Mathematics Stack Exchange
Mathematics Stack Exchange

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.

Statistical inference makes propositions about a population using data sampled from the population. To test a hypothesis about a population, a typical workflow is to select a statistical model of the process that generates the data and then deduce propositions from the model.

Statistical propositions include—

  • a point estimate, which is a particular value that best approximates some parameter of interest,

  • an interval estimate, for example, a confidence interval (or set estimate), which is an interval constructed using a data set drawn from a population so that, under repeated sampling of such data sets, such intervals would contain the true parameter value with the probability at the stated confidence level,

  • a credible interval, which is a set of values containing, for example, 95% of posterior belief,

  • rejection of a hypothesis, or

  • clustering or classification of data points into groups.

3930 questions
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Asymptotically normal but biased estimator

This is the problem 2.11 from Lehman book "Theory of point estimation" 2-nd edition. Construct a sequence $\{\delta_{n}\}$ of estimators of $g(\theta)$, satisfying $$ \sqrt{n}[\delta_{n} - g(\theta)]\stackrel{d}{\to}\mathcal{N}[0,v(\theta)], \;…
Lars
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Limiting Distributions and the Weak Law of Large Numbers

I have that $Y_1, Y_2, ..., Y_n$ are i.i.d. Poisson random variables with mean 1, and that $U_n = \sqrt{\frac{\sum_{i=1}^{n}{Y_i^2}}{n}}$. Given that I have a sequence $U_1, U_2, ..., U_n$, I'm trying to find the pdf/pmf of the limiting distribution…
Guest
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$\def\Var{\operatorname{Var}}\Var \left[\frac{(n-1)S^2}{\sigma^2} \right] = 2(n-1) \Longrightarrow \Var(S^2) = \frac{2\sigma^4}{(n-1)}$

$\def\Var{\operatorname{Var}}$ $$\Var \left[\frac{(n-1)S^2}{\sigma^2} \right] = 2(n-1) \Longrightarrow \Var(S^2) = \frac{2\sigma^4}{(n-1)}$$ I know that when you take a constant out of the variance, you square it... but this implies that $\sigma^2$…
user220549
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Likelihood Theory and finding maximum likelihood

Let $X_1,\,\ldots,\,X_n$ be independent random variables, each with probability density function $$f(x;\,\theta)=\frac{2x}{\theta^2}\qquad \text{for }0
user2850514
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Exponential distribution word problem involve watch

The amount of time that a watch will run without having to be reset is a random variable having an exponential distribution with $\theta=120$ days. Find the probability that x is the amount of time a watch will run without being reset. a. have to be…
Fernando Martinez
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Maximum Likelihood Estimator for a Poisson random variable

$X$ is distributed as a Poisson random variable with parameter $\lambda$ if it has the probability mass function: $f(x; \lambda) = \dfrac{e^{-\lambda}\lambda^x}{x!},\ x = 0,1,2,\ldots$ Find the maximum-likelihood estimator for the parameter…
Rosie E
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Population Proportions

50% of teens believe that we need to address the problem of climate change. If a random poll of 1500 teens was taken (about the size of the Gallup poll), what is the chance that its proportion would accurately reflect the population proportion…
Rosie E
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z alpha by 2 distribution-help!

How do we use z aplha by 2 distribution? I don't get it one bit. For the table it says z on lhs and second decimal place of z on the top in the rhs. Firstly if e.g apha is 0.05 then apha by 2 is 0.025 so we know that we have z subscript 0.025 so…
user134785
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simple condition for Asymptotic normailty of MLE

Is these condition sufficient for asymptotic normality of MLE? if it is pls. let me know the references. 1-First and second derivatives of $\ell(\theta,\eta)$ are defined. 2-The Fisher information matrix be non-singular and continuous with…
Sedi
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Do Chi^2 confidence intervals give same results as F-test for variances?

If I have two sample variances, and I compute their confidence intervals as $\left[\, \frac{(n-1)s^2}{\chi^2_{n-1,1-\alpha/2}},\ \ \frac{(n-1)s^2}{\chi^2_{n-1,\alpha/2}} \,\right]$, I could reject the null hypothesis if…
feetwet
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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. The same info can be found in this wikipedia…
skymonkey
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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 following problem - problem for me, that is, not…
Geoffrey Thomas
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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 want to infer $\theta$ in a sample-efficient way…
Kilean Hwang
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Cost Estimate Error

I'm working on a cost estimate for a project. I have six categories for the cost estimate. Of the six categories, I know exactly what the cost will be for three of them (50% certain on categories). Based on my best judgement, I have estimates for…
user1260014
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The Gamma function for non integers

I am trying to work on problem 5.18 on Casella Berger and have a doubt. I will write the main informations below and then will point out where I am having trouble. Let $X$ be a random variable with a Student's $t$ distribution with $p$ degrees of…
user1125373
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