# 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.

1,517
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### A result of the covering number

Suppose $\mathcal{F} = \{f_x : x \in \mathbb{R}^d \}$ and each $f_x$ shares the same law $P$. If $\mathcal{F}$ is a class of uniformly bounded functions satisfying $L_r$-continuity, i.e. $\forall f \...

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20 views

### rank-k PCA, relation between subspace angle and variance

I have asked a similar question before with the special case $k=1$, here is the link PCA, relation between the error and variance.
As is known, the rank-k PCA aims to solve the following optimization ...

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39 views

### PCA, relation between the error and variance

As is known, the rank-1 PCA aims to solve the following optimization problem
$$\min_{x\in\mathbb{R}^d}\quad -x^T \Sigma x\quad\quad\quad \text{s.t.}\quad \Vert x\Vert_{2}=1,$$
where $\Sigma\in\mathbb{...

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49 views

### Local limit theorems for circular/spherical distributions

Here are some of the classical density functions for spherical distributions (on the $\mathcal{S}^{d-1}$ sphere, living in the Euclidean space $\mathbb{R}^d$):
$$\mathbf{x}\mapsto \frac{(\kappa/2)^{d/...

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22 views

### How to calculate the R Squared given limited information? [migrated]

I am attempting to solve the following problem from my Statistics class:
WALC=β_0 + β_1 TOTEXP + β_2 AGE + β_3 EDUC + β_4 SIZE + e. Suppose RSS=2032.73 and SD(WALC)=12.4537. N = 45. What is R Squared?...

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109 views

### Kullback–Leibler chains

The following question was asked and then deleted by the post author:
Let $P$ and $Q$ be two probability distributions defined over the same space, with $KL(P \parallel Q) < \infty$. For $\epsilon ...

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61 views

### Average the covariance matrix over all orthogonal matrices

Let $M=O\Lambda O^\top$ be a positive semi-definite matrix, where $\Lambda\in \mathbb{R}^{p\times p}$ is a diagonal matrix with non-negative entries and $O\in \mathbb{R}^{p\times p}$ is an orthogonal ...

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27 views

### Covariance structure of $k$-statistics

How can I express the covariance structure of k-statistics, that is $Cov(k_i,k_j)$ for all $i,j$, including the variances when $i=j$ ?
Furthermore, How can I estimate it without biais ?
On this page : ...

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124 views

### Practical pseudorandom generators

It is known that existence of pseudorandom generators (PRGs) is equivalent to the existence of one-way functions. In turn, the latter is an open problem.
I am curious if someone developed kind of &...

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44 views

### Student's t-test for lognormally distributed samples

I want to understand whether the differences in the means of two independent samples (distributed lognormally) are statistically significant. In order to use Student's t-test, the data must be ...

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140 views

### Variance-based localized Rademacher complexity for RKHS unit-ball

Let $\mathscr X$ be a compact subset of $\mathbb R^d$ (e.g the unit-sphere). Let $K: \mathscr X \times \mathscr X \to \mathbb R$ be a positive kernel function and let $\mathscr H_K$ be the induced ...

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32 views

### Whether to use binomial or Poisson distribution, when events are not able to happen continuously, but an average rate is known [migrated]

Let us say there are on average 15 snowy days in December.
Let $X$ be the number of snowy days in a given December.
Is the binomial distribution, with $X \sim B (31, \frac{15}{31})$, or the Poisson ...

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32 views

### Tail probability for the sum of laplace RIVs

Suppose we have $n$ RIVs $X_1,X_2,..,X_n$ where $\forall i. X_i \sim Lap(\frac{1}{\epsilon})$.
For convenience, we denote $Y=\sum_{i=1}^{n}X_i$.
Also, it is known that -
$$
Pr[Lap(\frac{1}{\epsilon}) \...

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73 views

### Binary Regression : Is this an open problem in Mathematics/Statistics?

Let $X$ be a random variable which takes values from $\Omega = (0,1)^m$ with a probability distribution $p(x)$. Assume $p$ is a BV function with non zero total variation and $p(x)>0\forall x\in\...

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122 views

### Almost independence of $x^\top a$ and $x^\top b$ for $x$ uniform on the sphere in $\mathbb R^d$ and $a,b \in \mathbb R^d$ with $a^\top b = 0$

Let $d$ be a large positive integer. Let $x$ be uniformly distributed on the unit-sphere in $\mathbb R^d$ and let $a$ and $b$ be perpendicular vectors in $\mathbb R^d$, i.e such that $a^\top b=0$. Let ...

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85 views

### Concentration inequality for the sample covariance matrix

I'd like to know if there is a concentration inequality for the sample covariance matrix that don't assume the knowledge of the true mean.
Background.
Given a probability distribution $\mu$ on $\...

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**1**answer

94 views

### Convergence of quadratic form $y^T Q y$ where $y$ is a random iid sequence of length $n$ and $Q$ is an $n \times n$ random matrix independent of $y$

For each positive integer, let $Q_n=(q_{i,j})_{i,j \in [n]}$ be a random $n \times n$ psd matrix. In the limit $n \to \infty$, suppose the eigenvalues of this sequence of matrices are uniformly ...

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67 views

### Probability to cross an envelopp for 1D random walk?

Imagine we have an evolving sequence composed of 1 and -1 (ex: -1-11-111...) where the probability to get -1 or 1 is 1/2. n is the lengh of my sequence.
I can make an analogy with random walk: let ...

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71 views

### Independence between $X_{n-k:n}$ and $\sum\limits_i Y_{n-i:n}-Y_{n-k:n}$

If $(X_i,Y_i), i=1,\ldots,n,$ is i.i.d sample from the joint distribution $F$ and there is dependence between the two variables say $R$. Denote the order statistics for the two variables $X_{1:n},\...

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37 views

### Percentile interval Lemma

Let $\theta$ be a parameter and $\hat{\theta}$ the plug-in estimate, I need a proof of the following lemma, as given in [1], p. 173, in the form of a reference or of a direct argument:
Percentile ...

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60 views

### KL-divergence and sub-$\sigma$-algebras

I am trying to understand if the following claim is true:
Let $P$, $Q$ be probability measures on $\mathcal{X}$. For any $\sigma$-algebra $\mathcal{G}$, with countably many atoms (sets with $\...

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41 views

### Bootstrap-$t$ confidence intervals

I'm writing a dissertation about bootstrap methods and the main book I'm using is Efron, B., & Tibshirani, R.J. (1994), An Introduction to the Bootstrap (1st ed.), Chapman and Hall/CRC. Now I need ...

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79 views

### Are the first 4 statistical moments independent? [closed]

Are the first 4 statistical moments independent? Is there a mathematical demonstration that can show independence one from each other? Can the value of one moment influence the value of another? If so,...

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21 views

### Empirical covering number bounds of functions in rkhs ball

In this paper https://arxiv.org/pdf/1905.11882.pdf, the authors rely on showing the optimal "potential" functions lie in a set with a bounded Holder norm. They're able to show specific ...

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99 views

### Are there any function spaces with bounded gradients?

Are there any known function spaces where the gradients are uniformly bounded? For a problem I’m working on, I’ve been able to show my functions are bounded in a ball within an RKHS (reproducing ...

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26 views

### Non-parametric estimator of continuous process's law from one sample path?

Motivation: If $X$ is a random-variable defined on some probability space $(\Omega,\Sigma,\mathbb{P})$ then Glivenko-Cantelli lemma guarantees that the empirical distribution $\frac1{N}\sum_{n=1}^N \...

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35 views

### Asymptotic lower and upper bounds for the eigenvalues of hadamard product $W \circ W$, where $W$ is a large Wishart matrix

Let $n$ and $d$ be large positive integers with $n,d \to \infty$ such that $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ random matrix with iid copies of log-concave isotropic ...

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75 views

### Using $\delta$-method to “estimate” undefined moments of a random variable?

I posted this over on MSE without much luck. Not sure if posting here is considered cross-posting but I can remove it if it is.
Let $X\sim\mathcal N(\sqrt 2,1/x^2)$. The expected value $\mathsf EX^{-1}...

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68 views

### sub-exponential type upper bound on the Poisson probability

I posted this question on Math Stack Exchange, though I'm not satisfied with the answer I received.
Question:
For a Poisson random variable $Z$ with the parameter $\lambda,\,$ what would be a good ...

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71 views

### Testing uniformity for continuous probability distributions

Suppose I can sample from a random variable $X$ which is distributed on a compact interval, say, $[0,1]$. Fix a distance measure between distributions, say total variation. Let $\epsilon\in(0,1)$. How ...

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56 views

### On estimating Covariance between a random variable and its non-linear transform

Let $X$ be a random variable taking values on the real line.
Let $R(X) = max\{0, X\}$. Is it true that the covariance $Cov[X, R(X)] \ge 0$ irrespective of the distribution of $X$? Many experiments, as ...

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71 views

### Expected value of Tukey’s half-space depth for log-concave measures

Let ${\mathbb P}$ be a probability measure in ${\mathbb R}^n$. Let $x\in{\mathbb R}^n$ be an arbitrary point. Let ${\mathbb H}_x$ be the set of halfspaces of ${\mathbb R}^n$ containing $x$. Let
\begin{...

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128 views

### reverse KL-divergence: Bregman or not?

I am having a little trouble getting my head around the two "directions" of the Kullback-Leibler divergence:
Definition (Kullback-Leibler divergence) For discrete probability distributions $...

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54 views

### Column subset selection with least angle optimization

Given a matrix $A$ I need to select a "representative" subset of its columns so that the each non-selected column is as close as possible to a selected one.
Formally:
Given $A \in \mathbb{R}...

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27 views

### Get covariance from log-density function

Problem
Given a following log-density function
$$ \ln p(y| a, b) = a \cdot g(y) + b \cdot h(y) + k(a,b)$$
where $g(y), h(y), k(a,b)$ are difined function and $a,b$ are parameters.
Find $\Bbb Cov( g(Y),...

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76 views

### Upper bound on the ratio of Poisson CDFs [closed]

Suppose $X \sim Pois(\lambda)$. I'm interested in an upper bound on the ratio, $$\dfrac{P(X \leq n)}{P(X \leq n-1)}\,,\,\,\text{for $n=1,2,3,...$}$$ Observe that, the ratio is $>1$ & as $n \to \...

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128 views

### Expectation of product of random matrices

Let $X$ and $Y$ be independent random symmetric matrices. What can one say about $\mathbb{E} [X Y X Y]$ or $\mathrm{trace} \mathbb{E} [X Y X Y]$ in terms of properties of $X$ and $Y$?
In particular, ...

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184 views

### High-probability lower bound for norm of least squares solution when both design matrix $X$ and response vector $y$ are random (and independent)

Let $n,d \to \infty$ with $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ matrix independent rows uniformly distributed on the the unit-sphere in $\mathbb R^d$ and let $y$ be a ...

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102 views

### A basic property of maximal correlation

Let $𝑋$ and $𝑌$ be random variables. Then the maximal correlation $\rho_{m}(X;Y)$ is defined as:
$$\rho_{m}(X;Y):=\max_{f,g}\mathbb{E}[f(X)g(Y)],$$
where the maximization is taken over real-valued ...

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55 views

### Normalizing constants preserve metric entropy

Suppose $\mathcal{F}=\left\{f\in L^2([a,b]): 0<\underline{c}\leq f\leq\overline{c} \right\}$. Consider the following transformation
$$\tilde{\mathcal{F}} := \left\{\frac{f}{\int f d\mu}: f\in \...

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23 views

### Techniques to bound a regularized loss involving a maximum?

I am studying the following optimization problem
$DROT(a,b) = \max \int f(x)\,dP(x) + \int g(y)\,dQ(y) - \frac{1}{\gamma}(\phi(f)+\varphi(g))$ subject to $f(x)+g(y)\leq c(x,y)$ where $\phi$ and $\...

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73 views

### Intuition behind the noncentral chi square as Poisson mixing

It is known (cf. Wikipedia, Noncentral_chi_distribution) that the non-central chi-square distribution with k degrees of freedom is a Poisson weighted mixture of central chi-squared distributions).
...

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112 views

### Bounds for the extreme singular-values of random matrix with thresholded entries

Let $n,d,k$ be large positive integers such that $\max(n/d,k/d) =: \lambda < 1$. Let $X$ be a random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $W$ be a $k \times d$ random ...

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26 views

### Maximum entropy distribution in the hyperbolic plane with given “mean” and “variance”

On the Euclidean sphere, by defining suitable analogs of the mean and variance in terms of the first circular moment, it can be shown that the von Mises distribution is the maximum entropy ...

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23 views

### sampling the distribution W, the mapping of k ordered observations to N distributions

we have a set of N ordered distributions.
also, we have K<N samples from those distributions, without replacement, ordered according to the original set (we can imagine someone samples a ...

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76 views

### What probability distribution is this?

Thank you in advance for any suggestions or feedback.
I have a discrete 1D probability distribution represented as a vector $\textbf{p}$, $p_i = p(x_i)$.
I am interested in finding the Wasserstein (...

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193 views

### in Euclidean space defined by multivariate normal distribution, what fraction of points falls within n-ball (centered at origin) tangent to point p?

In a Euclidean space defined by the multivariate normal distribution, what fraction of all points falls within or are tangent to (as opposed to falling outside of) the n-sphere whose center is at the ...

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93 views

### Derivative of the function of random variable

Suppose we have a function $\phi(X)$ of random variable $X$. Suppose both of $\phi(X)$ and $X$ are random variables. If $\phi$ is differentiable, how to calculate the derivative of $\phi(X)$ w.r.t. $...

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54 views

### Bayes risk of binary classification problem with conditionally independet covariates

In the setting of this problem, $\eta(\vec{x})$ is $P(Y=1|\vec{X}=\vec{x})$, $Y \in {0,1}$, $X \in R^d$. Being the true probability know, the classification rule is simply $\eta(\vec{x})>0.5 \...

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112 views

### What is the ideal form of an h-curve?

This question concerns mathematical modelling of the citation curve, well-known in the sciencemetry.
The citation curve (or else the $h$-curve) of an individual researcher is the vector $(c_1,c_2,\...