.s

Let $X_i, Y_i$ be i.i.d Bernoulli $0/1$ random variables with
$\mathbb{E}[X_i] = p$ and $\mathbb{E}[Y_i] = q$.
Let
\begin{align*}
X &= X_1 X_2 + Χ_2 Χ_3 + \ldots +X_{n-2} X_{n-1}+ X_{n-1} X_n\\...

**4**

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73 views

### What is the entropy of binomial decay?

Let's play a game. I start with $N$ indistinguishable tokens, and I wait $T$ turns. Every turn, each token has probability $p$ of disappearing. I want an analytic formula for the entropy of this ...

**4**

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164 views

### Distributions over permutation groups $\mathcal{S}_n$

Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the ...

**4**

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85 views

### Is there an example that both Berry-Essen bound and DKW bound are attained?

The Berry-Essen bound stated that
$$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$
where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\...

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284 views

### a variation on Hanson-Wright inequality

The classic Hanson-Wright inequality states that for a Gaussian random vector $\mathbf{x}\in\mathbb{R}^n$ distributed as $\mathcal{N}(\mathbf{0},\mathbf{I})$ and $\mathbf{A}\in\mathbb{R}^{n\times n}$ ...

**4**

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98 views

### The asymptotic behavior of the ratio between the largest two of $n$ i.i.d. chi-square random variables

My question is about the asymptotic behavior of the ratio between the largest and second largest values of $n$ independent chi-square random variables.
Let $X_1, \ldots, X_n$ be $n$ independent and ...

**4**

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54 views

### Regularity Conditions for L1 convergence of maximum likelihood estimators

Let $X_1,\ldots, X_n$ be i.i.d. observations from a family of pdf or pmf $\{f_{\theta}: \theta \in \Theta \}$. We know that there are sufficient regularity conditions on the family $\{f_{\theta}: \...

**4**

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69 views

### How well does an estimator perform on another dataset?

Suppose $X \sim N(0, \Sigma)$ is a $d$-dimensional Gaussian random vector. And we have $2n$ $i.i.d$ sample $X_1, \ldots, X_{n}, \ldots, X_{2n}$.
Let $\hat{\Sigma}_1 = \frac{1}{n}\sum_{i=1}^nX_i X_i^\...

**4**

votes

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461 views

### An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$
where $dH$ is the unit invariant Haar measure on the group of unitary ...

**4**

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369 views

### Sum of the entries of the inverse covariance matrix

Let $T \in\left(0,1\right)$, $n\in\mathbb{N}$ and $e_n = [1,\ldots,1]\in\mathbb{R}^n$. Consider the covariance matrix $\mathfrak{A}_n = \left[sinc\left(\frac{T\left(r-s\right)}{n}\right)\right]^n_{r,s=...

**4**

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182 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:
$$h(\mu_{1},\ldots,\...

**4**

votes

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543 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

**0**answers

561 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 (...

**4**

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481 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 ...

**4**

votes

**0**answers

177 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 \...

**4**

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705 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 ...

**4**

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234 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 = a*e^{\...

**4**

votes

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81 views

### Importance sampling of finite path of stochastic difference equation

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...

**4**

votes

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983 views

### Concentration of sum of independent random variables

Let $X_1, ..., X_n$ be i.i.d. sub-Gaussian random variables with mean $0$ and variance $1$. That is, we have $\Pr[|X_i| > t] \leq \exp(1-t^2/K^2)$ for all $t>0$ and a parameter $K$.
Then we can ...

**4**

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342 views

### Bochner's theorem, in stages

Bochner's theorem (for the real line version) asserts an infinite tower of inequalities, as a positivity condition. Taking each one, what do they mean, in an elementary fashion (at least at the start)?...

**4**

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708 views

### Expected operator norm of inverse Wishart matrix

Let $ W\sim W_p(n,I)$ be a white $p\times p$ Wishart matrix, and assume $n>p+1$, which ensures that $W$ is invertible almost surely. Let $\|W^{-1}\|_{\text{op}}$ be the operator norm (maximum ...

**4**

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188 views

### Has anyone used reflection in bootstrapping methods for one parameter hypothesis tests?

Here's my idea for a bootstrapping method for testing hypotheses about one parameter. Please tell me if you have seen this somewhere before. If not, I'd appreciate pointers for direction of further ...

**4**

votes

**0**answers

778 views

### For what sub-$\sigma$-algebra are these two measures equivalent?

In two statistics papers (linked inline below) I have come across two definitions of certain probability measures. I conjecture that for particular choices of the construction that they are ...

**4**

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755 views

### Compressed Sensing with an Unusual Basis

I'm wondering if compressed sensing can be applied to a problem I have in the way I describe, and also whether it should be applied to this problem (or whether it's simply the wrong tool).
I have a ...