# 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.

1,376
questions

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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

**1**answer

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**

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**0**answers

26 views

### Simple Linear regression b_1 proof [closed]

How do I go from the second equation to the third one?
image

**-1**

votes

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

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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 $...

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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**

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**0**answers

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

**1**answer

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

**1**answer

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**

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**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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**

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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

**1**answer

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

**1**answer

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

**0**answers

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||...

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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**

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**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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, ...

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vote

**2**answers

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

**0**answers

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

**0**answers

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

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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

**1**answer

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)^...

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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 ...

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vote

**2**answers

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

**2**answers

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 $M = \frac{1}{2} [P+Q]$....

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vote

**0**answers

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**

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**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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}...

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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

**2**answers

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}...

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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 ...

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vote

**0**answers

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

**1**answer

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

**1**answer

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 ...