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 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 Truncated Normal
I am doing exercise 3.18 of "The Bayesian Choice":
Give a sufficient statistic associated with a sample $x_1,...,x_n$
from a truncated normal distribution $$ f (x|\theta) \propto \exp(-(x
...
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Is Pitman-Koopman-Darmois Theorem valid for discrete random variables?
I am interested in the Pitman-Koopman-Darmois theorem.
I'm having a hard time finding a simple rigorous version of this theorem as I struggle finding sources.
This helpful post provides three sources ...
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How does reparametrization of the Fisher information matrix change the variance expression for the sufficient statistics?
If I have an exponential family distribution of the form $$p_{\theta}(x) = e^{\theta^T\cdot t(x) - \psi(\theta)},$$ where $\theta$ is a vector of parameters, $t(x)$ is a vector of sufficient ...
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Usage of Sufficient statistic for a Gamma distribution
I need some help to understand how to utilize sufficient statistic from a data.
Suppose I observe some random process that produces $x\in X$, where all elements have a gamma distribution. As far as I ...
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Find a two dimensional sufficient statistic for $\theta$
Let $\{X_i\}_{i=1}^n$ be conditional independent given $\theta$ with distribution
$$p_{X_i | \theta} (x |\theta) = \frac{1}{2i\theta}, \ -i\theta<x<i\theta.$$
Find a two dimensional sufficient ...
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Likelihood principle and inference
I've been reading Casella and Berger's Statistical Inference. In section 6.3 the author stated the likelihood principle: if the likelihood functions from two samples are proportional, then the ...
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FInding a complete and sufficient statistic
I am attempting to learn how to find a complete and sufficient statistic. So, I am working on this problem for class:
Let $X_1, \cdot\cdot\cdot,X_n$ be a random sample from the pdf $f(x_i|u)=e^{-(x-\...
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