[EX] A Minimization Problem

Ok, here's a simple exercise...

Exercise:

Let $x_1, ..., x_N$ be $N$ (not necessarily pairwise different) real numbers.

1. Prove that $f(t):= 1/N sum_(n=1)^N |x_n - t|$ is convex in $RR$ and attains a global minimum.

2. Find all the values $x^**$ which mimimize $f$ and evaluate $f(x^**) = min_(t in RR) f(t)$.

3. Prove that $g(t):= 1/N sum_(n=1)^N (x_n - t)^2$ is strictly convex in $RR$ and attains its global minimum at a unique point.

4. Find the value $hat(x)$ which mimimizes $g$ and evaluate $g(hat(x)) = min_(t in RR) g(t)$.

5. Compare $f(x^**)$ and $g(hat(x))$: which value is greater?

6. Both functions $f$ and $g$ and both values $x^**$ and $hat(x)$ have straightforward statistical meanings. Can you tell what they are?