# Questions tagged [umvue]

UMVUE stands for Uniform Minimum Variance Unbiased Estimation.

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### Find UMVUE of Ratio of two parametric functions

Let T be UMVUE of $g(\theta)$ and S be UMVUE of $h(\theta)$. Is there any way to find UMVUE of ratio of $g(\theta)$ and $h(\theta)$ i.e , $g(\theta)/h(\theta)$?
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### UMP Test and UMVUE when there are nuisance parameters

Consider $X_1,...,X_n \sim Weibull(\theta, c)$ where $c>0$ is unknown. Several textbook examples consider when $c$ is known, but here, we consider when $c$ is unknown. Suppose now we wanted to find ...
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### How should one re-formulate hypothesis after hypothesis testing or whether or not to discard data after significance testing or archive it?

@Nalzook summarized what he thinks Ronald Fisher did. Ask a question. Propose a null hypothesis based on the question. Do some experiments, and collect some data. Assuming the null is true, calculate ...
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### Why we do not define the reciprocal variance of the Minimum Variance Unbiased Estimators as the FIsher information?

If I give you data on death rate of rats in China and ask you to estimate the population of Cuba based on that, you'll surely say that the data contains no information about the quantity to be ...
1 vote
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### UMVUE of $\theta^2 (1- \theta )$ X is random sample from bernoulli distribution

Let $X_1, X_2 ..... X_n$ be a random sample from bernoulli distribution with parameter $\theta$ , Obtain UMVUE of $\theta^2 (1- \theta )$ MY APPROACH I calculated that T = $\sum X_i$ is ...
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### Let X1,X2,...,Xn be a random sample from N(thetha,1). find the UMVUE of P(X>0) [duplicate]

I have tried this question and reached the unbiased estimator as the mean of Y when we define the variable Yi=1 if P(X>0) Now I am looking for the UMVUE of P(X>0)
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### Finding UMVUE for exponential sample [duplicate]

Let $X_1,...,X_n$ be a random sample of i.i.d. exponential distribution with probability density function $$f(x|\theta)=\frac{1}{\theta}exp(-\frac{x}{\theta}), \ x\geq0$$ Let $S_n=\sum_{i=1}^nX_i$ and ...
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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 ...
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### How many classmates does a freshman have?

The freshmen at East China Normal University has just received their student ID. Let the last three digits of a student ID be ABC, then A is the class he is in, whereas BC is his number in the class. ...
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### Conditional Expectation (Poisson) UMVUE

Suppose $X_1,X_2,\ldots,X_n$ is a random sample from a Poisson distribution with mean $λ$. How can I find the conditional expectation $E \left( X_1\times X_2\times X_3 \mid \sum_{i=1}^n X_i= z \right)$...
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### UMVUE for probability of cutoff

Let $X_i \sim N(\mu,1)$, i.i.d. We aim to find UMVUE for $p(\mu) = P_{\mu}(X_1 \leq u)$ for some fixed $u$. I have shown that $\bar{X}$ and $X_1 - \bar{X}$ are independent. ($\bar{X}$: sample mean). ...
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### Estimator with variance equal to Cramér-Rao lower bound in $N(x_i\theta,1)$-distribution

Let $Y_1,\ldots, Y_n$ be independent and $N(x_i\theta,1)$ distributed, with for each $Y_i$ a mean of $x_i\theta$ for known $x_1,\ldots,x_n$. In a previous section of this exercise I found that the ...
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I am unsure how to finish this problem in Lehmann's book. The problem asks to prove that among the class of all symmetric distributions $\mathcal{F}$, no UMVUE exists for the center of symmetry $\... • 2,296 1 vote 1 answer 574 views ### Use the Lehmann-Scheffé theorem to deduce that$\overline{X}$is an UMVUE estimator for$\theta$Let$X_1,X_2,\ldots,X_n$be a random sample whose distribution is$X\sim\operatorname{Bernoulli}(\theta)$. (a) Prove that$\sum_{i=1}^n X_i$is complete. (b) Use the Lehmann-Scheffé to deduce that$\... ### How to show a UMVUE exists only if $g(p)$ is a polynomial of degree at most $n$?
Let $X\sim Bin(n,p)$. The problem is to show that a UMVUE can exist for $g(p)$ only if $g(p)$ is a polynomial in $p$ of degree at most $n$. For the case when $g(p) = \frac{1}{p}$ we can show that it ...