# Questions tagged [sufficient-statistics]

A sufficient statistic is a lower dimensional function of the data which contains all relevant information about a certain parameter in itself.

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### Sufficient Statistics - Relating the Intuition with the Mathematical Definition

I believe the heuristic definition of a Sufficient Statistic makes sense to me - when you take a sample in order to make an inference about the parameter related to the probability distribution, and ...
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### When a function of sufficient statistic is itself sufficient?

I'm following notes at onlinecourses and I got confused on transformation of sufficient statistics. For example, if $X$ is a sufficient statistic for $\mu$, why $Y=X^2$ is not a sufficient statistic ...
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### Proving the MVUE is the following

I am stuck on the following question and I was wondering if can get some help. Let $f(x;\theta) = g(\theta)h(x),\ a(\theta) \leqslant x \leqslant b(\theta)$ with $a(\theta)$ decreases and $b(\theta)$...
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### Prove the maximum order statistic $X_{(n)}$ is a minimal sufficient statistic for the uniform$(0,\theta)$ family using a particular theorem

I'm doing Exercise 6.26 in Casella and Berger's Statistical Inference, and I'm trying to prove the following: "Use Theorem 6.6.5 to establish that, given a sample $X_1,...,X_n$, the maximum order ...
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### Finding the conditional distribution of single sample point given sample mean for $N(\mu, 1)$

Suppose that $X_1, \ldots, X_n$ are iid from $N(\mu, 1)$. Find the conditional distribution of $X_1$ given $\bar{X}_n = \frac{1}{n}\sum^n_{i=1} X_i$. So I know that $\bar{X}_n$ is a sufficient ...
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### Intuitive understanding of the Aldous-Hoover representation theorem for row-column exchangeable arrays

I would like to ask a couple of questions about the Aldous-Hoover theorem for the representation of probability distributions over (2D) arrays with exchangeable rows and columns. I'd be happy about ...
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### Are unbiased efficient estimators stochastically dominant over other (median) unbiased estimators?

General description Does an efficient estimator (which has sample variance equal to the Cramér–Rao bound) maximize the probability for being close to the true parameter $\theta$? Say we compare the ...
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### Intuitive understanding of the Halmos-Savage theorem

The Halmos-Savage theorem says that for a dominated statistical model $(\Omega, \mathscr A, \mathscr P)$ a statistic $T: (\Omega, \mathscr A, \mathscr P)\to(\Omega', \mathscr A')$ is sufficient if (...
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### Minimal sufficient statistics

Suppose we have data $X = X_1,\ldots,X_n$, $Y = Y_1,\ldots,Y_n$ that is i.i.d. generated by a distribution $\mathbb{P}_\theta$. Let $T$ be a test statistic such that that $T(X) = T(Y)$ if and only ...
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### Proving Order Statistics are Minimal Sufficient

I am given continuous i.i.d. random variables $X_1,\dots,X_n$ having an unknown p.d.f. $f$ and am trying to show that the vector of order statistics $(X_{(1)}, \dots, X_{(n)})$ is minimal sufficient ...
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### Does Fisher's factorization theorem provide the pdf of the sufficient statistic?

From Wikipedia Fisher's factorization theorem or factorization criterion provides a convenient characterization of a sufficient statistic. If the probability density function is $ƒ_θ(x)$, then $T$ ...
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### Intutitive meaning behind the formal definition of sufficient statistic?

According to the definition of sufficiency, a statistic is sufficient for a parameter if the conditional distribution of $X$ given a value of statistic does not depend upon the parameter. What I am ...
I have derived a likelihood function for $\theta$ as follows: $$L(\theta)=(2\pi\theta)^{-n/2} \exp\left(\frac{ns}{2\theta}\right)$$ Where $\theta$ is an unknown parameter, $n$ is the sample size, ...