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.

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If $P$ is the set of all distributions, the only sufficient subfield is the trivial one

According to an article by Bahadur, if $P=\left\{p\right\}$ is the set of all probability measures on the measurable space $\left(\Omega,\mathcal{A}\right)$, $\mathcal{A}$ is the only possible sufficient subfield. The claim is left unproved in…
Evan Aad
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existence of sufficient statistic for sum of two i.i.d. uniform variate

Let $X_1$ and $X_2$ be two i.i.d. Uniform variate on $[\alpha,\beta]$. Does there exist a sufficient statistic for $X_1+X_2$? Let us divide the problem in two parts. First part obviously consists to find the distribution that $X_1+X_2$ follows.…
am_11235...
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Can I use the maximum likelihood to show a statistic is complete?

Problem: Consider a random sample of size $n$ that follows a density probability function given by: $$f(x,\theta)=\frac{1}{\theta} x^{-\frac{\theta+1}{\theta}}\mathbb{1}_{(1,+\infty)},\:\:\theta>0$$ where $\theta$ is unknown. Provide an…
Pedro Gomes
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Non-Parametric nCRP VAE. How to condition the nCRP prior

I recently read the paper Nonparametric Variational Auto-encoders for Hierarchical Representation Learning http://openaccess.thecvf.com/content_ICCV_2017/papers/Goyal_Nonparametric_Variational_Auto-Encoders_ICCV_2017_paper.pdf And I am confused…
Boyang Zhang
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Need Check on Proof with Chebychev's Inequality: Statistical Theory

So I have a problem with my homework, and I just need to see if my proof and thinking are correct. The problem I have is this: Show that the sample variance $s^2$ is a consistent estimator for the variance of X, $\sigma^2$. I had to manipulate…
Perdue
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Trying to find consistency of an estimator: Stat Theory

So I have a homework problem that my study group and I are stuck on. The problem goes,"Let $X$ be a random variable with mean $\mu$, and variance $\sigma_2$, and we have a sample $\{X_1, X_2 , \ldots , X_n \}$. Show that $T=…
Perdue
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Mathematical Statistics (Significance level)

Let $X_1$, $X_2$ be a random sample of size $n=2$ from the distribution having pdf $$f(x;\theta)=\left( \dfrac 1{\theta} \right)e^{-\frac x{\theta}}, 0 \lt x \lt \infty$$ We reject $H_0: \theta=1$ is the observed values of $X_1$, $X_2$, say $x_1$,…
Paul Ash
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Reasoning behind the statement 'Consistency is a property of a sequence of estimators rather than one point estimator"

I just started learning statistical inference course and I am doing this topic called consistency. I am not able to understand this line 'Consistency is a property of a sequence of estimators rather than one point estimator" If anyone could explain…
Daman
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Test for a poisson distributed random variable

I need help with the following exercise: Assume the amount of apples falling to the ground from a single tree can be modeled by a poisson distributed random variable $X$ with expectation $m$. The apples from 4 trees are collected and counted every…
user25470
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Estimate parameters with moments method

I am trying to know how to estimate a parameter with the moments method. The wikipedia article and similar websites are too confusing and formal for me to understand. I'm looking for a more basic and school type "how-to". For example ; Let…
Dranna
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Confidence interval for exponential distribution

I'm having trouble with homework and hope somebody can help. The lifespan of a lightbulb is assumed to be a random variable $X$ with density function: $$f_X(x)=e^{-x/\theta}/\theta,\, 0\leq x $$ The lifespan of a single lightbulb have been measured…
user25470
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Likelihood ratio test of two sample normal distributed with known means(which is 0) and unknown variance

Let $X_1,X_2,\cdots ,X_n$ be random sample form $N~(0,\theta_1)$ and Let $Y_1,Y_2,\cdots ,Y_m$ be random sample form $N~(0,\theta_2)$. Determine the $\lambda$, the likelihood ratio test in testing $H_0 : \theta_1=\theta_2$ and $H_1 : \theta_1 \neq…
cavvot
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Why Hypothesis testing emphasizes the rejection

When I take a statistical inference course, the professor said that hypothesis testing emphasizes the rejection, and usually we would say that we cannot reject $H_0$ than accept $H_0$. I'm confused about this statement and interpretation of…
HCR
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co-variance matrix of discrete multivariate random variable

Let $\mathcal{S} = \{(\delta,0,0),(0,\delta,0),(0,0,\delta)\}$. If we consider a random variable $S$ defined over $\mathcal{S}$ such that \begin{equation} S = \begin{cases} s, & \text{ if } s\in\mathcal{S}\\ 0, & \text{otherwise}. …
Satya Prakash
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How to compute the consistency of an estimator

Can u help me? consider a sample $X_1,X_2,\dots,X_n$ from the following density function $$f_{\theta}(x)= \frac{1}{\theta}\exp\left[-\frac{1}{\theta}x\right],\quad x>0$$ where $\theta>0$ is an unknown parameter. Show that the following…
Angela Francesca Papa
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