# 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.

48 questions
Filter by
Sorted by
Tagged with
29 views

### Conditional Density Function on Underlying Exponential

Problem Statement: Suppose $Y_1, Y_2,\dots, Y_n$ are a random sample from an exponential distribution with mean $\theta.$ Let $\displaystyle U=\sum_{i=1}^n Y_i.$ Find the conditional density function ...
41 views

### Rao-Blackwellization in Black Box VI

In the paper, "Black Box Variational Inference," by Ranganath et al. (2013), the authors derive a Rao-Blackwellized estimator of the gradient of the evidence lower bound with respect to a ...
80 views

### 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)})=\...
168 views

58 views

57 views

### 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"...
698 views

### 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 ...
33 views

### 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 $\theta$....
172 views

### 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 ...
256 views

253 views

### maximum likelihood estimate and the UMVUE

I have been working on this question and I am little confused about how to solve it. To evaluate the prevalence of periodontal diseases in a population, suppose that $x_i$, $i=1,\ldots,n$ are the ...
82 views

### 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 ...
79 views

### 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 ...
219 views

### Rao-Blackwell for Minimum-Variance Unbiased Estimator

Let $X$ be an observation from a distribution with probability mass function:$f(x;\theta) = \left(\frac{\theta}{2}\right)^{|x|}(1-\theta)^{1-|x|}I_{\{-1,0,1\}}(x), \, \theta \in (0,1).$ Use Rao-...
425 views

71 views

### Understanding Rao-Blackwell [duplicate]

From Casella and Berger: Let $W$ be an unbiased estimator of $\tau(\theta)$ and let $T$ be a sufficient statistics for $\theta$. Define $\phi(T) = E[W|T]$. Then $E_{\theta}[ \phi(T)] = \tau(\theta)$ ...
38 views

### 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 $\theta$ ...
163 views

### How to improve an estimator for a Poisson sample

Given the statistical model $(\mathbb N_0^n, P(\mathbb N_0^n),\operatorname{Poi}(\vartheta)^{\otimes n}:\vartheta >0)$, $T(X)=X_1X_2$ is an unbiased estimator of $\vartheta^2$. I want to improve ...
300 views

### Why can Gibbs sampling outputs be used in Rao-Blackwellization?

I'm currently learning Chib (1995)'s method to calculate the marginal likelihood of a Bayesian model using Gibbs sampling outputs. I'm stuck in the Rao-Blackwellization step. Suppose $\mu$ and $\phi$...
199 views

### How to improve an estimator that is already unbiased?

I am solving a two part problem where the second part of the problem is to improve on the estimator in the first using the Rao-Blackwell theorem. In this case, the sufficient statistic is just Y ...
85 views

### How does Rao-Blackwellisation use only $X$ in $Y | X$ to produce the result?

How does Rao-Blackwellisation use only $X$ in $Y | X$ to produce the result? Since I read that what Rao-Blackwellisation does is find first $$\mathbb{E}[Y | X=x] := h(x)$$ and then draws an ...
71 views

### 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.
45 views

### Conditional Expectation of Order Statistic [duplicate]

Given a a random sample $X_1, X_2, X_3, X_4$ and family of densities $\mathcal{P} = \left\{ f_\theta: \theta \in \Theta \right\}$, where $f_\theta(x) = \frac{1}{2}\mathbb{I}_{[\theta-1, \theta + 1]}$, ...
2k views

783 views

### MVUE using factorization criterion and Rao-Blackwell theorems

Suppose that $Y_1, Y_2, ..., Y_n$ is a random sample from a distribution with density function  f(y) = \begin{cases} \theta y^{\theta - 1}\ \ \ \ 0 < y < 1, \\ 0\ \ \ \ \ \ \ \ \ \ \ ...
216 views

### Knowing both sample mean and sample range improves estimate of the the variance?

It is known that sample variance and sample mean are independent for normally distributed variables, which means knowing sample mean does not say anything about to estimate the variance of the ...
568 views

### Conditioning a biased estimator on a sufficient statistic

I'm afraid my awareness of the Rao–Blackwell theorem has been limited to textbook accounts and exercises, and those deal only with its application to unbiased estimators. Maybe it's properly called ...
3k views

### Rao-Blackwellization of Gibbs Sampler

I am currently estimating a stochastic volatility model with Markov Chain Monte Carlo methods. Thereby, I am implementing Gibbs and Metropolis sampling methods.Assuming I take the mean of the ...
303 views

75 views

### The normal and bernoulli distributions

I'm working on this problem and am a little stumped. I was wondering if someone could give me a hint? $x_1,...,x_n$ are iid $N(\mu,\sigma^2)$ where $\mu$ is unknown and $\sigma$ is known. $n>3$ ...
185 views

### How does one can guarantee that any unbiased estimator is MVUE due to it containing a minimal sufficient statistic?

Sufficiency is okay. But I don't really get it why the fact guarantees it has minimal variance? Can anyone explain to me somewhat intuitively?