Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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.

For more information see https://en.wikipedia.org/wiki/Completeness_(statistics) and for some intuition and examples see What is the intuition behind defining completeness in a statistic as being impossible to form an unbiased estimator of $0$ from it? and links therein

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Why do we not care about completeness, sufficiency of an estimator as much anymore?

When we begin to learn Statistics, we learn about seemingly important class of estimators that satisfy the properties sufficiency and completeness. However, when I read recent articles in Statistics I can hardly find any papers that address complete…
pineapple
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What is the intuition behind defining completeness in a statistic as being impossible to form an unbiased estimator of $0$ from it?

In classical statistics, there is a definition that a statistic $T$ of a set of data $y_1, \ldots, y_n$ is defined to be complete for a parameter $\theta$ it is impossible to form an unbiased estimator of $0$ from it nontrivially. That is, the only…
user1398057
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Meaning of completeness of a statistic?

From Wikipedia: The statistic $s$ is said to be complete for the distribution of $X$ if for every measurable function $g$ (which must be independent of parameter $θ$) the following implication holds: $$ \mathbb{E}_\theta[g(s(X))] = 0,…
Tim
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Jointly Complete Sufficient Statistics: Uniform(a, b)

Let $\mathbf{X}= (x_1, x_2, \dots x_n)$ be a random sample from the uniform distribution on $(a,b)$, where $a < b$. Let $Y_1$ and $Y_n$ be the largest and smallest order statistics. Show that the statistic $(Y_1, Y_n)$ is a jointly complete…
emlu
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Intuition Behind Completeness

The definition for completeness is that if a statistic $s(x)$ is complete, we have that for every measurable $g$, $$E_\theta(g(s(x))) = 0\,, \ \forall\,\theta\ \Rightarrow\ g(s) = 0 \text{ a.s.}$$ I've heard that we can think of completeness as…
Christopher Aden
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Sufficient Statistic for non-exponential family distribution

Question: Let $X_1,X_2,\ldots,X_n$ be an iid sample from $N(\theta , 4 \theta^2 )$. I want to show that this model is not a member of the exponential family and to find a sufficient statistic for $\theta$ Attempt:…
WeakLearner
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Whether the minimal sufficient statistic is complete for a translated exponential distribution

Let $X_1, X_2..., X_n$ follows iid negative exponential distribution with pdf $$f(x) = \frac{1}{\theta^2} \: e^{-\frac{(x-\theta)}{\theta^2}} \: \: I_{(x>\theta)} $$ I have to show whether the minimal sufficient statistic for this pdf is complete or…
n1234
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What is exponential family criterion to test the sufficiency and completeness of an estimator?

I am struggling to understand the following result from Casella and Berger about sufficiency and completeness for exponential families: Let $X_{1},X_{2},...,X_{n}$ be iid observations from an exponential family with PDF or PMF of the…
userNoOne
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Is a minimal sufficient statistic also a complete statistic

I know that if a statistic is both sufficient and complete then it must also be minimal sufficient. But on the other hand, could I say a minimal sufficient statistic must also be a complete statistic?
zqzwxec11
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Finding complete sufficient statistic

Let $X_1, \dots, X_n$ be iid. $\text{Uniform}[-\theta,\theta]$. I need to find the complete sufficient statistic for $\theta$ or prove there does not exist such. I know that $T = (X_{(1)}, X_{(n)} )$ is a sufficient statistic for $\theta$ but it is…
Sam88
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Complete sufficient statistic and unbiased estimator

I am now studying complete sufficient statistic. My question is: Is there any relationship between the existence of complete sufficient statistic and the existence of unbiased estimator? I know that by Lehmann-Scheffe Theorem, if an unbiased…
Mhr
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What is the difference between complete statistics and complete family of distributions?

I fail to understand when we call a family of distribution is complete and when a statistic is complete. What is the difference between both?, Is there a relation between them? Please provide examples along with your answer.
Vishaal Sudarsan
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Basu's Theorem Proof

I am having trouble with the proof of Basu's theorem... specifically, I'm not sure about the $\theta$s in the expectations below: Let $T$ be a complete sufficient statistic. Let $V$ be an ancillary statistic. Let $A$ be an event in the sample…
LotsofQuestions
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Are These Conjectures Regarding Sufficient Statistics True?

I have these conjectures that I cannot quite prove (unless I impose another regularity condition of parameter-independent support for distribution, in which case, the conjectures are trivially true --- I think). Would appreciate 1) confirmation (w/…
Shang Zhang
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Trying to make sense of claims regarding Rao-Blackwell and Lehmann-Scheffé for sufficient/complete statistics

I am currently trying to learn the two related concepts of the Rao-Blackwell theorem and the Lehmann-Scheffé theorem. Assume we have the random sample $X_1, \dots, X_n$ with mean $\mu$ and variance $\sigma^2 < \infty$. We have that $E[S^2] =…
The Pointer
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