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.

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26
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848 views

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 ...
19
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1answer
428 views

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{...
11
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0answers
246 views

Is there a sense in which the Chi-squared distribution is an inner product?

I have been self-studying statistics recently, and the apparent similarities between linear algebra (especially Hilbert spaces) and statistics have been popping out to me. Linear independence gets ...
11
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0answers
198 views

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. ...
10
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1answer
190 views

Hottest Days of The Year

Recently, there has been much talk in the media of it being the hottest day of the year so far. It has always seemed to me that there are likely many more of these in the northern hemisphere than the ...
10
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0answers
967 views

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 ...
9
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0answers
429 views

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" ...
9
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0answers
247 views

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 $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
9
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0answers
717 views

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 $...
9
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0answers
415 views

Does this calculation have a name, or a generic formulation?

Background Informatiom I would appreciate help in identifying or explaining this operation: To calculate each of the $n$ values of $f(\Phi)$: Sample from the distribution of each of $i$ parameters, ...
8
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0answers
183 views

The sum of eigenvalues of integral operator $S(f)(x)=\int_{\mathcal{X}} k(x,y)f(y)d\mu(y)$ is given by $\int_{\mathcal{X}} k(x,x) d\mu(x)$?

Setup: Let $(\mathcal{X},d_{\mathcal{X}})$ and $(\mathcal{Y},d_{\mathcal{Y}})$ be two separable metric spaces. Let $M^1(\mathcal{X})$ be the space of Borel probability measures on $\mathcal{X}$ with ...
8
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1answer
133 views

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 ...
8
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0answers
160 views

Finding an upper bound for $\frac{d}{d\theta}\beta^*(\theta)|_{\theta=\theta_0}$

Suppose that a random variable X has a distribution depending on a parameter $\theta$, $\theta \in \Theta$, and consider a test of hypothesis $H_0: \theta = \theta_0$ versus the alternative $H_1: \...
8
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0answers
542 views

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 ...
7
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0answers
176 views

Only three types of limit of distributions truncated to a finite interval in the upper tail?

Suppose random variable $X$ has a continuous probability distribution with an unbounded upper tail; that is, the CDF of $X$ (call it $F$) is absolutely continuous and $F(x)<1$ for all $x\in\mathbb{...
7
votes
1answer
109 views

Does the absence of explicit probability space hinder the empirical application of statistical theory based on measure theory?

To motivate my question, I start off with a very simple example of prediction problem. Let's say Mike is interested in predicting the crime rate, which we denote as the random variable $y$, in the ...
7
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1answer
358 views

Category theory and statistics

I've been juggling with some concepts from statistics revolving around properties of estimators and sufficient statistics, and I can't help but notice that they have a strong categorical flavor, e.g. ...
7
votes
1answer
1k views

Kullback-Leibler divergence of binomial distributions

Suppose $P \sim \mathrm{Bin}(n,p)$ and $Q \sim \mathrm{Bin}(n,q)$. Their Kullback-Leibler divergence is defined by $$D_{KL}(P||Q)=\mathbb{E}_{P}\left[\log\left(\frac{p(x)}{q(x)}\right)\right],$$ with $...
6
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0answers
117 views

Proof of a technical fact in the book of Schapire and Freund on boosting

I am currently looking at Exercise 10.3, Chapter 10 of the book on Boosting by Schapire and Freund. More precisely, in the middle of the exercise they propose to use, without proof, the technical fact ...
6
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0answers
97 views

Expected number of digits of the smallest prime factor of $77^{77}-18$

Let $X$ be the number of digits of the smallest prime factor of $$77^{77}-18$$ which is a composite $146$-digit number. ECM indicates that the smallest factor has more than $30$ digits. Assuming ...
6
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0answers
161 views

Area of a triangle where the three vertices are randomly chosen on a circle; also $3D$ version.

My teacher gave us an interesting problem today. Consider a circle of radius $1$, choose three points on that circle at random and make a triangle connecting the three. On average what will the area ...
6
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2answers
149 views

Estimate Grade Distribution Based on Performance of Each Question

As the title states, I would like to be able to estimate the grade distribution of an exam based on the mark distribution of each individual question. To give a quick example of what I mean, suppose ...
6
votes
1answer
103 views

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 ...
6
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0answers
788 views

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(...
6
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0answers
192 views

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 & \...
6
votes
1answer
136 views

About cutting Almonds

Every year, during Christmas baking, I chop almonds, which causes me to puzzle over the same question, and I don't quite know how to approach it. I start out with N almonds. Let's assume they are all ...
6
votes
1answer
606 views

A maximal Hoeffding's inequality?

Let $X_1, \cdots, X_n$ be real-valued independent random variables satisfying $|X_k|\le 1$ and $\mathbb EX_k=0$. Hoeffding's inequality tells us that for any $k=1,\cdots, n$ and $t>0$, $$\mathbb P\...
6
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0answers
222 views

Does Multiplicative Version of Azuma's Inequality Hold?

We know that there are multiplicative version concentration inequalities for sums of independent random variables. For example, the following multiplicative version Chernoff bound. Chernoff bound: ...
6
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0answers
115 views

Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?

I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants): $$ \log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm T}\boldsymbol\Sigma^{-1}(\mathbf{x}-\...
6
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2answers
3k views

What do angle brackets ($\langle\rangle$ ) mean in mathematics/statistics (autocorrelations)?

Okay, so the logarithmic return on a stock is given by: $$r_τ (t) = \ln P(t+τ) - \ln P(t),$$ where τ is the interval of time. I have no problem calculating that. My question comes to the following ...
5
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0answers
52 views

How easy is it to create false evidence for a biased coin?

I have a biased coin which comes up heads with probability $p$. I know the value of $p$, but I want to falsely claim that the coin has a different probability of heads, $q$, where $q > p$. To ...
5
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0answers
3k views

When to use cumulative moving average vs a simple moving average?

After reviewing the Wikipedia page on moving averages, the difference between the simple moving average and cumulative moving average are clear: 1) Simple moving average only considers the last n ...
5
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0answers
85 views

Probability of the existence of a specific pattern in one million coin flips

I came across this question while preparing for an interview. Given a coin with head-up probability p, flip it $N = 1,000,000$ times. What is the probability that a string of "HHHHHHTTTTTT" (i.e. 6 ...
5
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0answers
114 views

Proving a function of the empirical distribution is a Martingale

Let $X_1, \dots, X_n$ be a sequence of i.i.d. random variables with distribution function $G$, let $$ G_t = \frac{\# \{ k : X_k \leq t \}}{n} $$ define the empirical distribution relative to the ...
5
votes
1answer
415 views

Sum of best X dice in Y dice rolled (or roll X pick best Y) odds/calculation

Background: In many pen and paper RPGs there is often an option or bonus/penalty to rolls that incorporates rolling multiples of the required die and taking the best or worst of those rolls for your ...
5
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0answers
84 views

On average, where is the lift?

This started as a computing problem with several variables, and I'd like to know if there's a closed form formula for the average position of the lift. Context: there's a building with $N$ floors and ...
5
votes
1answer
360 views

Find the minimum-variance unbiased estimator for given $\tau(\theta)$

Let $X = (X_1, \dots, X_n)$ - a sample from the distribution $U (0,\theta)$. Prove that $T(X) = X_{(n)}$ is complete and sufficient estimation for $\theta$ and find the minimum-variance unbiased ...
5
votes
0answers
328 views

Interpreting the Lindeberg's condition

I know the Lindeberg's CLT but I don't have a good grasp of the intuition behind the Lindeberg's condition. Could you please give some intuition behind said condition via an example (or, perhaps, via ...
5
votes
1answer
114 views

Version of Conditional Expectation

I would like to proof the following theorem. Let $(X,Y)$ be a random variable with values in $\mathbb{R}^2$. Supposte that $\mathcal{L}(X,Y)$ has density $f(.,.)$ with respect to Lebesgue measure $\...
5
votes
2answers
98 views

Determining number of randomly picked people

Firstly I want to put big disclaimer here. This particular problem is a smaller part of my homework. Since even after discussion with my fellow classmates we are not sure how to handle it we decided ...
5
votes
0answers
88 views

Random sphere-valued fields

I would like to generate random functions from an $m$-sphere $S^m$ to an $n$-sphere $S^n$ that are not too wild, some kind of generalization of random Gaussian fields. More precisely, I want $f(x)$, ...
5
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0answers
56 views

single variable is significant but overall test is not

I do a multiple regression with 3 independent variables $X_1$, $X_2$ and $X_3$. The correlation between $Y$ and $X_1$, $Y$ and $X_2$, and $Y$ and $X_3$, are each large and statistically significant. ...
5
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0answers
144 views

Divergence based robust inference

The term 'divergence' means a function $D$ which takes two probability distributions $g,f$ as input and puts out a non-negative real number $D(g,f)$. I have learnt that the inference based on ...
5
votes
0answers
927 views

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}{\...
5
votes
1answer
1k views

Distribution of $\frac{X}{|Y|}$, where X and Y are standard normal r.v.'s

Let X and Y be independent standard normal random variables. What is the distribution of $\large \frac{X}{|Y|}$? Attempt: Let $\large U = \frac{X}{|Y|}$ and $ V = |Y|$. This transformation is not ...
5
votes
0answers
279 views

Rate of convergence of a martingale

I have a question related to convergence rates of martingales: Assume that there is a sequence of maximized likelihood ratios: $ \frac{f_{\hat{\theta}_{n}} \left ( Y_{1},Y_{2},\dots,Y_{n} \right ) }{...
5
votes
0answers
354 views

What is ${\rm cov}(e_i, \hat y_i)$ in simple linear regression?

The model is $y_i = \beta_0 + \beta_1x_i + \epsilon_i$ What is ${\rm cov}(e_i, \hat y_i)$? What is ${\rm cov}(\epsilon_i, \hat \beta_1)$? What is ${\rm cov}(e_i, \epsilon_i)$? For 1, I am writing $...
5
votes
0answers
161 views

Distribution for ratio of dependent quadratic forms.

Random vector $\mathbf{x}_{0}$ $\sim$ $\mathcal{N}\left(\boldsymbol{\mu}, \mathbf{\Sigma} \right)$ is a sum of two orthogonal random vectors: $\mathbf{x}_{0}$ = $\mathbf{x}_{1}$ + $\mathbf{x}_{1}^{\...
5
votes
0answers
82 views

Deconvolution of distribution of diffraction reflexes

I'm a chemist stuck in a mathematical problem. Please bear with me as I'm trying to express myself in Math language. Let me explain in short terms the experimental method I'm using: X-ray diffraction....
5
votes
0answers
476 views

Multilinear or Tensor Regression?

Given input data $x_t\in \mathbb{R}^n$ and output data $y_t\in\mathbb{R}^m$, the closed form solution to $\min_A \sum_t \|y_t - Ax_t\|^2_2$ is given by $A = (XX^T)^{-1}XY^T$ where $x_t$ form the ...