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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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

Suppose we have $X_1,\dots, X_n \overset{iid}{\sim} N(\mu = 0, \sigma^2 = 1)$, for a known $n$. And we want to calculate $E[X_{(1)} | \bar{X} = c]$, where $c \in \mathbb{R}$ is known, $X_{(1)}$ is the ...
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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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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 $\theta$ ...
117 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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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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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. $n>3$ ...