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1
vote
0answers
31 views
$X \sim $ Geometric$(p)$, $Y \sim$ NegativeBinomial$(2,p)$: find the MVUE for $p(1-p)$
Let $X$ and $Y$ be independent random variables with pdf
\begin{align}
f(x)&=p(1-p)^x,x=0,1,2,\ldots \\
f(y)&=(1+y)p^2(1-p)^y, y=0,1,2,\ldots
\end{align}
Find the MVUE for
$$
p(1-p).
$$
I ...
1
vote
0answers
99 views
Need help finding UMVUE for a Poisson Distribution
My fellow classmates and I are stuck on a homework problem that is a three part problem to find the UMVUE of a Poisson distribution.
The problem goes like this:
Let X ~ Pois$(\lambda$), and we want ...
0
votes
0answers
113 views
Let X1,X2,…,Xn be i.i.d. N(θ1, θ2), please prove that E[(x1-θ1)^4] = 3θ2^2
If x$_{1}$, x$_{2}$,...,x$_{n}$ is sampled from N($\theta$$_{1}$, $\theta$$_{2}$), how can I prove that E [(x$_{1}$ - $\theta$$_{1}$)$^{4}$] = 3$\theta$$_{2}$$^{2}$?
I started off this question ...
-2
votes
1answer
65 views
Finding UMVUE of parameter $\displaystyle a^{-\theta}$, $(0<a<1)$
Suppose $X_1,X_2,\ldots,X_n$ is a random sample of $\operatorname{Beta}(\theta,1)$. How can I find UMVUE of parameter $\displaystyle a^{\theta}$, $(0<a<1)$?
0
votes
0answers
79 views
Function of an efficient estimator
Let T be an efficient estimator for a parameter $\theta$. Let $g(\cdot)$ be a (real valued) non-linear function. Is $g(T)$ an UMVUE for $g(\theta)$?
4
votes
2answers
165 views
Sufficiency in Lehmann Scheffe
We are wondering what sufficiency in the Lehmann Scheffe Theorem is needed for.
Our reasoning was:
If an unbiased estimator is uncorrelated with all unbiased estimators of 0, it is UMVUE
If the ...
0
votes
1answer
233 views
Obtain the minimum variance unbiased estimators
Let $X_1, X_2,\cdots,X_n$ be a random sample from the distribution with the p.d.f $$f(x)=\frac{1}{\beta -\alpha},\alpha<x<\beta $$ where $0<\alpha<\beta<\infty$. Obtain the minimum ...
0
votes
0answers
356 views
Uniform minimum variance unbiased estimator
Let $X_1, X_2, ..., X_n$ be an iid random sample from a Poisson$(\lambda)$ distribution:
a) find the UMVUE of $\theta$ = $\lambda^k$ for $k > 0$ a known integer
b) find the UMVUE of $\tau = ...
5
votes
1answer
283 views
Finding a minimum variance unbiased (linear) estimator
Here is a basic question that perhaps has a simple answer, but one that I was not able to find by quickly scanning the literature.
Suppose that I have a collection of $n$ unopened boxes. Each box ...
7
votes
2answers
1k views
How do I know which method of parameter estimation to choose?
There are quite a few methods for parameter estimation out there. MLE, UMVUE, MoM, decision-theoretic, and others all seem like they have a fairly logical case for why they are useful for parameter ...
