# Questions tagged [umvue]

UMVUE stands for Uniform Minimum Variance Unbiased Estimation.

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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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### UMVUE of $\exp(t \mu)$ for $N(\mu, \sigma^2)$ where $\mu$ is not known but $\sigma^2$ is known [duplicate]

I am new to UMVUE calculations and struggling a bit with this question. I want a hint about how to approach the derivation of the UMVUE for $\exp(t [\mu)$ for $N(\mu, \sigma^2)$ where (\mu is not ...
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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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### 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 ...
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### Finding UMVUE of $\theta e^{-\theta}$ where $X_i\sim\text{Pois}(\theta)$

Suppose $X_1, X_2, . . . , X_n$ are i.i.d Poisson ($\theta$) random variables, where $\theta\in(0,\infty)$. Give the UMVUE of $\theta e^{-\theta}$ I found a similar problem here. I have that the ...
### Find UMVUE of $\frac{1}{\theta}$ where $f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$
Let $X_1, X_2, . . . , X_n$ be iid random variables having pdf $$f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$$ where $\theta >0$. Give the UMVUE of $\frac{1}{\theta}$ ...
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 $\theta$ ...