Questions tagged [mathematical-statistics]
Mathematical theory of statistics, concerned with formal definitions and general results.
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Posterior Distribution using a Normal Likelihood and Laplace Prior
I have the working out below but is this correct. I just want the posterior distribution of when mu=0 given x.
What I have tried is setting mu=0 after rewriting the first pdf with the summation ...
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How does Huber compute the $\operatorname{var}(s_n)/E[s_n]^2$ and $\operatorname{var}(d_n)/E[d_n]^2$?
(N.B. I am cross posting this question from math stackexchange since after
x days I have still not received any responses.)
How does Huber in book 'Robust statistical procedures' in chapter 1 compute ...
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How to deal with negative loss during deep learning training, when using negative log likelihood loss? [duplicate]
I have a question regarding how to deal with negative loss during the training of my deep learning model. I've observed many instances of negative loss generated by the data points as outlined below.
...
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Expectation of the realized volatility
I was reading Zhang and Wang 2023 and I have some doubts regarding it. The realized Stochastic Volatility Model is expressed as follows:
$$\begin{matrix}
y_t = \exp \big( \frac{h_t}{2} \big) \...
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Example probability model predicting vectors of ranks?
Background
I'll confess, I like Richard McElreath's lectures so much that I've been watching his backlog of lectures (even though I've already seen all of the recent lectures of the same course).
In ...
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Stability for algorithm implies no overfitting
We let $\mu$ be a distribution on the set $Z=X \times Y$. For any $S \in Z^n$ and $i$ in $[n]$, and we define $S^{(z,i)}$ as the vector in $Z^n$ that coincides with $S$ in all entries except the $i$-...
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The test point’s standard deviation from the origin (ESL’s Exercise)
This is Ex.2.4 from The Elements of Statistical Learning.
I don’t understand the sentence that I underline in the image.
I know that $\sqrt{10}$ is approximately equal to 3.1, but I don’t know how to ...
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If $P(A|D) > P(A)$ and $P(B|D) > P(B)$, then is $P(A \cap B|D) > P(A \cap B)$?
There are 3 events $A, B, D$ such that $D$ makes $A$ more likely and $D$ makes $B$ more likely. Does this mean that $D$ makes it more likely that both $A$ and $B$ occur? How can you prove this using ...
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Condition variance of $\hat{\beta}$ different under heteroskedasticity
I am following Bruce Hansen's Econometric textbook. Under the assumptions that $\{x_i,y_i\}$ are i.i.d., $E(e_i|x_i) = 0$, and errors are heteroskedastic, we derive the variance of the OLS estimator
$...
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Partial Differential Equation approach for ridgeplot
I have a series of probability density functions (pdfs) such that I can create a ridge plot with them evolving in time. Is it possible to know the intermediate function in between two of the functions ...
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Is there a simpler proof than mine for this obvious proposition about correlations?
$\newcommand{\e}{\operatorname E}$"Obviously" if $g$ is a weakly increasing function and $X$ and $g(X)$ are both random variables with finite variance, then the covariance (and hence the ...
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Pooled Standard deviation of means
I am reading the book Statistical Methods In Online A/B Testing.
I have two questions:
1]
Please Consider the scenario, an A/B test in which the variance of A and B groups are assumed to be same, and ...
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Two contradicting derivations of the Covariance Matrix for Linear Regression
I am looking to compute the variance covariance matrix for the standard linear regression coefficients $\hat{\beta}$ when:
$$Y = X \beta + \epsilon $$
and $\epsilon \sim N(0,\sigma^2)$. I have derived ...
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Convergence of variational posterior
Let $q_\lambda(z)\in\mathcal Q$ be the variational posterior approximation of $p(z|y)$.
Note that the optimal $\lambda^*$ is approximated by the following recursive sequence
$$
\lambda^{(k+1)}=\lambda^...
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Probability RV is min among several iid RVs [duplicate]
This question is inspired by this programming question.
Suppose I have 3 RVs which are independent.
$$X \sim N(25.5, 2.5)$$
$$Y \sim N(25.2, 3.5)$$
$$Z \sim N(24.9, 1.7)$$
I want to know what is the ...
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What is the cross product of two probability distributions $P \times Q$?
In many papers on machine learning and statistics, I encounter the following notation
Let $P$ be a distribution, and let $Q$ be another distribution.
Then the author creates an object $P \times Q$
...
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Ensemble mean of a fraction
I want to compute the ensemble mean of the term: $\frac{Y^2}{X}$
Both $X$ and $Y$ are random variables that are not independent. I want to compute $E[\frac{Y^2}{X}]$. I proceed as follows, (Using the ...
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How to obtain the local unbiased condition for an estimator from global unbiased condition?
A standard problem in classical statistics is to find a good estimator that minimizes a given cost function under certain conditions. Normally we want to require the estimator $\hat\theta$ to be ...
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Finding Sample Range of Fisher's z-distribution via Approximating Hypergeometric $\,_2F_1\left(\frac{1}{2},\frac{x+1}{2};\frac{3}{2};-z^2\right)$
Recently, I have encountered Hypergeometric function $\,_2 F_1\left(\frac{1}{2},\frac{x+1}{2};\frac{3}{2};-z^2\right)$ in the context of order statistics.
In particular, I am trying to evaluate an ...
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Probability distribution of measurements and Parameters of measurements
I am new to statistics and recently learned about ISO guidelines for Accuracy & Precision and Uncertainty & Error. I have tried to plot a graph for what I have learnt including all the ...
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How do I obtain the primal and dual for the estimator $\min _\beta\left[\|\beta\|^2+\sum_{i=1}^n \xi_i^2\right]$ s.t. $\xi_i=y_i-h(x_i)^\top \beta$?
I am working on a statistical learning exercise that requires some knowledge of convex optimization which I am unfortunately lacking.
Consider the linear regression model
$$y_i=h(x_i)^\top\beta+\...
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How does the formula of a negative binomial regression model look like?
For a term paper I use a zero-inflated negative binomial regression. I would like to viszualize the formula in my method section. Lets say I model the zero component based on variables v1 and v2, ...
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Is mathematical statistics an active area of research? [closed]
Is mathematical statistics an active area of research? Are there big unsolved problems in the area? I am interested in seeing areas of mathematical statistics where there is active research. For ...
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Time derivative of covariance terms
I have a few covariance expressions for two correlated/dependent RVs like $\mathrm{Cov}(X',Y')=E[X'Y']$, where $X'=X-E[X]$, where $E[X]$ is the mean of $X$ and $X'$ is the fluctuation. I do not have ...
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Recommendations for study material for mathematical statistics
I am currently preparing for a qualifying exam in mathematical statistics and am looking for good resources for self-study. I have access to old exams which I have been slowly working on; however, I ...
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How do I know which statistic is for which parameter when calculating joint sufficient statistics using factorization criteria?
For the normal distribution for example, after factorization we get
$\mathcal{L} = (2 \pi \sigma^2)^{-\frac{n}{2}}\exp\left(-\frac{n\mu^2}{2\sigma^2}\right) \exp\left(-\frac{1}{2\sigma^2}\left(\sum_{i=...
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did this paper make a mistake on anova?
The screenshot indicates that the lowest the score/value, the higher the fear level. In this case, a lower mean = higher fear, and higher mean = lower fear. Should it be that, Generation Y is ...
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How to obtain marginal density [x], given [y|x] and [y]
I came across a problem knowing density of Y, conditional density of Y given X, how would I obtain density of X? Or would this even unique?
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Best way of determining how unique an inspector's ratings are compared to the average inspector
Noob question.
I have a full population of every inspector rating for all items being rated. Each inspector has inspected a subset of the items and given them a rating, but different inspectors have ...
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Calculate posterior distribution and full conditional of a HMM
Set up a Bayesian analysis of an hidden Markov model and calculate the posterior distribution and the full conditionals, given this assumptions:
The state space of the hidden process has size m
$Z_t|...
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Cramér-Rao regular model
Why is the fact that the support of each $f$ belonging to a Cramér-Rao regular model does not depend on the parameter implied by the condition that the derivative with respect to the parameter of $f $ ...
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The difference of $\sum_{i=1}^{n}X_{i}$ and $\sum_{i=1}^{n}X_{(i)}$
Here is a exercise from *Mathematical Statistics. Jun Shao. Second edition. EX2.20
Let $X_1,..., X_n$ be $i.i.d.$ random variables having the exponential distribution $E(a,\theta)$, $a\in R$, and $\...
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Sufficient Statistic for a family of distributions consisting of Poisson family and Bernoulli family
Suppose $(X_1, . . . ,X_n)$ is an i.i.d. sample from the distribution $f_{\theta,k}(x)$, where $\theta \in (0, 1)$ and $k = 1, 2$. Assume that $$f_{\theta, k}(x)=\begin{cases} \text{Poisson($\theta)$},...
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Is there a law or theorem related to occurrence of an event with highest probability in a population with infinite size?
Assume, we have a key that appears in either of the three rooms randomly (red room, blue room, and green room). We have the following probability distribution:
...
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In Bland Altman Analysis, does it make sense to have bias as a primary endpoint, but 95% LoAs as a secondary endpoint?
In the context of Bland Altman analysis, is it justifiable to take a mean difference as a primary endpoint, but the 95% limits of agreement as a secondary endpoint?
My understanding is that in B&A ...
Artful Dodger
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Unifying multiple sample data (meta-analysis)
I would like to unify multiple sample data (of different sizes) into some "unified sample" to evaluate its collective variance. Is this something statisticians do? I thought of unifying ...
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Covariance between sample mean and sample variance
I am trying to figure out the covariance between sample mean and sample variance from a population. We DO NOT know whether the population is normal (if it's normal, then the covariance is zero between ...
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Probability generating function [closed]
X and Y are independent random variables having poisson distribution with parameters a and b respectively. By using probability generating function, prove that X + Y have a poisson distribution and ...
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Intuition of Influence Function and Score function: $E[IF(X)S_{\beta}(X; \theta_0)]$
Question
I find a theorem regarding influence function and score function
\begin{align*}
E\left\{IF(Z) S_\beta\left(Z, \theta_0\right)\right\}&=1\\
E\left\{IF(Z) S_\eta^T\left(Z, \theta_0\right)\...
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Expected absolute deviation greater than standard Laplace
Could there exist a distribution, other than standard Laplace (probability density of the form $1/2e^{-|x|}$), on $\mathbb{R}$ such that $E[x]=0,E[|x|]=1$ and that
\begin{equation*}
E[|x-a|] \geq |a|+...
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Minimizing cross entropy over a restricted domain?
Suppose $f(x;q)$ is the true distribution. The support of the random variable $X$ is $\Omega$. Suppose, I am interested in a particular subset of $\Xi \subset \Omega$. I would like to minimize the ...
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Mixture of Conditional Random Variables by Sampling
I am struggling to put my transformation of data into mathematical contexts. My goal is to define a mapping that transforms the original data into some awkwardly mixed data. In my simulation study, I ...
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In Sutton & Barto, what is meaning of weight given to actual final return after termination in the Delta return? (equation 12.3)
The equation 12.3, in the book RL An Intro by Sutton & Barto, is shown below with its "diagramatic" explanation in figure 12.2:
Once the episode has terminated at time T, what's the ...
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Why does the reparameterization trick work when some components are still stochastic? [duplicate]
I am trying to understand the reparameterization trick. I got some intuition while looking at this popular question, but I still feel largely confused. I am putting my understanding and doubts here ...
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Show local asymptotic distribution of the Maximum Likelihood estimator
from a problem set I have the following exercise:
Let $X_1, \ldots, X_n$ be iid with $X_1 \sim N(\theta, 1)$, where $\theta \geq 0$. The ML estimator of $\theta$ is $\hat{\theta}_n = \max (\bar{X}_n, ...
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Wald Test on two exponential samples
Based on independent samples of data $X_i \sim$ Expon$(\lambda_1)$, $i = 1, \cdots, n,$ and $Y_j \sim$ Expon$(\lambda_2)$, $j = 1, \cdots, n$ (both samples of the sample size $n$), find the Wald test ...
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Simple linear regression parameter estimation
In the method of least square I have the following
$\hat{B_1}=\frac{\sum (Y_i-\bar{Y})X_i} {\sum (X_i-\bar{X})X_i} \\
~~~~= \frac{\sum (Y_i-\bar{Y})(X_i-\bar{X})} {\sum (X_i-\bar{X})^2} \\
~~~~= \...
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33
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Natural Exponential family with polynomial partition function
Consider a natural exponential family:
\begin{equation}
f(x|\theta) = h(x) e^{ x \theta -A(\theta) }
\end{equation}
We assume that we are supported on $\mathbb{R}$ (i.e., $h(x)>0$ for $x \in \...
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Yes or no: is Pearson’s r is a measure of goodness of fit to an affine function? [duplicate]
Is the statement "Pearson’s r is a measure of goodness of fit to an affine function" literally true? Why or why not?
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What are the undefined constants and functions in Stern's 2011 paper?
I'm reading the 2011 paper on ranking called Moderated Paired Comparisons by Steven E. Stern and there are no definitions given for some of the constants and functions in equation 1.
As you can see, ...
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