# Questions tagged [rao-blackwell]

The Rao-Blackwell Theorem is a result which makes possible to better an estimator by conditioning on a sufficient statistic, preserving the expectation of the estimator.

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### Rao-Blackwell for Poisson parameter: is $\text{E} \big( \bar{X} \mid \sum_{i = 1}^n X_i \big)$ tractable?

I searched for "rao blackwell poisson", "umvu poisson", and "umvue poisson" on this website but didn't find anything that specifically answered my question. I also searched on Google for "umvu poisson"...
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### Finding UMVUE for a function of a Bernoulli parameter

Given $m$ i.i.d. Bernoulli( $\theta$ ) r.v.s $X_{1}, X_{2}, \ldots, X_{m},$ I'm interested in finding the UMVUE of $(1-\theta)^{1/k}$, when $k$ is a positive integer. . I know $\sum X_{i}$ is a ...
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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 $\theta$....
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### Rao-Blackwell part of the Lehmann-Scheffe theorem

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 ...
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### Minimum-variance unbiased estimator to estimate quantiles when the errors are normal distributed

What is the minimum-variance unbiased estimator to estimate quantiles when the errors are normal distributed? median When we wish to estimate the median, $\mu$, of a normal distributed variable then ...
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### Rao-Blackwellization in variational inference

The Black box VI paper introduces Rao-Blackwellization as a method to reduce the variance of the gradient estimator using score function, in section 3.1. However I don't quite get the basic idea ...
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### What is the necessary condition for a unbiased estimator to be UMVUE?

According to the Rao-Blackwell theorem, if statistic $T$ is a sufficient and complete for $\theta$, and $E(T)=\theta$, then $T$ is a uniformly minimum-variance unbiased estimator (UMVUE). I am ...
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### Rao-Blackwell exponential distribution

Let $X_1,..,X_n$ be a random sample of $X\sim\text{Exp}(\lambda)$ with $f(x;\lambda)=\frac{1}{\lambda}e^{-\frac{1}{\lambda}x}I_{[0,\infty]}(x)$ i) Find a unbiased estimator of $\lambda$ based ...
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### Finding UMVUE of Bernoulli random variables

Given i.i.d. Bernoulli$\left(\theta\right)$ r.v.s $X_1, X_2, ...,X_n$, I'm asked to solve for the UMVUE of $\left(1−\theta\right)^2$ for the case $n=4$. I think that $\sum X_i$ is my sufficient ...
Suppose that $T_1$ is sufficient and $T_2$ is minimal sufficient, U is an unbiased estimator of $\theta$, and define $U_1=\mathbb{E}(U|T_1)$ and $U_2=\mathbb{E}(U|T_2)$ a)Show that \$U_2=\mathbb{E}(...