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.

Filter by
Sorted by
Tagged with
-1
votes
0answers
12 views

expectation of the function of Wishart matrix eigenvalues

For Given a $N×M$ random complex gaussian matrix $X$ where $M=XX^H$, let $\lambda_1>\lambda_2>....>\lambda_N$ be the ordered eigenvalues of $M$ my objective is to get an estimation of \begin{...
2
votes
1answer
67 views

Test for OU-Process

Suppose that I'm given a sample from time-series $(x_n)_{n=1}^N$ and want to decide if it comes from an OU process or not. Is there a (rigorous) test I can use? So far, everything I've seen is hand-...
-3
votes
0answers
26 views

Simple Linear regression b_1 proof [closed]

How do I go from the second equation to the third one? image
-1
votes
0answers
87 views

On a concentration bound without i.i.d. assumptions

Pick uniform independent random integers $x_1,\dots,x_k\in\{0,1,\dots,2^t-2,2^t-1\}$ and let $x_1$ be minimum and $x_k$ be maximum. Pick $k-1$ non-negative integers $g_1=x_1-x_2$ to $g_{k-1}=x_{k-1}-...
-5
votes
1answer
49 views

How to get the E[XY] when X and Y are both binary variables? [closed]

Suppose $W_i \in \{0,1\}$, then the textbook said $$E[W_iW_{i'}]=Pr(W_i=1)Pr(W_{i'}=1|W_i=1)$$ Why this equation holds?
-3
votes
0answers
31 views

Stats probability question [closed]

In a game of craps, you roll two fair dice. Whether you win or lose depends on the sum of the numbers occurring on the tops of the dice. Let x be the random variable that represents the sum of the ...
-2
votes
0answers
55 views

Distribution of gaps between uniform random variables

Pick $k$ uniform independent random integers $x_1,\dots,x_k\in\{0,1,\dots,2^t-2,2^t-1\}$ and denote $y_{\sigma(i)}=x_i$ where $\sigma$ is a permutation in $S_n$ such that $y_1\leq y_2\leq\dots\leq y_{...
-1
votes
0answers
20 views

(Conditional) Independence of additive Gaussian noise disturbed sensor

Let's assume a random variable $Z\in\mathbb{R}^n$ being $Z_{i} = X + Y_i$, where $X\in\mathbb{R}^n$, $Y_i\in\mathbb{R}^n$ and $i\in\mathbb{N}$ being an index. It is assumed that $Y_i\perp Y_j$ (where $...
0
votes
0answers
29 views

What is known about convergence of empirical extrema?

VC theory provides an answer to Problem 1 specified below. I am wondering what is known about a similar issue, Problem 2. $$ ~ $$ Problem 1 Let $X$ be a set, let $\mathcal{D}$ be a distribution ...
-1
votes
0answers
32 views

Correlate two circular correlation coefficients?

I have a question regarding the relationship of two circular correlations. Given I have a circular correlation coefficient of brain synchronizations and a circular correlation coefficient of turn-...
0
votes
1answer
39 views

How to combine global standard deviation given several sample statistics?

Is there any approximation formula (best guess?) to calculate global std given multiple set statistics (size, mean, std)? I have an aggregated statistics from several sets. ...
6
votes
1answer
430 views

Which books should I read in order to be prepared to study information geometry?

At the moment, I am preparing my master's thesis (in statistics) and I intend to keep studying in order to pursue a doctoral degree. To be precise, I am mainly interested in studying Information ...
-1
votes
0answers
9 views

How to calculate Chi-Square density value only known P-value? [migrated]

Everywhere online there is how to calculate the Chi-Square density value given a confidence level: $\alpha$/p value; but I can not find how one calculates the inverse? How to calculate the $\alpha$/p-...
-1
votes
0answers
84 views

How to avoid using a probability distribution that doesn't exist?

I have this problem, of which I know the solution, but I'm looking for the mathematically proper way of writing it. Say I have a (infinite) population of people, where each individual is labeled by ...
0
votes
0answers
28 views

expectation of the exponential of the inverse of variable with Marchenko–Pastur distribution

This question is related to another answered before distribution on the inverse Wishart matrix eigenvalues summation my question is, is their finite expression for the expectation of \begin{align} {\...
2
votes
1answer
74 views

distribution on the inverse Wishart matrix eigenvalues summation

Let $\lambda_1>\lambda_2>....>\lambda_N$ be the ordered eigenvalues of Wishart matrix my objective is to find if it is possible the distribution of: \begin{align} s = \sum\limits_{i = 1}^...
-1
votes
1answer
35 views

Approximating expectation of exponential of Wishart matrix

I am trying to obtain an Approximating expectation of exponential of Wishart matrix $X (N,N)$ with $\operatorname{rank} (X)=K$defined as: \begin{align} J = E[{e^{{v^H}Xv}}] \end{align} where $v$ is $...
0
votes
0answers
46 views

Concentration (or two sided tail bounds around expectations) of maximum and minimum of $n$ iid, subgaussian random variables

I asked this on MSE, but got no answer, hence asking here now. Help appreciated! My question is motivated by this question and this question, where the first was aimed for giving a one sided tail ...
1
vote
1answer
49 views

Approximating expectation of the trace of inverse of a Gaussian random matrix combination

In order to characterize the performances of MIMO systems that depend directly on the distribution of the eigenvalues of random Hermitian matrix so I would like to feature the quality of some ...
2
votes
1answer
72 views

Expectation of the trace of inverse of a Gaussian random matrix

Given a $N×M$ random complex gaussian matrix $X$ and $N×K$ random complex gaussian matrix $Y$ I'm interested in approximating the expectation expressed as: \begin{align} E[trace({(aX{X^H} + I)^{ - ...
3
votes
0answers
182 views

Can information be extracted more precisely using more random trials?

Write $n$ iid draws of $(x,y)$ as $(x^n, y^n)$. Fix $R\in (0,H(y))$. What is the min of $n^{-1}H(x^n|f(y^n))$ over $f$ with $H(f(y^n))\leq nR$, taking $n\to \infty$?
0
votes
1answer
88 views

Independence in a sequential problem with observations getting added to buckets

Consider a sequence of random observations $(O(t))_{t\geq 1}$, with $O(t)=(D(t),J(t),Y(t))$. Denote $\mathcal{F}(t) := \sigma(O(1),\ldots,O(t))$, the filtration induced by the first $t$ observations. ...
-1
votes
0answers
36 views

Can we write down the density of this distribution?

Simple version: I am looking for the density of the random vector $(X+Z,Y+Z)$, where $X,Y,Z$ are independent gamma random variables (with non-restricted parameters). Next step: Actualy, i'm looking ...
2
votes
1answer
63 views

Non-asymptotic results for M-estimators?

Does anyone know if there are any standard non-asymptotic results for M-estimators? I'm looking for finite-sample guarantees. Figured maybe someone here might know.
0
votes
1answer
77 views

Is “$\mathbb{E}(T_n|X)\rightarrow 0 $ a.s.” equivalent to a statement that does not involve the Radon–Nikodym derivative as a black box?

Let $\{T_n\}_n$ be a sequence of random variables, and let $X$ be another random variable. Each $\mathbb{E}(T_n|X)$ is a random variable, therefore the statement "$\mathbb{E}(T_n|X)\rightarrow 0$ ...
0
votes
0answers
45 views

What is the distribution of the norm of the multivariate $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d?$

Let $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d$ follow a multivariate normal distribution. Then what's the distribution (PDF, CDF etc.) of $X?$ When $\mu = 0, \Sigma = I_d,$ we know that $||X||...
1
vote
0answers
61 views

Dependence rank: what is the size of the largest subcollection of random variables which is statistically independent?

Let $X_1,\ldots,X_p$ be random variables on the same space. Define their dependence rank, denoted $rank(X_1,\ldots,X_p)$ as the largest nonnegative integer $k$ such that there is a subcollection of $k$...
0
votes
1answer
34 views

Good upper-bound for $\mathbb E_A[e^{-t\|A\|_2}]$, for $t\ge0$ and random m by n matrix with iid entries with law $N(0,1)$

Let $A$ be a random $m$-by-$n$ matrix with iid $N(0,1)$ entries, $m$ and $n$ large with $n/m \longrightarrow \alpha \in (0, 1)$ . Let $\|A\|_2$ be the largest singular value of $A$ (i.e the spectral ...
4
votes
1answer
266 views

Inverse marginal property of a collection of $\sigma$-algebras

In my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space" I introduced the Inverse marginal property (IMP) for a collection of $\sigma$-algebras. Let $(\...
1
vote
1answer
141 views

Does kernel regression preserve monotonicity?

Consider the Kernel regression estimator: $$\hat{y}(x)=\frac{\sum_{i=1}^n{K(x-x_i)y_i}}{\sum_{i=1}^n{K(x-x_i)}},$$ where $x,x_1,\dots,x_n\in\mathbb{R}^d$, $y_1,\dots,y_n\in\mathbb{R}$, where $K:\...
7
votes
1answer
120 views

Books to develop a unified view of statistics and information theory?

I hope to understand the connection between statistics and information theory in a deep philosophical sense. I suppose the best place to start would be David MacKay's Information Theory, Inference, ...
1
vote
2answers
48 views

Non-parametric regression and curvature

Given a finite set of points $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$ in the plane, Linear Regression tells us how to find the straight line "$y=a+bx$" best approximating the given points, in the ...
0
votes
0answers
30 views

Non asymptotic error bound for non parametric estamation $f(x)=\mathbb{E}[Y|X=x]$

I am considering the following model: $(X_i,Y_i)_{i=1}^n$ are iid random pairs with $(X_i,Y_i)\in[0,1]^2$. Let $f(x)=\mathbb{E}[Y|X=x]$. Consider an estimate $\hat{f}_n$ of $f$. Under some hypothesis ...
3
votes
0answers
209 views

Upper-bound for eigenvalues of $E [UU^T]$, where $U$ is uniformly distributed on the unit $n$-sphere

Let $X$ be a $\sigma$-subGaussian random vector on $\mathbb R^n$ (for large $n \ge 3$), meaning that the random variable $X^Tv$ is $\sigma$-subGaussian for every unit vector $v \in \mathbb R^n$. ...
2
votes
0answers
93 views

Inequality on the Kullback-Leibler divergence

Let us define the arithmetic, geometric, and harmonic means of $x,y \in \mathbb{R}$ weighted by $\alpha =(\alpha_x,\alpha_y) \in [0,1]$, respectively as \begin{equation} a_\alpha(x,y) = \frac{\...
0
votes
1answer
107 views

Local behavior of the Vandermonde convolution

An interesting combinatorial identity is the Vandermonde convolution identity: $$ \sum_k {n\choose k}{m\choose s-k} = {n+m \choose s},$$ which can be proved by considering the coefficients in $(x+1)^...
0
votes
0answers
30 views

restriction of a formula with matrix inverse multiplied by a vector

I'm trying to reproduce a proof from this paper but I'm stuck in one point (Lemma 6). The general subject is bayesian model for multi-armed bandit problem solved with Thompson sampling. I think I ...
1
vote
2answers
59 views

What is the uncertainty on the (Pearson) correlation coefficient?

Do you know what is the uncertainty on the Pearson correlation coefficient as a function of the uncertainty on the measurement in the data set. I know of an expression giving the uncertainty related ...
0
votes
2answers
138 views

Statistical divergence

Does anyone know about a statistical divergence of this type? \begin{equation} \text{D}(P||Q) = \frac{1}{2} \left[\text{KL}(M||P) + \text{KL}(M||Q)\right] \end{equation} where $M = \frac{1}{2} [P+Q]$....
1
vote
0answers
53 views

Comparison of concentrations of different $L^p$-norms of (sub) Gaussian distributions

It's well-known that the Euclidean $2$-norm of subgaussian random vectors concentrates in high dimensions, e.g. when $X \sim \mathcal{N}(0,I_n),$ (or in general $X$ is subgaussian with independent co-...
0
votes
0answers
12 views

Statistics of trees in vertex-covering forests

What can be said about the properties of graphs $F$ that are generated from the edges of complete symmetric graphs $G(V,E)$ in the following way: fix an enumeration process for the edges in $E$ ...
3
votes
1answer
133 views

Prove or disprove the linearity of expectiles

For expectation (mean), there are many useful properties such as Linearity of Expectation: $\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y]$ $\mathbb{E}[\alpha X]=\alpha\mathbb{E}[X]$ (The 2 equations ...
0
votes
1answer
53 views

Tail bounds for the absolute difference of a coupled pair of sub-Gaussian random variables

Let $P$ and $Q$ are sub-Gaussian distributions on $\mathbb R$, and $(X,X')$ be a coupling of $P$ and $Q$, i.e $(X,X') \sim \pi$ for some distribution on $\mathbb R^2$ with marginals $P$ and $Q$. ...
2
votes
0answers
65 views

Tightness of Hilbert-space-valued arrays

Let $\mathcal{H}$ be a separable Hilbert space. Assume we have some triangular array $W_{n,j}, j=1, \ldots ,n $ of $\mathcal{H}$-valued random elements with $\mathbb{E} \Vert W_{n,j} \Vert_{\mathcal{H}...
0
votes
0answers
29 views

What is the number of iterations needed for the message passing algorithm to converge when applied to an acyclic factor graph?

I understand that the message passing algorithm (Belief Propagation algorithm), when applied to a factor graph consists in an exchange in messages between the factor nodes and the variable nodes, ...
0
votes
2answers
61 views

Martingale optional stopping before a stopping time

Here’s an easy one, I hope: Suppose $\tau$ is a stopping time and $(M_t)$ is a martingale which together satisfy the hypotheses of the optional stopping theorem so that $\mathbb{E}[M_\tau]= \mathbb{E}...
0
votes
0answers
60 views

Reference request for concentration on measure, following Vershynin's “High dimensional Probability” book, referred often in this question

I'm new to "concentration of measure" phenomenon that I need to learn quickly (started already, but would like to pick up the remaining basic results all within a week or two to get a working ...
1
vote
0answers
24 views

Bayesian posterior consistency when prior distribution is induced by a diffusion

Let $\Pi_{b,\sigma}$ be a prior distribution on $\{z_t\}_{t<T}\in C_0[0,T]$ induced by the following diffusion: \begin{align} d\tilde z_t&=b(\tilde z_t,t)dt+\sigma(\tilde z_t,t) dW_t, ~...
2
votes
1answer
87 views

Convergence of estimator given by a fixed point

Let $X$ be a non-negative random variable with cdf $F$ and define $$G(s) = E[\max(0,u(X)-sX)],$$ where $u$ is some real function. Let $s_0$ be the unique fixed point of $G$. Now let $X_1,\dots,X_t$ ...
1
vote
1answer
53 views

Conditional density for random effects prediction in GLMM

I am currently working on generalized linear mixed models (GLMM) and need some help concerning the prediction of the random effects. More specifically, I don't understand the given representation of ...

1
2 3 4 5
28