# Questions tagged [umvue]

UMVUE stands for Uniform Minimum Variance Unbiased Estimation.

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### MLE and UMVUE from an ordered sample of the exponential distribution

I am having a lot of trouble with every part of the problem below. Now, finding MLE's is simple in principle. I just find the distribution for $Y$ and then use calculus to find the value of $\sigma$ ...
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### Proving an Estimator of the sample variance to be MVUE

Question: Prove that $\hat{\sigma}_x^2=\displaystyle\frac{1}{N-1}\sum_{i=1}^N(X_i-\overline{X})^2$, with $\overline{X}=\frac{1}{N}\sum_{i=1}^N X_i$ is an unbiased, minimum variance estimator of the ...
1 vote
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### What are the uniformly minimum variance unbiased estimators (UMVUE) for the minimum and maximum parameters of a PERT distribution?

I believe the answers to this question are the sample minimum and the sample maximum, but I have not been able to find a reference or proof of this.
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### In linear models, why are we focused on BLUE rather than UMVUE?

It seems more natural to talk about UMVUE (following the basic statistical estimation theory). But when we turn to lm, we only care about BLUE, why? Are there any insurmountable difficulties here?
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### UMVUE of $\theta$ for $\mathrm{Uniform}(0,\theta)$ where $\theta \in[1, \infty)=\Theta$

Let $X_1, \ldots, X_n$ be a random sample from $\mathrm{U}(0, \theta)$, where $\theta \in[1, \infty)=\Theta$, say. Here I tried to find complete-sufficient statistics for $\theta$ as my main target is ...
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### UMVUE for $g(\theta)=\theta^2$ of Poisson random variables

Let $X_1,...X_n$i.i.d.~$Pois(\theta)$ with unknown $\theta>0$, I want to find the UMVUE for $g(\theta)=\theta^2$. I know that $T(x)=\Sigma_{i=1}^{n}x_i$ is complete and sufficient for $\theta>0$....
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### Can a Bayesian estimator perform better than an MVUE?

According to wikipedia: In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any ...
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### How to show if $c=X'w$ has an solution, then there exists uniqu $w_0$ such that $var(w_0'Y)\leq var(w'Y)$?

The whole question is the following: Consider the linear model $\mathbf{Y}=\mathbf{X}\mathbf{\beta}+\varepsilon,$ where $\mathbf{X}$ is a known $n \times p$ matrix, $\mathbf{\beta}$ is a $p$ -vector ...
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### Find the UMVUE of $P\{{X_i < Y_i\}}$ for $X_i,..X_n \sim N(\mu, \sigma^2)$ and $Y_i,..Y_n \sim N(\nu, \sigma^2)$

I have to find the UMVUE of $P\{{X_i < Y_i\}}$ for $X_i,..X_n \sim N(\mu, \sigma^2)$ and $Y_i,..Y_n \sim N(v, \sigma^2$). I know that I have to find the joint distribution of $X$ and $Y$ which I ...
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### UMVUE of the probability of a conditional poisson probability $P_{\lambda} (X=r )$ [duplicate]

Consider a $X_1, ... X_n \sim~ Poisson(\lambda)$, I want to obtain the UMVUE of $P_{\lambda} (X=r)$. This is my approach: $\operatorname{\mathbb{E}}_{\theta}[h(t)] = P_{\lambda} (X=r)$. The ...
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### Let $X_1,X_2,\dots,X_n$ be random sample from Poisson($\theta$). Find MVUE of $e^{-2\theta}$

Question: Let $X_1,X_2,\dots,X_n$ be random sample from Poisson($\theta$). Find MVUE of $e^{-2\theta}$ My attempt has been by modifying the answer from this question: The Poisson distribution is a one-...
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1 vote
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### Techniques for finding UMVUEs

I'm learning about the different techniques available to find the UMVUE such as Rao-Blackwell and Lehmann-Scheffé theorems. My question is how to know when is better to use one method from the other ...
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### How can I find the BUE of $\theta$ in the simple linear relationship $Y_i=\theta x_i^2+\epsilon_i$?

Let $Y_1,...,Y_n$ be described by the relationship $Y_i=\theta x_i^2+\epsilon_i$, where $x_1,...,x_n$ are fixed constants and $\epsilon_1,...,\epsilon_n$ are iid $N(0,\sigma^2)$. How can I find the ...
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### How do I find the UMVUE of $\sqrt{\alpha}$ here?
new user here self-studying some mathematical statistics. I came across this problem and am stuck. Problem: Suppose that for $i = 1, ... , n$, the positive random variables $X_i$ are independent and ...