1. Estimator question

I'm having some problems understanding this question related to the bootstrap method:

Generate a sample from $\displaystyle X_1,X_2, \dots X_{25}$ from the uniform distrubition on $\displaystyle [0,10]$

We pretend not to know $\displaystyle \theta$ of $\displaystyle [0,\theta]$ (which is in fact 10) and try two estimators for this parameter:

$\displaystyle T_1 = 2\bar{X}$
$\displaystyle T_2 = \frac{(n+1)M}{n}$

To choose between $\displaystyle T_1$ and $\displaystyle T_2$ we are interested in the MSE $\displaystyle E(T_1-\theta)^2$ and $\displaystyle E(T_2-\theta)^2$. We can estimate these by inserting an estimator $\displaystyle \theta$ in the expressions of their expectation value and variance. Determine a $\displaystyle \theta$ based on the data to do this and compare the answers.
------------------------------

I found the expectation value of both $\displaystyle T_1$ and $\displaystyle T_2$, but I have no idea what is meant by inserting an estimator $\displaystyle \theta$, and how to determine this from a given 25 draws from the distribution. Could someone help me by explaining this?

2. 1 I don't see where the bootstrap comes in
2 I guess M is the largest order stat.
3 Is n=25 or do you want this for any sample size?

3. The question is under the header parametric bootstrap, because the next question is estimating the MSE by drawing from $\displaystyle [0, \hat{\theta}]$. That question is easy I think.
M is indeed the largest order stat, and an explanation for n=25 or really any any would be great. I just don't understand the part where I decide an estimator from the data and fill it in in the expectation and variance of $\displaystyle T_1$ and $\displaystyle T_2$.

Err, well I though I had the expectation and variance right, but I don't think I got em. Could someone help me with these for both T_1[/tex] and $\displaystyle T_2$?

The expectation of both would be simply $\displaystyle \theta$ but how do I get the variance? o_O

4. $\displaystyle E(2\bar X)= 2E(\bar X)=2\biggl({\theta\over 2}\biggr)=\theta$

$\displaystyle V(2\bar X)= 4V(\bar X) =4\biggl({V(X_1)\over n}\biggr) =4\biggl({\theta^2\over 12n}\biggr)={\theta^2\over 3n}$

show me your density for M and I'll look it over.
I still don't know if you want n or 25.
Both estimators are unbiased.
The point is we want the one with the smaller variance.
I still don't see the point of the bootstrap here.

5. Well, the question is about n = 25, but for general n I need to know the variance.
Is this correct?

$\displaystyle V(\frac{(n+1)M}{n}) = \frac{(n+1)^2}{n^2}V(M) = \frac{(n+1)^2}{n^2}*\frac{n}{(n+1)^2(n+2)} = \frac{1}{n(n+2)}$

The M is the maximum out of the n variables. For the uniform distribution on $\displaystyle [0,1]$The epectation value for this is $\displaystyle \frac{n}{n+1}$ and the variance is $\displaystyle \frac{n}{(n+1)^2(n+2)}$
For the uniform distribution with $\displaystyle \theta = 10$ instead of 1, I guess this translates to the expectation value times theta, although I don't know what to do with the variance.
The propability function of M is the beta distribution with paramters a and 1 I think.

What remains even more vague to me is how I deduce an estimator from a dataset with 25 X's, and how I should fill that in in the expectations and variances...

The bootstrap comes in in the next question: determine the MSE of $\displaystyle T_1$ and $\displaystyle T_2$ by use of parametric bootstrap. First determine an estimator of $\displaystyle \theta$ and then simulate $\displaystyle E(T_1-\hat{\theta})$ and $\displaystyle E(T_2-\hat{\theta})$

6. the variance of M_n will have a factor of $\displaystyle \theta^2$

You can use the uniform(0,1) and multiply it by theta

then use $\displaystyle V(aX+b)=a^2V(X)$

BUT I would derive the variance directly from the largest order stat of a $\displaystyle U(0,\theta)$