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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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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Show that the sample Mean is not complete
Suppose that $X_1, ..., X_n$ is a sample from a $\mathcal{N}(-\frac12 \sigma^2, \sigma^2)$ density. Show that the statistic $\bar{X}$ = $n^{-1} \sum_{i=1}^n X_i$ is NOT complete.
I am struggling to ...
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Sufficient/complete statistic $\leftrightarrow$ injective/surjective map?
I can't understand the paragraph in Completeness (statistics) - Wikipedia:
We have an identifiable model space parameterised by $\theta$, and a statistic $T$. Then consider the map $f:p_{\theta }\...
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When is sufficiency and completeness of a statistic preserved? [duplicate]
This question has been asked in math stack but no one has replied.
I have been given these definitions in my statistical inference class:
Let $(X_1,...,X_n)$ be a simple random sampling of $X\...
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Complete Statistic for a family with finite r-th moment
Consider the family of all continuous distributions with finite $r$-th moment (where $r \geq 1$ is a given integer). We denote this family as, $$\mathscr{P}_r=\left\{f:f \ \text{is a pdf and} \int|x|^...
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Show minimal sufficient statistic is not complete in normal distribution
Let $Z_i$ for $1 \leq i \leq n$ be a sample from the $N(ap, bp(1-p))$ density, where $a \gt 0, b \gt 0$ are known but $p \in (0,1)$ is an unknown parameter.
I have shown that $T = (\sum^n_{i = 1} Z_i, ...
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Showing incompleteness of density
You observe a sample of 100 independent observations $X_i$ from a population with the density
$$
g(x)=C \sqrt{\lambda} \exp \left(-\lambda x^2-\lambda^2 x^4\right), \quad-\infty<x<\infty
$$
...
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Proving Incompleteness of joint sufficient statistic
Let $X_1, ..., X_n$ be a sample from the continuous density $C~exp(-(x-\theta)^4)$ (for $ -\infty < x < \infty$) with $\theta$ as unknown parameter. Show that the minimal sufficient statistic is ...
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How to show that $\{N(\theta,1):\theta \in \Omega\}$ is not a complete family of distributions when $\Omega$ is finite?
Consider the $\{N(\theta,1):\theta \in \Omega\}$ family of distributions where $\Omega=\{-1,0,1\}$. I am trying to show that this is not a complete family. That is, if $X\sim N(\theta,1)$, I need to ...
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UMVUE of $\theta$ for $\mathrm{Uniform}(0,\theta) $ where $\theta \in[1, \infty)=\Theta$
Let $X_1, \ldots, X_n$ be a random sample from $\mathrm{U}(0, \theta)$, where $\theta \in[1, \infty)=\Theta$, say. Here I tried to find complete-sufficient statistics for $\theta$ as my main target is ...
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Verifying the statistics are complete and sufficient for two parameter Pareto distribution
Let$(X_1,...,X_{n})$ be a random sample from the Pareto distribution
with pdf density $\theta a^{\theta} x^{-(\theta+1)}I_{(a,\infty)}(x),$ where $\theta>0$ and $a>0$
$\textbf{(i)}$ Show that ...
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