# 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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### Help understanding sufficient statistic proof

I am having a hard time following this proof, maybe the solutions have jumped some steps but was wondering if someone could help me follow it. The question: let $X_{1},...,X_{n}$ be independent and ...
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### How can I prove that $T=\sum_{i=1}^n X_i^6$ is sufficient for $\theta$?

I’m revising for an exam in Mathematical Statistics and I have found the following problem in one of the previous exams (I don’t know the source): Let $X_1,…,X_n$ be a random sample that follows the ...
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### Given n iid Pareto distributed random variables, find the UMP one sided test of the first moment

Given $X_1,...,X_n$ ($n\geq 2$) are iid and each have density: $f_X(x) = \frac{c^\theta \theta}{x^{1+\theta}}\mathbb{1}(x> c)$ for known $c$ and $\theta > 1$ then we can easily find the first ...
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### Conditional expectation of product of Normal variate given their sum

Given $$X_1,\ldots,X_n\stackrel{\text{iid}}{\sim}\mathcal N(0,1)$$ I would like to compute the conditional expectation $$\mathbb E\Big[\prod_{i=1}^n X_i \Big| X_1+\cdots+X_n=x\Big]$$ for statistical ...
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### Tricky exercise on sufficient statistics (undergraduate level)

During my semester, my class and I were subjected to an interesting exercise that challenged my entire understanding of sufficient statistics (It turns out that the median of this exercise was 0). ...
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### Showing order statistic is sufficient

Suppose the model $P_θ$ is the class of all continuous distributions; this is called a ‘nonparametric family’, where the unknown parameter $θ$ is the whole distribution function. Let $x_1,...,x_n$ be ...
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### An exercise in "Mathematical Statistics Jun Shao" about the completeness of a 'modified' exponetial family

It is not the first time meeting this problem in StackExchange and I have read the answer to it(the original solution is copied at the bottom, also available in Show a statistic is complete but not ...
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### How can I formally reason that the data itself is a sufficient statistic? [closed]

Intuitively, it is very clear that the entire data itself is a sufficient statistic for a parameter of interest. Formally, if random variable $X$ represents data, $S(X)$ is a sufficient statistic ...
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### Is the range of a sample of size n a "sufficient statistic"?

I know that every order statistic themselves are sufficient statistic. The range is the max minus the minimum. Is the range also a sufficient statistic because it is a function of two sufficient ... 1 vote
Let $X_1,X_2,...X_n$ be a random sample from $f(x,\theta)=\frac{1}{2 \theta}e^{\frac{-|x|}{\theta}}$.We know by Factorisation theorem that $\frac{\sum |X_i|}{n}$ is sufficient for $\theta$. But can ... 