# 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**

votes

**1**answer

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

**0**answers

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

**3**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**4**answers

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

**0**answers

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

**1**answer

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**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**3**answers

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

**3**answers

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

**2**answers

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**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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**

vote

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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,...

**0**

votes

**0**answers

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

**1**answer

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

**0**answers

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

**3**answers

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 ...