All Questions
Tagged with sufficient-statistics uniform-distribution
19
questions
1
vote
1
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63
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Find a two dimensional sufficient statistic for $\theta$
Let $\{X_i\}_{i=1}^n$ be conditional independent given $\theta$ with distribution
$$p_{X_i | \theta} (x |\theta) = \frac{1}{2i\theta}, \ -i\theta<x<i\theta.$$
Find a two dimensional sufficient ...
- 265
1
vote
2
answers
876
views
Show that the maximum of $x_1,...,x_n \sim \mathrm{Uniform}(0,\theta)$ is a sufficient statistic for $\theta$. (From definition)
Problem
Show that the maximum of $x_1,...,x_n \sim \mathrm{Uniform}(0,\theta)$ is a sufficient statistic for $\theta$.
Background
This question has been asked before, but most answers tackle the ...
4
votes
1
answer
365
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minimal sufficient statistic for $U(\theta, \theta+c)$. $(\theta,c)$ unknown
Suppose $X_1,\cdots,X_n$ are $i.i.d$ from a distribution with p.d.f
$$\delta_{(\theta,c)}(x)=\frac{1}{c}\mathbb{1}_{(x\in[\theta,\theta+c])},$$
where $\theta\in\mathbb{R}$ and $c\in\mathbb{R}^+$ ...
- 1,499
6
votes
1
answer
382
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2-dimensional minimal sufficient statistic for $U(-k\theta+k,k\theta+k)$
Find a two dimensional minimal sufficient statistic for $\theta$
from $n$ independent random variables $X_k\sim
> U(-k\theta+k,k\theta+k)$, $k\in\{1,\cdots,n\}$
Here is what I've attempted.
The ...
- 1,499
3
votes
1
answer
8k
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Sufficient statistics in the uniform distribution case
I am currently studying sufficiency statistics. My notes say the following:
A statistic $T(\mathbf{Y})$ is sufficient for $\theta$ if, and only if, for all $\theta \in \Theta$,
$$L(\theta; \mathbf{y})...
- 2,096
2
votes
1
answer
368
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How to find confidence interval for Uniform([a,1])?
Let $ U_1, \dots, U_n $ be a random sample of uniform distribution
over $ [a,1] $. Construct a confidence interval for $ a $ with $ 1-\alpha = 0.95 $.
I managed to show that $ T = \min\{U_i\} $ is ...
1
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0
answers
48
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What's wrong with this proof that the sample sum is sufficient for $\theta$ in $U(0,\theta)$?
So let's say $X_i ~ U(0, \theta)$, and let's consider the two-sample sample sum, $t = \bar{X_2} = (X_1 + X_2)/2$.
So we want to show that $p(x|t) = p(x,t)/p(t) = p(x)/p(t)$ is independent of $\theta$....
- 457
1
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1
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1k
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Sufficient Statistic of Uniform $(-\theta,0)$
Let $X_1, ... , X_n$ be i.i.d random variables Uniform $(-\theta,0)$ , with $\theta > 0$ parameter
\begin{align}f_{\theta}(x_1,x_2,\cdots,x_n)&=\prod_{i=1}^nf(x_i;\theta)
\\&=\frac{1}{(\...
- 213
3
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1
answer
2k
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Prove the maximum order statistic $X_{(n)}$ is a minimal sufficient statistic for the uniform$(0,\theta)$ family using a particular theorem
I'm doing Exercise 6.26 in Casella and Berger's Statistical Inference, and I'm trying to prove the following:
"Use Theorem 6.6.5 to establish that, given a sample $X_1,...,X_n$, the maximum order ...
- 211
3
votes
2
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1k
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Checking if a minimal sufficient statistic is complete
Let $X_1, \cdots, X_n$ be iid from a uniform distribution
$U[-\theta, 2\theta]$ with $\theta \in
\mathbb{R}^+$ unknown. Check if the minimal sufficient statistic of $\theta$ is complete.
I found ...
- 1,387
2
votes
1
answer
3k
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Finding 2-dimensional sufficient statistic via Fisher-Neyman factorization when marginal pdf functions for x don't contain x
Let $X_1,...,X_n$ be mutually independent with pdfs given by $f_i(X_i\mid\theta) = 1/(2i\theta) $ where $ -i(\theta - 1)<x_i<i(\theta +1) $ and $\theta>0.$ To find a two-dimensional ...
- 1,276
6
votes
1
answer
8k
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Sufficient statistics for Uniform $(-\theta,\theta)$
So, I know that $\max(-X_{(1)},X_{(n)})$ is a sufficient statistic for the parameter $\theta$. But can I also say that $(X_{(1)},X_{(n)})$ are jointly sufficient for the parameter $\theta$ ?
In other ...
- 307
2
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0
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143
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Sufficiency and completeness of distribution
Let $X=(X_1,...,X_n)$ be drawn from the distribution with pmf
$p(x_1,...,x_n)\propto \begin{cases} 1/ {\theta\choose n} & \text{if all } x_i \text{ are different and }1 \le\max(x)\le\theta \\
...
- 903
4
votes
0
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563
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The relationship between UMVUE and complete sufficient statistic [duplicate]
Let $X_1,...X_n$ $U(-\theta , \theta)$
I want to find the UMVUE of $\theta$ if it is exists.
My answer is , there is no UMVUE in this case.
Because there is no complete sufficient statistic that ...
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6
votes
2
answers
2k
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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 $T = (X_{(1)}, X_{(n)} )$ ...
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