The Hill estimator by hand

(a) Under an exact Pareto tail, prove $E[\hat\alpha_k] = k\alpha/(k-1)$ and propose an unbiased estimator. (b) A series has six largest absolute daily returns $0.139, 0.087, 0.079, 0.076, 0.070, 0.065$, and the seventh largest is $0.060$. Compute $\hat\alpha_6$, its unbiased version and the standard error $\hat\alpha/\sqrt{k}$. Comment on the precision. (c) Under a Pareto tail with $\alpha = 3$, compute $P(|r| > 10\sigma)/P(|r| > 4\sigma)$ and compare with the normal ratio.