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.

509 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
22 votes
0 answers
1k 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 ...
ght's user avatar
  • 3,616
19 votes
0 answers
3k views

What does a product of many Gaussian matrices converge to?

Let $A$ be a product of $n$ $d\times d$ matrices with IID standard Gaussian entries and consider the value of $g(x)=x f(x)$ where $f(x)$ is the density of squared singular values of $A/\|A\|$. Is ...
Yaroslav Bulatov's user avatar
11 votes
0 answers
222 views

Functional Weak Convergence of Maximum Likelihood Estimator

Let $\hat{\theta}_n$ be the Maximum Likelihood Estimator of parameter $\theta$, where $n$ is the sample size. It is well-known that under sufficient regularity conditions, we have the asymptotic ...
Uchiha's user avatar
  • 87
11 votes
0 answers
524 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 ...
MLS's user avatar
  • 119
10 votes
0 answers
300 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 \...
Michael Hardy's user avatar
10 votes
0 answers
389 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 ...
John's user avatar
  • 101
9 votes
0 answers
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 ...
Steve Huntsman's user avatar
8 votes
2 answers
408 views

Concentration inequality for minimal eigenvalue of sample covariance

I was reading an article of matrix completion and met the following lemma The concentration inequality for $\sigma_{\max}$ part is a standard result. However, I didn't find any results like the $\...
aurora_borealis's user avatar
8 votes
0 answers
3k 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 ...
madison54's user avatar
7 votes
0 answers
239 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 ...
john mangual's user avatar
  • 22.6k
7 votes
0 answers
178 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 ...
untitled459's user avatar
7 votes
0 answers
616 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 ...
Hedonist's user avatar
  • 1,269
6 votes
0 answers
123 views

Why wavelet methods are not popular anymore in nonparametric statistics?

Back in my master years, I took a nonparametric statistics class. In this class, a few nonparametric methods were presented, but I remember spending a lot of times on methods based on wavelet ...
BabaUtah's user avatar
6 votes
0 answers
243 views

Dimension-free sample complexity for estimating Gaussian covariance

(also asked on math.se, with no answers) Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need sample covariance $\Sigma_m$ to be $\epsilon$-close to true covariance $\Sigma$: $$...
Yaroslav Bulatov's user avatar
6 votes
0 answers
174 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 ...
Christian Chapman's user avatar
6 votes
0 answers
384 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 $E$ ...
Steve's user avatar
  • 1,085
6 votes
0 answers
523 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 $\mathbf{A}\in\mathbb{R}^{n\times n}$ ...
mohi's user avatar
  • 859
6 votes
0 answers
568 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(...
math-Student's user avatar
  • 1,109
6 votes
0 answers
703 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=...
Student's user avatar
  • 159
6 votes
0 answers
186 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 ...
Tom LaGatta's user avatar
  • 8,362
6 votes
0 answers
255 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 ...
Tom LaGatta's user avatar
  • 8,362
5 votes
0 answers
130 views

Functional inverse problem based on a variational principle

I am trying to solve an inverse problem based on variational principle. I will first present a forward problem that is already solved, and then present the inverse problem that I am trying currently ...
can't stop me now's user avatar
5 votes
0 answers
187 views

Divergence for Bhattacharya Information matrix

The Fisher information matrix (in the scalar parameter case) can be obtained from the Kullback-Leibler divergence by $$g(\theta) = -\frac{\partial}{\partial \theta}\frac{\partial}{\partial \theta'}D(...
Ashok's user avatar
  • 779
5 votes
0 answers
143 views

On $\ell_1$ to $\ell_1$ operator norm of matrix with inverse Wishart distribution

Consider a random $n\times p$ matrix $X$ with $n\ll p$ and all entries of $X$ i.i.d. standard normal. For this $X$, the system of linear equations $y=Xw$ has infinitely many solutions, and the one ...
Samir K.'s user avatar
  • 151
5 votes
1 answer
325 views

Bounding the sensitivity of a posterior mean to changes in a single data point

There is a real-valued random variable $R$. Define a finite set of random variables ("data points") $$X_i = R + Z_i \; \text{for } i\in\{1,\ldots,n\},$$ where $Z_i$ are identically and independently ...
Ben Golub's user avatar
  • 1,058
5 votes
0 answers
522 views

Concentration inequality for max component of a multivariate Gaussian in the general case

I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
ted's user avatar
  • 271
5 votes
0 answers
469 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/...
Minkov's user avatar
  • 1,117
5 votes
0 answers
135 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 ...
David White's user avatar
  • 28.7k
5 votes
0 answers
311 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 ...
user50460's user avatar
5 votes
0 answers
184 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 \...
H. Arponen's user avatar
5 votes
0 answers
274 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^{\...
Codiloo's user avatar
  • 81
5 votes
0 answers
153 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 ...
Alekk's user avatar
  • 2,133
5 votes
0 answers
523 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 $d_i,i=1,\ldots,L$....
Ye Li's user avatar
  • 51
5 votes
0 answers
204 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 ...
Anna Varvak's user avatar
5 votes
0 answers
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 ...
Anthony Leverrier's user avatar
5 votes
0 answers
494 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 ...
Aryeh Kontorovich's user avatar
5 votes
1 answer
360 views

Inverse marginal property of a collection of $\sigma$-algebras

In my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space" I introduced the Inverse marginal property (IMP) for a collection of $\sigma$-algebras. Let $(\...
Ivan Feshchenko's user avatar
4 votes
0 answers
118 views

Projection of log-concave distribution on unit sphere surface

Let $\mathbf X : \Omega \to \mathbb R^d$ be a random vector following a zero mean, identity covariance log-concave distribution. Is there any known upper bound for the probability density function of $...
entechnic's user avatar
  • 141
4 votes
0 answers
298 views

When is $\prod_{i=0}^\infty (I-x_i x_i^T)=0$ for zero-centered Gaussian $x_i$?

Suppose $x_i\in \mathbb{R}^d$ is sampled IID from $\mathcal{N}(0,H)$. Let $A_i=(I-x_i x_i^T)$ and assume $d$ is large. What are necessary conditions for the following to converge with probability 1? $...
Yaroslav Bulatov's user avatar
4 votes
0 answers
46 views

Quantifying error in the reconstruction of convex polytopes from moments

The problem of reconstructing a geometric object from its moments is of interest in a variety of fields. In the paper The Inverse Moment Problem for Convex Polytopes, the authors show that a convex ...
Lucas Blakeslee's user avatar
4 votes
1 answer
239 views

Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?

This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian $$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$ are ...
Learning math's user avatar
4 votes
0 answers
154 views

Convergence rates for kernel empirical risk minimization, i.e empirical risk minimization (ERM) with kernel density estimation (KDE)

Let $\Theta$ be an open subset of some $\mathbb R^m$ and let $P$ be a probability distribution on $\mathbb R^d$ with density $f$ in a Sobolev space $W_p^s(\mathbb R^d)$, i.e all derivatives of $f$ ...
dohmatob's user avatar
  • 6,706
4 votes
0 answers
116 views

Exponential families closed under affine transformations

Let $(\Omega,\Sigma,\mu)$ be a probability space and let $\mathcal{M}$ be an exponential family of probability distributions for $\mu$ of the following form: There are $\varphi_1,\dots,\varphi_n:\...
ABIM's user avatar
  • 4,989
4 votes
1 answer
302 views

Asymptotic limit of trace of random matrix $(aI_m + WW^\top)^{-1}$, where $W$ has iid rows from $N(0,\Sigma)$

Let $m$ and $d$ be positive integers with $m,d \to \infty$ such that $m/d \to \rho \in (0,\infty)$. Let $W$ be a random $m \times d$ matrix with iid rows $w_1,\ldots,w_m \sim N(0,\Sigma)$ for a ...
dohmatob's user avatar
  • 6,706
4 votes
0 answers
74 views

Marginalization of Wishart distribution

Consider the following Wishart distribution $$ f({\bf W}) = \frac{ |{\bf W}|^{(n-p-1)/2} \exp\big[-\frac{1}{2}\text{tr}({\bf V}^{-1}{\bf W} ) \big] }{2^{np/2} |{\bf V}| \Gamma_p(\frac{n}{2})} \tag{1} $...
RenatoRenatoRenato's user avatar
4 votes
1 answer
446 views

Gaussian process kernel parameter tuning

I am reading on gaussian processes and there are multiple resources that say how the parameters of the prior (kernel, mean) can be fitted based on data,specifically by choosing those that maximize the ...
john's user avatar
  • 141
4 votes
0 answers
235 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 ...
gogurt's user avatar
  • 91
4 votes
0 answers
143 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 ...
Kostya_I's user avatar
  • 8,632
4 votes
0 answers
1k views

Hoeffding's inequality for random vectors

Let $x_1, \ldots, x_n$ be $n$ i.i.d. samples of a bounded random variable $X \in [a, b]$. We know from the Hoeffding's inequality that : $$\mathbb{P} \left( \left| \frac{1}{n} \sum_{i=1}^n x_n - \...
mfrt's user avatar
  • 113
4 votes
0 answers
139 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 ...
Saptarshi Soham Mohanta's user avatar

1
2 3 4 5
11