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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Likelihood ratio theorem for Minimal sufficient statistic when the PDF is not pointwise defined
For a background, I am a pure mathematician and am looking for a fully rigorous statement / proof for results in mathematical statistics :). It is probably better to post this in the math page but I ...
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$X_1,...,X_m$ follows iid $\text{Bin}(n,p)$, $0<p<1$ . $T=\sum X_i $. What is the UMVUE of $q/p$ where $q=1-p$? [duplicate]
I know that $T \sim \text{Bin}(mn,p)$ and it's also complete and sufficient. For a UMVUE I need a function of $T$. But I'm bothered by $1/p$ situation. Would $mn/T$ be an UE of $1/p$ just as $T/mn$ is ...
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Is sufficiency of $T$ necessary in this theorem? [duplicate]
In our lecture notes, my professor has written the following theorem:
Let $\textbf{X} = (X_1, \dots, X_n)$ have joint density function
$f_{\textbf{X}}(\textbf{x} \mid \theta)$ and $T$ denote a ...
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conditional distribution and sufficiency
The question is related to
Puzzled by the definition of sufficient statistics in Mood, Graybill, and Boes
For a random sample $X_1, X_2, X_3, \dotsc, X_n$ from a distribution $f( ;\theta)$, a ...
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Prove that conditioning upon a sufficient and minimum sufficient statistic respectively , the expected value of an unbiased estimator gives same value [closed]
Suppose that $T_1$ is a sufficient and $T_2$ is minimal sufficient, $U$ is an unbiased estimator of $\theta$ and define $U_1=E[U| T_1]$ and $U_2=E[U|T_2]$.
a) Show that $U_2=E[U_1|T_2].$
b) Show that $...
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Minimal sufficient statistic for negative binomial with both parameters unknown
\begin{align}
\text{Let } & \binom{-r} {\phantom{+}x} = \frac{\overbrace{(-r)(-r-1)(-r-1) \cdots (-r-x+1)}^\text{$x$ factors}}{x!}, \\[8pt]
\text{and } & q = 1-p, \text{ where } 0<p<1, \...
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Sufficient statistics for hierarchical Poisson-Gamma model
I have the following model:
$X_i \mid W = w \sim \operatorname{Poisson}(w\lambda)$ where $W \sim \operatorname{Gamma}(1/\sigma,1/\sigma)$
I would like to calculate jointly sufficient statistics for $(\...
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How to show that a statistic is not sufficient?
We know that a statistic $T(X)$ is sufficient if and only if and only if its joint distribution can be written as product of only function of sample and function of statistic and parameters only i.e. $...
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Factorization Theorem for Two Parameters
The PDF of $X \sim\mathcal{N}(\mu, \sigma^2)$ is
$$
\begin{align*}
f(x) & = \frac{1}{\sigma\sqrt{2\pi}} \, \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \\
&= \frac{1}{\sigma\sqrt{2\pi}} \, \...
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Why is the distribution of sufficient statistic same as the random sample in exponential family?
The exponential family of distributions is defined as below in Ferguson's Mathematical Statistics: A Decision Theoretic Approach book:
A sufficient statistic, $\mathbf{T}$, is given as follows:
Then ...
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Weaker "Complete" Statistic Definition
Why wouldn't we define completeness for a sufficient statistic $T(x)$ as $g(T(x))$ is dependent on $\theta$ for all functions g?
The definition
$$\mathbb{E}_\theta \left[ g(T(X)) \right] = 0 \quad \...
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Difference between estimator and statistic in Rao-Blackwell theorem
Question: Is there a difference between the words 'estimator' and 'statistic' when both are used to find a parameter $\theta$? My understanding is that the estimator 'estimates' a parameter and a ...
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Factorization theorem for sufficient statistics in case of continuous random variables
By the factorization theorem (Fisher-Neyman), we have that a statistic $ T(X) $ is sufficient if and only if there exists a factorization: $ f(x\mid \theta) = g(T(x)\mid \theta)h(x) $. Notation ...
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Concept of complete sufficient statistic for the cdf $F(x)$
I am studying Hogg and McKean's "Introduction to Mathematical Statistics." At the end of section $7.7$ where they talk about completeness, sufficiency etc for multi-parameter case, theny ...
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Motivation behind the technique to find MVUE of $3\theta_2^2$
This question if from Hogg and McKean's "Introduction to Mathematical Statistics."
Exercise 7.7.11.
Let $X_1,X_2,\cdots,X_n$ be a random sample from a $N(\theta_1,\theta_2)$ distribution.
(a)...