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.
11,539
questions with no upvoted or accepted answers
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Expectation of maximum of minimums of permutations
Assume $n$ random permutations $\pi_1,\pi_2,\ldots,\pi_n: \lbrace 1,2,\ldots,m \rbrace \rightarrow \lbrace 1,2,\ldots,m \rbrace$. Let $X_i = \min(\pi_1(i),\pi_2(i),\ldots,\pi_n(i))$ and $Y = \max(X_1, ...
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Expectation of truncated log-normal
Let's assume that $y=e^x$, where $x\sim N(\mu,\sigma^2)$, that is, $y$ follows a lognormal distribution.
I'm interested in finding how $\mathbb{E}\left[y|y\geq a\right]$ varies with $\mu$ and $\...
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Find a function such that follows to normal in distribution
Suppose that $X_{n}\sim \text{Binomial}(n,\theta)$, where $n=1,2,\ldots$ and $0<\theta<1$. Find a function $g$ such that $\sqrt{n}(g(\frac{1}{n}X_n)-g(\theta))\xrightarrow{D} N(0,1)$ for each ...
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An estimator for the c.d.f $F$ at a point $x_0$?
Problem: Let $X_1,X_2,\ldots,X_n$ be independent identically distributed random variables (i.i.d's) with common CDF $F$. Fix $x_0\in\mathbb{R}$ and find an unbiased estimator for $F(x_0)$. Show that ...
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The distribution of the ith order statistic for discrete random variables
Assume $(X_i)_{i=1,...,n}$ are a sequence of real iid random variables with continuous density $p_x$.
We know that $$Y:=\sum_{i=1}^n 1\{X_i\leq u\}\sim Bin(n,F_x(u)),$$ since $1\{X_i\leq u\}\sim Ber(...
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Decrease of entropy when iterating a random discrete function
Let $m$ be a positive integer. Let $S$ be the set of non-negative integers $x$ less than $m$, with $|S|=m$. Let $X_0$ be the discrete uniform distribution over $S$, with $P(x)=\begin{cases}
1/m & \...
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What is the Fisher information for the parameter $\theta$ in a uniform distribution with likelihood $f(X,\theta)=\frac1\theta 1\{0\le x\le\theta\}$?
If X is U[$0$,$\theta$], then the likelihood is given by $f(X,\theta) = \dfrac{1}{\theta}\mathbb{1}\{0\leq x \leq \theta\}$. The definition of Fisher information is $I(\theta) = \mathbb{E} \left[ \...
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Smallest eigenvalue of matrix with random elements (non-central Wishart)
Suppose that $X \in \mathbb R^{d \times n}$ is a random matrix with independent entries, each of which follows the standard normal law $\mathcal N(0, 1)$, and that $M \in \mathbb R^{d \times n}$ is a ...
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The statistical average of a continuous value: $\overline{O} = \int O(x) \rho(x) dx$, but coordinate invariant
I am trying to solve a Lagrange multiplier problem for the following equation
$$
L= - \int_{-\infty}^\infty \rho(x) \ln \frac{\rho(x)}{q(x)} dx + \alpha \left( 1- \int_{-\infty}^\infty \rho(x) dx \...
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How to lower bound $\tau$ based on the expression of $H$?
Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ normalized symmetric Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \...
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Is the Fisher-Information even continuous in a regular statistical model?
Definition (Regular Model [1, p. 203]).
A standard statistical model $\big( X, \mathcal F, (\mathbb P_{\vartheta})_{\vartheta \in \Theta}\big)$, where $\Theta \subset \mathbb R$ is an open interval, $...
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Why doesn't the Borel-Kolmogorov paradox cause problems in practice?
The Borel-Kolmogorov paradox shows that the usual formula for conditional density $f_{X|Y}(x|y) = f_{X,Y}(x, y)/f_Y(y)$ can lead to inconsistent results depending on the coordinate system that is used ...
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Correct measure in concentration inequalities or hypothesis testing
In most discussions of concentration inequalities or calculations of rejection region in hypothesis testing, the measure used is left vague. For example, for independent random variables $X_1, \ldots, ...
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Is there a pattern to the coefficients in the piecewise equations of the Irwin–Hall distributions?
Intro and Problem Statement
The Irwin–Hall distribution is a probability distribution of the sum of $n$ independent, uniformly-distributed, continuous random variables in the interval $[0, 1]$. The ...
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Estimate nearly-singular Gaussian covariance matrix
Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need sample covariance $\Sigma_m$ to be $\epsilon$-close to true covariance $\Sigma$:
$$E\|\Sigma_m-\Sigma\| \le \epsilon \|\...
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