Consider an i.i.d. sample $\{X_1, \ldots , X_n\}$ from the uniform distribution on $[ 0,\theta]$ and the estimator
$$M_n = \max\{X_1,X_2,\ldots,X_n\} $$
What does the above statement mean?
I mean, I can think that the joint distribution is something like
$$\left(\frac1\theta\right)^n $$
But I dont get the max part? The main question in exercise is:
Prove that Mn p→ θ as n → +∞
A proxy question..I am trying to understand example 3.1 here (http://faculty.ksu.edu.sa/73125/Publications/%5B3%5D%20Convergence%20of%20Random%20Variables.pdf)
Why is $\ F_n(x) = x^n$
The max of these random variables is less than $x$ if and only if all of them are less than $x$, and since they are independent and identically distributed, the probability of that is the $n$th power of the probability that the first one is less than $x$.