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
22
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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 ...
19
votes
0
answers
3k
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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 ...
11
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0
answers
222
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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 ...
11
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0
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524
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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
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300
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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
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0
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389
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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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0
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1k
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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
votes
2
answers
408
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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 $\...
8
votes
0
answers
3k
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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 ...
7
votes
0
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239
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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
0
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178
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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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0
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616
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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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0
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123
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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 ...
6
votes
0
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243
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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 $\Sigma$:
$$...
6
votes
0
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174
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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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0
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384
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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$ ...
6
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0
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523
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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}$ ...
6
votes
0
answers
568
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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
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=...
6
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0
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186
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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
votes
0
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255
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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 ...
5
votes
0
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130
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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 ...
5
votes
0
answers
187
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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(...
5
votes
0
answers
143
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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 ...
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 ...
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 ...
5
votes
0
answers
469
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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
votes
0
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135
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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
votes
0
answers
311
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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
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 \...
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^{\...
5
votes
0
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153
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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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0
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523
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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
votes
0
answers
204
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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 ...
5
votes
0
answers
1k
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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
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 ...
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 $(\...
4
votes
0
answers
118
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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 $...
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?
$...
4
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0
answers
46
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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 ...
4
votes
1
answer
239
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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 ...
4
votes
0
answers
154
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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 $f$ ...
4
votes
0
answers
116
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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:\...
4
votes
1
answer
302
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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 ...
4
votes
0
answers
74
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Marginalization of Wishart distribution
Consider the following Wishart distribution
$$
f({\bf W}) = \frac{ |{\bf W}|^{(n-p-1)/2} \exp\big[-\frac{1}{2}\text{tr}({\bf V}^{-1}{\bf W} ) \big] }{2^{np/2} |{\bf V}| \Gamma_p(\frac{n}{2})} \tag{1}
$...
4
votes
1
answer
446
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Gaussian process kernel parameter tuning
I am reading on gaussian processes and there are multiple resources that say how the parameters of the prior (kernel, mean) can be fitted based on data,specifically by choosing those that maximize the ...
4
votes
0
answers
235
views
What happens in the martingale CLT if I norm by the conditional variance instead?
TLDR: I'm a statistician (bear with me!) trying to use the martingale CLT but I only can estimate the conditional variance instead of the unconditional one. Can I do anything to get a CLT with norming ...
4
votes
0
answers
143
views
Geometric meaning of the chi-square "measure of association"
In Statistics, there's a standard test of independence of two random variables taking values in finite sets $I,J$. It relies on the computation of $\chi$-square statistics,
$$
\chi^2:=\sum_{(i,j)\in ...
4
votes
0
answers
1k
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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 :
$$\mathbb{P} \left( \left| \frac{1}{n} \sum_{i=1}^n x_n - \...
4
votes
0
answers
139
views
Probability of having complete conversion in a box of three different object with interaction rules
Say there a 3 types of Objects A,B,C which randomly interact in pairs to form new objects following the below rules:
$$A + B = AB$$
$$B + C = BC$$
$$C + A = CA$$
$$AB + C = ABC$$
$$BC + A = ABC$$
$$CA ...