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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Gaussian sufficient statistic calculation
Consider the Gaussian model
$$
Y_i = \beta + \epsilon_i,\, i = 1, \cdots, n,\; \mbox{where}\; \epsilon_i
\stackrel{i.i.d.}{\sim} \mathcal{N}(0, \sigma^2),
$$
parametrized by $\beta$, with known $\...
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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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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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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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Why does a sufficient statistic contain all the information needed to compute any estimate of the parameter?
I've just started studying statistics and I can't get an intuitive understanding of sufficiency. To be more precise I can't understand how to show that the following two paragraphs are equivalent:
...
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Sufficient statistics for layman
Can someone please explain sufficient statistics in very basic terms? I come from an engineering background, and I have gone through a lot of stuff but failed to find an intuitive explanation.
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Sufficient statistic for the family of PERT distributions?
A beta distribution is one of the form
$$
\text{constant}\times x^{\alpha-1} (1-x)^{\beta-1} \, dx \quad \text{ for } 0<x<1.
$$
According to this Wikipedia article, the family of "PERT ...
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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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Bayesian definition of sufficient statistics [duplicate]
Some time ago I wrote a question about what I think/thought (up to my understanding) is an ambiguity of the common definition of sufficient statistics :
Conditioning in the definition of sufficient ...
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Sufficiency for Truncated Geometric
Here is a deviant of a question I feel like I have seen several times on truncated exponentials and similar distributions for finding sufficient statistics:
Let
$$\mathbb{P}(Y=y)=\theta^y(1-\theta)^{\...
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Are these statistics sufficient?
Question (Casella and Berger 6.5):
Let $X_1 \ldots X_n$ be independent random variables with pdfs:
$f(x_i|\theta)= \begin{cases} \frac{1}{2i\theta}, & -i(\theta - 1)<x_i<i(\theta+1) \\ 0,...
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Relating sufficient statistics to parameters
I'm studying sufficient statistics and I came across this problem:
A dataset consists of independent triples $(W_i,Y_i,Z_i)$ of independent random variables with distributions as follows,
$$ W_i \sim ...
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