# 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.

1,850
questions

3
votes

1
answer

47
views

### Maximum entropy probability distribution with fixed interval and variance?

What is the maximum entropy probability distribution if the support is a fixed interval (e.g. [-1,1]) with an already known variance?
If we know the support is a fixed interval, then the maximum ...

0
votes

0
answers

24
views

### UMVUE for $\rho$

Given $X=\left( X_{1},\cdots, X_{n}\right)$ and $X\sim N\left( 0,\Sigma \right)$, where
$$
\Sigma=\sigma^{2}
\begin{pmatrix}
1&\rho&\cdots &\rho\\ \rho &1&\cdots &\rho\\ \vdots ...

2
votes

1
answer

111
views

### Difference between deep neural networks and expectation maximum algorithm

Having had a short encounter with deep neural networks, it seems to boil down to the task of determining the values of a vast amount of parameters.
The expectation maximization algorithm, of which I ...

0
votes

0
answers

38
views

### Understanding CDF comparisons between normalized sums of hypergeometric and binomial distributions

I am analyzing a population composed of $N$ bits, containing $K$ ones ($1$s) and $N-K$ zeros ($0$s). When sampling $n$ bits without replacement, the scenario aligns with a hypergeometric distribution. ...

1
vote

0
answers

42
views

### The limit ratio of two Markov Chain Probability

Suppose there are two given SDE in $\mathbb{R}^d$:
$$
\begin{align}
\left\{
\begin{aligned}
dX_t&=\begin{bmatrix}-\nabla V(X_t)+2\beta^{-1}v_F^\theta(X_t)\end{bmatrix}dt+\sqrt{2\beta^{-1}}dW_t,&...

0
votes

1
answer

80
views

### Reverse Pinsker's inequality for smooth density classes

Suppose we are given a class of probability density functions $\mathcal{F}$ so that for every $f \in \mathcal{F}$ we have $\alpha \leq f \leq \beta$ for some positive $\alpha, \beta \in \mathbb{R}_+$ ...

0
votes

0
answers

37
views

### Quotient of estimators

Say $A$ is a set of a finite number of samples, and $\hat{\mu}_A$ and $\hat{\sigma}_A$ are unbiased estimators (computed over $A$) of $\mu$ and $\sigma$ which are some distinct population statistics. ...

2
votes

0
answers

49
views

### Concentration result for self-normalized empirical process

In Theorem 1.1 of this paper by Bercu, Gassiat and Rio, a concentration result is derived for the 'self-normalized' empirical process. Specifically, suppose that $(X,X_n)_{n \ge 1}$ is a sequence of i....

1
vote

0
answers

60
views

### Gain of a steady state Kálmán filter

It is well known that the state covariance of a steady-state Kálmán filter is the solution of a discrete Riccati equation.
$$P_\infty = F(P_\infty - P_\infty H^T(HP_\infty H^T+R)^{-1}HP_\infty)F^T + Q$...

0
votes

0
answers

33
views

### Limiting value of trace of resolvent matrix involving two independent Wishart random matrices

Let $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that
$$
d/n_k \to \phi_k \in (0,\infty).
$$
Let $X_1 \in \mathbb R^{n_1 \times d}$ and $X_2^{n_2 \times d}$ be independent ...

2
votes

1
answer

158
views

### How can one construct a confidence interval for the mean of a uniformly distributed random variable using a sample of size 2

Two numbers will be randomly (and independently) selected from a uniform distribution on an interval with length $L$ and center $M.$
It is very easy to estimate $M$ (just take the average of the two ...

0
votes

0
answers

68
views

### Inequality related with log-concave distributions

Fix any $n$-dimensional unit vector $\mathbf v$.
Let $\mathbf x$ be a random vector following the $n$-dimensional standard normal distribution. It has been shown (Analysis of Perceptron-Based Active ...

1
vote

1
answer

104
views

### Concentration inequalities for heavy-tailed distributions

Suppose $X_1,...,X_N$ are $N$ i.i.d random variables with heavy tailed distributions. For example, $E[X_i^p]\leq 1$ for some $p\geq 1$. Are there some concentration inequalities to bound the tail
$$P(\...

1
vote

1
answer

110
views

### A property of the distribution related to stochastic ordering

Let $X$ be a random variable with a symmetric support $S\subset[-M,M]$ for some $M>0$. (i.e., if x is a point of increase of CDF $F_X(\cdot)$, so is $-x$.)
Has the infimum value of $c$ such that
\...

0
votes

0
answers

62
views

### Asymptotic stochastic ordering for weighted sum of i.i.d. random variables

Are you aware of any literature focusing on the conditions such that for two i.i.d. sequences of discrete r.v.'s $\{X_n\}$ and $\{Y_n\}$,
\begin{equation}
a_1X_1+a_2X_2+\ldots+a_nX_n\geq_1 a_1Y_1+...

0
votes

0
answers

66
views

### Gibbs Priors form a Martingale

I am working on adapting variational inference to the recently developed Martingale posterior distributions. The first case, which reduces the VI framework to Gibbs priors, is proving hard to show as ...

1
vote

2
answers

244
views

### Joint moments like $\tau(XYXYXY)$ in terms of individual moments of free variables $X,Y$

Terry Tao RMT book has the following formula for joint moment of freely independent random variables $X,Y$ in Section 2.5
$$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$
...

0
votes

0
answers

76
views

### Some new questions on Rademacher complexity

For $A\subset R^n$,$A=(a_1,a_2,\dots, a_n)$, $\sigma_i$ are Rademacher random variable.
Is $|\mathbb{E}_\sigma \inf_{a\in A}\sum_{i=1}^n\sigma_ia_i| \le |\mathbb{E}_\sigma \sup_{a\in A}\sum_{i=1}^n\...

2
votes

2
answers

155
views

### Random partition of an interval – Dirichlet distributed?

Let $X_1, \ldots, X_N \sim \operatorname{Unif}[0,1]$ and consider the intervals between successive order statistics: $[0, X_{(1)}], [X_{(1)}, X_{(2)}], \ldots, [X_{(N)}, 1]$.
What is the distribution ...

1
vote

0
answers

71
views

### Estimates on number of observations and iterations depending on number of states, when applying the Baum-Welch algorithm to HMMs

Are there papers estimating how many observations $\mathcal{O}$ and how many random initializations $\ell$ one needs, to get an arbitrary good agreement between the model $\lambda = (\pi,A,B)$ that ...

0
votes

0
answers

37
views

### Multivariate delta method gradient calculation with mixed moments

Here's a slightly simplified version of my problem (using fewer dimensions than what I'm actually solving).
Take $X \in \mathbb{R}^2$. We already know that $\sqrt{n}(X - \mu) \overset{d}{\to} N_2(0,\...

0
votes

2
answers

65
views

### $n$-wise extension of covariance / correlation

Suppose we have $n$ random variables $X_1, \ldots, X_n$ and we want some sort of characterization about how "statistically" related they are as a whole, with the motivation being that for $n=...

0
votes

1
answer

178
views

### Expectation of top-K selection of squared Gaussian random variables

Let us have
$$
Z = [z_1, z_2, \dots, z_n],
$$ where $z_i \sim N(0, \sigma^2)$ and are iid. Additionally, consider
$$
X_k := \{ x \in \{0, 1\}^n : e^T x = k \}
$$ If $Y = \max_{X \in X_k} |Z^T X|^2,$ ...

1
vote

0
answers

60
views

### A small lemma on cache resets (Bloom filters in particular)

Assume a fixed set of message $D$ and an associated distribution for selecting each message $d_i$ such that the total probability $\sum_{i \in D} d_i = 1$. We create a cache with $M$ bits and $k$ ...

1
vote

2
answers

209
views

### Relationship between fixed points and inversions in permutations

Inversions in a permutation $Y$ are defined as pairs where $Y_a < Y_b$ but $a > b$, while fixed points in $Y$ are defined as elements where $Y_a = a$ (i.e., 1-cycles). Let $S_\alpha$ be the set ...

1
vote

0
answers

63
views

### Dimension-free sample complexity for the inverse of Gaussian sample covariance?

Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need the inverse of the sample covariance $\Sigma_m^{-1}$ to be $\varepsilon$-close to true inverse covariance $\Sigma^{-1}$ (in ...

2
votes

0
answers

180
views

### Statistical invariants of Riemannian manifolds

$\DeclareMathOperator\diam{diam}\DeclareMathOperator\rad{rad}\DeclareMathOperator\iso{iso}\DeclareMathOperator\com{com}\DeclareMathOperator\con{con}$A cheap way of defining invariants of Riemannian ...

0
votes

0
answers

75
views

### High probability bound on number of sparse solutions to Gaussian linear system

Suppose we have a random matrix $A \in \mathbb{R}^{m \times n}$ with all entries i.i.d. from the standard Normal distribution $\mathcal{N}(0, 1)$. Suppose $k$ divides $n$, and let $S \subseteq \mathbb{...

7
votes

2
answers

217
views

### Evolution of the empirical mean of a list as we remove elements proportional to their value

Consider a list of $N$ integers $k_1,k_2,\dots k_N$, drawn independently from some distribution $P(k)$ with $k_i \geq 1$. We denote its mean with $\langle k\rangle=\sum_{k=1}kP(k)$. The first two ...

0
votes

2
answers

94
views

### Points based partial ranking

I want to rank a population $P=\{P_1,\ldots,P_n\}$. I am given a set $R=\{R_1,\ldots,R_k\}$ of partial rankings. The partial rankings may have varying sizes (e.g. the first ranking ranks only 8 ...

3
votes

2
answers

285
views

### Minimax optimal multiple hypothesis test

Let us consider the following two-player game
between Chooser and Guesser.
There is a finite set $\Omega$
and $k$ probability distributions
on $\Omega$, denoted by $
\mathcal{P}
=\{P_1,\ldots,P_k\}
$.
...

2
votes

1
answer

167
views

### Sum of arrival times of Chinese Restaurant Process (CRP)

Suppose that a random sample $X_1, X_2, \ldots$ is drawn from a continuous spectrum of colors, or species, following a Chinese Restaurant Process distribution with parameter $|\alpha|$ (or ...

3
votes

1
answer

96
views

### When does the optimal model exist in learning theory?

In the context of learning theory, we usually have: data $(x,y)\sim P(x,y)$, with $x\in\mathcal{X}\subseteq\mathbb{R}^d$ and $y\in\mathcal{Y}\subseteq\mathbb{R}^k$, a hypothesis class $\mathcal{F}\...

2
votes

1
answer

95
views

### expectation of the product of Gaussian kernels and their input

I was wondering if anybody knows how to solve: $$\mathbb{E}{\mathbf{z} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})}\left[ (\mathbf{x}{i} - \mathbf{z})(\mathbf{x}{j} - \mathbf{z})^\top \exp\left( - (\...

0
votes

0
answers

18
views

### Analyzing point distributions in Voronoi tessellations from two probability sources $p$ and $\tilde{p}$

Let's suppose I have a probability distribution $p$ and another distribution $\tilde{p}$. Suppose I sample $K$ points from the distribution $p$ which will be my centroids for my Voronoi tessellation. ...

0
votes

0
answers

44
views

### Approximate CDF of integral using the Berry-Esseen theorem

I'm trying to approximate CDF of the integral $$\frac{1}{T}\int_0^T e^{\sigma W_t+\left(r-\frac{\sigma^2}{2}\right)t}dt,$$
where $W_t$ is the Wiener process, i.e. $W_t\sim N(0,t)$.
For this I use ...

1
vote

1
answer

101
views

### Analytical solution for a double integral involving logistic functions and Gaussian distributions

I am working on a mathematical problem involving the evaluation of a double integral, and I am seeking an analytical solution or techniques to solve it. The integral I'm dealing with is as follows:
$$...

7
votes

2
answers

405
views

### Upper bound on VC-dimension of partitioned class

Fix $n,k\in \mathbb{N}_+$.
Let $\mathcal{H}$ be a set of functions from $\mathbb{R}^n$ to $\mathbb{R}$ with finite VC-dimension $d\in \mathbb{N}$. Let $\mathcal{H}_k$ denote the set of maps of the ...

3
votes

0
answers

131
views

### Known relations between mutual information and covering number?

This is a question about statistical learning theory. Consider a hypothesis class $\mathcal{F}$, parameterized by real vectors $w \in \mathbb{R}^p$. Suppose I have a data distribution $D \sim \mu$ and ...

0
votes

0
answers

90
views

### Weighted least squares regression: Iterative modeling of variance

In chemical analysis, the instrument's signal are plotted as a function of chemical concentration. In general, higher the concentration higher is the response and the relationship is linear. At ...

0
votes

1
answer

77
views

### Existence and uniqueness of a posterior distribution

I am wondering about the existence and uniqueness of a posterior distribution.
While Bayes' theorem gives the form of the posterior, perhaps there are pathological cases (over some weird probability ...

0
votes

0
answers

50
views

### Tight Chernoff Concentration for Bernoulli(p) RV

I remember seeing a research paper on tight concentration of Bernoulli(p) random variable in terms of $p$.
What I mean is that they used a stronger upper bound for the MGF than $E[e^{s(X-p)
}]\leq e^{...

2
votes

0
answers

69
views

### Construct a Bregman divergence from Wasserstein distance

I was wondering whether one has studied the Bregman divergence arising from a squared Wasserstein distance.
More precisely, let $\Omega\subset \mathbb{R}^d$ be a compact set and $c\in \Omega\times \...

0
votes

1
answer

180
views

### Concentration inequalities for random sampling without replacement

Let a population $C$ consist of $N$ values $c_1, c_2, \cdots, c_N$, with $c_i\in \{0,1\}$. Let $X_1, X_2, \cdots, X_n$ denote a random sample without replacement from $C$ and let $Y_1, Y_2, \cdots, ...

2
votes

0
answers

55
views

### Continuous-time Wold decomposition

I'm looking for a reference for the Wold–Zasukhin decomposition in continuous time for stationary random processes on the real line.
I am aware of the classic result in the book from Rozanov, which ...

1
vote

0
answers

65
views

### Approximation of continuous function by multilayer Relu neural network

For continuous/holder function $f$ defined on a compact set K， a fix $L$ and $m_1,m_2,\dots,m_L$, can we find a multilayer Relu fully connected network g with depth $L$ and each $i$-th layer has width ...

1
vote

0
answers

142
views

### conjecture for general form of minimax estimator

I had previously posed an overly ambitious version of this conjecture here,
Form of minimax estimator,
which was quickly shot down by Václav Voráček (on twitter) and Iosif Pinelis (MO answer in the ...

0
votes

0
answers

29
views

### Probability related to record index that cross zeros in $[0, 1, \cdots, N]^{\mathbb{N}}$

This question is inspired by this paper. For those who are interested in more details and applications about record index and record values, you can find them in the paper.
Let $X=\{0, 1, 2, \cdots, N\...

1
vote

1
answer

325
views

### Form of minimax estimator

Let $\Delta$ be the set of all probability distributions over $\mathbb{N}=\{1,2,\ldots\}$ and fix some $\mathcal{P}\subseteq\Delta$.
Suppose additionally that $\Delta$ is endowed with some norm $||\...

2
votes

1
answer

165
views

### Law of iterated logarithm for quadratic variation of Brownian motion

Let $(\Omega, \mathcal{F}, \mathbb{P})$ denote a probability space supporting a standard Brownian motion $B$. Let $\Pi=\{\pi_n : n \ge 0\}$ denote the sequence of dyadic uniform partitions of the ...