The Stack Overflow podcast is back! Listen to an interview with our new CEO.

Questions tagged [mathematical-statistics]

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

1,474 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
3
votes
0answers
93 views

Likelihood with random censoring

Suppose to observe a random sample from a r.v. $Y_i=\min(T_i,C_i)$ where $T$ and $C$ are iid absolutely continuous distribution. I would like to inference about a parameter of $T$ (for example, $\...
3
votes
1answer
62 views

Is it valid to calculate propensity score for each treated individual separately?

I have temporal twitter data, and I want to calculate propensity score for the treatment and control group. The problem is, the treatment happened at different time for different user, and I want to ...
3
votes
0answers
55 views

Deriving Wilk's distribution for two normal independent variables

Let $X$ and $Y$ be two independent, normal random variables with known and equal variances, let $\{X_1,\ldots,X_m\}$ and $\{Y_1,\ldots,Y_n\}$ be random samples of size m and n respectively. We are ...
3
votes
0answers
185 views

minimum number of rolls necessary to determine how many sides a die has

For example, say you have a black box that has a number of n-sided dice in it. You have 4-sided dice, 6-sided, 8, 10, 12, 20 and so on. The die sides are all A except for one side that says B, e.g. ...
3
votes
0answers
39 views

Why to care about local uniform convergence of characteristic function in proving multivariate CLT?

Let $X_{1},X_{2}, \dots \sim_{i.i.d.}$ some Borel probability distribution on $\mathbb{R}^{k}$; let $\mathbb{E}X_{1} \equiv m$; let $\text{var} X_{1} \equiv V$. Then it can be shown that $\sqrt{n}(\...
3
votes
0answers
102 views

Monte carlo method and Convergence in Distribution

Monte Carlo method, from what I could gather, allows one to obtain observations/draws from a possibly unknown statistical distribution. Let's say $T(X)\sim F$, where $T$ is a statistic, $X$ is a ...
3
votes
0answers
94 views

Does MCMC method can be used to calculate the mean and variance of the distribution of random variable functions?

I am not professional in Probability & Statisticsin, in order to clearly describe my problem, please be patient of the long introduction.THANKS! Background of my question Assume I have several ...
3
votes
0answers
85 views

Deriving bias of local linear regression

I have been reading Elements of Statistical Learning, and in Chapter 6 on Local Regression, they state the following for fitting local regression at point $x_0$ from data $\mathbf{x}$ of size $\rm{dim}...
3
votes
0answers
483 views

How to use pooled results from multiple imputation?

I've been reading some posts about data imputation using multiple imputation, specifically the MICE R package. I get the main idea of creating multiple datasets with imputed data. The part that is not ...
3
votes
0answers
607 views

Beta Distribution Fitting with constrained location and scale

I'm fitting a set of beta distributions. My data is constrained to live on $[0,1]$, both theoretically and empirically. A typical output of scipy.stats.beta.fit ...
3
votes
0answers
40 views

Verifying long-range dependency in multi-variate time series

I am fairly new to the area of time series and I am trying to understand the notion of long-range dependence in time series. My goal is to characterize the same in the case of multi-variate time ...
3
votes
0answers
45 views

(When not assuming differentiability) what is the definition of Fisher information?

Here we assume (for simplicity) that the parameter $\theta$ is one-dimensional. When one has sufficient regularity and can push in partial derivatives to and pull out partial derivatives from the ...
3
votes
0answers
20 views

Difference between a translationary invariant and a stable distribution

I understand that a stable distribution is a distribution whose linear combination of two independent random variables with this distribution has the same distribution (ignoring location and scale ...
3
votes
0answers
111 views

Weakest assumptions for central limit theorem to hold?

Let $X_i$, $i=1,2,...$ be any sequence of random variables (i.e. not necessarily independent or identically distributed). What are some weak conditions you know of on $X_i$ that makes $\sum_i X_i$, ...
3
votes
0answers
286 views

Derive Conditional Variance from Covariance Matrix

Given the mean, variance, and covariance of two random variables $X$ and $Y$, how would I find $\operatorname{Var}(Y\mid X)$? I know that I can find $E(Y\mid X)$ using the definition of covariance ...
3
votes
1answer
1k views

Backpropagation proof and usage confusion

I've been taking Andrew Ng's course on Coursera, and although it has been great so far, I loathe his lack of supplementary documents on proofs. Thankfully, there are some great articles found pretty ...
3
votes
0answers
140 views

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

Let $Y_1\sim \operatorname{Poisson}(\lambda_1)$ and $Y_2\sim \operatorname{Poisson}(\lambda_2)$, where $Y_1$ and $Y_2$ are independent, and $\lambda_1, \lambda_2>0$. What is the variance of $\frac{...
3
votes
0answers
82 views

Finding the asymptotic distribution of $\frac{1}{n+k}\sum_{i=1}^{n}X_i$

Let $(X_1,X_2,...,X_n)$ be a random sample from a population distribution with $E(X_i)=\mu $ and $Var(X_i) =\sigma^2$ for all $i$ find the asymptotic distribution of $T= \frac{1}{n+k}\sum_{i=1}^{n}X_i$...
3
votes
0answers
469 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 $\...
3
votes
0answers
111 views

Prove that the joint density of independent multivariate normal variables is a matrix-normal

Let $X_1,...,X_n \sim N_p(\mu_i,\Sigma_i)$ be Multivariate Normal a.v. independent. Show that $W = (X_1,...,X_n) \sim MN(M,\mathbb{I},\Sigma)$ where $M = [\mu_1 \mu_2...\mu_n]$ and $\mathbb{I}$ ...
3
votes
0answers
38 views

Valuation of players across uneven games

I have a 'portfolio' of players that I need to evaluate and compare across many games. There is a common metric,let's call it value, I am using to evaluate a players performance in a single game. ...
3
votes
0answers
146 views

Check that a statistic is complete

I have a question regarding completeness of a statistic. So the problem is: $n$ numbers are chosen randomly and independently between $a$ and $b$ ($0 < a < b$) but the information about $a$ and ...
3
votes
0answers
117 views

Why is the norm of a linear function $\| w \|^2$ and not $\langle c, X X^T c \rangle $?

I asked Why is the regularization penality equal to $\langle c , Kc \rangle_{R^n}$ when using the kernel trick in Tikhonov regularization? and got good answers, however, there is still one detail that ...
3
votes
1answer
432 views

Is Complete Statistic Uncorrelated with Ancillary Statistic

By Basu's theorem, we know that any ancillary statistic is independent of a statistic that is both sufficient and complete. I was wondering if the assumption of sufficiency and completeness can be ...
3
votes
0answers
150 views

Does clustering lead to overdispersion?

TL;DR Clustering is often cited as a source of overdispersion in count data. However, I seem to arrive at the conclusion that clustering actually reduces the dispersion. Could someone confirm this ...
3
votes
0answers
62 views

Finding the limiting distribution of sample covariance $\hat{\sigma}_{XY}$?

How can I find the limiting / asymptotic distribution of $\sqrt{n}(\hat{\sigma}_{XY}-\sigma_{XY})$, provided $\sigma_{XY}=E[(X-\mu_X)(Y-\mu_Y)]$ and $\hat{\sigma}_{XY} = n^{-1}\sum(X_i-\bar{X})(...
3
votes
1answer
793 views

Asymptotic Distribution of the Wald Test Statistic

I am trying to understand the asymptotic distribution of the Wald test statistic, specifically under the alternative hypothesis which I've found little reference to. For clarity, the binary ...
3
votes
0answers
109 views

Adversarial learning gradient derivation

I'm working through Convolutional Neural Network paper here on adversarial learning and I'm having trouble with the derivative proof of adversarial logistic regression. The correct answer presented (...
3
votes
0answers
220 views

Limiting distribution of $\frac{\bar{x}-p}{\frac{pq}{n}}$ from mean of $Bin(1,n)$?

I found some difficulties in here. We know that if $X$ has Binomial distribution with $1$ trial and $p$ success, or what we called $X$~$Bin(1,p)$, we have $\mu=p$ and $\sigma^2=p(1-p)=pq$. From that, ...
3
votes
0answers
197 views

Distribution of the ratio of two shifted generalized gamma random variable

$X \sim \mathrm{GG}\left(p,d,\theta_{1},\mu\right)$ where $p$ is power, $d$ is shape, $\theta_1$ is scale and $\mu$ is location parameter. Also Consider $Y \sim \mathrm{GG}\left(p,d,\theta_{2},\mu\...
3
votes
0answers
576 views

How can I use gradient descent on the dual form of the linear SVM problem?

I understand that this is the dual form of the linear SVM problem (with a hard margin): $J(\mathbf{\alpha}) = \dfrac{1}{2}\sum\limits_{i=1}^{m}{ \sum\limits_{j=1}^{m}{ \alpha_i \alpha_j y_i y_j {\...
3
votes
0answers
99 views

Markov model parameter concentration and Fisher Information Matrix

For iid data, the posterior on the parameter $$ p(\theta \mid x_{0:T}) = \prod_{t=0}^T p(x_t \mid \theta) p(\theta) $$ is known to become independent of the prior which is the Bernstein-von Mises ...
3
votes
0answers
104 views

Mahalanobis distance for highly multivariate random variable

I have to compute the Mahalanobis distance for a $10^6$ dimensional multivariate random variable. What is the best (and fastest) way to do this? I am currently taking cholesky decomposition of the ...
3
votes
0answers
103 views

Confidence bounds on mean of a limited normal distribution

Suppose we are sampling from an underlying normal distribution with mean $\mu$ and variance $\sigma^2$, i.e $\mathcal{N}(\mu, \sigma^2)$. However, whenever we find a sample value that is greater than ...
3
votes
0answers
81 views

How does the prior enter the calculation when estimating the evidence using thermodynamic integration?

I am attempting to perform Bayesian model comparisons. To calculate the evidence, or marginal likelihood, for each model I use so-called thermodynamic integration having fitted the model using a ...
3
votes
0answers
94 views

Derivative of a Riemann-Stieltjes integral (quantile regression)

(Moved from the mathematics site as I didn't receive any response there) In pp. $5−6$ of Roger Koenker's Quantile Regression, the author minimizes the function $(\tau-1)\int_{-\infty}^\hat{x}(x-\hat{...
3
votes
0answers
141 views

Prove $\sigma^2(aX+b)=a^2\sigma^2(X)$ for discrete random variable

I'm a total newbie to statistics/math in general, so please bare with me. I'm currently reading the book The Cartoon Guide to Statistics, and stumbled upon the following statement (p. 69): $$ \...
3
votes
0answers
578 views

Does the UMVUE have to be a minimal sufficient statistic?

I'm studying point estimation and I have found this question that seems pretty tricky to me. If $T$ is a minimal sufficient statistic for $\theta$ with $E(T) = \tau(\theta)$, can you say that $T$ ...
3
votes
0answers
128 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 ...
3
votes
0answers
339 views

Finding the uniformly most powerful test

Let $X_1,X_2,...,X_n$ denote a random sample from density, $$f(x;\theta)={1\over 2\theta}, \quad 0<x<2\theta.$$ Find the uniformly most powerful test for testing $H_0:\theta \le \theta_0$ vs ...
3
votes
0answers
143 views

How does using Tukey's test correct for multiple comparison problem?

I am curious about the intuition behind the Tukey's HSD. I know that it is designed for post-hoc test(WHEN and HOW part), but I want to know underlying theory that justifies its usage(WHY part). To ...
3
votes
0answers
108 views

Distributions of eigenvalues of random matrices: what can they be used for in data mining?

I've accidentally come across some papers discussing distributions of principal components of the sample covariance matrices. An example of such a paper is Johnstone, 2001, On the distribution of the ...
3
votes
0answers
143 views

Local Version of Bernstein Von-Mises Theorem?

The Bernstein-Von Mises theorem says that, under reasonable conditions, the posterior distribution $p(\theta | x_{1},\ldots,x_{n})$ converges weakly to the normal distribution after suitable rescaling....
3
votes
0answers
83 views

Properties of the KL topology [reference request]

I'm trying to understand better what are the implications of a sequence of random variables $X_n$ converging toward some limit $X$ in the KL topology, ie the probability density functions are such ...
3
votes
0answers
90 views

A Fisher information metric which doesn't refer to any exponential family

If $\mathcal E$ is some exponential family of distributions, then we can view it as a Riemannian manifold with local metric the Fisher information matrix. We can then define the Fisher metric: the ...
3
votes
1answer
33 views

Estimating the number of classes from a sample

Suppose I have N smarties, each of which is one of C distinct colours. Suppose further that N is known and largish (10,000) but C is not, and that for each colour C there are $c_i$ smarties of that ...
3
votes
0answers
106 views

What statistical test should I use between two scenarios?

Scenario: I have a test data of response time of two websites in two different servers. Let's say website A and website B are tested on server A and server B not respectively but website A and website ...
3
votes
0answers
348 views

Estimating probability distribution function of data stream

Although a similar question exists, I couldn't find my answer. I'm not a statistician hence please neglect if some terminologies aren't correct and let me know if I am interpreting something wrong. ...
3
votes
0answers
31 views

Bound on the total change using Pearson's r

I am given an increasing series $(x_1,....x_n)$ and I know the pearson correlation between $(x_1,....x_n)$ and some (unknown) increasing series $(y_1,....y_n)$. Can I derive an upper and a lower ...
3
votes
0answers
257 views

Sufficient statistics and UMVUE for joint poisson, bernoulli

Given a pair $(X,Y)$ of r.v.s such that: $$X \sim \text{Poisson}(\lambda)\quad \text{and}\quad Y \sim B(\frac{\lambda}{1+\lambda})$$ with $X,Y$ independent, determine a one-dimensional sufficient ...