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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), \...
tychonovs-scholar's user avatar
1 vote
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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....
harrydiv321's user avatar
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}...
pbb's user avatar
  • 345
5 votes
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 ...
Ho-Oh's user avatar
  • 917
0 votes
0 answers

How do I apply the Rao-Blackwell Theorem to find MVUE of parameter theta?

Let Y1, Y2, . . . , Yn be independent and identically distributed random variables having the same population distribution with density: f(y; θ) = ( θ(3^θ)/y^(θ+1) , y ⩾ 3; 0, elsewhere.) where θ is a ...
VoidzenNullscape's user avatar
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Given $X_1,...X_n$ is an iid sample from $Binomial(2,\theta)$ distribution, find the UMVUE for $\theta^2$.

I'm looking to validate my solution or understand why it's incorrect. Attempt: By writing the pdf of the distribution in the form $$f(x;\theta)=\exp\left(x\log\left(\frac{\theta}{1-\theta}\right) + 2\...
Alborz's user avatar
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1 vote
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: $...
Nothing's user avatar
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4 votes
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 that: ...
Kadmos's user avatar
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