Questions tagged [complete-statistics]
A complete statistic T (in some statistical model) is such that for all functions g, if E g(T)=0 for all parameter values, then g is identically zero.
122 questions
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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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Can conditional expectation of a non-constant function not depend on a conditioning variable under a completeness condition?
My coauthors and I have been struggling with the following question:
Let $(X,Y,Z,W)$ be random variables with support on $\mathbb{R}^{4}$ such that the conditional expectation operator with respect to ...
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Why exactly Is conditional inference impossible, when conditioning on a non-ancillary statistic?
Suppose we define an estimator $U(X)$ of parameter $\theta$ by "manually" mapping each individual value in the sample space of sample $X$ to some value in the sample space of $U$. Obviously ...
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Non-complete statistics for $f_\theta(x) = \frac{1}{\theta}e^\frac{\theta-x}{\theta}\mathbb{1}_{[\theta,\infty]}(x)$
I have a probability density function of the form $$f_\theta(x) = \frac{1}{\theta}e^\frac{\theta-x}{\theta}\mathbb{I}_{[\theta,\infty]}(x)$$ and I have found that $$T = \left(\sum_{i=1}^{n}X_i, X_{(1)}...
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Weaker "Complete" Statistic Definition
Why wouldn't we define completeness for a sufficient statistic $T(x)$ as $g(T(x))$ is dependent on $\theta$ for all functions g?
The definition
$$\mathbb{E}_\theta \left[ g(T(X)) \right] = 0 \quad \...
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Define "ancillary information" without referencing ancillary statistics
I'm looking for a definition of ancillary information that isn't essentially "information in an ancillary statistic." Interestingly, definitions for ancillary information seem hard to come ...
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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)...
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Complete and sufficient statistic when only the maximum of the data is observed
I found the joint PDF, but I was unable to apply factorization afterwards.
How can i solve this problem?
The joint PDF is given by:
$
F(z, \Delta) = \left(f_X(z) F_Y(z)\right)^\Delta \left(f_Y(z) F_X(...
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Motivation behind this exercise problem on complete sufficient statistic
This is from Hogg and McKean's "Introduction to Mathematical Statistics" Chapter 7 (Sufficiency), section 7.4 (Completeness and Uniqueness).
Exercise 7.4.10.
Let $Y_1 < Y_2 < \cdots &...
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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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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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Why are complete statistics named "complete"?
I get why sufficient statistics are named "sufficient", but what about "complete" statistics?
I have this definition from F.J. Samaniego, Stochastic Modelling and Statistical ...
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Structural (causal) interpretation of the completeness condition
Consider two random variables $X,Y$. We say the joint distribution of $P(X,Y)$ is complete w.r.t. $X$ if the following condition holds:
For all $y$, $E\{g(x)|y\}=0$ almost everywhere if and only $g(x)...
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Completeness of Gamma family
Let $X_1,...,X_n$ has a Gamma$(\alpha,\alpha)$ distribution. Find the minimal sufficient statistics. Is this a complete family?
My attempt: I found the Minimal sufficient statistics is $T(x)=(\...
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