Hi,

I'm preparing a lesson with a student, and since he's not a major in maths, I'm looking for a better solution than mine, because it looks too "mathematical"

We have a sample $\displaystyle M=\max(|X_1|,\dots,|X_n|)$ (iid) where Xi follows a uniform distribution over $\displaystyle [-\theta,\theta]$, where $\displaystyle \theta>0$ is a parameter.

Previously, it was asked to find the mean and the variance of Xi. And the cdf of |Xi| (y/theta for y in [0,theta])

Question 3) asks for the cdf of M, which is $\displaystyle G(u)=\frac{u^n}{\theta^n}$, for $\displaystyle u\in[0,\theta]$

It's ok from here.

Then, question 4) asks for the mean of M.

What I did is taking the derivative of G, and then $\displaystyle \mathbb{E}(M)=\int_0^\theta u G'(u) ~du=\dots=\frac{n}{n+1}\cdot \theta$

I'm already concerned that this is a too complicated method... So if you have a better one, denote it (1)

Second part of question 4) asks k such that W=kM is an unbiased estimator for $\displaystyle \theta$

Nothing magic here, $\displaystyle k=\frac{n+1}{n}$

Third part of question 4) asks to show that W converges in mean square.

It is not mentioned that it converges to $\displaystyle \theta$. So how can I explain to the boy that it should converge to theta ?

This is the most awful part, because what I did is to use the property that :

$\displaystyle W$ converges to a constant c $\displaystyle \Longleftrightarrow$ $\displaystyle \lim_{n\to\infty}\mathbb{E}(W)=\theta$ and $\displaystyle \lim_{n\to\infty} \mathbb{V}\text{ar}(W)=0$

But calculating the variance of W is a really ugly... So if you have a better solution, please denote it (2)

And if you think these are the only ways to solve the questions, just eat a T-bone steak for dinner

Thanks