Questions tagged [statistics]
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory and other branches of mathematics such as linear algebra and analysis.
11,536
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Does the average primeness of natural numbers tend to zero?
Note 1: This questions requires some new definitions, namely "continuous primeness" which I have made. Everyone is welcome to improve the definition without altering the spirit of the question. Click ...
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Kähler Geodesics
Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric
$$\begin{pmatrix} \frac{1}{1-|a|^2}&\frac{1}{1-a\bar{b}}\\\frac{1}{1-\bar{a}b}&\frac{1}{1-|b|^2}\end{...
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Asymptotic behavior of recurrence $x_{n+1}=\mbox{Stdev}(x_1,\dots,x_n)$
Here $x_1>0$ is the initial condition and $x_{n+1}$ is defined by
$$x_{n+1}=\Big[\frac{1}{n}\sum_{k=1}^n x_k^2 -\frac{1}{n^2}\Big(\sum_{k=1}^n x_k\Big)^2 \Big]^{1/2}.
$$
Clearly, $x_n=\lambda_n \...
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Why is the partition function able to describe the whole system?
No matter what the real system or subject is, if there is a partition function $Z$, then these kind of identities hold
$$\langle X\rangle=\frac{\partial}{\partial Y}\left(-\log Z(Y)\right).$$
If one ...
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Rigorous Proof of Slutsky's Theorem
I was hoping to type up my proof of Slutsky's Theorem and get confirmation on the excruciating details being all correct...
Statement of Slutsky's Theorem:
$$\text{Let }X_n, \ X,\ Y_n,\ Y,\text{ share ...
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Looking for references related to an inequality in order statistics
I was reading the paper "on the minimum of several random variables". In example 10 item (ii) it states:
Let $1\leq k\leq n$. Let $g_i,i\leq n$, be independent $N(0,1)$ Gaussian random variables. ...
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Random walks in $\mathbb{Z}^2$
Consider a random walk on the integer lattice in the plane. If a “particle” making a random
walk arrives at a lattice point $p = (k_1,k_2)$ at the time $t$, then one of the four neighbors
$(k_1±1, k_2 ...
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Donsker's Theorem for triangular arrays
Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
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Idempotence and the Rao–Blackwell theorem
Original question:
In the Wikipedia article on the Rao–Blackwell theorem, we read:
In case the sufficient statistic is also a complete statistic, i.e., one which "admits no unbiased ...
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Estimating Parameter - What is the qualitative difference between MLE fitting and Least Squares CDF fitting?
Given a parametric pdf $f(x;\lambda)$ and a set of data $\{ x_k \}_{k=1}^n$, here are two ways of formulating a problem of selecting an optimal parameter vector $\lambda^*$ to fit to the data. The ...
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What is the variance of self-information (or surprisal)?
The self-information of an outcome $x_i$, or surprisal, is defined as:
$$
I(x_i)=-\log P(x_i),
$$
where $P$ means probability. This way, the Shannon entropy can be seen as the "average" or "expected" ...
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Exponential distribution unbiased estimator
Let $$X_1, \ldots, X_n \overset{iid}{\sim} Exp(\lambda), \quad \lambda > 0$$
The Maximum-Likelihood-Estimator is given by $$\widehat{\lambda} = \frac{1}{\frac{1}{n}\sum_{i=1}^{n}{X_i}} = \frac{n}{\...
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the parametrization of a Gumbel in terms of a Gaussian
Extreme Value Distribution From a Gaussian. I was wondering how the parametrization of $\alpha$ and $\beta$ of a Gumbel $e^{-e^{-\frac{x-\alpha }{\beta }}}$ was done in terms of a cumulative Gaussian $...
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Covering number/Metric Entropy of the unit ball with respect to Mahalanobis distance
Let $B$ denote the unit ball on $\mathbb{R}^d$ and $N(\epsilon, B, d)$ be the cardinality of the smallest $\epsilon$-cover of $B$. An epsilon cover is a set $T \subset B$ such that for any $x \in B$, ...
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Distributions with 'Gaussian Tails'
In a paper I was reading, the following seemingly artificial assumption is used:
suppose $f$ is some probability density function on $\mathbb{R}^d$, and let $\phi$ denote the density of a $N(0,I_d)$ ...
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