# Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

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### Expected squared distance from origin of training points vs. test points

This is from Exercise 2.4 (Page 39) of Elements of Statistical Learning: The edge effect problem discussed on page 23 is not peculiar to uniform sampling from bounded domains. Consider inputs drawn ...
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### Time evolution of a Bayesian posterior

I have a question regarding the time evolution of a quantity related to a Bayesian posterior. Suppose we have binary parameter space $\{ s_1, s_2 \}$ with prior $(p, 1-p)$, The data generating ...
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### Intuitive explanation for Marchenko-Pastur law

I am looking for an intuitive reasoning behind the Marchenko Pastur law, which is described as a law of large numbers analog for random matrices. I know the law gives the probability density function ...
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### quantile of standardized t distribution

How to show that, for any given left tail probability, the corresponding quantile of standardized t distribution is increasing in degree of freedom for left tail probability less than 0.5? This is ...
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### Cox's Theorem: ignorance, objective priors, and the Mind Projection Fallacy

I've been trying to understand Cox's Theorem and the problems surrounding it. There's so much information on this topic that I've become confused as to the exact state of the theorem. I've gathered ...
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### Reference Request: Information Geometry for Ridge Regression

I am reading the book "regression estimators" by Gruber 2010 where he uses this technique to compare Ridge Regressors, however he concentrates on deriving the mathematical results without ...
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### Why is the $\chi^{2}$ approximation for deviance GLM $\sim \operatorname{Binomial}(n_{i},\pi_{i})$ not valid when $n_{i} = 1$?

I know from McCullagh & Nelder's text (p.118) that the $\chi^{2}$ approximation for deviance for the binomial family is based on a limiting operation in which $n$, the number of observations, is ...
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### I'm not asking for a conjugate prior. Is there a distribution $p(x|y)$ that satisfies $\int p(x|y)Beta(y|a,b) dy = Beta(x| a', b')$?

I know the result of integrating a Gaussian against another Gaussian is still Gaussian, $$\int N(x|\mu_y,\sigma_y)N(y|\mu,\sigma) dy = N(x|\mu',\sigma')\quad.$$ Can I get the same form for Beta ...
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### Cox's Theorem: the necessity of (un)countably additivity

I've been trying to understand Cox's Theorem and the problems surrounding it. There's so much information on this topic that I've become confused as to the exact state of the theorem. I've gathered ...
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### Why low rank expansions can exploit the redundancy that exist between different feature channels and filters?

I read Jaderberg et al., 2014 paper about Speeding up Convolutional Neural Network with Low Rank Expansions. In the introduction, it is written in bold font: Our key insight is to exploit the ...
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### What is an example of data where the permutation test succeeds but a normal t-test fails?

In literature, I normally see authors use a two sample permutation test on normal data to show that it works as well as the two sample t-test. However, the real power for permutation tests should be ...
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### Kurtosis, bias, unbiased and statistics

I apologize ahead of time if this is too vague or meta to be a valid question. I've been looking at Algorithms (Sedgewick & Wayne). They define a class stdstats. In that they define min, max, ...
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### Help with a proof of Bayes classifier optimality

I have a class assignment to provide a proof that Bayes classifier for the two label version is optimal in that it's error rate is always ${\le}$ any other classifier. I've never worked through a ...
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### Derivation of prediction intervals for a normally distributed population with unknown population standard deviation

I have via the ISO standard 16269 found the solution to a problem that I've been working on. Based on a couple of independent samples from a normally distributed population, I would like to determine ...
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### Is there a commonly-accepted/used notion of parametric statistical model equivalence?

To fix notation, let a set of possible data $X$ and a set of admissible parameter values $\Theta$ be given. Let $\mathscr P(X)$ be the set of probability distributions on $X$. A parametric ...
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### Which is a better estimator, averaged functions vs. A function of an average?

Problem: Assume that we want to estimate $f(\theta)$ with a pre-specified strictly increasing function $f$ and a parameter $\theta$. Let $\hat{\theta}_1$ and $\hat{\theta}_2$ be unbiased estimators ...
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### Deriving spectral measure

While reading this book, I got stuck on page 266 where the authors found the spectral measure $F(du)$ of the generalized covariance function $K(h) = \Gamma(-\alpha/2) |h|^{\alpha}, ~0<\alpha<2.$ ...
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### A consistent estimator with infinite expectation?

Typical (or common) approaches to prove an estimator is consistent require finite mean and variance. The proofs usually follow from concentration bounds, e.g. Markov, Chebyshev, etc. I'm wondering ...
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### The difference of normal means is also minimax?

Let $X_i \sim N(\xi, \sigma^2)$ and $Y_i \sim N(\eta, \tau^2)$ for known $\sigma^2$ and $\tau^2$. I know that $\bar{X}$ and $\bar{Y}$ are minimax under squared error loss since their variance is ...
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$y_{i}$ is a kx1 matrix, $x_{i}$ is a kxk matrix, $\beta$ is a 1xk matrix of coefficients and $u_{i}$ is a kx1 matrix of error terms. $y_i=\max\{0, x_{i}^{\prime}\beta + u_{i}\}$ and $med(u_{i}|x_{i}... 0answers 83 views ### Why is this statistic F-distributed? A book I'm reading claims that the statistic:$\frac{(RSS_0 - RSS_1) / (p_1 - p_0)}{RSS_1 / (N - p_1 - 1)}$has an F distribution. Why is this? I know that an F distribution is something like$\frac{\...
I'm trying to formulate the solution for the following problem: I was thinking in finding the equivalent distribution on $X_i$ based on $Y_i$, but I think I'm cheating. I think that the autor wants ...