Skip to main content

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.

Filter by
Sorted by
Tagged with
1 vote
1 answer
44 views

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 ...
Onofrio Olivieri's user avatar
1 vote
1 answer
72 views

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 ...
stat_student123's user avatar
0 votes
1 answer
120 views

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 ...
johnsmith's user avatar
  • 345
5 votes
2 answers
245 views

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 ...
0xbadf00d's user avatar
  • 303
0 votes
1 answer
250 views

Poisson: finding UMVUE for $\lambda + \lambda^2$ [closed]

Let $X_1,..., X_n$ be iid sample from the Poisson distribution with parameter $\lambda$. Find the UMVUE of $\lambda + \lambda^2$. I know $T := \sum\limits_{i=1}^n X_i$ is complete and sufficient for $\...
Pramesh Pudasaini's user avatar
0 votes
0 answers
364 views

What is the difference between MVB UMVUE and MVUE.?

Cramer Rao inequality gives MVB and if MVB exist it is MLE. Rao Blackwell gives UMVUE, but isn’t when we have MVB estimator for unbiased it is UMVUE? Then what is MVUE? MVB minimum variance bound ...
User0405's user avatar
4 votes
1 answer
194 views

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 ...
abhishek's user avatar
  • 236
1 vote
3 answers
676 views

Rao-Blackwell and unbiased estimators of zero

In Casella & Berger we are trying to prove that any estimator $\phi$ based on a complete sufficient statistic T is the unique best unbiased estimator of its expectation. However, in the preceding ...
Artem Moskalev's user avatar
1 vote
0 answers
130 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 ...
Adrian Keister's user avatar
4 votes
1 answer
115 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 ...
Ethan S's user avatar
  • 41
3 votes
1 answer
177 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)})=\...
boyaronur's user avatar
  • 143
6 votes
1 answer
573 views

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 $\...
The Pointer's user avatar
  • 2,086
3 votes
1 answer
1k views

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 ...
Academic005's user avatar
0 votes
1 answer
245 views

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 ...
smaillis's user avatar
  • 133
3 votes
1 answer
1k views

Rao Blackwell theorem on Bernoulli distribution

I need help with the following Problem: Let $X_1,...,X_n$ be a random sample of iid random variables, $X_i\sim Ber(p), p\in (0,1)$. We want to estimate $\theta = p^2$. It is known, that $\hat{\theta}(...
stats19's user avatar
  • 61
1 vote
1 answer
249 views

Cramer-Rao Lower Bound Proof (fuzzy step)

The following is the derivation of the Cramer-Rao lower bound as detailed on p.336 of Casella and Berger's Statistical Inference: $\frac{d}{d\theta}E[W(\bf{X})|\theta] = \int_{\chi}W(\bf{x})\left[\...
tvbc's user avatar
  • 154
3 votes
2 answers
230 views

Variance of Rao Blackwellization for MC Estimate of Expectation

from https://arxiv.org/abs/1401.0118 If we have a function $J(X,Y)$ of two random variables $X$ and $Y$ and we want to compute the expectation $\mathbb E_{p(X,Y)}[J(X,Y)]$. We define $\hat J(X)= \...
jzin's user avatar
  • 337
0 votes
0 answers
270 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"...
Novice's user avatar
  • 581
5 votes
2 answers
2k 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 ...
wanderer's user avatar
  • 214
1 vote
0 answers
48 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 $\theta$....
user49404's user avatar
  • 457
0 votes
0 answers
217 views

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 ...
user avatar
3 votes
1 answer
406 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 ...
Maverick Meerkat's user avatar
7 votes
1 answer
1k views

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 $\...
asdf's user avatar
  • 384
7 votes
2 answers
364 views

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, ...
0xbadf00d's user avatar
  • 303
2 votes
1 answer
127 views

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, $...
Juan Chong's user avatar
2 votes
1 answer
338 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 ...
Eugeniah Arthur's user avatar
1 vote
1 answer
239 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 ...
honeybadger's user avatar
  • 1,572
4 votes
1 answer
260 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 ...
honeybadger's user avatar
  • 1,572
2 votes
1 answer
523 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-...
Andrew's user avatar
  • 53
0 votes
1 answer
551 views

Rao-Blackwell Problem [closed]

Let $X_1, .. X_n$ be iid Bernoulli($\theta$). I want to estimate $\lambda = \theta(1-\theta)$ using $\delta$ (unbiased estimator for $\lambda$) where $\begin{cases} \delta = 1 & X_1 =1 \...
j doe's user avatar
  • 21
5 votes
0 answers
316 views

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 ...
Sextus Empiricus's user avatar
9 votes
1 answer
543 views

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 ...
avocado's user avatar
  • 3,633
2 votes
2 answers
611 views

Variance of Rao Blackwellization for Monte Carlo Estimate of Expectation

from https://arxiv.org/pdf/1401.0118.pdf If we have a function $J(X,Y)$ of two random variables $X$ and $Y$ and we want to compute the expectation $\mathbb E_{p(X,Y)}[J(X,Y)]$. We define $\hat J(X)= ...
chris elgoog's user avatar
1 vote
0 answers
83 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 $E_{\theta}[ \phi(T)] = \tau(\theta)$ ...
Marcel's user avatar
  • 1,390
1 vote
0 answers
49 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 $\theta$ ...
diadochos's user avatar
  • 175
1 vote
1 answer
223 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 ...
user avatar
2 votes
1 answer
390 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 $\phi$...
Ding Li's user avatar
  • 453
0 votes
0 answers
362 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 ...
user3707850's user avatar
2 votes
1 answer
92 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 ...
mavavilj's user avatar
  • 4,119
4 votes
2 answers
670 views

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 ...
DanRoDuq's user avatar
  • 586
2 votes
1 answer
99 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.
0 votes
0 answers
65 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 $f_\theta(x) = \frac{1}{2}\mathbb{I}_{[\theta-1, \theta + 1]}$, ...
Mitch Baker's user avatar
1 vote
1 answer
4k views

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 $...
jmoore00's user avatar
  • 389
3 votes
1 answer
2k views

Why does the MSE version of Rao-Blackwell theorem require $T$ to be a sufficient statistic?

The proof for the MSE version appears not to depend on $T$ being a sufficient statistic. I provide a minimal version here: Let $\hat\theta,T$ be observable random variables, and let $\theta \in \...
ziggystar's user avatar
  • 1,614
2 votes
2 answers
1k 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\ \ \ \ \ \ \ \ \ \ \ ...
JaviOverflow's user avatar
2 votes
1 answer
330 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 ...
oercim's user avatar
  • 689
2 votes
1 answer
775 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 ...
Michael Hardy's user avatar
11 votes
2 answers
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 ...
mscnvrsy's user avatar
  • 545
2 votes
2 answers
502 views

Rao-Blackwellizing: Is there any difference conditional on different sufficient statistics

Suppose I have two different sufficient statistics $a_1$ and $a_2$ while $a_1$ summarizes information more efficient than $a_2$. For example, if the sample space is $\left\{y_1,y_2,y_3,y_4,y_5,y_6\...
purod's user avatar
  • 305
11 votes
2 answers
790 views

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 $\...
Stan Shunpike's user avatar