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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.

4
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1answer
116 views

an application of nth moment of Poisson distribution with stirling number

I was reading the paper on arixv. I was confused the equation of nth moment of Poisson distribution. The detail and partial paper as follow: ... For large N, this connection probability takes ...
2
votes
0answers
40 views

How sensitive is ML-estimation of the expected value if the covariance matrix is not correct?

Suppose, we have a random variable $Y \sim \mathrm{N}\left( Ax, \, \Sigma \right)$ and realisations $y$. I would like to estimate $x$, the parameter of the expected value. The loglikelihood function ...
5
votes
3answers
144 views

Removing outliers from circular average data

I'm trying to find the average from a set of circular data and am using the following which is doing what I'm expecting. $$a = \arctan\left(\frac{\sum\limits_{i=1}^N \sin(a_i)}{\sum\limits_{i=1}^N\...
3
votes
1answer
88 views

integral involving hypergeometric function of matrix argument

This conjecture comes from an observation on simulations of the matrix variate noncentral Beta distribution (similar to this observation, but I open a new question because yet I'm not sure it is ...
1
vote
1answer
44 views

Noncentral matrix beta distributions of type I and II

In Gupta & Nagar's book Matrix variate distributions, the noncentral Beta type I(B) distribution with parameters $a$, $b$ and noncentrality parameter $\Theta$ is defined by $U={(S_1+S_2)}^{-\...
1
vote
1answer
301 views

Is there any “fundamental” distinction between min-plus, max-plus, min-product, and max-product algebras?

In the paper Faster Algorithms for Max-Product Message Passing by McAuley and Caetano (see e.g. here or here), several statements are made which seem mathematically questionable to me. For ...
4
votes
1answer
62 views

How to simulate the fractional noncentral Wishart distribution?

I already asked this question on math.stackexchange but got no answer. For a non-integer number of degrees of freedom $\nu > p-1$, one can simulate the central Wishart distribution $W_p(\nu, \...
3
votes
2answers
223 views

Concentration inequality for sum of iid random variables that involve KL distance

Conider $X \in \mathbb{R}^d$ and $Y \in \{0,1\}$, and a joint distribution $p_{XY}(x,y)$, and a set of $N$ i.i.d. samples $\{(X_i,Y_i)\}_{i=1}^{N}$. Define $p_{X0} = p_{XY}(x,0)$ and $p_{X1} = p_{XY}(...
12
votes
4answers
695 views

Good introduction to statistics from a algebraic point of view?

There are already lots of questions on this subject like Is there an introduction to probability theory from a structuralist/categorical perspective? Is there a combinatorial/topological treatment ...
1
vote
0answers
34 views

RMHMC sampling in non-parametric setup

The aim is to sample distributions using Fisher information (as mass matrix in Hamiltonian MCMC sampling). Details can be found in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.190.580&...
3
votes
1answer
197 views

Wasserstein convergence of conditional measures

Suppose $W_r(\mu_n,\mu)\to0$, where $\mu_n$ and $\mu$ are discrete probability measures on some metric space $\Omega$, and that all measures have the same number of atoms $d$ (but not the same atoms): ...
0
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0answers
48 views

Change of covariance under Lipschitz maps

Consider a finite point set $X$ in $\mathbb{R}^n$ and a $1$-Lipschitz map $\,f:\mathbb{R}^n \to \mathbb{R}^m$. Is it true that the maximum eigenvalue of the centered covariance matrix of $f(X)$ is ...
1
vote
0answers
28 views

Matrix variate t-distribution and product of Beta distributions

This is a reference request for the following result. Let $X$ be a random matrix following the matrix variate $t$-distribution $T_{p,m}(\nu, M, U, V)$ (as defined in Wikipedia). Then $$ \frac{\det(U)}{...
1
vote
0answers
53 views

Central Limit Like theorem for the distribution of F-statistics on all possible partitions?

I'd be happy for simply a reference or even search terms as I feel like this has to be known*. Suppose we have a known probability distribution $X$ and a fixed integer $n$. I am interested in the ...
8
votes
2answers
771 views

Lower bounds on Kullback-Leibler divergence

This was originally a question on Cross Validated. Are there any (nontrivial) lower bounds on the Kullback-Leibler divergence $KL(f\Vert g)$ between two measures / densities? Informally, I am ...
2
votes
2answers
98 views

Extremal Covariance Duality

Given real vectors $v$ and $r$ of the same size, what are the following? $\inf\{v'R^{-1}v ~ \colon ~ R>0 \, , \, \text{diag}(R)= r\}$ $\sup\{v'Rv ~ \colon ~ R>0\, , \, \text{diag}(R)= r\}$ ...
3
votes
1answer
58 views

Maximizing the $\alpha$-moment of a distributution

Given $\alpha$ and constant $\mu$, $$\begin{array}{ll} \text{maximize} & \displaystyle\int_0^\infty p(x)x^\alpha \,\mathrm d x\\ \text{subject to} & \displaystyle\int_0^\infty p(x)\,\mathrm d ...
2
votes
0answers
68 views

Stochastic Approximation Algorithms Converging to Local Equilibriums

Consider the stochastic iterative updates \begin{align} \theta_{t+1} \leftarrow \theta_t + \alpha_t \cdot \left [ h(\theta_t) + M_t \right ], \end{align} where $\theta_t \in \mathrm{R}^d$, $h \colon ...
7
votes
0answers
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 ...
6
votes
3answers
241 views

Bound on probabilities of the sum of uniform order statistics

Let $X_1,...,X_n$ be i.i.d. random variable with a uniform distribution on [0,1]. Denote by $X_{(1)}\leq X_{(2)} \leq \ldots \leq X_{(n)}$ their order statistics. Given $k\geq 1$ and $u\in[0,k]$, I ...
3
votes
3answers
296 views

Automatic vs numerical differentiation of a function known from samples

Suppose I have $n$ samples $(x_i, f(x_i))_{i=1}^n$ from an unknown function $f$. I need to approximate (estimate) the derivative $f'(x^*)$ at some new test point $x^*$, that is not necessarily one of ...
2
votes
2answers
273 views

An alternative proof of Bayesian Cramer-Rao

My question is: Are there an alternative proof of Cramer-Rao lower bound that does not use Cauchy-Swartz inequality? Let me outline the classical proof and explain why I am interested in this ...
0
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0answers
22 views

Can an exponential family MRF lose its exponentiality when adding CPDT constraints?

I am reading the book "Graphical Models, Exponential Families, and Variational Inference" by Wainwright and Jordan. In there, they show that any Markov Random Field (MRF) with exclusively finite ...
2
votes
0answers
35 views

increasing inter-class distances results in decreasing linear regression error

Let $\{\mathbf{x}_i, y_i \}$ be a set of binary-labeled samples ($\mathbf{x}_i \in \mathbb{R}^d, y_i \in \{a,b\}, a,b\in\mathbb{R}$). Let $\{ \mathbf{x}'_i, y_i \}$ be also such a set. Define $\mathbf{...
1
vote
0answers
95 views

Anti-concentration bounds for folded normal and inverse of gaussian variables

Are there any easy to use bounds on sums of the following kind : $$ \sum_{i = 1}^{i = N} |a_i| \geq P \\ a_i \sim \mathcal{N}(0, 1) \\ $$ and also for sums of the form : $$ \sum_{i = 1}^{i = M} \...
2
votes
0answers
45 views

Rate of $L_1$ loss in estmating density on $[0,1]$

Let $f$ be a density on $[0,1]$ and let $X_1,X_2,\ldots$ be $\textit{iid}$ $f$-distributed. Also, let $f_n$ denote the kernel density estimator, i.e. $$f_n(x) = \frac{1}{nh_n} \sum_{i=1}^n K\left(\...
5
votes
1answer
145 views

Expectation of max of Gaussian multiplied by a functional of Gaussian

Let $X \in \mathbb{R}^{d}$ follows the standard Gaussian distribution $N(0, I_d)$. Let $Y = \max_{j\in[d] } X_j$. It is not hard to see that \begin{align} \mathbb{E}\left [ Y \cdot X\right] = \sum_{j=...
1
vote
1answer
151 views

On concentration of a sum random variable

Take a random variable defined as $$r=u_{11}v_{1}v_{1}+u_{12}v_{1}v_{2}+\dots+u_{n,n-1}v_{n}v_{n-1}+u_{nn}v_{n}v_{n}$$ where $v_{i}$ are independent uniform random variables from $\{0,\dots,b\}$, $u_{...
3
votes
1answer
89 views

A $t$-test for ordered pairs

Suppose I have random variables $$ W_i = \begin{cases} w_1 &\text{with prob. } p_1, \\ w_2 &\text{with prob. } p_2, \\ w &\text{with prob. } 1-p_1-p_2,\end{cases} \qquad i = 1, \dots, 2n+1....
6
votes
1answer
188 views

How close $k$-sums of a random set of numbers are on average?

Consider a set of random iid variables $x_1, \ldots x_n$ uniformly distributed on $[0, 1]$. For each $S \subset [n]$ with $1 \leq |S| = k < n$ take $\sigma_S = \sum_{i \in S}x_i$. Obviously $\...
1
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0answers
34 views

Equivalent of linearity for the Fréchet mean under translations

Following Pennec (2006), the Fréchet mean for a set of draws $\{x_i\}_{i=1,\dots,n}$ from a metric space $M$ with distance function $d$ is defined as the minimiser of the variance of the draws, $$ E[\...
1
vote
1answer
93 views

Fisher information with vanishing probability

I am confused about the definition of the Fisher information and the case when probability is 0. Consider discrete set $\epsilon$ of possible measurement outcomes. Fisher information is defined as: $$...
6
votes
1answer
107 views

Existence of distribution for certain order statistics

This is an open question: given a sequence of $n$ real numbers $x_1<x_2<\dots<x_n$, does there always exist a probability distribution, such that $\{x_i\}$ happens to be the $n$ expected ...
1
vote
1answer
70 views

Is there a name for the sample variance process of a Lévy process?

Let $X_t$ be a Lévy process which is known to have mean zero and finite variance $t \cdot \sigma^2$, but for which the value of $\sigma^2$ is unknown. How do we estimate $\sigma^2$? One approach would ...
0
votes
1answer
54 views

Looking for a specific kind of a compactly supported one dimensional distribution

I am looking for a sequence of probability distributions (parameterized by $h \in \{1,2,3,4,..\}$) supported on the compact interval $x \sim [a(h),b(h)]$ such that, $a(h) > \frac{b(h)}{h^{\nu^2}} ...
3
votes
2answers
135 views

Breaking the rotate-then-substitute alphabetic cipher

My question is not typical for MathOverflow, and arises in my teaching rather than research, but I think there will be readers who can give interesting answers. Identify $\{\mathrm{A}, \ldots, \...
2
votes
0answers
28 views

Orthogonal polynomial expansion for bivariate noncentral chi-square and bi-variate noncentral student t distribution

This is a research question for which I am not able to find any existing reference. So, I am reaching out for help. The project is related to studying the sequence of rejections in multiple hypothesis ...
0
votes
0answers
93 views

log-like distance between probability distributions

Given two probability density functions (PDF) $f$ and $g$, both defined over the same set $X$, there are many ways to describe/measure the distance between them, e.g., KL divergence and Hellinger ...
1
vote
1answer
53 views

A problem with elementary inequality involving probabilities and Brier scoring rule

I am trying to prove certain relations between certain values of the so called Brier inaccuracy measure (Brier scoring rule). Given a vector $p = (p_1, \ldots p_n)$, where $p_1 + \ldots p_n = 1$ and $...
2
votes
0answers
143 views

Rank of Hadamard product with random matrices

I do research in statistics and am not sure whether the following is considered research level or not in mathematics. If it isn't, I'm happy because that means the answer is probably known and I can ...
3
votes
1answer
798 views

Upper bound total variation by Wasserstein distance for continuous distance

I am reading the survey of the relationships between metrics of distributions (see https://arxiv.org/pdf/math/0209021.pdf for the paper). The general results show that for general distributions, we ...
3
votes
1answer
106 views

Multivariate normal concentration

If $X\sim N(0,\Sigma)$ for some $d$-dimensional normal distribution, then $X = \Sigma^{1/2} Z$ where $Z\sim (0,I)$. How to compute the following quantity? $$ \operatorname{var} (X^T X) = \...
2
votes
1answer
452 views

Confidence intervals for the endpoints of the uniform distribution

Consider $n$ iid observations $X_1,X_2,\dots ,X_n$ from a $Uniform(a,b)$ distribution, where $a$ and $b$ are both unknown. How do we construct a joint confidence interval for $(a,b)$? I would prefer ...
2
votes
1answer
103 views

Expand the pdf of Wishart distribution into power series via orthogonal polynomials

In the univariate case ($\chi^2$ distribution), I know we can expand the pdf into power series of the variance $\sigma^2$ with Laguerre polynomials. Indeed, since the Laguerre polynomials are related ...
6
votes
1answer
603 views

Brownian motion and its maximum and its minimum

Let $W_u, 0\leq u \leq t$ be Brownian motion. Let $m_t= min_{0\leq u\leq t} W_u$ and $M_t = max_{0 \leq u \leq t} W_u$. The fact that $(M_t , W_t)$ is absolutely continuous with respect to Lebesgue ...
1
vote
1answer
70 views

Conformal prediction for the case of single tailed events

I'll start with a motivating example and only then proceed to the question. Consider a list of total packages of milk that were purchased on 9 consecutive days on a given store, $z_1,\ldots,z_9 = 1,...
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0answers
92 views

concentration inequalities on RKHS?

Let $\mathcal{H}$ be an RKHS and $\phi$ is defined such that $\langle \phi(x), l\rangle_{\mathcal{H}} = l(h(x),h'(x))$, where $h$ and $h'$ are two functions on a function space $\mathcal{F}$. I'm ...
2
votes
1answer
107 views

Square integrable conditional expectations as projections

I see this page Ordinary least square and random projection, and I am thinking that how $L^2$ integrable random variables be regarded as projections over a defined filtration sequence $\mathcal{F_n}$ ?...
3
votes
0answers
169 views

Second-Order Taylor Expansion of Wasserstein Metric and Related Metrics

Suppose that we have a parametric distribution $P_{\theta}$, which is indexed by the parameter $\theta \in \mathbb{R}^d$. Let $W\{\cdot,\cdot\}$ be the Wasserstein Metric between two distributions. ...
5
votes
3answers
102 views

Looking for a certain kind of a distribution

Is there any probability distribution supported on a compact or a half-open interval (of $\mathbb{R}$) such that if a vector $\vec{x} \in \mathbb{R}^n$ is sampled by sampling its coordinates like that ...