# 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 Theorem

I'm having problems on understanding the Rao-Blackwell theorem. In particular I don't understand why the resulting estimator is the one with minimum variance between ALL unbiased estimators of the ...
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### Using Rao-Blackwell to improve the estimator of P(X/Y < t)

X and Y are independent N (0, 1) random variables, we want to approximate P (X/Y ≤ t), for a fixed number t. The first part of the problem was to describe a naive Monte Carlo estimate. I described ...
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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 ...
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### How does Rao-Blackwellization of the Metropolis-Hastings algorithm work?

I've read the paper A vanilla Rao-Blackwellization of Metropolis-Hastings algorithms, but I don't get what their actually suggested estimator is. To give some detail, we are considering the following ...
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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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### Finding UMVUE of function of poisson parameter

I am to estimate $\exp(-\lambda)\lambda^2/2$ from the distribution $Exp(\lambda) \sim \frac{e^{-\lambda}\lambda^x}{x!}$ I used the indicator function $W=\mathbb I_{2}(X_1)$ as an initial unbiased ...
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### Using all Metropolis-Hastings proposals to estimate an integral

Suppose we run the Metropolis-Hastings with target distribution $\mu$ to compute the integral $\int f\:{\rm d}\mu$. Usually, we use the estimator $$A_n:=\frac1n\sum_{i=0}^{n-1}f(X_i).$$ However, ...
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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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### Why does the Rao-Blackwell Theorem require $\Bbb E(\hat{\theta}^2) < \infty$?
The Rao-Blackwell Theorem states Let $\hat{\theta}$ be an estimator of $\theta$ with $\Bbb E (\hat{\theta}^2) < \infty$ for all $\theta$. Suppose that $T$ is sufficient for $\theta$, and let \$\...