# Questions tagged [sufficient-statistics]

For questions about sufficient statistics. A statistic is sufficient for a parametric model if the distribution of the data conditioned on the statistic is parameter-free. For more general questions about statistics and estimators, please use "statistical-inference".

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### $\mathbb E(X_1 X_2|Y)$ when $Y = X_1 + \cdots + X_n$

Find $\mathbb E(X_1 X_2|Y)$ when $Y = X_1 + \cdots + X_n$ for a Bernoulli distribution for coin flipping with $n$ flips. Here heads = $1$, and tails = $0$. I understand that $X_1 X_2$ equals $1$ if ...
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### Basu's theorem and completeness

Recently, I was reading up on the Basu's theorem and what i gathered of it was that if a statistic $T$ is complete and minimal sufficient then it is free from Ancillary statistics. My question is why ...
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### Proving that $T_{n}(X)=\sum_{i=1}^{n}X_i$ is a sufficient statistic for $p$ given a sample of i.i.d random variables

I am asked to show that the statistic $T(X):=\sum_{i=1}^{n}X_i$ is a sufficient statistic for $p$, where $X_i\sim Geom(p)$ are i.i.d random variables. Given a sample $x=(x_1,x_2,\dots,x_n)$ I have to ...
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### why should conditioning on a random variable subtract information from it?

The defintion of a sufficient statistic is as follows: A statistic $t = T(X)$ is sufficient for underlying parameter $θ$ precisely if the conditional probability distribution of the data $X$, given ...
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### prove that maximum and minimum is sufficient statistic for uniform($\theta$ , $\theta$+1)

I'm trying to understand the proof that $T(X_1,..., X_n)=(max\{X_1,..., X_n\}, min\{X_1,..., X_n\})$ , is a sufficient statistic for $\theta$, given that $f_{\theta}=Uniform(\theta, \theta+1)$ from ...
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### An example of sufficient statistic and how to understand it.

I am learning sufficient statistics, and the general idea seems to be straightforward but it turns out to be really confusing when I take a closer look (and it is probably because of the conditional ...
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### Sufficient statistic for normal distribution by a bijective map

For $X_i$~$\mathcal{N}(\mu,\sigma^2)$, I know that since $T=(\Sigma^{n}_{i=1}x_i,\Sigma^{n}_{i=1}x_i^2)$ is a sufficient statistic for $(\mu,\sigma^2)$, $\bar{T}=(\bar{X}_n,S_n^2)$ ($\bar{X}_n$ is the ...
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### Nonexistence of UMVUE

In Mathematical Statistics written by Jun Shao(2003), exercise 3.22 claims that Exercise 3.22. Let $\left(X_{1}, \ldots, X_{n}\right)$ be a random sample from $P \in \mathcal{P}$ containing all ...
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### Sufficient statistic and the maximum likelihood estimator of the probability of having an infectious disease when people are grouped and tested

Suppose N students arriving at a college are all equally likely to have a particular disease with an unknown probability p. The disease status (affected / not affected) of all students are independent....
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