# 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.

1,754
questions

2
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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 $...

1
vote

1
answer

43
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### Poisson Process x SIR model

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 $b$...

1
vote

1
answer

84
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### 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),...

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0
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### Using Dudley Integral to estimate maximum singular value of Gaussian random matrices [migrated]

On Exercise 5.14 of Wainwright, it provides a way to estimate maximum singular value of Gaussian random matrices using the one-step discretization bound and Gaussian comparison inequality as shown.
...

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0
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48
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### Upper bound for the partial sum of weighted binomial coefficients

Suppose $n$ is a natural number and $n_0 = q n$ for some positive $q \in (0, p)$. I am interested in finding a tight upper bound for the following partial summation:
$$
\sum_{x > n_{_0} }^n \...

1
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2
answers

140
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### 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
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0
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29
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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, ....

5
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2
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195
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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, ...

2
votes

1
answer

148
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### 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
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1
answer

51
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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 ...

2
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0
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71
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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 ...

2
votes

1
answer

81
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### 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 ...

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0
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31
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### Bound the variance of the largest order statistics among discrete random variables

I have a question about bounding the variance of order statistics.
Given that for $i \in \{1,\cdots,\lambda\}$, denote $Bin(s,\frac{1}{n})$ to be a binomial random variable with success probability $\...

1
vote

1
answer

34
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### 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
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47
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### 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
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0
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51
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### 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

41
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### 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

67
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### 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

95
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### 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
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0
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263
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### 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?
$...

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1
answer

167
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### 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^{\...

1
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1
answer

90
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

207
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### 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
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0
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29
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### 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

218
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### 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 ...

3
votes

1
answer

187
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### 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
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0
answers

74
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### 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

40
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### 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

72
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### 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

51
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 $...

0
votes

2
answers

135
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### 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

73
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

53
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### 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

48
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### 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
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0
answers

89
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### 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

99
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
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78
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### 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

40
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### 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

128
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### 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 = ...

3
votes

1
answer

115
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### 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

69
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### 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

76
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### 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 ...

3
votes

1
answer

167
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### 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 ...

1
vote

1
answer

93
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### Local maxima of the sum of Gaussian functions in *one dimension* are always strict local maxima - proof?

Motivated by this question asked earlier, I was wondering whether one can prove easily that the local maxima of the sum of Gaussians:
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, \quad x_1 < x_2 < \...

0
votes

0
answers

55
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### Support function of the intersection of a hyper-ellipsoid and a Euclidean ball

Le $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_d$ be positive numbers. For any $x \in \mathbb R^d$ and $r \ge 0$, define $\gamma(x,r) := \sup_{z \in E(r)}x^\top z$, where
$$
E(r) := E \cap B_2^d(r)...

3
votes

2
answers

140
views

### Precise asymptotics for moments of order statistics of normal distribution

Let $X_1, \cdots, X_n \sim N(0,1)$ be i.i.d. normal random variates. I am interested in understanding the first two moments of the quasi-range $X_{(n)}-X_{(n-1)}$ (i.e., the maximum value minus the ...

2
votes

0
answers

41
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### Probability bounds of some ranked version of Dirichlet distribution

Recently I have come across a distribution defined on the open ranked simplex $\nabla^{n-1}_+ = \{\vec x \in \mathbb{R}^n:\sum_{k=1}^n x_k =1, x_1 \geq x_2 \geq \cdots \geq x_n > 0\}$, whose ...

9
votes

1
answer

485
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### Popular mistakes in probability

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...

-1
votes

1
answer

59
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### Under which conditions Mean Square Continuity implies Sample Continuity for Gaussian Processes?

First, let us give the setting.
Let $(\Omega, \Sigma, \mathbf{P})$ be a probability space, let $T$ be some interval of time, and let $X: T \times \Omega \rightarrow S$ be a stochastic process.
By Mean ...

0
votes

0
answers

31
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### Verifying if this Gaussian mixture is a regular parametric model

*I am not sure if this question is better-suited for here or the MSE.
According to A Tutorial on Fisher Information (pg. 40), for so-called regular parametric models we have
$$
\sqrt n(\tilde\theta-\...