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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Bayesian Network for wordle

Intro In preparation for stuyding AI, I'm currently studying probability and bayesian inference. As a first challenge in the subject, I want to model and train a Bayesian network that is able to solve ...
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Causal discovery for pairwise independent joint dependent variables

Consider the standard example for variables that are pairwise independent but joint dependent. $$ (x,y,z)= \begin{cases} (0,0,0) & \text{probability 1/4} \\ (1,1,0) & \text{probability 1/4} \\ ...
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Rao Score test and Neyman-Rao test

I've got the Laplace location/scale model where $X_1,...,X_n$ are iid random variables with common density \begin{equation} p_{\theta,\eta}(x)=\frac{1}{2\eta}e^{-|\frac{x-\theta}{\eta}|}. \end{...
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Expectation of the largest order statistic from uniform random variables

If $X_1, ..., X_n$ are iid random variables from the Uniform[$0,\theta$] distribution, where $\theta >0$, compute the expectation of the largest order statistic denoted $X_{(n)}$. I am looking to ...
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I have to determine $d_{\alpha}$ in terms of critical values of a chi squared disdtribution, significance level $\alpha$ and sample size n

So far I have the following: Assuming the Null hypothesis $X_{i} \thicksim$ EXP$(1)$ $Y: = \sum_{i = 1}^{n} X_{i}$ $Y \thicksim$ GAM$(1,n)$ $\bar{X} = Y/n$ $P(\bar{X} \leq$ $d_\alpha) = 1 - \alpha$ $P(...
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Datasets for optimal experimental design.

I am looking for some real-world datasets/benchmarks for optimal experimental design problems. Nomatter which criteria (A, D, E-optimal) we use for the optimal design, the information matrix is the ...
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How to take the derivative with respect to a function without a clear substitution?

In Statistical Inference, the Karlin-Rubin Theorem requires that a given statistical model has a Monotone Likelihood Ratio with respect to a sufficient statistic $T(X)$. In order for the model to ...
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Should I reject the null hypothesis or not?

EDIT: My apologies, I had a coding error. I accidentally used the same standard deviation for both samples. Now that I fixed that, both the normal and Student's confidence intervals are stupidly ...
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Fisher-Neyman Factorisation Theorem and sufficient statistic misunderstanding

Fisher Neyman Factorisation Theorem states that for a statistical model for $X$ with PDF / PMF $f_{\theta}$, then $T(X)$ is a sufficient statistic for $\theta$ if and only if there exists nonnegative ...
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Maximum likelihood for two dependent variables

Suppose you have a box containing 10 balls, of which $\theta$ are white and the rest are green. Suppose we take two balls for without replacement and let $X_i = 1$ if the i-th drawn ball is white and $...
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How to prove if $\mathbb{E}[X]>\mathbb{E}[Y]$ then $\mathbb{E}[\frac{1}{X-Y}]>0$ [closed]

This statement seems pretty intuitive and seems to be true. I tried using Jensen's Inequality but have failed. Can anyone prove this?
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choosing an index for points given to playoff teams [closed]

Short brief:I am doing a project about the correlation of the success of teams(for example) in champions league and their revenues. I want to build a linear regression for example based on the success....
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Difference between MVB and UMVU estimators

I am trying to understand the difference between the UMVUE (uniformly minimum-variance unbiased estimator, also known as minimum-variance unbiased estimator (MVUE)) and the MVBE (minimum variance ...
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Variational Inference for Item Response Models Estimation

I should work on a project about Variational Inference for Item Response Models Estimation. This is my university project but I don´t find relevant information around this topic so I would be very ...
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With a $Gamma(2, \frac{1}{2} )$ use the CLT to prove the random variables $ \sqrt{n}( \overline{X}_{n} - 1) \rightarrow_{d} N(0, \frac{1}{2}$

I'm currently noodling through a proof as to why a Gamma distribution of $Gamma(2, \frac{1}{2} )$ converges as per the Central Limit theorem: $$ \sqrt{n}( \overline{X}_{n} - \mu) \rightarrow_{d} N(0,...
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Stochastic Ordering of multivariate normal distribution

Let $$X\sim\mathbb{N}([0,1,0]^{\rm T}, \mathrm{\Sigma})$$ and $$Y\sim\mathbb{N}([1,0,0]^{\rm T}, \mathrm{\Sigma}).$$ For some real constant $c$ what can be said about $$\mathbb{P}[\cap_{i=1}^3|X_i|\...
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Change of variables formula for mathematical statistics

I'm trying to prove or falsify Proposition 1.8 in Shao, Mathematical Statistics: It suffices that we show for any $A\in \mathbb{B}^k$ and $i=1,2,...m$ $\int_{A\cap g(A_{i})} f_{Y}(y) dy=\int_{g^{-1}(...
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Is $T(X) = X$ a trivial sufficient statistic?

I know that the First Theorem for Sufficient Statistics states the following: For a given statistical model for the random vector X $=(X_1, . . ., X_n)$ with pdf/pmf $f_{\theta}$, then $T(X)$ (with ...
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Can the chi-square statistic in Kruskal-Wallis test be compared to determine the most appropriate to distinguish the groups?

I have a dataframe in R which format is similar as follows: v1 v2 v3 group 1 3.5 100 a 3 5 200 a 10 5.5 150 b 8 7.5 210 b 4 4.5 300 c 9 2.5 200 c ... My ...
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UMP test-why is it wrong to check it this way?

So, I want to know if a test is a UMP test. for: $ θ_0=3 $ $ θ_1<θ_0 $ using neyman-pearson lemma suppose I got something like this where our estimator for $ θ_0 $ is the mean and for some ...
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The latest best approach to determine the order of the system model

The classical maximum likelihood estimation using Akaike's criteria is defined by $$\text{AIC}=-2\log^-\text{(maximum likelihood)} + 2 \text{(no. of independently adjusted parameters within the model)}...
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How does One Come Up with Confidence Intervals For Fermi Estimation Problems?

A common question is estimate some quantity $X$, and give a 95% confidence interval for it. The estimation part, I understand how to think about and attack, but I'm always lost when it asks for a ...
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Bagging Linear Model?

I have a question regarding bagging linear models. Suppose you wanna do linear regression on data X and y. Alice directly implements (OLS) Linear Regression on it. The model is A1. Bob applies bagging....
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1 answer
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Statistical estimator of expected value of the gradient of an unknown function

Fix a probability space $(\Omega, \mathcal{A}, \Bbb P),$ a continuously differentiable function $f:\Bbb R^n \rightarrow \Bbb R,$ and a random vector $X: \Omega \rightarrow \Bbb R^n.$ Furthermore, we ...
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Simplifying a Kullback-Leibler divergence

I want to find an approximation of a mixture of probability distributions that minimises the Kullback-Leibler divergence (KLD). I need to verify my result, as it seems suspect. We have a joint ...
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Find UMVUE of ${\rm e}^{-\theta \tau}$ in which $\theta$ is parameter of ${\rm Exp}(\theta)$

Suppose $X_1,... ,X_n\ {\rm i.i.d.\sim Exp}(\theta)$, i.e. $X_i \sim f(x) = \theta{\rm e}^{-\theta x}I_{(x>0)}$. Find UMVUE of ${\rm e}^{-\theta \tau}$ in which $\tau > 0$ is given. Hint: ${\rm ...
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Discrete uniform rejection region

I am currently studying for an exam. There was one problem listed that I currently am puzzled about how to approach. I would like some guidance as to what can be done in part a). I will attach a ...
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Find UMVUE of $\sigma$ and $3\sigma^2$ in which $\sigma$ is parameters of $N(0, \sigma^2)$.

Assume $X_1,...,X_n$ is sample of $N(0,\sigma^2), \sigma > 0$, find (1) complete and sufficient statistic of $\sigma^2$, (2) UMVUE(uniformly minimum variance unbiased estimation) of $\sigma$ and $...
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Confused about interpretation and intuition behind Mcnemar's Test

I've always been under the impression that Mcnemar's Test is for paired categorical data, just like paired t-tests are for paired continuous data. However, I was looking at the Wikipedia article for ...
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Is this a binomial experiment?

I ran a survey where I asked respondents to say how much they liked a certain game. They could choose one of five possible options. I calculated the sample proportion of all five options (say A, B, C, ...
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Method of Moment Estimator of Normal Distribution

The sample r.v. $X_1, X_2,\ldots, X_n$ i.i.d. $N(\mu, \sigma^2)$, find the moment estimator of $P\{X>1\}$, where $X\sim N(\mu, \sigma^2)$. Here is my answer. I want to know whether it is right, thx....
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Let a random experiment be the casting of a pair of fair six-sided dice and let X equal the minimum of the two outcomes . [closed]

the problem is I am suffering a lot in my journey,I cant even determine where I must start solve mathematical problems: Let a random experiment be the casting of a pair of fair six-sided dice and let ...
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How to prove multivariate Bayesian Cramér-Rao inequality?

I came cross multivariate Bayesian Cramér-Rao inequality as follow recently, but I don't know how to prove it. Let $\{f(\cdot ; \theta): \theta \in \Theta\}$ be a family of probability density ...
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Proof of the information bottleneck equations

In The Information Bottleneck Method, the third term of Eq.(31) is $P_{t+1}(y|\tilde{x})=\sum_yp(y|x)p_t(x|\tilde{x})$, which minimizes the term $D_{KL}[p(y|x)|p(y|\tilde{x})]_{<p(x,\tilde{x})>}$...
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2 votes
3 answers
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How can we stay confidence replacing the population standard deviation by it's estimate?

So imagine we take $n$ random samples from a Bernoulli Trial. Thus our random samples are composed by binary random variables $X_1, X_2, ..., X_n$. So by central limit theorem we know that the ...
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1 answer
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Lehmann–Scheffé theorem's statement

In my notes I have the following L-S theorem statement: Let $T(X_1,...,X_n)$ be an estimator for $\theta \in \Theta$. If $T$ is: unbiased a function of complete and sufficient statistic $S_c(X_1,......
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2 votes
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Why are causal inference diagrams so useful or effective?

Is there a short explanation of why Pearl's casual inference diagrams are so highly-regarded, useful or effective? I can't help but think it's just so simple an idea that I can't tell why it could be ...
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Weighted Chi-square

If X~$N(0,I_n)$ and A is a symmetric nxn matrix Let $Q=X^TAX$. Then we know Q is chi-square iff A is idempotent matrix. However can Q be weight chi square if A be any symmetric nxn matrix? Weight Chi ...
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Calculating the posterior distribution - missing dependency

I am reading an old paper from https://www.jmlr.org/papers/volume1/tipping01a/tipping01a.pdf. In that paper, specifically, Equation 10 says $$ p(w|t,\alpha,\sigma^2) = \frac{p(t|w,\sigma^2)p(w|\alpha)}...
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Why don't n-1 for df in this scenario

Simple statistics question, but can't figured out. What is the critical value t* for constructing a 99%, percent confidence interval for a mean with 11 degrees of freedom? My answer for this is 3....
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Probabilistic bounds on approximation of nonlinear functions via Volterra functionals (and related methods)

I'm working on a nonlinear systems identification problem, and as far as I can tell variations on Volterra functionals are the best approach known for this problem - barring deep learning. The latter ...
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Power and size of a statistical hypotheses testing

I would like to know here on the page 132 why the pink (the power of the test) is vertically aligned with the blue (the Type I error). What is the intuition behind it ? Also, how the formula (7.2) in ...
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Number of estimated parameters in gaussian mixture model

Considering a gaussian mixture model with $n$ components, the model contains $n-1$ weight parameters to estimate and $2\cdot n$ parameters for mean and and variance to estimate. In total this model ...
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1 answer
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Find the maximum likely estimator of $\frac{1}{\sigma^n}e^{-\frac{\sum_{i=1}^{n}x_i-n\mu}{\sigma}}$ [duplicate]

Find the maximum likely estimator of $L(\sigma,\mu)=\frac{1}{\sigma^n}e^{-\frac{\sum_{i=1}^{n}x_i-n\mu}{\sigma}}\mathbb{I}_{(-\infty,X_{(1)}]}(\mu)$ We differentiate $L$ and equate to zero to find the ...
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Interpreting a linear regression model

The relationship between annual average temperature over 10 years in various towns and the area of the UK in which the town is located was investigated. Area is described as being one of five ordered ...
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EM algorithm derivation

In the original derivation of the Expectation Maximization (EM) algorithm by Dempster et al. J. Royal Stat. Soc. B (1977), distribution of the incomplete data is given as (Equation 1.1) $$g(\mathbf{y};...
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1 vote
1 answer
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Unit-Root Test Alternative Hypothesis - Dickey-Fuller Test

For a simple autoregressive model satisfying ${p_t = \phi_0 + \phi_1 p_{t-1} +\epsilon_t}$, when you want to test whether the series has a unit root (non-stationary) why is the alternative hypothesis ...
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5 votes
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Farmer wants to know how wet their field is

Problem A farmer wants a better understanding of rainfall on their field. Assuming rain falls randomly and with equal likelihood over the entire field, the farmer thinks they can model the volume of ...
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How to standardize obs of a sample mean?

Given a population with a non-standard normal distribution, we know that the sample mean x' is non-standard normally distributed with average "a" and standard error "b" that is the ...
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Does this probability density function linked to the Normal distribution have a name?

In the process of deriving a confidence interval for the 'natural parameter'$\frac{\mu}{\sigma^2}$ of the Normal distribution, a (conditional) density is derived with a particularly simple form,$$g(y;\...
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