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.
1,848
questions
2
votes
2
answers
785
views
Defining a measure of uniformity for measurable subsets of $[0,1]^2$ w.r.t dimension $\alpha\in[0,2]$
Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,2]$ and $\text{dim}_{\text{H}}(A)$ is the Hausdorff ...
3
votes
0
answers
52
views
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 ...
7
votes
1
answer
353
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 ...
0
votes
0
answers
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, ...
2
votes
2
answers
334
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 ...
2
votes
0
answers
97
views
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 ...
0
votes
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}...
0
votes
0
answers
40
views
When wavelet estimates fail?
I am interested in some models studied in non-parametric estimation, more precisely the Gaussian white noise model,
$$dX_{t_{1},...,t_{d}}=f(t_{1},...,t_{d})dt_{1}...dt_{d}+\theta dW_{t_{1},...,t_{d}}$...
3
votes
1
answer
165
views
Variance lower bound for natural exponential family
Let $Q$ be a probability measure on $\mathbb{R}$. Let $$Q_h(dy) = e^{y \cdot h} Q(dy) / M(h) \quad \text{where} \quad M(h) = \int e^{y \cdot h} Q(dy)$$ defined for $h \in (-c,\infty)$ with some $c &...
0
votes
1
answer
70
views
Convergence in expectation of a discontinuous function
Consider a random variable $X\in \mathbb{R}^d$. Let ${\theta_m}$ be a sequence of real numbers that converge to $\theta$. Let $f(x,y)$ be a function that is not continuous. To be specific, fix, $x=a$, ...
4
votes
0
answers
118
views
Projection of log-concave distribution on unit sphere surface
Let $\mathbf X : \Omega \to \mathbb R^d$ be a random vector following a zero mean, identity covariance log-concave distribution.
Is there any known upper bound for the probability density function of $...
0
votes
1
answer
222
views
Poisson Process x SIR model [closed]
Consider the simplest SIR model:
$$S'=-a SI$$
$$I'=a SI - b I$$
$$R'=b I$$
It is known that the waiting time of an infeccious person in the compartment $I$ follows an exponential behavior with rate $b$...
0
votes
1
answer
101
views
Positivity of linear combination of gaussian variables
Consider a collection of independent standard Gaussian variables $w_i$ for $i = 1, 2, \ldots, N$. Define its linear combination $f:=\sum_{i=1}^Na_iw_i+b_i$, where $a_i=pb_i$ ($p$ is a fixed parameter),...
1
vote
2
answers
215
views
Anti-concentration of gaussian variable
Let $X$ be $\mathcal{N}(\mu,\sigma^2)$ gaussian. Its expectation $\mu$ is positive. Can we derive a lower bound on
$$\mathbb{P}(X\geq\epsilon)\geq g(\epsilon,\mu,\sigma) \text{ where } \epsilon\leq\mu$...
0
votes
0
answers
36
views
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, ....
6
votes
2
answers
506
views
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, ...
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) ...
0
votes
1
answer
59
views
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 ...
2
votes
0
answers
84
views
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 ...
2
votes
1
answer
162
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 ...
5
votes
1
answer
777
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 ...
1
vote
1
answer
48
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) ...
0
votes
1
answer
161
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 ...
0
votes
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 $...
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.
$$\...
0
votes
1
answer
148
views
Spectral norm of matrices of bounded random variables
Assume $A\in \mathbb{R}^{n\times n}$ with each entry being i.i.d. bounded r.v. in $[a,b]$, is $\Vert A\Vert_2$ is sub-Gaussian?
Intuitively, since $\{A_{ij}\}_{i,j=1,...,n}$ is bounded, then
$$\Vert A ...
2
votes
2
answers
218
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 ...
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?
$...
1
vote
1
answer
286
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
$$
H(e^{\...
2
votes
1
answer
227
views
Bounding Kullback-Leibler
Suppose we have a probability distribution $P$ on a finite set $S$. We draw $N$ i.i.d. samples according to $P$ and use these samples to define an empirical distribution $R$. We measure the Kullback-...
0
votes
2
answers
245
views
What mathematical formalism might be used to disprove natural selection, on the basis that there are too many independent genetic parameters? [closed]
I have nagging doubts that the random genetic mutation process of natural selection is sufficient to explain evolution, even when coupled with sexual selection (Darwin proposed that evolution is ...
4
votes
0
answers
46
views
Quantifying error in the reconstruction of convex polytopes from moments
The problem of reconstructing a geometric object from its moments is of interest in a variety of fields. In the paper The Inverse Moment Problem for Convex Polytopes, the authors show that a convex ...
7
votes
1
answer
272
views
A reference for a sum found in Gould's Combinatorial Identities book
On p. 49 in Gould's book Combinatorial Identities, the author states that the sum $$\sum_{k=0}^{n-1}(-1)^k\binom{n}{k}\binom{2n}{2k}^{-1}$$ "... arises naturally in a statistical problem; it ...
4
votes
1
answer
242
views
When is $\prod_{i=0}^\infty (I-x_i x_i^T)=0$ for isotropic Gaussian $x_i$?
Suppose $x_i$ is sampled IID from isotropic zero-centered Gaussian random variable in $d$ dimensions with covariance $\Sigma=c*I$. When is the following true with probability 1?
$$\prod_{i=0}^\infty (...
1
vote
0
answers
106
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 $X=(X_1,X_2,\cdots, X_n)$ ...
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 $\|f^*\|_{L_2(\mu)} < \infty$ ...
0
votes
1
answer
125
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, ...
0
votes
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 $...
1
vote
2
answers
670
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, \...
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 ...
0
votes
0
answers
71
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:\...
2
votes
1
answer
79
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 $\{...
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, ...
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, ...
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
$$
...
3
votes
0
answers
86
views
Make inference on parameter $\lambda$ in Box-Cox transformation by MLE method
The original form of the Box-Cox transformation, as appeared in
their 1964 paper, takes the following form: $$y(\lambda )=\begin{cases}\frac{y^{\lambda}-1}{\lambda}, & \lambda \neq 0\\ \log(y), &...
2
votes
1
answer
512
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 = ...
4
votes
1
answer
182
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 ...
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 $\mathbb{P}$. ...
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 ...