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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Prove the convergence of the LASSO model in the presence of limited eigenvalues

I am researching the properties of the Lasso model $\hat \beta:= \operatorname{argmin} \{\|Y-X\beta\|_2^2/n+\lambda\|\beta\|_1\}$, specifically its convergence when the data satisfies restricted ...
GGbond's user avatar
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7 votes
1 answer
343 views

Singular Fisher information matrix and existence of unbiased estimators

I'm doing some research into the Cramer-Rao bound for time of arrival localization and have come across a rather strange result: the FIM is singular, but there exists an unbiased estimator. My ...
JNL's user avatar
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28 views

Adjust X to strengthen the linearity to Y, in regression model

Assume that we have 2 series X and Y, and obvious we can fit a linear regression model and get all the statistics. I am seeking for some transformation / adjustment which will adjust the value of X, ...
Pique's user avatar
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2 votes
2 answers
326 views

Minimal sufficient statistic: a measurability issue in a well-known theorem

Given a statistical model $\{\mathbb{P}_\theta\mid\theta\in\Theta\}$ on $(\Omega,\mathscr{F})$, and given a real-valued random variable $X$, we say a real-valued random variable $T$ is a sufficient ...
No-one's user avatar
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2 votes
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Estimation of eigenfunctions of a Covariance operator

I'm currently working with the book "Inference for Functional Data with Applications" by Horáth and Kokoszka and trying to understand Hilbert space models for functional data. At the moment ...
kelik's user avatar
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1 answer
72 views

Estimation on rotationally-disturbed random vectors

During developing a new statistical estimator, I faced the following problem. Let $\mathbf{x}_i$ be a sequence of i.i.d. $d$-dimensional random vectors with \begin{align*} \mathbf{x}_i = \mathbf{O}...
Seung Hyeon Yu's user avatar
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40 views

When wavelet estimates fail?

I am interested in some models studied in non-parametric estimation, more precisely the Gaussian white noise model, $$dX_{t_{1},...,t_{d}}=f(t_{1},...,t_{d})dt_{1}...dt_{d}+\theta dW_{t_{1},...,t_{d}}$...
BabaUtah's user avatar
3 votes
1 answer
165 views

Variance lower bound for natural exponential family

Let $Q$ be a probability measure on $\mathbb{R}$. Let $$Q_h(dy) = e^{y \cdot h} Q(dy) / M(h) \quad \text{where} \quad M(h) = \int e^{y \cdot h} Q(dy)$$ defined for $h \in (-c,\infty)$ with some $c &...
Oxonon's user avatar
  • 143
0 votes
1 answer
69 views

Convergence in expectation of a discontinuous function

Consider a random variable $X\in \mathbb{R}^d$. Let ${\theta_m}$ be a sequence of real numbers that converge to $\theta$. Let $f(x,y)$ be a function that is not continuous. To be specific, fix, $x=a$, ...
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Projection of log-concave distribution on unit sphere surface

Let $\mathbf X : \Omega \to \mathbb R^d$ be a random vector following a zero mean, identity covariance log-concave distribution. Is there any known upper bound for the probability density function of $...
entechnic's user avatar
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1 answer
221 views

Poisson Process x SIR model [closed]

Consider the simplest SIR model: $$S'=-a SI$$ $$I'=a SI - b I$$ $$R'=b I$$ It is known that the waiting time of an infeccious person in the compartment $I$ follows an exponential behavior with rate $b$...
Quiet_waters's user avatar
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1 answer
101 views

Positivity of linear combination of gaussian variables

Consider a collection of independent standard Gaussian variables $w_i$ for $i = 1, 2, \ldots, N$. Define its linear combination $f:=\sum_{i=1}^Na_iw_i+b_i$, where $a_i=pb_i$ ($p$ is a fixed parameter),...
happyle's user avatar
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1 vote
2 answers
209 views

Anti-concentration of gaussian variable

Let $X$ be $\mathcal{N}(\mu,\sigma^2)$ gaussian. Its expectation $\mu$ is positive. Can we derive a lower bound on $$\mathbb{P}(X\geq\epsilon)\geq g(\epsilon,\mu,\sigma) \text{ where } \epsilon\leq\mu$...
M.K's user avatar
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0 answers
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Stationarity of ARMA-like time series

It is well known that for $X_t \sim ARMA(p,q)$ where $\phi(B)X_t = \theta(B)Z_t, Z_t\sim WN(0, \sigma^2)$, if $\phi(z)\neq0$ in the unit circle, $\{X_t\}$ is stationary. Now assume $\{Y_t, t=0, \pm1, ....
dc3506's user avatar
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6 votes
2 answers
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Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win where $T$ is the number of cumulative tails flipped

I have a weird question which probably seems out of place here but it has proven more difficult than anticipated. I am going to describe the game without showing work toward a solution. Numerically, ...
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2 votes
1 answer
183 views

Approximation to ratio distribution

Recently I was thinking if there is a way to do the following: assuming I have some sampled points of distribution $\mathcal{X}$ and distribution $\mathcal Z$ (whose MGF I do not have in closed form) ...
Hazards's user avatar
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1 answer
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What can we say about the order of convergence of a critical point of Gaussian mixture density to its limit when the parameter $h$ goes to $0?$

Density of Gaussian mixture with $n$ components is given by: $$f(x):=C \sum_{i=1}^{n}e^{-\frac{1}{2}||\frac{x-x_i}{h}||^2}, x_i \in \mathbb{R}^d, h > 0$$ where $C$ is a normalization constant ...
Learning math's user avatar
2 votes
0 answers
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A complex problem involving densities (likelihood functions) and optimization

Consider the following autoregressive process with normal errors: \begin{equation}\label{7YlUV4i8nuO}\tag{I} y_t = \phi y_{t-1}+ u_t, \quad u_t \overset{iid}{\sim} N(0,\sigma^2) \end{equation} We ...
PSE's user avatar
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2 votes
1 answer
160 views

Weighted sum of two random variables ranked by first order stochastic dominance

Suppose $X$ and $Y$ are two non-negative, independent random variables such that $X \succcurlyeq_{st} Y$. That is, $X$ first-order stochastically dominates $Y$. Suppose that $X$ and $Y$ have smooth ...
avk255's user avatar
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5 votes
1 answer
749 views

Mathematics research relating to machine learning

What branch/branches of math are most relevant in enhancing machine learning (mostly in terms of practical use as opposed to theoretical/possible use)? Specifically, I want to know about math research ...
Artus's user avatar
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1 vote
1 answer
47 views

Covariance inequality for left skewed distributions

Consider a left skewed random variable $X$ with mean $1$, median $>1$ and support on $[0,2)$. Suppose we have a class of functions $\mathbf{G}$ and each of it's members satisfy $G(x): [0,\infty) ...
Dejan Evisal's user avatar
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1 answer
156 views

Optimal hypothesis testing uses sufficient statistics?

Cross post Optimal hypothesis testing uses sufficient statistics?. In statistical estimation with any convex risk for a model with a sufficient statistic, in seeking optimal estimators it suffices to ...
Kweku A's user avatar
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0 answers
57 views

The limit spectral distribution of the random matrix $(\hat{\Sigma}_1+\hat{\Sigma}_2)^{-1}\hat{\Sigma}_1$

Let $S_1$ and $S_2$ be the collection of i.i.d. copies of $X\sim\mathcal{N}(0,I_p)$, where $|S_1|=n_1,|S_2|=n_2$. Let $\hat{\Sigma}_1$ and $\hat{\Sigma}_2$ be the covariance matrix using samples in $...
aurora_borealis's user avatar
1 vote
1 answer
64 views

Is the main part of certain exponential family sub-Gaussian?

$X$ is in the form of exponential family i.e. $$\mathbb{P_\theta}x = h(x)e^{\langle \theta,T(x)\rangle-\phi(\theta)}$$ where $\theta\in \mathbb{R}^d$. If $\nabla\phi(\theta)$ is L-Lipschitz i.e. $$\...
dc3506's user avatar
  • 71
0 votes
1 answer
148 views

Spectral norm of matrices of bounded random variables

Assume $A\in \mathbb{R}^{n\times n}$ with each entry being i.i.d. bounded r.v. in $[a,b]$, is $\Vert A\Vert_2$ is sub-Gaussian? Intuitively, since $\{A_{ij}\}_{i,j=1,...,n}$ is bounded, then $$\Vert A ...
dc3506's user avatar
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2 votes
2 answers
212 views

Continuity of Nash equilibrium for a family of games

The question may be too vague, but ultimately in search of various (counter)examples or theorems to exhibit the following: Do continuous families $t\mapsto G_t$ of "games" (say each $G_t$ is ...
Chris Gerig's user avatar
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4 votes
0 answers
298 views

When is $\prod_{i=0}^\infty (I-x_i x_i^T)=0$ for zero-centered Gaussian $x_i$?

Suppose $x_i\in \mathbb{R}^d$ is sampled IID from $\mathcal{N}(0,H)$. Let $A_i=(I-x_i x_i^T)$ and assume $d$ is large. What are necessary conditions for the following to converge with probability 1? $...
Yaroslav Bulatov's user avatar
1 vote
1 answer
279 views

Converse of the Herbst argument?

Background For a real-valued random variable $X$, define its entropy by $H(X) = E[\phi(X)] - \phi(E[X])$, where $\phi(u) = u \log u$. It can be shown that, if the entropy satisfies the bound $$ H(e^{\...
aest's user avatar
  • 143
2 votes
1 answer
222 views

Bounding Kullback-Leibler

Suppose we have a probability distribution $P$ on a finite set $S$. We draw $N$ i.i.d. samples according to $P$ and use these samples to define an empirical distribution $R$. We measure the Kullback-...
Bill Bradley's user avatar
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0 votes
2 answers
243 views

What mathematical formalism might be used to disprove natural selection, on the basis that there are too many independent genetic parameters? [closed]

I have nagging doubts that the random genetic mutation process of natural selection is sufficient to explain evolution, even when coupled with sexual selection (Darwin proposed that evolution is ...
user501885's user avatar
4 votes
0 answers
46 views

Quantifying error in the reconstruction of convex polytopes from moments

The problem of reconstructing a geometric object from its moments is of interest in a variety of fields. In the paper The Inverse Moment Problem for Convex Polytopes, the authors show that a convex ...
Lucas Blakeslee's user avatar
7 votes
1 answer
271 views

A reference for a sum found in Gould's Combinatorial Identities book

On p. 49 in Gould's book Combinatorial Identities, the author states that the sum $$\sum_{k=0}^{n-1}(-1)^k\binom{n}{k}\binom{2n}{2k}^{-1}$$ "... arises naturally in a statistical problem; it ...
Sela Fried's user avatar
4 votes
1 answer
242 views

When is $\prod_{i=0}^\infty (I-x_i x_i^T)=0$ for isotropic Gaussian $x_i$?

Suppose $x_i$ is sampled IID from isotropic zero-centered Gaussian random variable in $d$ dimensions with covariance $\Sigma=c*I$. When is the following true with probability 1? $$\prod_{i=0}^\infty (...
Yaroslav Bulatov's user avatar
1 vote
0 answers
105 views

Distribution of norm over projected unit vectors

I am interested in the distribution of norms of projected unit vectors, for a particular class of projections. We first draw an $n$-dimensonal unit vector $v=X/||X||$ where $X=(X_1,X_2,\cdots, X_n)$ ...
galoistr93's user avatar
2 votes
0 answers
50 views

Convergence of minimiser of empirical risk to minimiser of population risk

Let $X_1, \dots, X_n \sim \mu$ be some random elements of a space $\mathcal{X}$. Let $H$ be a Hilbert space of functions $f: S \to \Re$ with norm $\|\cdot\|_H$. Let $\|f^*\|_{L_2(\mu)} < \infty$ ...
user27182's user avatar
  • 315
0 votes
1 answer
123 views

Analogues of Kac-Bernstein characterisation theorem for non-normal distributions

Let $X,Y$ be two independent random variables. The Kac-Bernstein theorem states that if $X+Y,X-Y$ are also independent, then $X,Y$ are Normal. Are there analogues of this theorem for non-normal, ...
TheSimpliFire's user avatar
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0 answers
58 views

Norms of Wigner matrices under power law decay

Suppose $\Sigma=\operatorname{diag}(h)$ where $h=(1^{-p},2^{-p},3^{-p},\ldots,d^{-p})$ and $p> 1$ $X$ is a matrix with $b$ rows sampled independently from $\operatorname{Normal}(0,\Sigma)$ Suppose $...
Yaroslav Bulatov's user avatar
1 vote
2 answers
647 views

Upper bound about Gaussian tail bound

From the definition of sub-Gaussian distribution $X$ w.r.t. $\sigma$ i.e. $$\mathbb{P}(|X-\mathbb{E}(X)|\geq t) \leq 2 \exp(-\frac{t^2}{2\sigma^2}).$$ It's natural that when $X \sim \mathcal{N}(\mu, \...
dc3506's user avatar
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3 votes
0 answers
87 views

Question on an integral inequality

I am reading van de Vaart and Weller, Weak Convergence and Empirical Processes With Applications to Statistics. And I am stuck in the proof of Theorem 2.6.7 on page 141. For simplicity I restae the ...
newbie's user avatar
  • 53
0 votes
0 answers
70 views

Kernel density estimation is sub-gaussian

Let $X_1, ..., X_n$ be i.i.d. samples drawn from a pdf $f(x)$ on the real line. The kernel density estimator is defined as follows, $$\hat{f_n}(x) = \frac{1}{nh}\sum_1^n K(\frac{x-X_k}{h})$$ where $K:\...
dc3506's user avatar
  • 71
2 votes
1 answer
78 views

p.d.f. of exponential family

I have a question about the p.d.f. of exponential family. Suppose $(X,\mathcal{F})$ is a measurable space and $\{F_{\theta},\theta\in \Theta\}$ is a distribution family on $(X,\mathcal{F})$. When $\{...
newbie's user avatar
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1 vote
0 answers
109 views

When is the solution to a Fredholm integral equation a PDF?

I have two questions about inhomogenous Fredholm integral equations of the first kind: $$f(x) = \int_a^b K(x,t) g(t) dt$$ where $f, K$ are known and $g$ is not. If a unique solution for $g$ exists, ...
Wilbur's user avatar
  • 111
0 votes
0 answers
114 views

Using projections to determine equidistribution

Suppose I have a collection of points on $\mathbb{S}^{n-1} \subset \mathbb{R}^n.$ I want to know that they are equidistributed (if you want to be more precise, you have a sequence of such collections, ...
Igor Rivin's user avatar
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3 votes
0 answers
86 views

Asymptotic approximation of Fisher information matrix for small Gaussian perturbation

Let $$ X=Y/a+b+\epsilon Z, $$ where $Y\sim\operatorname{Poisson}(\lambda)$ and $Z\sim\mathcal N(0,1)$ are independent. Also define $\theta=(\lambda,a,b,\epsilon)$. The Fisher information matrix $$ ...
Aaron Hendrickson's user avatar
3 votes
0 answers
84 views

Make inference on parameter $\lambda$ in Box-Cox transformation by MLE method

The original form of the Box-Cox transformation, as appeared in their 1964 paper, takes the following form: $$y(\lambda )=\begin{cases}\frac{y^{\lambda}-1}{\lambda}, & \lambda \neq 0\\ \log(y), &...
M.Ramana's user avatar
  • 1,172
2 votes
1 answer
491 views

Can we use Bernstein's inequality without knowledge of variance?

I have a question about Bernstein’s inequality for bounded random variables. Its statement is the following. Let $X_1, \ldots, X_N$ be independent, mean zero random variables with $|X_i| \leq K \ (i = ...
aest's user avatar
  • 143
3 votes
1 answer
152 views

Existence of disintegrations for improper priors on locally-compact groups

In wide generality, the disintegration theorem says that Radon probability measures admit disintegrations. I'm trying to understand the case when we weaken this to infinite measures, specifically ...
Tom LaGatta's user avatar
  • 8,362
2 votes
1 answer
82 views

VC-based risk bounds for classifiers on finite set

Let $X$ be a finite set and let $\emptyset\neq \mathcal{H}\subseteq \{ 0,1 \}^{\mathcal{X}}$. Let $\{(X_n,L_n)\}_{n=1}^N$ be i.i.d. random variables on $X\times \{0,1\}$ with law $\mathbb{P}$. ...
Math_Newbie's user avatar
1 vote
1 answer
108 views

Asymptotic expansion on the following integral of exponential function

I wish to obtain the asymptotic for the following integral $$ \int_{r: \|r\|\le 1} \exp(M\cdot a^Tr) \, dr, $$ where $a$ is a given vector in $\mathbb{R}^d$, $\|\cdot\|$ is a general norm function and ...
user497696's user avatar
4 votes
1 answer
239 views

Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?

This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian $$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$ are ...
Learning math's user avatar

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