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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Prove that $T$ is a complete statistic and find a UMVUE for $p$
While preparing for my prelims, I came across this problem:
Let $X_1, X_2,\cdots, X_n$ be a sequence of Bernoulli trials, $n \geq 4.$ It is given that, $X_1,X_2,X_3 \stackrel{\text{i.i.d.}}{\sim} Ber(\...
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Birnbaum's Theorem: Strong belief in a model $\implies$ the likelihood function must be used as a data reduction device?
Working through understanding section 6.3.2 (pg. 292-294) in Casella and Berger's Statistical Inference (2nd-ed).
The following definitions and principles are given:
Definition (Experiment): An ...
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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 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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