# 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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### 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{...
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### 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-...
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### Simple Linear regression b_1 proof [closed]

How do I go from the second equation to the third one? image
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### 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 ...
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### 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 ...
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### 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)^{ - ...
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### 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 $n\to \infty$?
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### 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. ...
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### 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 ...
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### 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.
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### 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 "$\mathbb{E}(T_n|X)\rightarrow 0$ ...
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### 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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### 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 ...
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### 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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### 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-...
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### 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 $E$ ...
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### 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 ...
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### 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 $Q$. ...
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### 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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### 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, ~...
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 $X_1,\dots,X_t$ ...