Questions tagged [statistical-inference]

The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

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Compare the speed at which $(\bar{X})^2$ converges to zero to the speed at which $\sqrt{n}$ diverges to infinity

Setup Assume the following formation $$ \bar{X} \overset{P}{\to} 0 \text{ as } n\to\infty. $$ Lemma We want to show $$ \sqrt{n}\cdot\left(\bar{X}\right)^2 \overset{P}{\to} 0. $$ This is indeterminate ...
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Does the standard sum-product message passing algorithm save any computation over just doing this?

In MacKay's "Information theory, inference, and learning algorithms" book, in chapter 26 he covers the sum-product algorithm for calculating marginal probabilities, partition functions, etc. ...
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proof of CEF decomposition theorem

The proof of CEF decomposition theorem. I don't know how it comes here. Thank you very much! E{E[h(x)e|x]} = E{h(x)E[e|x]}
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Exercise - Student t distribution / Normal distribution - determine the proportion of samples exceeding some value

Is my reasoning on the following exercise correct? I am not sure if I correctly used Student t distribution instead of normal distribution and whether my calculation of std makes sense. Exercise: &...
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Probability problem - finding the missing number.

I am a retired philosopher, familiar with some of the philosophical problems about probability (e.g. Hume's problem of induction) but at a loss in calculating probabilities. I recently came across the ...
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Randomized and nonrandomized statistical tests

I've been reading recently about randomized and non-randomized statistical tests and their differences. If I understand the definition correctly, a nonrandomized test for all values of the test ...
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How to combine two probabilities for the same event? Context: error correction codes / decoding

I'm learning the maths behind error correction codes. For this purpose I made this question for myself: Assume there are two random bits $x_0$, $x_1$, which are both i.i.d. and have a 50% chance of ...
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How to estimate the best variance-proxy of a sub-Gaussian distribution from data?

Suppose we have $N$ independently identically distributed (i.i.d.) samples $X_1,\cdots,X_N$ generated from a sub-Gaussian random variable $X \sim \mathbb{P}$. Then by definition there exists the ...
Asce's user avatar
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Find Logarithmic Maximum Likelihood Estimation (Log MLE) for a piecewise Probability Density Function (PDF) with zero in one of it's rules.

Consider random variable Y having PDF $$ f_{Y}(y|\theta) = \begin{cases} \\ \frac{1}{\theta}ry^{r-1}e^{ \frac{-y^r}{\theta}}, & y , \theta> 0 \\\\ 0, & \text{otherwise} \end{cases} $$ ...
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Distributing the error in a frequency table so that the $\chi^2$ statistics are distributed according to the actual distribution?

When using the $\chi^2$ statistic, if the errors (difference between observed and expected) are too low, the resulting statistic will be low. If we repeat the experiment several times with similar ...
Chris Vilches's user avatar
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Calculating MSE of MOM and MLE of a Uniform Distribution

Let $X_1, X_2, X_3$ be a random sample of size three from a $uniform(θ, 2θ)$ distribution, where $θ > 0$ I solved to get $\tilde{\theta}_{MoM}$ to be $\frac{2\bar{x}}{3}$. Also, I got $\hat{\theta}...
Maale Faustus's user avatar
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Optimal Calibration for a Test Set

I want to calibrate the parameters $\theta$ of a known forward model $y=f(\theta, x)$, i.e., I want to identify offsets of around +- 10% from a nominal model $\theta_0$ I can not measure $y$ directly, ...
scleronomic's user avatar
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Why is the formula $f_{X|Y}(x|y) = \frac{P_{Y|X}(y|x) f_X(x)}{P_Y(y)}$ true?

Suppose we have a random sample $X=(X_1,\dots, X_n)$ which depends on $\theta$, treated as a value of the random variable $\Theta$. The posterior pdf is given by $$f_{\Theta | X}(\theta|x)=\frac{f_{X|...
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Consistency of Biased Estimators

In Statistical Inference, we were taught this theorem, Consider an estimator $T_n$ of population parameter $\theta$, using $n$ samples. $T_n$ is a Consistent Estimator of $\theta$ if $$E[T_n] \to \...
Harry's user avatar
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Interpreting $P(\alpha|\text{data})\propto P(\text {data} | \alpha)\cdot P(\alpha)$ [closed]

In the context of posterior and prior probabilities, one has $P(\alpha|\text{data})\propto P(\text {data} | \alpha)\cdot P(\alpha)$. What confuses me here is that probability is defined for events, ...
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Entropy of Gaussian Mixture Models (GMM) and Conditions for Maximum Entropy

Background In my exploration of Gaussian Mixture Models (GMMs) within the scope of statistical learning, I have encountered the concept of entropy as a measure of uncertainty or randomness in a ...
Alireza Ghazavi's user avatar
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Understanding a measurabiliy statement from Section 6.3 in Lehmann and Romano

This paragraph is at the end of Section 6.3 in the book Testing Statistical Hypotheses by Lehmann and Romano: In most applications, $M(x)$ is a measurable function taking on values in a Euclidean ...
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Large amount of sets in Venn Diagrams and inference

Let $U$ be the universe of a multiple choice questionnaire. We can say $|U| = 1$. We have $k \geq 3$ sets of answers in $U$ such that for all $k$, we have $ S= \sum_{u_i \in U} |u_i| > 1$. The ...
Eemil Wallin's user avatar
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PDF Random Variable Statement

Is it true that if the probability density function of a continuous random variable is an even function, then the continuous random variable is symmetric?
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Lognormal linear regression

I would like to fit a linear regression model with a lognormal distribution that has a linear expectation in the $X$, $Y$ space. So, I need a model with the following properties: $$ Y|X \sim Lognormal ...
Amav's user avatar
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Calculate standard deviation of 3D Gaussian distribution in spherical coordinates [closed]

Let's say I define a spherically symmetric 3D Gaussian PDF, centered around $\mu = 0$, in spherical coordinates, $$ f(r, \theta, \phi) = f(r) = \frac{1}{(2 \pi \sigma^2)^{3/2}} \exp{\bigg( - \frac{r^2}...
Manuel Ballester's user avatar
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How do I compare the variabilities between groups with extremely different ranges?

It's hard to fit it all in the title but my problem is basically finding a measure of variability or spread that I could use to compare groups where some have really high values while others have ...
ConfusedConfucius's user avatar
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How to sample data in sample efficient way to make posterior single modal

I have a deterministic function $f(x;\theta)$ and measured observation $y=f(x;\theta) + \epsilon$. Here, $\theta$ is unknown, $x$ can be varied, $y$ is observation, and $\epsilon$ is a random error. I ...
Kilean Hwang's user avatar
5 votes
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Distribution of the maximum point defined by a sequence of random variables

Suppose $X_1,\cdots,X_n,\cdots$ are i.i.d. and follow the uniform distribution on $(0,1)$: $U(0,1)$. Define $T_n$ to be the maximum point of the function $$f_n(t)=\sum_{i=1}^{n}\frac{\log(1+t^2X_i)}{t}...
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Closed-form expression for multivariate survival function of the probability distribution itself

Is there a closed-form expression (or simpler expression) for the survival function of the distribution itself: $$V(z)=\int\mathbb{1}_{\{\mathbf{x}\mid p(\mathbf{x})>z\}}p(\mathbf{x})\mathrm{d}\...
FizzleDizzle's user avatar
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Computing the $p$-value under a $T$-student distribution without statistical software

Assume $n = 25$ observations from a normal distribution have a sample mean $\bar{x} = 21.2$ and $s_{n} = 0.5$. We wish to test whether this observed mean is lower than a population mean with $\mu_{old}...
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Continuity of the power function in UMPU testing for two parameter uniform distribution

Suppose $X_1 , \dots, X_n$ are i.i.d. $U(\theta_1, \theta_2)$, with $\theta_1 < \theta_2$. We want to test for $$ H_0: \theta_1 \leq 0 \quad \text{versus} \quad H_1: \theta > 0. $$ It can be ...
ムータンーオ's user avatar
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Do quantiles of quantiles converge to quantiles?

Is there any truth to the statement "the median of medians converges to the true median?". While it's false that the median of medians is the median, is there a way to make this true ...
user125763's user avatar
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Joint sufficient statistics for normal distribution (denominator n - 1)

Example 24-6 here, for i.i.d. $X_i$ from a normal distribution $(\theta_1, \theta_2)$, expresses the joint density $$ f(\textbf{x}; \theta_1, \theta_2) = \exp\bigg[\frac{-1}{2\theta_2}\sum_{i = 1}^n ...
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When does the optimal model exist in learning theory?

In the context of learning theory, we usually have: data $(x,y)\sim P(x,y)$, with $x\in\mathcal{X}\subseteq\mathbb{R}^d$ and $y\in\mathcal{X}\subseteq\mathbb{R}^k$, a hypothesis class $\mathcal{F}\...
rick's user avatar
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Can you help me understand the Independence assumption of a Chi Squared Test for Independence? (Thought Experiment)

Ok, I'm going to preface my question with letting you know that it's more of a weird thought experiment. I’m trying to understand the independence assumption of a Chi Square Test for Independence by ...
Hector's user avatar
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Assumptions used in U-test: "dependent variable" vs "independent observations"

I am reading this link [https://statistics.laerd.com/spss-tutorials/mann-whitney-u-test-using-spss-statistics.php][1] in which they describe the 4 assumptions of applying U-test. I am not clear about ...
weidade3721's user avatar
2 votes
1 answer
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How to reconcile "conditioning reduces entropy" with certain phenomena in Bayesian inference

From information theory we know that conditioning random variable X on Y will not increase its entropy (i.e., $H(X|Y) \leq H(X)$). However, in Bayesian inference, we know that if a prior is strong (i....
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6.36 of Theory of Point Estimation, second edition

the question is: $$ X_1, X_2, ..., X_n $$ i.i.d random variables of uniform distribution U(a,b), where a<b. Show that $$ Z_i = \frac{X_{(i)}-X_{(1)}}{X_{(n)} - X_{(1)}}$$ ,i = 2,...n-1, are ...
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Fisher information with known moments

I have a sequence $X^n$ of length $n$, where each $X_i$ takes a value from a finite set with probability vector $\mathbf{p} = [p_1, \ldots, p_K]^T$, i.e., $X_i \in [K]$, where $p_{X_i}(k) = p_k, k = 1,...
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Can someone help me find/ give me a proof of the PRESS formula for the Predicted Residual Sum of Squares in K-fold Cross Validation?

Similar to the formula for the n-fold CV case (LOOCV), $$\mathrm{PRESS} = \sum_{i = 1}^{n}\left(\frac{y_i-\pmb{x}_i^\top \mathbf{X}^+\pmb{y}}{1-p_i}\right)^2$$ Where $p_i$ is the ith leverage (this ...
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About Calculate fisher information of normal distribution [closed]

Suppose $X_1, \ldots, X_n$ are iid $\mathrm{N}(0, \exp (2 \gamma))$; that is, the density of $X_i$ is $$ (2 \pi)^{-1 / 2} e^{-\gamma} \exp \left(-x^2 e^{-2 \gamma} / 2\right) . $$ I want to calculate ...
trivial_fish's user avatar
13 votes
2 answers
611 views

Probability - Interview Question - Hidden Assumptions and Phrasing Issues

I’ve encountered the following seemingly simple probability interview question in my workplace: Two reviewers were tasked with finding errors in a book. The first had found 40 errors and the other ...
Yonatan Harari's user avatar
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Use Variational Inference to find an approximate of distribution

I have learnt about Variational Inference at this website. In my understanding, it is a method that sampling random variables from an easy distribution $q$ which is most similar to an intractable ...
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Find MAP and LMS estimate with given PDFs [closed]

If we have $f_X(x)=\left\{\begin{array} 11/4, 0\leq x\leq 1\\3/4, 1\leq x \leq 2 \\0, otherwise\end{array}\right.\quad\text{and} \; f_{Y|X}(y|x)\quad\left\{ \begin{array}{A}\frac{y-x} 2, x\leq y \leq ...
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Can anyone help to solve this task ?In a multiple-choice test with m options, a student knows the correct answer with probability p,...?

"In a multiple-choice test with m options, a student knows the correct answer with a probability p, and in the absence of knowledge, chooses randomly one of the available options. What is the ...
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Derive Cramer-Rao lower bound for $Var(\hat{\theta})$ given that $\mathbb{E}[\hat{\theta}U]=1$

I am trying to derive the Cramer-Rao lower bound for $Var(\hat{\theta})$ given that we already know $\mathbb{E}[U]=0$, $Var(U)=I(\theta)$ and $\mathbb{E}[\hat{\theta}U]=1$. I am struggling with using ...
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Verifying change of variable in published work

In the proof of Proposition A.11 on page 32 of this paper, the author takes the following step: $$ \int_U \exp(-\|D_Gu\|^2_2/2 - \omega \|D u\|^2/2)du = \text{det}(I+\omega D_G^{-1} D^2 D_G^{-1})^{-1/...
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Is the bias form an estimator a real number or we need to define it as a positive number/negative number?

Is the bias form an estimator a real number or we need to define it as a positive number/negative number? It seems to me that it can be any real number from the following. From the proof of the ...
Oliver Queen's user avatar
1 vote
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56 views

Statistical inference for the integral equation

Consider a integral equation $$ \begin{aligned} \mathbb{E} \left[ Y|A \right] &=\mathbb{E} \left[ g\left( W \right) |A \right]\\ \int{yp\left( y|a \right) dy}&=\int{g\left( w \right) p\left( ...
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Finding Posterior Distribution for Simple Linear Regression

A simple linear model is given as $$ Z_i = \gamma_1 + \gamma_2 y_i + \epsilon_i$$ $i=1, \ldots,n.$ Let $\mu = (\gamma_1, \gamma_2)'$. Assuming that $\epsilon_i \sim N(0,1)$ and using a noninformative ...
holala's user avatar
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Finding the posterior and Bayes estimator with a beta prior

Let $Y_i, \ i =1,2,\ldots n$ be a random sample from the probability function $$f(y\mid p) = \frac{2y}{p^2}, \quad 0 < y \le p$$ where $p\sim Beta(2n+1, 1)$ is the prior, find the posterior ...
holala's user avatar
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Fisher Information Matrix Weibull Distribution

Is there a book or website that guide us to proof the Fisher Information Matrix for Weibull Distribution? No matter how I just have direct answer instead of the step-by-step proof...
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A proof for increasing the Evidence Lower Bound always results in increment in the log marginal likelihood

I have the confusion about the $\mathrm{ELBO}$. Let's say if we have the observed data $x$, the hidden variable $z$ with underlying parameters $\theta$. The log marginal likelihood function is defined ...
Fellow InstituteOfMathophile's user avatar
1 vote
1 answer
53 views

Interpreting a concentration inequality

In the following paper I am slightly confused about the way they use a concentration inequality derived in Lemma A1. In Lemma A1, under the assumption that $(n ,p)$ satisfies $\log p/n^{1/4} \to 0$ as ...
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