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.
1,847
questions
2
votes
1
answer
151
views
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 ...
0
votes
1
answer
69
views
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 ...
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 ...
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....
1
vote
0
answers
73
views
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
$$Q(\textbf{x}) = \sum_{1 \leq i \leq n} ...
1
vote
0
answers
263
views
Tail bounds for random Gaussian chaos?
Let $g = (g_1, \dots, g_d)$ be a sequence of independent standard Normal random variables, and suppose $\Sigma$ is a $d \times d$ (deterministic), real, symmetric, positive definite matrix. The Hanson-...
2
votes
2
answers
265
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Integral of product of Hermite polynomials w.r.t marginal distribution of first two-coordinate of random vector on unit-sphere
This question is related to: https://math.stackexchange.com/q/4270522/168758
Let $H_n(x) \in \mathbb R[x]$ be the probabilist's $n$th Hermite polynomial. This an $n$th degree polynomial given by the ...
1
vote
1
answer
367
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Approximate the singular values of a certain random dot-product kernel matrix (in the sense of El Karoui, Cheng-Singer, etc.)
Let $g:\mathbb R \to \mathbb R $ be a continuous function which is
"sufficiently smooth" (e.g $\mathcal C^3$) around $0$, and
"sufficiently integrable" (e.g integrable w.r.t $N(0,...
1
vote
1
answer
343
views
Occupation times for two-state Markov processes
Consider a two-state Markov process in continuous time, with states labelled $A$ and $B$. The transition rates for going from state $A$ to $B$, and state $B$ to $A$ are $\alpha$ and $\beta$ ...
0
votes
0
answers
95
views
Books on limiting properties of matrices with growing size
This question has been posted on Math-Se previously.
I am studying asymptotic properties of the Projection Matrix
$$
H_n=X'(X'X)^{-1}X
$$
By the Gerschgorin disc theorem, the bounds on the ...
3
votes
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 ...
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$. ...
1
vote
1
answer
80
views
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 $...
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+\...
2
votes
1
answer
112
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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.
...
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$...
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>\...
0
votes
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{\...
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), ...
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 ...
2
votes
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/...
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 ...
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 ...
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 ...
1
vote
0
answers
44
views
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$ ...
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 ...
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 ...
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 ...
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 ...
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}}_{\...
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 ...
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:
$${\bf m}_{\bf x} = \mathbb{E} ({\bf x}) \\
{\bf C}_{\...
0
votes
1
answer
90
views
Realizations of alternative configurations
Consider a discrete distribution $P(\mathbf{X},Y)$ with $\mathbf X = \{ X_1, \dotsc, X_N \}$. I use the shorthand notation $p(\mathbf{x}, y)$ for $P(\mathbf{X}=\mathbf{x}, Y=y)$. Consider $P_\text{ind}...
4
votes
0
answers
74
views
Marginalization of Wishart distribution
Consider the following Wishart distribution
$$
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}
$...
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\...
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 ...
0
votes
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,...
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
$\mu_n = \frac 1n \sum_{i=1}^n \delta_{X_i}$, $n \geq 1$, ...
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 ...
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 $\...
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 ...
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}\...
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} \...
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 ...
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 ...
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 $...
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 ...
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}...
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 $\mu$, ...
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 ...