part-ii/2016-114/

[tripos-comments[bot]]

Part II 2016 Principles of Statistics: Paper 2, Section II,

https://questions.tripos.org/part-ii/2016-114/

[ArthurConmy]

b)

Compute (tis painful) $\mathbb{ E}[S^2] = \theta$ hence the result follows from Cramer-Rao as the estimator is unbiased.

c)

Adjust Cramer-Rao proof to show that $ \text{Var}( \tilde{\theta}_n ) \ge \frac{(1+B'(\theta))^2}{nI(\theta)} \ge \frac{1}{n} $ hence as MSE = Bias^2 + Variance, the result follows.