Mathematics Stack Exchange
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Questions tagged [umvue]

In statistics, an uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

In statistics, an uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

10 questions
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1 answer

UMVUEs for the means of $3$ independent normal distributions with the sum of means being $1$

Let $\theta_1, \theta_2$ and $\theta_3$ be nonnegative parameters with the constraint $\theta_1+\theta_2+\theta_3=1$. We observe $X_{i 1}=\theta_1+\epsilon_{i 1}, X_{i 2}=\theta_2+\epsilon_{i 2}, X_{i 3}=\theta_3+\epsilon_{i 3}$ for $i=1,2, \ldots,…
Ho-Oh
  • 956
4
votes
1 answer

UMVU estimator of $\lambda^2$ via Rao-Blackwell

I have been working on a problem, which goes as follows: Given the statistical model $(\mathcal{X},\mathcal{B},\mathcal{P})$, where $\mathcal{P}=\{P_{\lambda}^{\otimes}:P_{\lambda}=Pos(\lambda), \lambda>0\}$, use the Rao-Blackwell theorem, to find…
tychonovs-scholar
  • 783
4
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2 answers

Find UMVUE for $1/\lambda^3$ in gamma distribution.

I was taking a look at old probability qualifiers and this is from one of them: Suppouse $\alpha$ is known in $X\sim Gamma(\alpha, \lambda)$. Find the UMVUE for $1/\lambda^3$. It is well known…
Kadmos
  • 3,243
2
votes
1 answer

UMVUE of $\mu^4$ when $(X_{i})_{i=1 \ldots n} \sim N(\mu, \sigma^2)$

I am trying to find UMVUE of $\mu^4$ when $(X_{i})_{i=1 \ldots n} \sim N(\mu, \sigma^2)$, both $\mu, \sigma^2$ are unknown. I know that the complete sufficient statistics for $\mu, \sigma^2$ and $\bar{X}_n, S_n^2$. And I want to find an unbiased…
PrimeFactorial
  • 23
1
vote
1 answer

UMVUE for the Product of Means from Independent Samples

Let $ x_1, \ldots, x_m $ be i.i.d. samples drawn from a distribution $P$, and $ y_1, \ldots, y_n $ be i.i.d. samples drawn from a distribution $Q$. Assume that the samples $x_i$ and $y_j$ are independent of each other. Suppose $\bar{X}$ is the…
milad jalali
  • 37
1
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0 answers

UMVUE of $\mu^p$ where $X_1,\cdots,X_n\sim\mathcal{N}(\mu,\sigma^2)$

$\newcommand{\E}{\mathbb{E}}$ $\newcommand{\V}{\mathbb{V}}$ Let $p\in\mathbb{N}$. Is there a nice general expression for the UMVUE of $\mu^p$, where $X_1,\cdots,X_n\sim\mathcal{N}(\mu,\sigma^2)$ are i.i.d.? In the case of $p=2$, we know that…
harrydiv321
  • 59
1
vote
1 answer

UMVUE of $\mathbb{E}[X^2]=\lambda^2 + \lambda$ where $X\sim\mathrm{Pois}(\lambda)$.

This is the same question as this: UMVUE of $E[X^2]$ where $X_i$ is Poisson $(\lambda)$. Here, I restate the problem for completeness: Let $X_1, \ldots, X_n \overset{\text{i.i.d.}}{\sim} \mathrm{Pois}(\lambda)$, find the UMVUE of…
pbb
  • 385
1
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0 answers

Disproving the regularity condition of Cramer-Rao Lower bound

Let $X = (X_1,\cdots, X_n)$ where $X_1,\cdots,X_n$ be i.i.d from the uniform distribution $U(0,\theta)$ with $\theta>0$. I was asked to show the regularity condition of the Cramer-Rao lower bound: $$\frac{\partial}{\partial\theta}\int…
Nothing
  • 1,768
0
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How to prove the UMVUE of $ET^k$ is $T^k$?

Here is the problem. Suppose $T$ is the UMVUE of $\theta$, please prove the UMVUE of $ET^k$ is $T^k$(k=1,2,3,...), where $ET^{2k}<\infty$ holds. I have a fake proof by using L-S theorem: Because T is the UMVUE of $\theta$, T should be sufficient and…
UMVUE
  • 1
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1 answer

UMVUE for geometric distribution when counting the number of failures before success

I have encountered the following problem: Let $X_1$, $\ldots$, $X_n$ be an independent random sample from the distribution with p.d.f. $$f(x;p)=p(1-p)^x, ~~x=0,1,2,\ldots,~~0
Lumos
  • 157