Questions tagged [mathematical-statistics]

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

2,159 questions with no upvoted or accepted answers
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17
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806 views

Is there a general expression for ancillary statistics in exponential families?

An i.i.d sample $X_1,\dots,X_n$ from a scale family with c.d.f. $F(\frac{x}{\sigma})$ has $S(X)$ as an ancillary statistic if $S(X)$ depends on the sample only through $\frac{X_1}{X_n},\cdots,\frac{X_{...
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Are there unbiased, non-linear estimators with lower variance than the OLS estimator?

Consider an ordinary least squares model, $$y = \beta X + \epsilon \qquad \epsilon\sim N(0, \sigma)$$ The Gauss-Markov theorem tells us that the ordinary least-squares (OLS) estimator is the minimum-...
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1k views

What does it mean to take the expectation with respect to a probability distribution?

I see this expectation in a lot of machine learning literature: $$\mathbb{E}_{p(\mathbf{x};\mathbf{\theta})}[f(\mathbf{x};\mathbf{\phi})] = \int p(\mathbf{x};\mathbf{\theta}) f(\mathbf{x};\mathbf{\phi}...
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2k views

Taylor Series and Multivariate Delta Method

I asked this question on https://math.stackexchange.com/ but did not get any answer. Sorry for cross posting. I'm trying to understand delta method for matrices and vectors to find the variance-...
7
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1answer
10k views

What does "def" above an equals sign mean?

I am reading this: https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf and on equation (17), there is a def on top of the equal sign. What does this mean?
7
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993 views

Proof of Kolmogorov-Smirnov test

Could someone provide me a reference, preferably a book, where I can find detailed proofs and explanations of the Kolmogorov-Smirnov test (including the two-sample variant) and the derivation of the $...
7
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1answer
227 views

Finding the distribution of sample range for a Beta population

Let $X_1,X_2,\ldots,X_n$ be i.i.d random variables having density $$f(x)=2(1-x)\mathbf1_{0<x<1}$$ I am trying to derive the distribution of the sample range $R=X_{(n)}-X_{(1)}$. The usual way I ...
6
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44 views

Simulate correlate random variables with given marginal distribution where one is always larger

Is it possible to simulate pairs of random variables with a given marginal distribution and population correlation where one random variable is larger than the other? More formally, I need to simulate ...
6
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0answers
107 views

How do we call a more extreme case of fat tails than a power law?

According to Wikipedia the most extreme case of a fat tail follows a power law: The most extreme case of a fat tail is given by a distribution whose tail decays like a power law. That is, if the ...
6
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276 views

Strange connection between Bernouilli, Uniform and Geometric distributions

Final update on 11/29/2019: I have worked on this a bit more, and wrote an article summarizing all the main findings. You can read it here. Let us consider $Z = X_1 + X_1 X_2 + X_1 X_2 X_3 +\cdots$ ...
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91 views

French website Providing Instruction/Tutorials on Statistical Theory

This is somewhat of an odd question for CV, but since it's a question about statistical education, I think it falls within the scope of CV. Several years ago I stumbled across a French website that ...
6
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1answer
118 views

Asymptotic joint distribution of the sample medians of a collection and a sub-collection of i.i.d. random variables

Let $X_1,\dots, X_n$ denote i.i.d. random variables, with a smooth density $f$ having unique median $m$. It is known that the sample median $M_n$ of $X_1,\dots, X_n$, $M_n = \operatorname{med}(X_1, \...
6
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208 views

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 ...
6
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673 views

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 ...
6
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221 views

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 ...
6
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1answer
366 views

minimizer weighted linear regression

In a regression problem, with $y=X\theta+\epsilon$ and $X$ is an $n$ by $p$ matrix the ‘weighted least squares estimate is the minimizer $\theta^{*}$ of $f(\theta)=\sum_{i=1}^{n}\omega_{i}(y_i-x_i^{'}\...
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2k views

Deriving the maximum likelihood for a generative classification model for K classes

In Christopher Bishop's book "Pattern Recognition and Machine learning", there is the following question: Consider a generative classification model for $K$ classes defined by the prior class ...
6
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264 views

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 ...
6
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158 views

Hypothesis test on the Euclidean length of an unknown vector

Question Suppose I observe a vector $\mathbf{x}=[X_1 \ldots X_n]$, where each $X_i=m_i+n_i$, with $n_i$ being an independent zero-mean Gaussian random variable with variance $\sigma^2$ (i.e. $n_i\sim\...
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134 views

Orthogonal intersection in a Riemannian manifold

Let $S$ be the set of all probability distributions on $\mathbb{R}$ and $S_n=\{p_\theta\}$ be an $n$ dimensional submanifold of parameterized family of probability distributions on $\mathbb{R}$ where $...
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70 views

Proof: Nearest Neighbor classifier achieves Bayes rate asymptotically on countable domains

I am trying to understand in which situations the 1-NN classifier asymptotically attains the Bayes error rate. My intuition is that if the domain is countable, then 1-NN will asymptotically do as well ...
5
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74 views

Symbolic Formulae for Linear Mixed Models

I would like to understand how to create a good formula for a linear mixed model, using the Machines data set from the package ...
5
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0answers
75 views

Does Fisher scoring always outperform Newton optimization?

My understanding is that Fisher scoring has several advantages over Newton raphson optimization such as Computational efficiency: if certain conditions are met (example:During MLE estimation, if link ...
5
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0answers
87 views

Inequalities on Fisher Information / expected second derivative?

Under some regularity conditions we can compute fisher information as $ - \mathbb{E}_{\theta_0} [\frac{\partial}{\partial \theta^2} \ln f(x;\theta_0)] $ I was wondering if there are some kind of ...
5
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2answers
94 views

How to win this dice probability game?

The game is a variation of Pig. Here is how the game works: There are about 20 players. Each round, a single six sided die is rolled. All players add that rolled number to their "bank." However, if a ...
5
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386 views

Why we really need the concept of "Local" Rademacher complexity?

Recently, I have been studying High-Dimensional Statistics: A Non-Asymptotic Viewpoint written by Martin J. Wainwright. In this book, the author uses a special complexity measure which is called Local ...
5
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246 views

Is an inadmissible estimator necessarily dominated by some admissible estimator

Basic example: $X$ has a $p$-variate iid standard Normal distribution; the sample mean is not admissible if $p>2$ and is dominated by the Stein shrinkage estimator. However, the Stein shrinkage ...
5
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1answer
2k views

How do I get the p value of AD test using the results of scipy.stats.anderson()

I am conducting an Anderson Darling test for normality using scipy.stats.anderson() command in python. I am getting test statistic, critical values at various significance levels as the output. The ...
5
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0answers
446 views

Why can't the complete class theorem be easily generalized to all locally-compact spaces?

So I was reading Christian P. Robert's The Bayesian Choice, going through the constellation of results related to complete class theorems, and I don't see why all of them are necessary. In particular, ...
5
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1answer
242 views

Covariance of order statistics convergence?

Suppose I have a sample $(X_1 \dots X_n)$ and $(Y_1 \dots Y_n)$, all of which are $N(0,1)$ random variables. I am interested in the asymptotic behaviour of $$\frac{1}{n} \sum_{i=0}^n X_{(i)}Y_{(i)} $$...
5
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0answers
2k views

Calculating a confidence interval for a weighted sample

In a nutshell, I'd like to compute a confidence interval for some weighted sample day where the final value I'm seeking is a sum of different weighted samples. Illustrated here with a toy example set ...
5
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0answers
154 views

Finding the Cauchy Principal Value Mean of the pdf for $Z=XY$ where $\mathbb{E}[Y]$ does not exist

Let's first define the Cauchy Principal Value Mean (PVM). For a continuous random variable $V$ with pdf $f_{V}(v)$, the PVM of $f_{V}$ is \begin{equation} \mathrm{PV}(\mathbb{E}[V])=\lim_{a\to\infty}\...
5
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0answers
293 views

Variance of quotient of Poisson random variable and sum of the Poisson sample

Let $$Y_1\sim \operatorname{Poisson}(\lambda_1)\\Y_2\sim \operatorname{Poisson}(\lambda_2),$$ where $Y_1$ and $Y_2$ are independent, and $\lambda_1, \lambda_2>0$. What is the variance of $$\frac{...
5
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777 views

Under what conditions will a Bayesian posterior fail to converge to a point mass?

Let's say you have a Bayesian model: $$\theta' \sim g(\theta|\mu) $$ $$ y \sim p(y|\theta')$$ And we have some data ($n$ data points) $\mathbf{y}_n$, which we will use to perform inference on $\...
5
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0answers
63 views

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 ...
5
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0answers
337 views

Calculating Formula with logs and Weighted Average

I am trying to figure out an equation from the following paper by Cadena and Kovak (2016): http://pubs.aeaweb.org/doi/pdfplus/10.1257/app.20140095 on pages 264 and 265. I don't think the context is ...
5
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0answers
259 views

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 ...
5
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0answers
221 views

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 ...
5
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1answer
403 views

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 ...
5
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0answers
1k views

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, ...
5
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0answers
3k views

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 ...
5
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0answers
983 views

Conditional expectation everywhere non-zero while unconditional one zero?

I have a real-valued random variable $X$ that takes on positive and negative values, and $$E(X)=0 \tag{A}$$ There is also another real-valued random variable $Y$, not independent from $X$, neither $...
5
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0answers
203 views

Distribution of $X'X$ if $X\in\mathbb{R}^{T \times N}$ and $X_i'\sim N(\mu,\sigma^2I_N)$

Let $x_i\in\mathbb{R}^N$ be multivariate normal distributed with mean vector $\mu\in\mathbb{R}^N$ and no correlation: $x_i\sim N(\mu,\sigma^2 I_N)$. Given $T$ iid samples, define the matrix $$X:=\left(...
5
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0answers
128 views

Relation between asymptotic relative efficiency for tests and estimators

The asymptotic relative efficiency for unbiased estimators is the limit of the ratio of the variances as the $n\rightarrow \infty$. Is there a relation to asymptotic relative efficiency according to ...
5
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0answers
309 views

Restriction matrix for a VAR

In New Introduction to Multiple Time Series Analysis by Luetkepohl (2005), section 5.2.1, it says that one can specify linear restraints for a VAR, $Y = \beta X + U$, in the form $$ \operatorname{vec}{...
5
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0answers
673 views

Sum of independent Wishart with same degrees of freedom but different scale matrices

Is there any result showing that a sum of independent Wishart with same degrees of freedom but different scale matrices is a Wishart? For example, if I have two random variables: $$ Y \sim W_p(n,\...
5
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0answers
349 views

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 ...
4
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0answers
89 views

What conditions are needed for $a_n = O_p(n^d) \implies E[a_n] = O(n^d)$?

Let $X_n$ be a uniformly integrable sequence of random variables. In a recent question I asked about the possibility of converting Big $O_p$ convergence in probability of the sequence $X_n$ to Big $O$ ...
4
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0answers
132 views

What is it called when a random variable is weakly greater than another for all elements of the sample space?

Suppose I have random variables $(X_1,X_2)$ defined on a probability space $(\Omega, \mathcal{F},P)$ such that for any element $\omega \in \Omega$, $X_1(\omega) \geq X_2(\omega)$. I'm looking to work ...

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