# 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.

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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 ---...
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1 vote
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### If a statistic $T(X)=\Sigma_{i=1}^n X_i$ is sufficient does that imply the mean is also sufficient?

I've been working on some problems, the question asked me if the mean of a sample is a sufficient statistic for poisson distribution. I've already proved that $T(X)=\Sigma_{i=1}^n X_i$ is a sufficient ...
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### How to prove or disprove that a complete sufficient statistic exists?

We have a discrete random variable which takes values with probabilities $p, q, p+q$ and $r$. I want to construct a complete sufficient statistic based on a single observation from this distribution, ...
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### Completeness of normal sample values

I have a small query. We know that sample values are always sufficient. Can we say the same for completeness property? Let us say I have $X_1,X_2,...,X_n$ following $N(0, \sigma^2)$. Hence, if we take ...
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### Jointly complete and sufficient statistics for multivariate normal distribution [duplicate]

Consider the random sample X from the multivariate normal distribution where xi are i.i.d as N(µ,Σ). *Show that the sample mean x̄ and Sample covariance matrix S are jointly complete and sufficient ...
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### UMVUE of two-parameter exponential family distribution

Suppose $\{X_{i}\}_{i=1}^n\overset{i.i.d}{\sim}X$, where $X$ has density $$f_{X}(x)=\frac{1}{b}\exp\left\{\frac{x-a}{b}\right\},x>a$$ What is the UMVUE of $\mathbb{P}(X_1<u)$? Here is what I've ...
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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 ...
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### Reparametrization and its effect on sufficient/complete/minimal statistics

Suppose $X_1 \sim Pois(\lambda_1), X_2 \sim Pois(\lambda_2), X_3 \sim Pois(\lambda_1+\lambda_2)$. Separately I can find a sufficient, complete and minimal statistic for each of them. But considering ...
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### Conditional distribution of complete sufficient statistics being ancillary of $\alpha$

Regarding the distribution and statistics as described here, I need to show that the conditional distribution of $\overline{X}$ given $X^*=x^*$ does not depend on $\alpha$. I remember my professor ...
1 vote
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Let $X_1,...,X_n$ be iid and follow $Gamma(\alpha, \beta)$, where $$f(x,\alpha, \beta)=\frac{x^{\alpha-1}e^{-x/\beta}}{\Gamma(\alpha)\beta^\alpha}$$ I already showed that $\overline{X}$ and $X^*=\... 30 votes 2 answers 3k views ### Why do we not care about completeness, sufficiency of an estimator as much anymore? When we begin to learn Statistics, we learn about seemingly important class of estimators that satisfy the properties sufficiency and completeness. However, when I read recent articles in Statistics I ... 1 vote 1 answer 70 views ### Statistics Theory Question Casella& Berger Theorem 6.2.28: If a minimal sufficient statistics exists, any complete statistics is minimal sufficient. So let's suppose$X_1...X_n$are iid$Bernoulli(p)p\in (0,1)$, then$\...
I'm trying to understand the proof of this theorem. An unbiased estimator $T$, that is a function of a complete statistic $S$, is unique, i.e. there can't be other unbiased estimators that are ...