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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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 ...
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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 ...
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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)})=\...
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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 $\...
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200 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 ...
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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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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}(...
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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[\...
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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)= \...
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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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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 $\...
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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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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, $...
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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 ...
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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 ...
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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 ...
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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-...
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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 \...
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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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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)= ...
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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 $E_{\theta}[ \phi(T)] = \tau(\theta)$ ...
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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 $\theta$ ...
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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 ...
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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 $\phi$...
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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 ...
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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 ...
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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.
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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 $f_\theta(x) = \frac{1}{2}\mathbb{I}_{[\theta-1, \theta + 1]}$, ...
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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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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 \...
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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\ \ \ \ \ \ \ \ \ \ \ ...
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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 ...
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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 ...
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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 ...
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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\...
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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 $\...
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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. $n>3$ ...
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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?
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Find the joint distribution of $X_1$ and $\sum_{i=1}^n X_i$

This question is from Robert Hogg's Introduction to Mathematical Statistics 6th version question 7.6.7. The problem is : Let a random sample of size $n$ be taken from a distribution with the pdf $...
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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 ...
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Unbiased estimator based on minimal sufficient statistic has smaller variance than one based on sufficient statistic

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}(...
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Rao-Blackwellization of sequential Monte Carlo filters

In the seminal paper "Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks" by A. Doucet et. al. a sequential monte carlo filter (particle filter) is proposed, which makes use of a ...