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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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 ...
Sarah Rune's user avatar
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 ...
Chen Samuel's user avatar
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 ...
Manfred Weis's user avatar
  • 12.6k
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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. ...
Dotman's user avatar
  • 105
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,&...
Frank's user avatar
  • 193
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 $\alpha, \beta \in \mathbb{R}_+$ ...
spacetimewarp's user avatar
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. ...
CWC's user avatar
  • 389
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....
WeakLearner's user avatar
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$...
Bernard 's user avatar
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 ...
dohmatob's user avatar
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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 ...
Aaron Hill's user avatar
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 ...
entechnic's user avatar
  • 149
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(\...
jkfds's user avatar
  • 7
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 \...
Ben's user avatar
  • 19
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+...
Ben's user avatar
  • 19
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 ...
BayesRayes's user avatar
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 $$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$ ...
Yaroslav Bulatov's user avatar
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\...
Hao Yu's user avatar
  • 185
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 ...
gusl's user avatar
  • 57
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 ...
Ben123's user avatar
  • 163
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,\...
James Ilken's user avatar
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=...
KGR's user avatar
  • 11
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 $Y = \max_{X \in X_k} |Z^T X|^2,$ ...
Alireza Khayatian's user avatar
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 $k$ ...
Birdy Nam Nam's user avatar
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 ...
virtuolie's user avatar
  • 173
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 ...
axk's user avatar
  • 517
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 ...
Alex's user avatar
  • 139
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{...
anon's user avatar
  • 43
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 ...
Sam's user avatar
  • 290
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 ...
Max's user avatar
  • 9
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\} $. ...
Aryeh Kontorovich's user avatar
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 ...
Grandes Jorasses's user avatar
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}\...
rick's user avatar
  • 121
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( - (\...
wsz_fantasy's user avatar
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. ...
Jose de Frutos's user avatar
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 ...
Paul R's user avatar
  • 39
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: ​$$...
Charles's user avatar
  • 21
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 ...
Math_Newbie's user avatar
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 ...
Tanishq Kumar's user avatar
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 ...
AChem's user avatar
  • 803
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 ...
CoilyUlver's user avatar
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^{...
Black Jack 21's user avatar
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 \...
John's user avatar
  • 483
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, ...
Dotman's user avatar
  • 105
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 ...
arknas's user avatar
  • 21
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 ...
Hao Yu's user avatar
  • 771
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 ...
Aryeh Kontorovich's user avatar
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\...
Sanae Kochiya's user avatar
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 $||\...
Aryeh Kontorovich's user avatar
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 ...
user6384's user avatar

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