# 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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### When are Bayes estimators injective as a function of sufficient statistics?

I know that Bayes estimators can be written only as a function of sufficient statistics. When are those functions injectives? That is, when can I say that, given a bayes estimator $\delta (\cdot)$ and ...
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### Help developing intuition behind sufficient statistics (Casella & Berger) [duplicate]

Migrated from MSE I am trying to understand the following intuition for sufficient statistics in Casella & Berger (2nd edition, pg. 272): A sufficient statistic captures all of the information ...
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### Sufficient statistic as iso-surfaces in the distribution density. Is it possible to generalise to multiple parameters?

For continuous distributions, there is a geometric intuition behind sufficient statistics that regards a multivariate probability density as several iso-surfaces. This works at least for cases where a ...
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### How does knowing the sign of the population correlation affect the sufficiency of its statistic?

As noted here, the sufficient statistic for the correlation under bivariate normality is Pearson's $r$, the maximum likelihood estimate of $\rho$. I suppose, however, this does not guarantee that $r$ ...
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### Find minimal sufficient statistic of this random sample with cursed support

Suppose $X_1,X_2,...,X_n$ is a i.i.d random sample with probability mass function $p(x_i,\theta)$ where $x_i \in \{\theta,\theta+1,\theta+2,...\}$ and $\theta \in \mathbb{R}$. I claim that minimal ...
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### Is the Sufficiency Principle an axiom?

Sufficiency Principle as defined in Casella: Where Sufficient Statistic is defined as: Question: Is the Sufficiency Principle an axiom? My thoughts and research so far: I'm uncertain if the ...
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### Does $f : p_\theta\mapsto p_{T\,\mid\,\theta}$ being injective imply statistic $T$ is sufficient?

Wikipedia says ... consider the map $f:p_{\theta }\mapsto p_{T\,\mid\, \theta }$ which takes each distribution on model parameter $\theta$ to its induced distribution on statistic $𝑇$. The ...
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### Why does the sufficient statistic for the bivariate normal not imply a sufficient statistic for the correlation under bivariate normality?

This question links to a document by Jon Wellner that defines the sufficient statistic for the multivariate normal (p. 7, Example 2.7). The result follows from the factorization theorem and is proven ...
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### Reference request for the existence of minimal sufficient statistics

I'd like a recent paper or book that shows in what conditions we can guarantee the existence of a minimal sufficient statistic. I know the paper "Sufficiency and Statistical Decision Functions&...
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### Formal definition of sufficient statistic

Let $(\Omega_X,\mathcal{F}_X)$ and $(\Omega _T,\mathcal{F}_T)$ be measurable spaces. Let $\mathfrak{M}$ be a family of probability measures on $(\Omega_X,\mathcal{F}_X)$. Let $X:\Omega\to \Omega _X$ ...
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### Sufficient Statistic for a finite family of Normal distributions

Suppose we have a finite family of normal distributions $P=\{N(0, 1), N(0, 2), N(1, 2), N(2, 2)\}$ and we want to find a sufficient statistic for this family. Intuitively it is clear that as the means ...
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### Can the dimension of a (potentially) sufficient statistic exceed the dimension of the parameter it estimates?

I understand that if the dimension of a sufficient statistic exceeds that of the parameter it estimates, then that particular sufficient statistic won't be minimal. Now, in the following case, I ...
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### How to prove that this statistic is not sufficient? [duplicate]

Problem. Given $X_1,X_2,X_3$ a random sample from the Bernoulli distribution with success $\theta$, show that the statistic $T= X_1+2X_2+3X_3$ is not sufficient. My attempt When I try to apply the ...
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### Prove that the sum is sufficient using using the definition of sufficiency

If $X_1,\ldots,X_n$ is an IID random sample, with $X_i\sim\,\text{Ber}(\theta)$, prove that $Y = \sum_i X_i$ is sufficient using the definition of sufficiency (not the factorization criterion). Now ...
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