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.

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Matrix-valued cumulant generating function for Wishart matrices

Suppose we have an axis-aligned Gaussian vector $v \sim \mathcal{N}(\mu, \sigma^2 I_{d \times d})$, and consider the Wishart matrix $W = vv^\top$. Is there a simple closed form/"Lowener order ...
user113925's user avatar
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1 answer
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The distribution of number of reverse order pairs in a randomly permuted array

There is an array $a_1,\dotsc,a_n$ whose elements are pairwise distinct. We define a reverse order pair to be an ordered pair $(a_i,a_j)$ such that $i < j$ and $a_i > a_j$. Consider the total ...
Ruiyuan Huang's user avatar
2 votes
1 answer
195 views

Measurability of maximum likelihood estimator under conditions from Lehmann's "Theory of point estimation"

I'm trying to prove that MLE from the proof of one theorem in Lehmann's "Theory of point estimation" (the theorem is below) is a measurable function. I know that under some regularity ...
Botnakov N.'s user avatar
1 vote
1 answer
48 views

tail probability of max of Gaussians

I'm trying to follow an argument in C. Giraud's "High Dimensional Statistics" (2nd Ed, p. 11 / $\S$ 1.2.3). The specific page is accessible via Google Books here but the formatting is awful....
AsBrB's user avatar
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Canonical representation of the a probability distribution for Hammersley Clifford Theorem

I'm reading the following paper http://www2.stat.duke.edu/~scs/Courses/Stat376/Papers/GibbsFieldEst/BesagJRSSB1974.pdf On page 7 they give the result that $$Q(\textbf{x}) = \sum_{1 \leq i \leq n} ...
Pavan Sangha's user avatar
1 vote
0 answers
263 views

Tail bounds for random Gaussian chaos?

Let $g = (g_1, \dots, g_d)$ be a sequence of independent standard Normal random variables, and suppose $\Sigma$ is a $d \times d$ (deterministic), real, symmetric, positive definite matrix. The Hanson-...
Drew Brady's user avatar
2 votes
2 answers
265 views

Integral of product of Hermite polynomials w.r.t marginal distribution of first two-coordinate of random vector on unit-sphere

This question is related to: https://math.stackexchange.com/q/4270522/168758 Let $H_n(x) \in \mathbb R[x]$ be the probabilist's $n$th Hermite polynomial. This an $n$th degree polynomial given by the ...
dohmatob's user avatar
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1 answer
367 views

Approximate the singular values of a certain random dot-product kernel matrix (in the sense of El Karoui, Cheng-Singer, etc.)

Let $g:\mathbb R \to \mathbb R $ be a continuous function which is "sufficiently smooth" (e.g $\mathcal C^3$) around $0$, and "sufficiently integrable" (e.g integrable w.r.t $N(0,...
dohmatob's user avatar
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1 vote
1 answer
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Occupation times for two-state Markov processes

Consider a two-state Markov process in continuous time, with states labelled $A$ and $B$. The transition rates for going from state $A$ to $B$, and state $B$ to $A$ are $\alpha$ and $\beta$ ...
StatisticalMechanic's user avatar
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Books on limiting properties of matrices with growing size

This question has been posted on Math-Se previously. I am studying asymptotic properties of the Projection Matrix $$ H_n=X'(X'X)^{-1}X $$ By the Gerschgorin disc theorem, the bounds on the ...
chuck's user avatar
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1 answer
351 views

Well-definedness of maximum likelihood estimation

Consider a family $\{\mu_\theta:\theta\in\Theta\}$ of probability measures on a measurable space $X$. Given $x\in X$, the maximum likelihood estimate is the value of $\theta$ which maximizes the ...
Quarto Bendir's user avatar
2 votes
1 answer
85 views

Asymptotics of $w^\top G^2 w$, where $w$ is a unit-vector, $G:=X^T(XX^T+t I_n)^{-1}X$, $t > 0$, and $X$ is an $n\times d$ gaussian random matrix

Let $X$ be an random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $w$ be a unit-vector in $\mathbb R^d$. With $\lambda>0$, and define $G:=X^\top(XX^\top + \lambda I_n)^{-1}X$. ...
dohmatob's user avatar
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Jeffreys' priors as coefficients of a linear estimator

I asked the following question in a forum more suitable for statistics, but I didn't get any answer; I hope, someone could shed light on my question: I have three random variables, $X_1$, $X_2$, and $...
user avatar
1 vote
1 answer
195 views

A problem related to stochastic ordering

Let $\boldsymbol{X} = (X_1,X_2)^{\rm T}\sim \mathcal{N}_2(\boldsymbol{\mu}, \mathrm{\Sigma})$, where \begin{eqnarray*} \boldsymbol{\mu} = (\mu_1, \mu_2)^{\rm T}& = &(\sqrt{\xi_1\xi_2/(\xi_1+\...
Satya Prakash's user avatar
2 votes
1 answer
112 views

A question on the applicability Chebyshev inequality for sequence of random quantities

Let $(X_n)_n$ and $(Y_n)_n$ be two mutually independent sequences of random tensors (i.e scalars, vectors, matrices, etc.) defined on the same probability space, and let $f$ be a measurable function. ...
dohmatob's user avatar
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2 votes
1 answer
109 views

Lower bound on likelihood of binary outcomes

I am wondering about the following: does there exist a stochastic process $(X_n)_{n \ge 1}$ with values in $\{0,1\}$ on a probability space $(\Omega, \mathcal F, \mathbb P)$ such that for all $n \ge 1$...
Tartrate's user avatar
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1 vote
1 answer
120 views

Stochastic ordering of absolute multivariate normal random variables

Let $X\sim\mathcal{N}(\boldsymbol{\mu}_1,\mathrm{\Sigma}_1)$ and $Y\sim\mathcal{N}(\boldsymbol{\mu}_2,\mathrm{\Sigma}_2)$. Then it is know that $\mathbb{P}(X>\boldsymbol{t})\leq\mathbb{P}(Y>\...
Satya Prakash's user avatar
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0 answers
49 views

Equal likelihoods for any two samples with equal standard deviations (normal distribution)

I have a conjecture that for all $x,y \in \mathbb{R}^n$, each with non-zero but finite sample standard deviations, $$\sum_{i = 1}^n{\log{\varphi(\frac{x_i-\overline{x}}{s_x})}} = \sum_{i = 1}^n{\log{\...
sircolinton's user avatar
3 votes
0 answers
461 views

Eigenvalues of Matérn covariance function

Recall that Matérn covariance function $C_\nu(d)$ is defined as $$ C_\nu(d)=\sigma^2\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{d}{\rho}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{d}{\rho}\right), ...
Zuofeng Shang's user avatar
1 vote
1 answer
173 views

Johnson-Lindenstrauss with Orthogonalization

I have been looking at constructions satisfying the Johnson-Lindenstrauss Lemma (e.g., projections onto random subspaces, random Gaussian matrices, random Rademacher matrices, etc.). It seems that ...
B Merlot's user avatar
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1 answer
280 views

An inequality in the optimality of Bayes' theorem

$\DeclareMathOperator\Ent{Ent}\newcommand{\prior}{\mathrm{prior}}\newcommand\Data{\mathrm{Data}}$I came across this paper on the optimality of Bayes' theorem https://sinews.siam.org/Portals/Sinews2/...
Chp's user avatar
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3 votes
1 answer
267 views

Divergence-free Gaussian vector field with given mean magnitude and correlation function

My general question is how to construct an isotropic random vector field $\vec f: \mathbb{R}^3 \to \mathbb{R}^3$ with a given mean magnitude $\mathbb{E}[\|\vec f(\vec x)\|]=\mu$ and with vector ...
math_lover's user avatar
2 votes
1 answer
291 views

An approximation problem w.r.t marginal distribution of coordinates of uniform random vector on high-dimensional unit-sphere

Let $X=(X_1,\ldots,X_d)$ be uniformly distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. Fix a "sufficiently integrable" function $h:\mathbb R \to \mathbb R$, and define ...
dohmatob's user avatar
  • 6,706
2 votes
1 answer
160 views

Compute the limit of trace of inverse of square of rank-1 perturbation of Wishart matrix

Let $a \ge 0$, $b,c>0$ be fixed constants, and let $X$ be an $m \times d$ random matrix with entries drawn iid from $N(0,1/d)$. Consider the random psd matrix $S := a 1_m 1_m^\top + b XX^\top + c ...
dohmatob's user avatar
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1 vote
0 answers
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Is there a local limit theorem for functions of Gaussian random vectors?

Assume that $\sqrt{n} (\boldsymbol{Z}_n - \boldsymbol{\mu}) \stackrel{\mathcal{D}}{\longrightarrow} \mathcal{N}(\boldsymbol{0},\Sigma)$, as $n\to \infty$, for some $\boldsymbol{\mu}\in \mathbb{R}^d$ ...
Aftermath 12345's user avatar
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
0 votes
1 answer
469 views

New experiments involving Ramanujan primes: Benford's law

I know that in the literature there are interesting articles involving the sequence of Ramanujan primes, I refer the Ramanujan Prime from the online encyclopedia Wolfram MathWorld. This week I ...
user142929's user avatar
3 votes
1 answer
154 views

Donsker class and law of the iterated logarithm

Let $P$ be a probability measure on a measurable space $(E, \mathcal {E})$, and let $\mathcal {F}$ be a countable collection of measurable functions $f : E \to \mathbb {R}$ which is a Donsker class ...
Rob McCuster's user avatar
0 votes
0 answers
88 views

Empirical estimation of Brenier map from data

Let $f:\mathbb R^d \to \mathbb R$ be a "nice" (say, continuous) function define $A = A_f := \{x \in \mathbb R^d \mid f(x) \ge 0\}$ and $B =B_f:= \{x \in \mathbb R^d \mid f(x) \le 0\}$, and ...
dohmatob's user avatar
  • 6,706
0 votes
2 answers
302 views

Conditions for existence of a distribution with full support

Consider a $6\times 1$ continuous random vector $$ \eta\equiv (\eta_1,\eta_2,..., \eta_6) $$ satisfying the following property: $$ \underbrace{\begin{pmatrix} \eta_1\\ \eta_2\\ \eta_3 \end{pmatrix}}_{\...
Star's user avatar
  • 88
0 votes
2 answers
475 views

Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x_1$ when $x=(x_1,\ldots,x_d)$ is uniform on the unit-sphere in $R^d$

Let $x=(x_1,\ldots,x_d)$ be uniformly distributed on the unit-sphere in $\mathbb R^d$. Question. What is a good upper-bound for Wasserstein distance between $N(0,1/d)$ and the marginal distribution ...
dohmatob's user avatar
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1 vote
1 answer
334 views

How is 4th order cumulant of a complex random vector defined?

Suppose that ${\bf x} \in\mathbb C^n$ is a complex random vector, we know the mean vector and covariance matrix of $\bf x$ are defined as follows: $${\bf m}_{\bf x} = \mathbb{E} ({\bf x}) \\ {\bf C}_{\...
Milad A's user avatar
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0 votes
1 answer
90 views

Realizations of alternative configurations

Consider a discrete distribution $P(\mathbf{X},Y)$ with $\mathbf X = \{ X_1, \dotsc, X_N \}$. I use the shorthand notation $p(\mathbf{x}, y)$ for $P(\mathbf{X}=\mathbf{x}, Y=y)$. Consider $P_\text{ind}...
Cesare's user avatar
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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
1 vote
1 answer
196 views

How to normalize an Inverse Wishart random matrix?

Background: Let $d\in \mathbb{N}$. Define the space of (real symmetric) positive definite matrices of size $d\times d$ as follows: \begin{align} \mathcal{S}_{++}^d := \big\{\mathbb{M}\in \mathbb{R}^{d\...
Aftermath 12345's user avatar
1 vote
0 answers
39 views

Conditional independence testing with generalized covariance measure

I am trying to understand the arxiv paper "The Hardness of Conditional Independence Testing and the Generalized Covariance Measure", specifically the use of the generalized covariance ...
vluzko's user avatar
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0 answers
163 views

Multinomial additive property

I am studying multinomial distribution and found additive property. But, the text gives me concise explanation. Suppose we focus on one particular category j, then you can easily show that Xj ∼ Bin(n,...
otheng's user avatar
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2 votes
1 answer
748 views

Convergence of empirical measures in Wasserstein distance

Let $X_1, X_2, \ldots$ be iid random variables with common distribution $\gamma$, the standard Gaussian distribution on $\mathbb {R}$, and let $\mu_n = \frac 1n \sum_{i=1}^n \delta_{X_i}$, $n \geq 1$, ...
Josh Ibrahim's user avatar
9 votes
2 answers
592 views

Induction arising in proof of Berry Esseen theorem

I've been studying the paper An estimate of the remainder in a combinatorial central limit theorem by Bolthausen, which proves the Berry Essen theorem using Stein's method: Let $\gamma$ be the ...
colin's user avatar
  • 143
1 vote
0 answers
179 views

Weak convergence of Cesaro means of weakly converging infinite-dimensional distribution

Suppose we have sequences of random variables $\{X_{n,m},n \in \mathbb{N}\}$ where the distribution of $(X_{n,m})_{n\in\mathbb{N}}$ converges weakly to an infinite-dimensional normal distribution $\...
moe.dancer's user avatar
0 votes
1 answer
211 views

Decomposing a standard deviation [closed]

I am trying to "decompose" a standard deviation of an economy-wide variable into sectoral components. I have data for the year 2010 on the dispersion (standard deviation) of total economy ...
user319004's user avatar
2 votes
1 answer
228 views

About a mixture

Consider the following mixture model for a univariate density function $$ (1) \quad f(x)=\int_{(m, \sigma^2)\in D} g(x; m, \sigma^2) \mu(d(m, \sigma^2)) $$ where $D$ is a compact subset of $\mathbb{R}\...
Star's user avatar
  • 88
3 votes
0 answers
60 views

Tuning parameters of PDEs given a set of data

I am interested in doing statistical inference in the context of PDEs. Loosely speaking, the kind of problem I have in mine is the following. Problem setting Let $(t_i, x_i, y_i) \in \mathbb{R} \...
Onil90's user avatar
  • 823
2 votes
0 answers
64 views

Distribution of unbiased estimator of covariance matrix with missing values

Initial setup Assuming $X_1, ..., X_n \in \mathbb{R}^m$ are iid, sampled from $\mathcal{N}(\mu, V)$, one can define the estimators for the sample mean $\hat{\mu} = \frac{1}{n} := X^T 1_n$, and sample ...
user43389's user avatar
  • 245
2 votes
1 answer
414 views

Mistake in Karl Pearson's 1900 paper introducing the chi-squared distribution

Background I'm reading Karl Pearson's 1900 paper titled: On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be ...
Kimmel's user avatar
  • 41
2 votes
1 answer
62 views

Asymptotic bound of quotient of absolute and squared deviation from mean

The following fraction shows up when trying to show consistency of the OLS slope estimator in a simple linear regression on a log-log scale where the window of observation changes as the sample size $...
AlbertRapp's user avatar
1 vote
1 answer
115 views

Probabilistic lower and upper-bounds for a certain random quartic form involving gaussian random matrices

Let $d,m \to \infty$ (integers) with $m/d \to \rho \in (0, \infty)$. Let $C$ be a $d \times d$ psd matrix with $trace(C)=\mathcal O(1)$, and let $w_1,\ldots,w_m$ be iid uniformly distributed on the ...
dohmatob's user avatar
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0 votes
1 answer
201 views

Factorisation of Gaussian random matrix into random Hermitian and correction factor

By the Bartlett decomposition, one has that for $k \leq n$ and $\mathbf{\Gamma}_{n\times k} \in \mathbb{R}^{n\times k}$ a standard Gaussian matrix with independent entries $$\mathbf{\Gamma}_{n\times k}...
user avatar
2 votes
0 answers
108 views

L1 error of estimators

I came across the following problem and I have no clue how to approach it. I am looking for help with directions or references. Consider the $\alpha$-stable distribution with unknown true mean $\mu$, ...
Robert's user avatar
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1 vote
0 answers
121 views

Help to use Statistics and algebra books for community [closed]

My father has 2000 statistics and higher algebra books (schaum series etc). Need to use these for community since he passed away (India) kindly guide me I just need to know if we can donate these ...
user3414674's user avatar

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