All Questions
Tagged with sufficient-statistics rao-blackwell
17
questions
0
votes
1
answer
120
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Unbiased estimator for parameter of random variables following a uniform distribution [duplicate]
Suppose $X_i$ are i.i.d. and have density $f_\theta(x) = \frac{1}{\theta}$ if $x \in (\theta, 2\theta)$ for positive $\theta$.
$(\min_iX_i, \max_iX_i)$ is a sufficient statistic for $\theta$?
To ...
- 345
4
votes
1
answer
194
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Rao-Blackwellisation using non-sufficient statistics
The following is given as a remark in chapter 7 of Introduction to mathematical statistics Hogg and Craig, 8th edition. (It is mentioned as "Remark 7.3.1")
Now, I do understand that the ...
- 236
3
votes
1
answer
177
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Rao–Blackwellization of Metropolis–Hastings
I am trying to achieve a Rao–Blackwellization of Metropolis–Hastings algorithm. In the paper by Robert et al. 2018, the following is given.
\begin{align}
ℑ=&\frac{1}{T}\sum_{t=1}^Th(\theta^{(t)})=\...
- 143
6
votes
1
answer
575
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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 $\...
- 2,096
3
votes
1
answer
1k
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Understanding the Rao-Blackwell Theorem
I've been reading up a lot on the practical applications of the Rao-Blackwell theorem. I do understand how the Bias and Variance and MSE aspects of the theorem fall in place (i.e. the mathematical ...
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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
0
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0
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217
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Rao blackwell theorem but the unbiased estimator is a function of the sufficient statistic
The Rao-Blackwell Theorem states the following:
Let $T(\mathbf X)$ be a sufficient statistic for the statistical model $(S, \{f_{\theta}: \theta \in \Theta\})$ and $\hat \theta(\mathbf X)$ be and ...
user255658
7
votes
1
answer
1k
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Relationship between completeness and sufficiency
hopefully this isn't a duplicate of another question (at least I didn't find one).
Here is a question I have about completeness and sufficiency:
Problem: Suppose $T(x)$ is complete sufficient for $\...
- 384
2
votes
1
answer
127
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Obtaining the expected value $E[X_{(1)} \mid\overline X = c]$
Suppose we have $X_1,\dots, X_n \overset{\text{iid}}{\sim} N(\mu = 0, \sigma^2 = 1)$, for a known $n$. And we want to calculate $E[X_{(1)} \mid \overline X = c]$, where $c \in \mathbb{R}$ is known, $...
- 21
1
vote
1
answer
240
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Sufficient Statistic and Unbiased Estimate in Exponential Family
I am reading this classic paper (Information and the Accuracy Attainable in the Estimation of Statistical Parameters) by CR Rao where he deals with sufficient statistics in exponential distributions ...
- 1,572
4
votes
1
answer
260
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Proof of Rao Blackwellization
I am reading this classic paper (Information and the Accuracy Attainable in the Estimation of Statistical Parameters) by CR Rao where he introduces the idea of minimizing the variance of an unbiased ...
- 1,572
1
vote
0
answers
49
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Sufficient statistic for the mean of a generic distribution?
Is there such thing as a "sufficient statistics for the expectation?"
From what I understand, a sufficient statistic is defined only when there is a family of distributions parametrized by $\theta$ ...
- 175
4
votes
2
answers
671
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Sufficient Statistic and Maximum likelihood
This is more a conceptual question, but it seems to me that a sufficient statistic for a parameter is a concepts that applies only if we want to estimate the parameter via maximum likelihood. Is this ...
- 586
2
votes
1
answer
99
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Reference book for practice problems on Inference
I was wondering if there is any book which has loads of problems on statistical inference.
Desired topics are
Unbiasedness
Consistency
Sufficiency
Completeness
Rao Blackwell Theorem
etc.
1
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1
answer
4k
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Finding sufficient statistic for Weibull density function
I am given the follow problem and am having trouble finding the sufficient statistic.
Suppose that Y$_1$, Y$_2$, ..., Y$_n$ denote a Weibull density function, given by:
f ( y | $\theta$ ) =
Let $...
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