# 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.

1,264
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### Overview of interpretations of classical probability

The Stanford Encyclopedia of Philosophy has a nice overview of numerous different interpretations of probability (classical as opposed to quantum) with an extensive bibliography.
What books would ...

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### Concentration or distribution of the scaled $l_p$ norm of a correlation matrix

Background:
Among Hermitan random matrices, correlation matrix has a lot of applications in statistics. People have studied the "empirical spectral distribution (ESD)" of a correlation matrix, the ...

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### Low rank matrix recovery — generalization of spectral method to nonlinear noise model

Suppose $X^{\star}$ is an $n \times n$ matrix with rank $r$, where $r \ll n$. We aim at recovering $X^{\star}$ from a noisy observation $Y$.
People have throughly studied the noise model $Y_{ij} = X^...

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### How to perform regression with different error variances? [migrated]

I have two series of measurements values, first series is X and second is Y.
I need to model Y as a function of X, where I know the method that was used to measure X is two times better then the ...

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50 views

### Reference: Maximum Entropy Principle

It seems that the derivations of the maximum entropy distributions is a "well-known" fact and so it is in the continuous and discrete cases.... However, I can't seem to find a proof/formal statement ...

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57 views

### Convexity of exponential family

It is known that (given a $\sigma$-finite Borel reference measure $\nu$ on $\mathbb{R}$) the parameter space of an exponential family is convex in Euclidean space. However, my question is, for an the ...

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45 views

### On $\ell_1$ to $\ell_1$ operator norm of matrix with inverse Wishart distribution

Consider a random $n\times p$ matrix $X$ with $n\ll p$ and all entries of $X$ i.i.d. standard normal. For this $X$, the system of linear equations $y=Xw$ has infinitely many solutions, and the one ...

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46 views

### Information theoretic lower bounds for sparse recovery

For the well-known problem of sparse recovery using $\ell_1$ minimization, it was shown in this paper that for any random measurement matrix, a recovery procedure that succeeds with constant ...

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22 views

### Minimal representation of t(y)

I have an extended F-distribution with density
$$\frac{1}{B(0.5d,0.5d)}y^{d/2-1}(1+y)^{-d}$$
for $y \geq 0$ and $d>0$ and ${B(0.5d,0.5d)}$ is the beta function.
I have in canonical form the ...

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160 views

### Bounding the sensitivity of a posterior mean to changes in a single data point

There is a real-valued random variable $R$. Define a finite set of random variables ("data points") $$X_i = R + Z_i \; \text{for } i\in\{1,\ldots,n\},$$ where $Z_i$ are identically and independently ...

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53 views

### Physical meaning of dividing the mean square by variance of a distribution

In the field of chromatography, the so-called "efficiency" a Gaussian peak or at times an exponentially modified Gaussian peak is expressed as the mean squared divided by the variance of the peak. The ...

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168 views

### Find $\inf_{P_{X_1,X_2}}P_{X_1,X_2}(\|X_1-X_2\| > 2\alpha)$ , where $\alpha > 0$ and inf is over couplings

Let $\mathcal X$ be a seperable Banach space with norm $\|\cdot\|$, and let $X_1$ and $X_2$ be random vectors on $\mathcal X$ with finite means.
Question. Given $\alpha > 0$, what is value of, ...

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41 views

### Expectation of $\left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2$, with complex Gaussians?

Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...

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65 views

### Concentration inequality for minimal eigenvalue of sample covariance

I was reading an article of matrix completion and met the following lemma
The concentration inequality for $\sigma_{\max}$ part is a standard result. However, I didn't find any results like the $\...

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26 views

### Linear combination of constrained random variables, and its convergence

I have positive random variables X1, X2, X3, ..., Xn such that their sum=1 (so they are random, subject to constraints that each Xi is positive their sum has to be 1.. so all are fractions).
Now, I ...