Conceptual question about estimators A random number generator produces uniformly distributed random numbers on the intervall $[0,a]$ where $a>0$ is unknown. We can draw $n$ independent $\mathcal U_{[0,a]}$ random numbers $X_1,...,X_n$ for the estimation. Find a mathematical formulation for the following estimators and check if they are biased and consistent.

*

*"Twice the arithmetic mean of the observations"

*"The maximum of the observations"

We just started estimators and I am having difficulties getting to the mathematical formulation and showing if they are consistent/bias.
My attempt:
The definitions:

*

*An estimator is unbiased if $E[\hat\theta_n]=\theta$

*An estimator is consistent if $\forall\epsilon>0$, $\lim_{n \to \infty}\Pr(\vert\hat \theta_n-\theta \vert>\epsilon)\to0$
Calculations:

*

*Based on the hint and answer from Milten I want to estimate $\alpha. $Let $\hat\theta=2\bar{X}_n$ then:

$$\begin{split} \implies E[\hat \theta]-\alpha&= E[2\bar{X}_n]-\alpha\\&=E\left[\frac{2}{n}\sum_{i=1}^n X_i \right]-\alpha\\ &=\frac{2}{n} \sum_{i=1}^nE[X_i]-\alpha \\ &= \frac
{2}{n}n \frac{\alpha}{2}-\alpha \\ &=\alpha-\alpha=0\checkmark \end{split}$$
My question:
I don't understand what  $\theta$ is supposed to be here. Am I trying to estimate twice the mean with the sample I have drawn? Or is $\theta$ supposed to be the mean? More generally, I don't quite get the relationship between $\hat \theta$ and $\theta$.
 A: Hint:
$\theta$ is the parameter we wish to estimate (i.e. the true, unknown value), and $\hat \theta$ is the estimator, i.e. the approximation of the parameter. In your case $\hat \theta$ will be the two estimators given.
So what paramter in the set-up do you think the two estimators will approximate?
Example: Say $X_1,\ldots, X_n$ were i.i.d. Bernoulli distributed with $P(X_i=1)=p$. Then the observed mean would be an estimator for $p$, and it would be unbiased since $E(\bar X)=p$. So here $\theta=p$ and $\hat\theta=\bar X$.
EDIT: I'll add a bit about esimators in general. There are often several possible choices of estimators for the same parameter, like in this problem. Another common example is uncorrected/biased sample variance (which is an MLE) versus corrected/unbiased sample variance. In theory you could declare any function $(\mathbb R_{\ge0})^n \to \mathbb R_{\ge0}$ an estimator for $\alpha$ (e.g. you could always guess $\alpha=0$), but we of course want estimators that approximate the parameter in some sense. Hence the special names for unbiased/consistent/etc. estimators.
For consistency, Chebyshev or Law of Large Numbers (if you have it available) will work for the first estimator. For the second one, I'd calculate
$$
P(|\alpha - \max(X_i)| > \varepsilon) = P(\max(X_i) < \alpha - \varepsilon)
$$
directly.
